Vector Autoregressions and Structural VARs for Monetary ...

Vector Autoregressions and Structural VARs for Monetary Policy Analysis Using Eviews

User Guide

Vector Autoregressions and Structural VARs for Monetary Policy Analysis Using Eviews

Prepared by

Dr Thomas Bwire

Senior Principal Economist

Research Department, Bank of Uganda

Published by

COMESA Monetary Institute (CMI)

First Published 2019

COMESA Monetary Institute (CMI) C/O Kenya School of Monetary Studies P.O. Box 65041 – 00618 Noordin Road Nairobi, KENYA TEL: +254-20-8646207 http://cmi.comesa.int

Copyright © 2019, COMESA Monetary Institute

All rights reserved. No part of this publication may be reproduced, stored in retrieval system, or transmitted in any form or by any other means, electronic, mechanical, photocopying, recording or otherwise without prior permission from the COMESA Monetary Institute.

Disclaimer

The Author is solely responsible for the views expressed herein. These opinions do not in any way represent the official position of the COMESA, its Member States, or the affiliated institution of the Author.

Typesetting and Design

Mercy W. Macharia

[email protected]

TABLE OF CONTENTS

List of Tables ..............................................................................................................vii

List of Figures .......................................................................................................... viii

List of Acronyms ........................................................................................................ ix

Preface ........................................................................................................................... x

Acknowledgements .................................................................................................... xi

1. Introduction to Modelling for Analysis of Monetary Policy Transmission Mechanism .............................................................. 1

1.1 Introduction to Modelling ................................................................................. 1

1.2. EViews Operation Environment, Basics and Data ...................................... 3

1.2.1 EViews Software ........................................................................................... 3

1.2.2 Getting familiarity with EViews Window ...................................................... 3

1.2.3 Getting Data into EViews ............................................................................. 4

1.2.4 Viewing the Data ........................................................................................ 11

2. Theoretical Aspects of Monetary Policy Transmission Channels ....................................................................................... 13

2.1 Introduction...................................................................................................... 13

2.2 Interest Rate Channel...................................................................................... 15

2.3 Money Channel ................................................................................................ 17

2.4 The Wealth Channel........................................................................................ 17

2.5 The Exchange Rate Channel ......................................................................... 18

2.6 Bank Based Channels ...................................................................................... 18

2.6.1 The traditional bank lending channel ........................................................... 19

2.6.2 Bank capital channel ................................................................................... 19

2.7 Balance Sheet Channel .................................................................................... 20

2.8 Expectations Channel ..................................................................................... 21

3. Motivating VAR Modelling for Analysis of MTM ....................... 23

3.1 Introduction to VAR ...................................................................................... 23

3.2 Exploring the VAR(p) methods for the analysis of MTM ....................... 24

|vi

4. Taking MTM Theories and the VAR Model to the Data ........... 27

4.1 Adjustments for Seasonal Effects ................................................................. 28

4.2 Deriving the Output GAP ............................................................................. 34

4.3 Unit Root Testing ............................................................................................ 42

4.3.1 Demonstration of unit root testing using ADF test ....................................... 51

4.4 Setting up a MTM VAR model ..................................................................... 62

4.4.1 Determination of lag length .......................................................................... 62

4.4.2 VAR(2) residual statistical properties ......................................................... 67

4.4.3 Stability of VAR(2) ................................................................................... 71

5. Structural Vector Autoregressive Models (SVARs) ...................... 73

5.1 Motivation ......................................................................................................... 73

5.2 VAR Identification .......................................................................................... 73

5.3.1 Imposing a recursive Identification Scheme .................................................... 76

5.4 Generating Impulse Response Functions and Forecast Error Variance Decomposition ................................................................................................ 88

5.5 Non-recursive Identification Scheme ........................................................... 94

5.6 Comparing Recursively and Non-recursively Identified SVAR ............... 97

6. VAR and Vector Error Correction Model (VECM) .................... 101

6.1 Deriving the VECM Framework ................................................................ 101

6.2 Demonstration Determination of Cointegration and Estimation of VECM ............................................................................................................. 105

6.3 Estimating a VECM ...................................................................................... 109

6.4 Long-run Exclusion Tests ............................................................................ 112

6.5 Long-run Weak Exogeneity Tests............................................................... 113

6.6 Granger Non-causality Test ......................................................................... 116

7. Granger Causality Test ............................................................... 119

8. References ................................................................................... 120

vii |

List of Tables

Table 1: Unit test in CPI (levels) ...................................................................................... 53

Table 2: Unit root test in LCPI (difference) ...................................................................... 54

Table 3: Unit root test in exr ........................................................................................... 55

Table 4: Unit root test in lgdp ......................................................................................... 56

Table 5: Unit root test in lM2 .......................................................................................... 57

Table 6: Unit root test in loil_price .................................................................................. 58

Table 7: Unit root test in lpsc .......................................................................................... 59

Table 8: Unit root test in lrate ......................................................................................... 60

Table 9: Unit root test in tb91 ......................................................................................... 61

Table 10: VAR(2) Estimates ............................................................................................. 64

Table 11: VAR Lag order selection criteria ...................................................................... 66

Table 12: Residual Serial Correlation LM test ................................................................. 67

Table 13: VAR Residual Normality Tests ......................................................................... 70

Table 14: VAR Residual Heteroskedasticity Test............................................................. 71

Table 15: VAR (2) Roots of the characteristic polynomial .............................................. 72

Table 16: Just-Identified SVAR estimates (short-run text form) ..................................... 81

Table 17: Over-identified SVAR estimates (short-run text form) ................................... 83

Table 18: SVAR estimates (short-run pattern matrix option) ......................................... 87

Table 19: Variance decomposition ................................................................................. 93

Table 20: Just-identified Non-recursive SVAR estimates (matrix form) ......................... 96

Table 21: Johansen's trace test results ......................................................................... 108

Table 22: Vector Error Correction Estimates ................................................................ 111

Table 23: A test of monetary policy neutrality ............................................................. 113

Table 24: A test of weak exogeneity ............................................................................. 115

Table 25: VEC Granger Causality/Block Exogeneity test ............................................... 117

|viii

List of Figures

Figure 1: Theoretical Monetary Transmission Mechanisms ........................................... 14

Figure 2: Quarterly GDP, CPI and oil_price in levels ....................................................... 31

Figure 3: Comparing seasonally unadjusted and seasonally adjusted series ................. 34

Figure 4: Output gap: Hodrick-Prescott Filter (λ=1600) ................................................. 36

Figure 5: Variables in levels and first differences ........................................................... 38

Figure 6: Level, first difference autocorrelations ............................................................ 46

Figure 7: Residual plots ................................................................................................... 69

Figure 8: IRF to one standard-deviation structural shock in tb91 (recursive structural factorisation) .................................................................................................... 91

Figure 9: IRF to one standard-deviation structural shock in tb91 (non-recursive structural factorisation).................................................................................... 97

ix |

List of Acronyms

ADF Augmented Dickey Fuller

AIC Akaike Information Criteria

CPI Consumer Price Index

DGP Data Generating Process

FEVD Forecast Error Variance Decomposition

GDP Gross domestic product

HP Hodrick-Prescott

HQ Hannan-Quinn

IFS International Financial Statistics

IMF International Monetary Fund

IRF Impulse Response Function

LR Likelihood Ratio

MTM Monetary Transmission Mechanisms

OLS Ordinary Least Squares

PSC Private Sector Credit

PWT Penn World Tables

SBC Schwarz Bayesian Criterion

UN United Nations

VAR Vector Autoregressive

VECM Vector Error Correction Model

WB World Bank

|x

Preface

The preparation of this User Guide followed a directive to COMESA

Monetary Institute (CMI) by the 23rd Meeting of the COMESA Committee of

Governors of Central Banks which was held in March, 2018 in Djibouti.

Governors noted that understanding and analysing monetary policy

transmission mechanism is crucial since COMESA member central banks are

currently moving gradually to inflation targeting monetary policy frameworks.

The overall objective of this User Guide is therefore to equip users with skills

on modelling and analyzing monetary policy transmission mechanism.

The User Guide utilizes strictly time series data, for the analysis of monetary

policy transmission mechanisms. It demonstrates all steps from data

organization to results interpretation using EVIES software. The key important

questions which the analysis in the User Guide addresses are: (i) Which

transmission channels, or combination of channels (interest rate, bank lending,

exchange rate, asset prices) are likely to be the most effective in transmitting

monetary policy; and (ii) What is the timing and magnitude of the effects of

policy changes on macroeconomic variables. The User Guide therefore,

provides an analytical guide on application of VAR, SVAR and VECM for

analysis of transmission mechanism of monetary policy.

It is hoped that the Guide will enable users to have a firm understanding of the

process involved in the analysis of transmission mechanism of monetary policy.

It is also hoped that the Guide will be used by COMESA member central

banks as a reference material to train their staff.

Ibrahim Zeidy Director and Chief Executive Officer

xi |

Acknowledgements

The Author was grateful to the COMESA Monetary Institute (CMI) for the

opportunity given to prepare the User Guide. He acknowledged technical and

professional support from the Director of CMI, Mr. Ibrahim Zeidy and the

Senior Economist, Dr. Lucas Njoroge. The Author also acknowledged the

earlier COMESA training material on the subject by Dr. Ole Rummel of Bank

of England’s Centre for Central Banking Studies.

The Author was especially grateful for the comments from the participants of

the Validation Workshop held from 7th – 11th May 2018 in Nairobi, Kenya that

provided the final inputs to the User Guide. The workshop was attended by

participants from the following COMESA member countries’ Central Banks:

Burundi, Djibouti, DR Congo, Eswatini, Kenya, Malawi, Mauritius, Sudan,

Uganda, Zambia, and Zimbabwe.

Chapter 1

Introduction to Modelling for Analysis of Monetary Policy Transmission Mechanism

1.1 Introduction to Modelling

The magnitude of economic influences is unobservable, so their modelling is

necessary. Measuring economic influences is done in a wider area called

econometrics and utilizes the inputs of mathematicians, statisticians and

economists (theoretical and applied). Uncovering economic influences makes

use of economic data, which in general bears three key characteristics. First,

unlike natural sciences, economic data is non-experimental and is in raw form

gathered by local government agencies - such as National Statistics agencies,

Central banks, Finance, Planning and Economic Development ministries, etc.

and various international agencies such as the World Bank (WB), International

Monetary Fund (IMF) – International Financial Statistics (IFS), United Nations

(UN), Penn World Tables (PWT), etc. Second, the data is random, so it

involves modelling the randomness and may require undertaking some

transformations. Thirdly, in addition to measurement errors, randomness of

data brings into play the error term – which we explore in due course.

Economic data falls under three broad categories of: cross sectional, i.e., data

collected at a point in time for a number of economic agents or institutional

units e.g. households consumption in a given year, GDP data collected for a

given number of countries at each given date (annually, semi-annually, and

quarterly); time series, data collected sequentially in time at regular intervals e.g.

Uganda’s annual GDP data for the period, 2000 – 2017; and panel/pooled data

- which combines both time series and cross section data. In general, economic

data can be disaggregated or aggregated. Disaggregated data is usually for

microeconomic modelling (a discipline called micro econometrics) while

aggregated data is usually for macroeconomic modelling (a discipline called

macro econometrics).

Vector and Structural Vector Autoregressions

|2

This User Guide utilizes strictly time series data, which is applied to modern

time series models suitable for the analysis of monetary policy transmission

mechanism using EViews econometric software. Analysis of monetary policy

transmission mechanism begins with model building – which, by and large, is

an art - perfected through continuous research, i.e., the knowledge of the

theory of what is to be modelled. Based on the constructed model(s), economic

relationships and outcomes can both be measured and predicted. In principal,

econometric models are essential for statistical inferences - estimation,

hypothesis testing and forecasting and policy analysis.

In practice, the analysis of monetary policy transmission mechanism has been

done using a variety of software packages, of which EViews is just one. Other

common ones have been Microfit, Stata RATS, CATS in RATS, OxMetrics

(PcGIVE), Gauss, Matlab, Dynare, Iris etc. Importantly, the software chosen

depends on the complexity of the Model to be estimated. The good news

though is that almost any of this econometric software can perform most of

the econometric estimations, except that they may have different speed of

program execution depending on the complexity of the model. In addition,

some are more users friendly and easy to calibrate than others. Similarly, others

have clear advantages in the estimation of models related to certain disciplines

Title bar

Main menu & menu items

Command window

Work space

Status line

Grayed menu items not available

Black menu items available


3 |

than others. In the following, we familiarize ourselves with EViews operational

environment.

1.2. EViews Operation Environment, Basics and Data

1.2.1 EViews Software

EViews – short form of Econometric Views is a user-friendly and powerful

statistical software package designed to provide sophisticated data analysis and

forecasting tools. The software comes with an extensive user User Guide which

contains many useful examples/explanations/ tutorials and is extremely well-

presented. Once you have gained basic familiarity with the basic concepts and

operations of the program, you should be able to perform most operations

without consulting the User Guide. Moreover, the program has a very

extensive help menu (one of the entries in the Main menu) which is simple to

use and sufficient for most users, so one may not again need to consult the

User Guide when conducting statistical analysis. This introductory chapter aims

to familiarise users with the fundamentals of working with EViews pertinent to

the purpose at hand, i.e., analysis of monetary transmission mechanisms

(MTM). The detailed and in-depth exposition of EViews fundamentals are

given in the User Guide I which is in the drop-down menu of the help menu in

the earlier versions of EViews or among the documents in PDF Docs file in

the drop-down menu of the help menu in the later versions of EViews, notably

EViews 10. All illustrations and output in this User Guide derive from EViews

9.0

1.2.2 Getting familiarity with EViews Window

Launch the program (i.e., double click on EViews icon) - this assumes the

program has been properly installed on your computer (the installation is

usually done by authorized IT staff). This brings forth the EViews window as

in the screen print below. We want to familiarize ourselves with the title bar,

main menu, command window, work space and status line, in there.

At the very top of the main window, is the title bar, labelled EViews and is

generally whiter (with later versions). Immediately below the title bar is the main

menu. If you move the cursor to an entry in the main menu, say Object and

click on the left mouse button, a drop-down menu will appear. Some of the items

in the drop-down menu, as shown in the print screen herewith are listed in black

whilst others are in gray. Black items are executable while the gray items are


|4

not or are simply unavailable. Clicking on a black listed entry in the drop-down

menu selects the highlighted item.

Below the menu bar is the command window space in which EViews

commands such as log, first difference transformations are entered or typed.

Each of the entered commands is executed as soon as you press the ENTER

button on the key board. The area in the middle of EViews window is the

work space area and is where various objects that EViews creates are

displayed. It is actually analogous to a sheet of paper in an exercise book or

work book on your desk. To close EViews, select File from the main menu

and in the drop-down menu, choose and execute Exit. Alternatively, and

obviously much more directly, click on the x button in the upper right-hand

corner of the EViews window. If necessary, EViews will warn you and provide

you with the opportunity to save any unsaved work. EViews work file is saved

in pretty the same way as any other computer-generated document. Select File

in the main menu, then Save As… in the drop- down menu and save the file.

Subsequently, click on Save through the File menu or as usual through the key

board operations (pressing Ctrl button, and whilst holding it down, press the S

button) to save any changes to the file. It is advisable that you save your work

continuously.

1.2.3 Getting Data into EViews

In this demonstration, we use quarterly time series data for Uganda provided in

excel sheet in MTM_data folder. The folder includes data points for a suite of

variables which are usually used in the analysis of various monetary policy

transmission mechanisms: Gross domestic product (GDP) in 2009/10 constant

prices, Core consumer price index (core_CPI), mid-rate Uganda shillings USD

nominal exchange (exr), international oil pump prices (oil_price), 91 day

Treasury bill rate (tbill_91day) – a measure of monetary policy, end period

stock of private sector credit (PSC), lending rate (lr) and a measure of broad

money (M2), all observed for the period 2000q1 – 2017q3, i.e., some 71

seasonally unadjusted observations. Given the data, we would like to read it into

an EViews work file so then it can be subjected to statistical analysis. EViews

obviously provides sophisticated tools for reading in data from a variety of

common data formats and sources, but here I will demonstrate about four,

among the many different ways of doing this from an Excel file, starting with

the easiest – a simpler copy and paste command type.


5 |

Launch the EViews program to see the EViews window we’ve already

described above. On the

Main menu, left click on

File and in the drop-down

menu, point the cursor at

NEW and navigate through

to Work file. Click on Work

file and it’s in the Work file

dialog box which pops

(given in the screen shot

below) that the user uses to

supply critical data

information to the software.

Starting with the work file

structure type, choose from the drop-down menu, the type of series at our

disposal. You notice from MTM_data folder our data is dated (2000 – 2017)

and is at a regular frequency (quarterlies), so given this, we choose dated-

regular frequency (which incidentally is EViews default entry). Similarly,

under Date specification, choose from the drop-down menu, Quarterly. We then

supply the Start date and the End date (in the form: 2000Q1 for start date

and 2017Q3 for end data). Although Work file name is optional, here we

provide - for convenience COMESA2018 (for WF) and Demo (for Page).

It is important to provide for page names, just as it is with sheets in the excel

book because in the process of executing the analysis or handling multiple

tasks, it is often the case that we end up with multiples of pages/sheets within

the same work file. These described entries are what we see in the EViews snap

shot hereof.

With these entries, press OK to open a new window, given here in the screen

print, with two variables; C and resid. As we all know, these two, namely

constant, C (also known as intercept) and residual, form an integral part of an

econometric model. They serve to capture important information about the

regressor or dependent variable. The C means, if we were to hold everything

else constant (the famous ceteris paribus notion in economics), the minimum

value that the dependent variable would take is equal to the magnitude of the

constant (be it positive or negative), subject to its statistical significance and the

model meeting standard statistical criteria for evaluating the estimated results.


|6

Moreover, because it cannot claimed that the whole of economics or even the

whole of economic theory can be encompassed in a model, but a well devised

model can bring out certain features of interdependence among economic

quantities that are not easily comprehended without its help (Beach, 1958), resid

captures, among other things, the unexplained component of the dependent

variable. Importantly, note that C and resid are not observable a priori, but are

generated with execution or estimation of a regression model.

The window gives both the Range and Sample of the data, spanning 2000Q1 to

2017Q3 - some 71 quarterly observations. In addition, the entries we made for

the work file and page

name in the previous

step are also reflected,

as COMESA2018 and

demo, respectively.

Note that if we had

not provided for the

page name upfront (as

in the previous screen

shot), the software

would instead baptise

the page (in the place

of demo-in this screen

print) a default name -

Untitled.

Because as stressed above, it is very important that work file pages are named,

if the user has not provided for the page name upfront, but finds it reasonable

to do so, it is equally straight forward to rename the untitled page. All we do is

to point the cursor at untitled – a page that we want to rename, right click the

mouse and select from the options, rename Work file page…whereof, we provide

the page name of choice.

It is at this point that we load the data. To do so, interactively open the excel

data file MTM_data folder. In the excel file, select and copy the series data for

the eight variables, the cell variable labels inclusive – ensure, for emphasis, that

both the series name and the series data have been copied. The column for

sample period should be excluded in this process, because, already, this has

been declared. In the open EViews window, check on the Quick command in


7 |

the Main menu and choose Empty group (edit series). A click on this

produces a spread sheet similar to that in excel .

The first column displays

the sample range period,

spanning from the start to

the end date that we have

supplied in the preceding

steps. As can be seen, the

first two rows are by default

empty, but unlike the

second, the first is

highlighted blue. Either of

these can be used to paste in

the data, albeit with

different implications, which,

if curious, you might want to

practically check. In the sheet,

(though not exactly printed to

reflect this) sample period

starts from the third row of

the first column.

Click on the second cell in the

second column, and while

maintaining the cursor in

there, make a right hand click

on your mouse and paste in

the copied data. As you will

see, this simple way of

importing the data also brings on board the series names, which then

automatically become visible in the work space window.

As can be seen in the screen print, the number of entries in the work space

window, in addition to C and resid, grows by the number of series we have

imported. Once sure the copy and paste command is complete, click on the x -

a small box like command - highlighted in red at the top most right-hand side

corner of the open excel-like sheet window (containing the copied in data). In


|8

the prompt message, select Yes. A window with all the series and C and resid

appears as shown in the screen print.

Note that any mismatch between the actual series length and length between

the start and end date may prompt the software to reject the data paste

command. Also, if during the copying, the series initial is omitted, the software

will assign the omitted initial a default name, usually SER01 (SER for series) for

the first series column, or in general SER0i, i=1, 2, ..., k for k multiple series.

This however can be easily replaced and/or renamed with the actual series

names through the Genr (generate) command (see pointer in the print screen

above) in the work space window. Alternatively, highlight the variable/series

you want to rename and while keeping the cursor in there, right click on the

mouse and choose rename, then provide the name for the variable/series in

question. Also, take note that at times, actually quite often, the initial given to

the series may matter, and may be rejected by EViews especially if the initial is

a preserve of the

software. In the event

that this is so, try

changing the initial

until it is acceptable.

The other easy way

involves direct

importation of data

from the data source, a

process we next

describe in detail, as

before.

To proceed, double

click on the EViews icon for the EViews window described earlier. Check File

and among the entries in the drop-down menu, point the cursor at Import and

following the arrow, choose Import from file…selecting this takes us to the

computer window, whereof, we choose the path where our MTM_data folder

is stored. On my PC, my MTM_data folder is stored on the desk top, so I

chose the desk top, and then navigated through to and opened COMESA

folder, then 2018 and finally double checked MTM_data.


9 |

Executing the steps

above should generate

an EViews screen as

displayed herein. As

can be seen from the

screenshot, this is a 3-

step procedure (see at

the top: Excel Read –

Step 1 of 3), in the

order, next (Step 1 of

3), next (Step 2 of 3),

next (Step 3 of 3) and

finish. We can also see

under Predefined

range (highlighted),

our data is being

picked from sheet2 of the excel workbook, noting that appropriate choices of

the excel book sheets can still be done from the arrow in the dialogue box.

Clearly the screen gives data as was in excel, arranged in columns. We also have

the option to transpose the incoming data to rows, but only if we have a

sufficient reason to do so.

Click next (Step 2 of 3), and then next (Step 3 of 3). In step 3, which is

Structure of the Data to be Imported, adjust for the Basic structure, from

the drop-down menu, to Dated – regular frequency (EViews default is

Unstructured/Undated). With this choice, we need to declare Frequency/date

specification. In the dialogue box for Frequency, we chose Quarterly (from the

drop-down menu thereof) and type 2000q1 in the dialogue box for Start date

(EViews default is 1), as shown in the screen print.


|10

With these entries, click finish, which gives a screen shot, like the final screen

we had under copy and paste command. The work file page is automatically

named by the excel folder name, MTM_data.

The other, and possibly the last way of reading in data into EViews I would like

to demonstrate in this

User Guide is the file

option.

Like in the previous

Import option, double

click on the EViews

icon for the EViews

window described

earlier. Check File and

among the entries in

the drop-down menu,

point the cursor at Open and following the arrow, navigate through to

Foreign Data as Workfile… Selecting this takes us to the computer window,

whereof, we choose the path where our MTM_data folder is stored, pretty

much in a similar way as already described under direct importation.

Like in the previous description, this file option also loads data in a 3-step

procedure, in the order, next (Step 1 of 3), next (Step 2 of 3), next (Step 3 of 3)

and finish. Do not forget that as describe above, Step 3, i.e., Structure of the

Data to be Imported, involve quite some editing, involving adjustments for

the Basic structure, which from the drop-down menu, has to be changed from the

EViews default of Unstructured/Undated) to Dated – regular frequency, and


11 |

consequently, we have got to

declare Frequency/ date

specification in the respective

dialogue boxes. Accordingly,

In the, we chose Quarterly

(from the drop-down menu) of

the dialogue box for

Frequency and type 2000q1 in

the dialogue box for Start date

(EViews default is 1), indeed as

has been described before and

also shown in the screen print

in the text. We realize, at the

end of the procedure, the work

file page is automatically

named by the excel folder name, MTM_data, as shown in the screen shot.

As mentioned earlier, there are many ways of importing data into EViews, but

the simplicity and complexity associated with each of these ways is user

specific. It is important that as users, we aim to choose the easiest way

possible– and anyone of the above is simpler in my judgement.

1.2.4 Viewing the Data

Viewing data in EViews is

important, not least because

we must verify and be

satisfied that the data now in

EViews work space is the

actual data that has been

called in from excel as the

data may have been distorted

during the process of reading

it into EViews.

To view the data within

EViews work space, click on

and highlight any one series,

and while maintaining the


|12

cursor in the highlighted series, hold down the Ctrl key on your key board

(using an alternative hand) and select the rest of the series, but one at a time.

Once all the series of interest are highlighted, and while maintaining the cursor

in the highlighted series, right click on your mouse and choose open as a

group. This road map is shown in the adjacent screen print.

Execute the open as a group command to display the variable’s data as is

shown in the EViews print screen herewith.

Examine the data to ensure it represents the actual data that was intended to be

read into EViews. Always remember, whenever you enter data into EViews, to

check it thoroughly and

carefully so to ensure it is

what you expect it to be

and that it has not been

distorted in the process

of calling it into EViews.

For emphasis, checking

your data is boring, but

extremely vital.

The open window can

then be closed using the

x command - highlighted

in red at the top most right-hand side corner of the open spread sheet-like

window. In the prompt message, select Yes to revert to the EViews work

space in the previous screen print. You are now ready to undertake any data

manipulation and analysis. In what will follow, we would want to explore the

data, including adjusting for seasonality and log transformations where

necessary, and to generate the output gap – a variable common in monetary

policy making circles.

In the next section, we discuss the known channels of monetary transmission

mechanism to inform the variable and data needs of this extensive exercise.

Chapter 2

Theoretical Aspects of Monetary Policy Transmission Channels

2.1 Introduction

Monetary transmission mechanism (MTM) describes the process through

which monetary policy decisions impact an economy. Simply put, when a

central bank decides on a monetary policy action, it sets in motion a sequence

of events, with the initial impact on financial markets, which in turn, slowly

(with a lag) permeates its way through to changes in aggregate demand (private

consumption and investment), which then influences current production levels,

wages and employment, and in the process steer domestic prices in the desired

direction of the monetary authority – low and stable inflation, which is the

ultimate pursuit/goal of monetary policy. It is this chain of developments,

triggered by monetary policy actions, which constitute monetary policy

transmission mechanism. As we delve into this discussion, it is order that I put

a disclaimer – a great majority of the discussion in this section is adapted from

Mugume (2011, 5-10), with express permission and is slightly modified only

where necessary.

Regardless of the monetary policy framework used in practice, a successful

implementation of monetary policy requires an accurate assessment of how fast

the effects of policy changes propagate to other parts of the economy and the

timing and size these effects. Cecchetti (1995), Mishkin (1995), and Christiano

et al. (1997) give a comprehensive survey of the literature on monetary

transmission mechanism. Whilst a consensus on the monetary transmission

mechanism has not emerged from this literature (Mugume 2011), the general

thinking is that traditionally, monetary policy impulses are thought to be

transmitted via money or credit channels—the so-called money versus credit

view of monetary policy (Davoodi et al., 2013).


|14

In the former, changes in the nominal quantity of money affect spending

directly. The transmission in the latter case is indirect. Monetary policy

transmission is triggered when the central bank changes the monetary base

through an open market operation, to withdraw or inject liquidity in the

banking system with implications for the interest rates. This eventually

permeates to changes in prices for a variety of domestic and foreign assets. The

literature also agrees to the existence of nominal rigidities that both currency

and bank reserves are nominally denominated and that their quantities are

measured in terms of the economy’s unit of account. This suggests implicitly

that if policy‐induced movements in the nominal monetary base are to have

real effects, nominal prices must respond with a lag to the policy movements in

a way that leaves the real value of the monetary base unchanged (Mugume,

2011, emphasis mine). Such sources of nominal rigidities, which limit the ability

of households to participate in financial markets, include sticky prices and

wages and market imperfections. Figure 1 depicts a black box of an eclectic view

of monetary policy transmission identifying the major channels common in the

literature.

Figure 1: Theoretical Monetary Transmission Mechanisms

Source: Adopted from Adam (2011, p.9)


15 |

The process begins with the transmission of open market operations to market

interest rates, either through the commercial banks excess reserves or through

the supply and demand for money more broadly. From there, transmission may

proceed through any of several channels. In what follows, we cover in more

detail how each channel works.

2.2 Interest Rate Channel

This is the primary mechanism at work in conventional macroeconomic

models, and hinges on the notion that the transmission of monetary policy

depends on private expenditures being interest elastic. The basic idea is that

given some degree of price stickiness, an increase in nominal interest rates, for

example, translates into an increase in the real rate of interest and the user cost

of capital – changes which in turn lead to a postponement in consumption or a

reduction in investment spending and, hence, the output level and prices. Note

for emphasis that in the channel, while the central bank on the contrary has

direct grip only over the short-term nominal interest rate, effectiveness of

monetary policy will depend on its ability to affect the real interest rate and the

sensitivity of consumption and investment to changes in the price of

intertemporal substitution.

This is the mechanism embodied in the New Keynesian macro models

developed by Rotemberg and Woodford (1997) and Clarida, Galí, and Gertler

(1999). The New Keynesian model consists of three equations: the aggregate

demand (IS) curve, aggregate supply (Philips) curve and the Uncovered Interest

Rate Parity (UIP) equation, involving three variables, output 𝑦𝑡 , inflation

𝜋𝑡 and short-term nominal interest rate 𝑖𝑡 . These are given in equations 1-3.

The aggregate demand curve for an open economy is represented as:

y

tttt rerryy 3211 (1)

Where ty is output,

tr is the real interest rate, rer is the real exchange rate, y

t

is an aggregate demand shock and coefficients 21, and 3 are the

persistence of output, impact of the interest rate on output and the impact of

the exchange rate on output, respectively.

The aggregate supply curve is defined as:


|16

tttt rery 3211 (2)

Where t is inflation, t is an aggregate supply shock and coefficients

21,

and 3 are the persistence of inflation, impact of output on inflation and the

impact of exchange rate on inflation, respectively.

The uncovered interest rate parity equation that captures the relationship of the

domestic economy with the rest of the world can be represented as:

s

ttttt premiiSS )( *

11 (3)

Where tS is the nominal exchange rate,

ti is the domestic nominal interest

rate, *

ti is the foreign nominal interest rate, prem is the risk premium, s

t is

the exchange rate shock and coefficient 1 is the persistence of exchange rate

movement.

The real exchange rate can be derived as iw

i

n

itt pp

Srer

1

1* where p is the

domestic consumer price index, ip is the consumer price index of country i,

iw is the weight attached to country i in the basket of countries that trade most

with the domestic economy and n is the number of trading partners.

The fourth equation is monetary policy reaction function which follows the

Taylor rule (Taylor 1993).

𝑖𝑡 = 𝜌 + 𝜗𝜋𝑡+1 + 𝛾𝑦𝑡 + 휀𝑡 (4)

Consistent with the behaviour of inflation targeting central banks, the central

bank systematically adjusts the short-term nominal interest in response to

movements in expected inflation (𝜋𝑡+1), and output gap (𝑦𝑡).

This description of monetary policy in terms of interest rates reflects the

observation, noted above, that most central banks today conduct monetary

policy using targets for the interest rate as opposed to any of the monetary

aggregates. In this New Keynesian model, monetary policy operates through

the traditional Keynesian interest rate channel. A monetary tightening, in the

form of a shock to the Taylor rule, that increases the short-term nominal

interest rate translates into an increase in the real interest rate as well when


17 |

nominal prices move sluggishly due to costly or staggered price setting. This

rise in the real interest rate then causes households to cut back on their

spending as summarized by the IS curve. Finally, through the Phillips curve,

the decline in output puts downward pressure on inflation, which adjusts only

gradually after the shock.

However, as Bernanke and Gertler (1995) have pointed out, the

macroeconomic response to policy-induced interest rate changes is

considerably larger than that implied by conventional estimates of the interest

elasticities of consumption and investment. This observation suggests that

mechanisms other than the narrow interest rate channel may also be at work in

the transmission of monetary policy. Such other alternative paths include;

2.3 Money Channel

This channel effectively assumes changes in reserve money are transmitted to

broad money via the money multiplier; that banks are in the business of

creating inside money. The money view of monetary policy assumes aggregate

demand and price levels move in line with money balances used to finance

transactions. It is this idea that forms the basis for broad money representing

the intermediate target in many central bankers’ money-focused monetary

policies (Mishkin, 1998).

2.4 The Wealth Channel

The wealth channel is built on the life-cycle model of consumption developed

by Ando and Modigliani (1963), in which households’ wealth is a key

determinant of consumption spending. The link to monetary policy comes

through the link between interest rates and asset prices. A policy-induced

interest rate increase and/or reduces the value of long-lived assets (stocks,

bonds, and real estate), shrinking households’ resources and leading to a fall in

consumption.

Asset values also play an important role in the broad credit channel developed by

Bernanke and Gertler (1989), but in a manner distinct from that of the wealth

channel. In the broad credit channel, asset prices are especially important in

that they determine the value of the collateral that firms and consumers may

present when obtaining a loan. In ―frictionless credit markets, a fall in the

value of borrowers’ collateral will not affect investment decisions; but in the


|18

presence of information or agency costs, declining collateral values will increase

the premium borrowers must pay for external finance, which in turn will

reduce consumption and investment. Thus, the impact of policy-induced

changes in interest rates may be magnified through this ―financial accelerator

effect. A direct effect of monetary policy on the firm’s balance sheet comes

about when an increase in interest rates works to increase the payments that

the firm must make to service its floating rate debt. An indirect effect arises,

too, when the same increase in interest rates works to reduce the capitalized

value of the firm’s long-lived assets. Hence, a policy induced increase in the

short-term interest rate not only acts immediately to depress spending through

the traditional interest rate channel, it also acts, possibly with a lag, to raise each

firm’s cost of capital through the balance sheet channel, deepening and

extending the initial decline in output and employment.

2.5 The Exchange Rate Channel

This is an important element in conventional open-economy macroeconomic

models. The chain of transmission here runs from interest rates to the

exchange rate via the uncovered interest rate parity condition relating interest

rate differentials to expected exchange rate movements. Thus, under floating

exchange rates and perfect capital mobility, arbitrage between domestic and

foreign short-term government securities causes incipient capital flows, which

change the equilibrium value of the exchange rate required to sustain

uncovered interest parity. With sticky prices, this change in the nominal

exchange rate is reflected in a real exchange rate depreciation that induces

expenditure switching between domestic and foreign goods. The effectiveness

of this channel depends on the central bank’s willingness to allow the exchange

rate to move, on the degree of capital mobility, on the strength of expenditure

switching effects (this depends on the commodity composition of production

and consumption), on the importance of currency mismatches, and on the

degree of exchange rate pass through.

2.6 Bank Based Channels

There are two distinct bank-based transmission channels. In both, banks play a

special role in the transmission process because bank loans are imperfect

substitutes for other funding sources.


19 |

2.6.1 The traditional bank lending channel

According to this view, banks play a special role in the financial system because

they are especially well suited to solve asymmetric information problems in

credit markets. Because of banks’ special role, certain borrowers will not have

access to credit markets unless they borrow from banks. As long as there is no

perfect substitutability of retail bank deposits with other sources of funds, the

bank-lending channel operates as follows. Expansionary monetary policy,

which increases bank reserves and bank deposits, increases the quantity of bank

loans available. Because many borrowers are dependent on bank loans to

finance their activities, this increase in loans will cause investment and

consumer spending to rise. An important implication of the bank lending

channel is that monetary policy will have a greater effect on expenditure by

smaller firms, which are more dependent on bank loans, than it will on large

firms, which can get funds directly through stock and bond markets (and not

only through banks).

2.6.2 Bank capital channel

In this channel, the state of banks’ and other financial intermediaries’ balance

sheets has an important impact on lending. A fall in asset prices can lead to

losses in banks’ loan portfolios; alternatively, a decline in credit quality, because

borrowers are less able or unwilling to pay back their loans, may also reduce the

value of bank assets. The resulting losses in bank assets can result in a

diminution of bank capital, as has occurred during the recent financial crisis.

The shortage of bank capital can then lead to a cutback in the supply of bank

credit, as external financing for banks can be costly, particularly during a period

of declining asset prices, implying that the most cost-effective way for banks to

increase their capital to asset ratio is to shrink their asset base by cutting back

on lending. This deleveraging process means that bank-dependent borrowers

are now no longer able to get credit and so they will cut back their spending

and aggregate demand will fall. Expansionary monetary policy can lead to

improved bank balance sheets in two ways. First, lower short-term interest

rates tend to increase net interest margins and so lead to higher bank profits

which result in an improvement in bank balance sheets over time. Second,

expansionary monetary policy can raise asset prices and lead to immediate

increases in bank capital. In the bank capital channel, expansionary monetary

policy boosts bank capital, lending, and hence aggregate demand by enabling

bank-dependent borrowers to spend more.


|20

2.7 Balance Sheet Channel

Like the bank-lending channel, the balance sheet channel arises from the

presence of asymmetric information problems in credit markets. When an

agent’s net worth falls, adverse selection and moral hazard problems increase in

credit markets. Lower net worth means that the agent has less collateral,

thereby increasing adverse selection and increasing the incentive to boost risk

taking, thus exacerbating the moral hazard problem. As a result, lenders will be

more reluctant to make loans (either by demanding higher risk premia or

curtailing the quantity lent), leading to a decline in spending and aggregate

demand. A particularly convenient, and widely adopted, model of this type is

he financial accelerator framework of Bernanke and Gertler (1989) and

Bernanke, Gertler and Gilchrist (1999), in which lower net worth increases the

problems associated with asymmetric information in debt financing, thereby

increasing the external finance premium.

Monetary policy affects firms’ balance sheets in several ways. Contractionary

monetary policy leads to a decline in asset prices, particularly equity prices,

which lowers the net worth of firms, which leads to a decline in lending,

spending and aggregate demand. Another way is through cash flow, the

difference between cash receipts and cash expenditure. Contractionary

monetary policy, which raises interest rates, causes firms’ interest payments to

rise, thereby causes a fall in cash flow. With less cash flow, the firm has fewer

internal funds and must raise funds externally. Because external funding is

subject to asymmetric information problems and hence an external finance

premium, additional reliance on external funds boosts the cost of capital,

curtailing lending, investment and economic activity. An interesting feature of

the cash flow channel is that nominal interest rates affect firms’ cash flow, in

contrast to the role of the real interest rate emphasized in neoclassical channels.

Furthermore, the short-term interest rate plays a special role in this

transmission mechanism, because interest payments on short-term (rather than

long-term) debt typically have the greatest impact on firms’ cash flow.

Additional asset price channels are highlighted by Tobin’s (1969) q‐theory of

investment and Ando and Modigliani’s (1963) life‐cycle theory of consumption.

Tobin’s q measures the ratio of the stock market value of a firm to the

replacement cost of the physical capital that is owned by that firm. All else

being equal, a policy‐induced increase in the short‐term nominal interest rate


21 |

makes debt instruments more attractive than equities in the eyes of investors;

hence, following a monetary tightening, equilibrium across securities markets

must be re-established in part through a fall in equity prices. Faced with a lower

value of q, each firm must issue more new shares of stock in order to finance

any new investment project; in this sense, investment becomes costlier for the

firm. Therefore, in the aggregate, across all firms, investment projects that were

only marginally profitable before the monetary tightening go unfunded after

the fall in q, leading output and employment to shrink as well. Meanwhile,

Ando and Modigliani’s life‐cycle theory of consumption assigns a role to

wealth as well as income as key determinants of consumer spending. Hence,

this theory also identifies a channel of monetary transmission: if stock prices

fall after a monetary tightening, household financial wealth declines, leading to

a fall in consumption, output, and employment.

2.8 Expectations Channel

Expectations are so central to monetary transmission that they deserve to be

analysed in detail. Assuming rational expectations, the precise effect of a policy

change on expectations can vary at different points in time or in the business

cycle. The market’s response will depend on the external and internal

environments and on the policy regime. The uncertainty on the impact of the

policy change on the economy enhances the need for a credible and

transparent regime. Central bank credibility will play a leading role, permitting

agents to evaluate more clearly the consistency of a specific policy decision.

With a credible inflation target, for example, monetary policy will be anchored

to the target in the medium term, allowing agents to generate a clearer and less

erratic expectation of the future behaviour of the policy rate, and diminishing

the impact of temporary disturbances that are likely to reverse in the future. If

the nominal objective is credible, the term structure associated to a reduction in

the policy rate, for example, must be consistent with the fact that future

expected policy rates—that partially determine long-term interest rates today—

reflect compliance with the policy goal. If, on the contrary, the target is not

credible or—more generally—there is no clarity on the central bank’s objective,

the effect on the rate structure will be more ambiguous. The market shall infer

future policy actions by looking at the currently available information. Thus,

the impact of a policy decision today on the global rate structure of the

economy should be more predictable—with a given financial structure— the

greater the degree of credibility in the goals of the central bank.

Chapter 3

Motivating VAR Modelling for Analysis of MTM

3.1 Introduction to VAR

Monetary economics focuses on the behaviour of prices, monetary aggregates,

nominal and real interest rates and output. Vector autoregressive (VAR)

methods with origins to the seminal work of Sim’s (1980), have become 'a

workhorse' in much of the empirical analysis of the interrelationship between

these variables and for uncovering the impact of monetary policy on the real

economy – the so called monetary policy transmission mechanism. The novelty

of VAR methods stems from its structure (Hamilton 1994: 326-7), which offers

both empirical tractability and a link between data and theory in economics, as

it uncovers and describes data facts and characteristics. The technique takes

into account the interactions between macro variables over time, and as shall

neatly be shown, allows a distinction in estimating the long-run (equilibrium)

and short-run (adjustment to the equilibrium) relations – a popularity tied to

the concept of cointegration (Engle and Granger, 1987). There is one equation

for each and every variable, so all variables in the system are treated as

potentially endogenous. Each variable is explained by own lags and lagged

values of the other variables.

Reduced-form VAR models treat the economy as a black box and aim only to

identify the linkages between some key inputs and key outputs of interest.

Specifically, assumptions about Exogeneity are tested for directly avoiding

making strong a priori assumptions, thus by design, a key advantage of the

reduced form VAR models is that they allow the data to speak freely about the

empirical content of the model. The approach is much more a theoretical, i.e.,

one does not have to maintain the existence of, estimate or test specific

theoretical formulations of any economic relationship, rather it invokes


|24

economic theory to choose the variables to include in the analysis, select the

appropriate normalization and to interpret the results.

Based on extensive empirical research, Christiano et al. (1999) derived stylized

facts about the effect of contractionary monetary policy shocks1 in a closed

economy setting. They conclude that plausible models of the MTM should be

consistent with at least the following evidence on prices, output and interest

rates: i) the aggregate price level initially responds very slowly; ii) interest rates

initially rise; and iii) aggregate output initially falls, with a J-shaped response,

and a zero long-run effect of monetary policy shock (long-run monetary policy

neutrality). However, VAR models of monetary policy shocks have exclusively

concentrated on simulating the response of the economy to shocks, leaving out

the systematic component of monetary policy – as formalized in the monetary

policy reaction function. In what follows, we first explore the theoretical

aspects of reduced VAR methods which is a building block to the formulation

of Structural VAR applied to the analysis of MTM.

3.2 Exploring the VAR(p) Methods for the Analysis of MTM

For ease of understanding, consider a Bivariate VAR system, for 𝑧1and 𝑧2, each

lagged once i.e. a VAR(1) set out as in eqn. 5

𝑧1𝑡 = 𝑎10 + 𝑎11𝑧1𝑡−1 + 𝑎12𝑧2𝑡−1 + 휀1𝑡𝑧2𝑡 = 𝑎20 + 𝑎21𝑧1𝑡−1 + 𝑎22𝑧2𝑡−1 + 휀2𝑡

𝐼𝑛 𝑚𝑎𝑡𝑟𝑖𝑥 𝑛𝑜𝑡𝑎𝑡𝑖𝑜𝑛

(𝑧1𝑡𝑧2𝑡)

⏟ 𝐙t

= (𝑎10𝑎20)

⏟ 𝐀0

+ (𝑎11 𝑎12𝑎21 𝑎22

)⏟

𝐀1

[𝑧1𝑡−1𝑧2𝑡−1

]⏟ 𝐙t−1

+ (휀1𝑡휀2𝑡)

⏟ 𝛆t

𝐶𝑜𝑚𝑝𝑎𝑐𝑡𝑒𝑑 𝑡𝑜𝐙t = 𝐀0 + 𝐀1𝐙t−1 + 𝛆t

(5)

In these all these varied expressions, each observed variable is expressed

equivalently as a linear function of their past values (own lag) and past values

(the lag) of the other variable plus white noise residuals, with no current dated

variables on the right-hand side (RHS), i.e. all regressors are predetermined. In

other words, all variables are potentially endogenous. Each of the 휀𝑖𝑡 is

1 Monetary policy shocks are defined as deviations from the monetary policy rule that are obtained by

considering an exogenous shock which does not alter the response of the monetary policy-maker to macroeconomic conditions.

Motivating VAR Modelling for Analysis of MTM

25 |

assumed to be stationary or more technically, integrated of order zero - I(0),

with zero mean and homoskedastic variance, that is, 𝐸[휀1𝑡] = 𝐸[휀2𝑡] = 0 and

𝐸[휀1𝑡2 ] = 𝐸[휀2𝑡

2 ] = 𝛿𝑖2, mean and variance are time invariant. A further

assumption is that the auto Covariances are equal to zero, i.e. 𝐸[휀1𝑡휀1𝑡−1] = 0

and 𝐸[휀2𝑡휀2𝑡−1] = 0. Critically in the VAR set up, shocks such as 휀1𝑡 and 휀2𝑡

can be contemporaneously correlated across equations. In this

case, 𝐸[휀1𝑡휀2𝑡] ≠ 0, which guarantees the structural/causal relationships to be

embedded in the data, although not explicitly modelled – at least at first. More

generally, a VAR(p) system of k-autoregressive equations becomes:

𝐙t = 𝐀0 + 𝐀1𝐙t−1 + A2Zt−2 +⋯+ Ap𝐙t−p + 𝛆t (6)

Where: 𝒁𝑡 is a (k1) vector of variables (= equations) at time t; A0 is a (k1)

vector of constants; Ai are (kk) matrices of coefficients; t is a (k1) vector of

errors at time t, which are assumed to be identically and independently

distributed, i.e., have zero mean (𝐸(휀𝑡) = 0), and homoskedastic variance,

(𝐸(휀𝑡2 = 𝛿2)), are serially uncorrelated (𝐸(휀𝑡휀𝑡−𝑝

′ ) = 0) for 𝑝 ≠ 0) and have a

time-invariant positive definite variance-covariance matrix, Ω, i.e. (𝐸(휀𝑡휀𝑡′) =

Ω). Thus, the error terms follow a white noise process, i.e., t ~ (0, Ω). The

residual covariance matrix, Ω has dimensions k x k , and contains information

about possible contemporaneous effects. Showing this is straight forward.

Consider a four variable VAR, such that in equation 6;

𝛆t = (

1𝑡2𝑡3𝑡4𝑡

)

We want to show that 𝐸(휀𝑡휀𝑡′) = Ω and that Ω has dimensions k x k , i.e.,

contains information about possible contemporaneous effects.

Given 𝛆t above,

𝐸(휀𝑡휀𝑡′) = Ω = 𝐸 (

1𝑡2𝑡3𝑡4𝑡

) [1𝑡 2𝑡 3𝑡 4𝑡]


|26

Ω = 𝐸

(

1𝑡2 1𝑡2𝑡 1𝑡3𝑡 1𝑡4𝑡

2𝑡1𝑡 2𝑡2 2𝑡3𝑡 2𝑡4𝑡

3𝑡1𝑡 3𝑡2𝑡 3𝑡2 3𝑡4𝑡

4𝑡1𝑡 4𝑡2𝑡 4𝑡3𝑡 4𝑡2)

Exploring the knowledge that 𝐸(휀𝑡2) = 𝛿2, and 𝐸(휀𝑡휀𝑡−𝑝

′ ) = 0 for 𝑝 ≠ 0),

then

Ω=

(

𝛿12 0 0 0

0 𝛿22 0 0

0 0 𝛿32 0

0 0 0 𝛿42)

We also know that 𝐸(휀𝑡2) = 𝛿2 = 1, thus

Ω = (

1 0 0 00 1 0 00 0 1 00 0 0 1

)

⏟ 4 𝑋 4

= 𝐼𝑘 (7)

Eqn. 7 shows that an innovation in any one variable has contemporaneous

effect only to itself, signalling a key flaw of unrestricted VAR in impulse

response analysis. The fact that 𝐸(휀𝑖𝑗) = 0; ji devoids the innovations in

the model from interactions among the variables – the interaction that is the

very essence of VAR. Indeed, as can be seen, 휀𝑖𝑖 or 휀𝑗𝑗 shock affects itself but

not 휀𝑖𝑗 or 휀𝑗𝑖 . Put otherwise, the VAR does not describe economic ‘structure

form’, although we may be able to recover it via testing the VAR. Thus,

traditional VARs, such as one in eqn. 6 are of limited use. They have been a

subject of Lucas critique (Lucas, 1972), primarily for lack of knowledge of

complete structural economic structure (atheoretical), and moreover, are prone

to spurious regression problem when data are non-stationary.

Chapter 4

Taking MTM Theories and the VAR Model to the Data

To demonstrate as many of the monetary transmission mechanisms, discussed

in section 2 above, we have provided data in the COMESA2018 EViews work

file on seven variables, for data covering the period 2000Q1-2017Q3 (some 71

observations), all on Uganda, except international oil price index. The variables

are quarterly time series observations on core consumer price index

(Core_CPI), nominal exchange rate (exr)-UGX/USD rate, gross domestic

product (GDP) – in 2009/10 constant prices, weighted average bank lending

rate (lrate), a measure of monetary aggregates – broad money (M2), credit to

the private sector (PSC) and three months Treasury bill rate (tb91) – which is

the key short-term interest rate used by the Bank of Uganda to signal its

monetary policy stance. Two of these variables, i.e., output and price are non-

policy variables and are, as discussed earlier, a focus in the transmission of

monetary policy, irrespective of the monetary policy regime in force. The rest

of the variables capture most of the most likely channels in COMESA region

for which the data is readily available, but on Uganda. These include the

interest rate (tb91) channel, the exchange rate (exr) channel, the credit and/or

bank lending (lrate and/or PSC) channel and the money (M2) channel. An

index of international commodity prices (oil_price) is included, but as an

exogenous variable.

Given the data and the purpose of this compilation, it is important, to mention

upfront, that we must focus on the following data related technical issues:

i. Seasonal adjustment and transformation of series to natural

logarithms

ii. Pre-testing for order of integration

iii. Determining the lag length of VAR and suitability of VAR

iv. Stability of the VAR


|28

The data being of relatively high frequency, it is important that before anything

else, the data is adjusted for seasonal effects.

4.1 Adjustments for Seasonal Effects

Economic analysis is focused on business cycles. As such performing an

analysis on variables with seasonality in them is a recipe for incorrectly

characterizing cyclical behaviour, and the ensuing results would be spurious

(Dejong and Dave, 2007; Nyanzi and Bwire, 2017). Testing and adjusting the

series for seasonality effects in EViews is done using the popular X12 method

by the Census Bureau of Statistics. Like any other built in routine procedures in

EViews, X12 method is an automated procedure, but is available only for

monthly and quarterly series for at least 3 full years.

The series at hand are of a quarterly frequency and spans for 17 years, making

it a suitable candidate for seasonal effects adjustment. In applied time series, it

is a common suspicion that quarterly GDP, CPI, oil_price and M2 series are

embedded with strong seasonal components. This claim can easily be verified

with series plots, a process we briefly describe.

To plot these series in EViews, click

on any one of the four series, which

then gets highlighted. While holding

the control key down, click on each of

the remaining three series, but one at a

time. This simple procedure highlights

all the four series of interest. Once

these are highlighted, and while

maintaining the cursor in the

highlighted area, right click on the

mouse, then from the drop-down menu

items, choose Open, and following

the arrow, choose, from the drop-down

menu items, as Group, as shown in the accompanying road map.

Executing the above procedure opens the series in a spread-like sheet, shown

in the screen print. While in this spread-like sheet, click on View (at the

extreme left had side corner), and from the drop-down menu, navigate to

Graph…as highlighted.


29 |

Click on graph, and by default, EViews highlight Basic type under Option

Pages and Line & Symbol under graph type as in the screen here.


|30

Click OK and you will see the graphs in the screen print. We could as well use

the Quick icon at the top- most window for the same results, except that here,

instead of opening the series data points, we would be required to highlight the

series name.

However, as can be seen, there are scaling issues, especially with cpi and oil

price series, to the extent that, relative to GDP and M2 series, they are hardly

noticeable. Given this, we would have to allow for dual axis scaling, whereof,

we can plot both CPI and oil price series on the secondary axis. Achieving this

is also straight forward.

While in the above EViews graph window, double click in the legend to yield a

screen reprinted here. Here we can do all sorts of statistical manipulation of the

graph (s), including choices of

secondary axis, which we explore.

We want first to specify upfront

from the legend, which variable

we want to put on the secondary

axis or rhs. As we can see, under

Legend, Attributes is highlighted,

and we can easily edit legend

entries by clicking on the legend

of interest, e.g. CPI and edit to

include (rhs) and so is oil as

shown, noting that CPI is ordered

first while oil is in the fourth

position.

Click on Axes & Scaling-one of the

entries under Option Pages

(appeared at the left-hand side of the

window), and as we clearly see from

the extreme right-hand side of the

screen that pops up in the dialogue

box for Series axis assignment, all

our series are, by default, assigned to

the left hand-side axis (primary axis). Recall that as per ordering above, CPI

came first, and this is the exact order EViews is following – the first series is

highlighted and indicated as #1 CPI (rhs). We want to transfer this and the


31 |

one in position 4 (oil (rhs)) to the Right axis. Check Right (providing 1 Left is

still highlighted) and click on 4 Left and check Right, and in the Dual axis

scaling dialogue box, change to Overlap scales (lines cross) – to get to the

following screen print. Click OK to yield Figure 2.

Figure 2: Quarterly GDP, CPI and oil_price in levels

Indeed, as can be seen, GDP and to some extent, oil prices are rugged,

reflecting a degree of seasonality. Within years, GDP tended to be highest in

the third quarter and lower in the first quarter while oil prices peaked in the

first and second quarters (and sometimes, to a lesser extent in the third

quarter). It is still possible, whilst in the graph window to retrieve the series

spread sheet. Click on View and in the drop-down menu, select Spread Sheet.

While nominal exchange rate could as well be embedded with strong seasonal

components, it has been argued in empirical setting that the seasonality here is

not regular and that adjusting it for seasonal effects would contradict the

assumption of rational behaviour in financial markets (Bwire, Opolot and

Anguyo 2013). In what follows, we describe the implementation of X12

procedure in EViews on quarterly GDP series, a procedure which then can be

replicated for the rest of the qualifying series. The null hypothesis for the test is

that seasonality is present.


|32

We implement this in the

COMESA2018 work file, using data in

the Demo page. In the workspace,

highlight and open GDP series. In the

open series window (excel like sheet),

click Proc, and point the cursor on

Seasonal Adjustment in the drop-down

menu of Proc and select Census X12... -

a road map highlighted the print screen

herewith.

Select Census X12...This brings forth the X12 Options screen, which is also

replicated in the screen print below. In the box for X11 Method – under

Seasonal Adjustment, a choice must be made between the EViews default of

Multiplicative and Additive – the

most popular two adjustment methods

in applied time series. Multiplicative

method applies when the series to be

adjusted is nonstationary and the series

values are strictly non-negative.

Additive method applies when the

series to be adjusted is in stationary

form, but in addition, the series values

must be positive. GDP in our

application here is nonstationary and

with no negative entries.

Given this, we retain the default of Multiplicative, retaining all the other default

options as given. Observe that under Component Series to Save, the Base

name for the series to be adjusted is given as GDP and the Final seasonally

adjusted series will appear in the work space window with an extension_SA

(where suffix SA is used to mean seasonally adjusted). This box is by default

checked. Click OK to implement the procedure.

The resulting results tables are usually too large, but for purposes of

demonstration, an extract of the three broadly available F- tests for seasonality

in the large results table: 1) Tests for the presence of seasonality assuming stability; 2)

Nonparametric test for the presence of seasonality assuming stability; and 3) Moving

seasonality test, performed on quarterly GDP series is provided.


33 |

D 8.A F-tests for seasonality Test for the presence of seasonality assuming stability. Sum of Dgrs.of Mean Squares Freedom Square F-Value Between quarters 716.9275 3 238.97584 26.200** Residual 611.1111 67 9.12106 Total 1328.0386 70 **Seasonality present at the 0.1 per cent level. Nonparametric Test for the Presence of Seasonality Assuming Stability Kruskal-Wallis Degrees of Probability Statistic Freedom Level 34.1783 3 0.000% Seasonality present at the one percent level. Moving Seasonality Test Sum of Dgrs.of Mean Squares Freedom Square F-value Between Years 185.1699 16 11.573117 2.005 Error 277.0066 48 5.770970 Moving seasonality present at the five percent level.

As can be seen, both the test for the presence of seasonality assuming stability

and nonparametric test for the presence of seasonality assuming stability find

evidence that seasonality is present at the 0.1 and one percent levels,

respectively, and so is moving

seasonality, which is present at the five

percent level. Replicating this procedure

for CPI and for oil reveals as expected,

presence of seasonality.

And as can be seen in the screen print

here, all these series bear the extension

_SA. Note however that even if the null

hypothesis of presence of seasonality is

rejected, the series in question will still

have this extension, providing the test

has been implemented. How to use the

series going forward is the user’s cup of technical business, but generally, when

a variable has no seasonality component, the user must revert to the unadjusted


|34

series, but use adjusted series where seasonality is present, in economic

modelling and analysis thereafter.

The corresponding pairs of the three seasonally adjusted series are plotted, in

Figure 3, for visual appreciation of the difference it makes between true

economic cycle and economic seasons – which is characterized by up- and

down- turns.

Figure 3: Comparing seasonally unadjusted and seasonally adjusted series

0

20

40

60

80

100

120

140

00:1

00:3

01:1

01:3

02:1

02:3

03:1

03:3

04:1

04:3

05:1

05:3

06:1

06:3

07:1

07:3

08:1

08:3

09:1

09:3

10:1

10:3

11:1

11:3

12:1

12:3

13:1

13:3

14:1

14:3

15:1

15:3

16:1

16:3

17:1

17:3

OIL

OIL_SA

4,000

6,000

8,000

10,000

12,000

14,000

16,000

18,000

00

:10

0:3

01

:10

1:3

02

:10

2:3

03

:10

3:3

04

:10

4:3

05

:10

5:3

06

:10

6:3

07

:10

7:3

08

:10

8:3

09

:10

9:3

10

:11

0:3

11

:11

1:3

12

:11

2:3

13

:11

3:3

14

:11

4:3

15

:11

5:3

16

:11

6:3

17

:11

7:3

GDP

GDP_SA

40

60

80

100

120

140

160

180

00:1

00:3

01:1

01:3

02:1

02:3

03:1

03:3

04:1

04:3

05:1

05:3

06:1

06:3

07:1

07:3

08:1

08:3

09:1

09:3

10:1

10:3

11:1

11:3

12:1

12:3

13:1

13:3

14:1

14:3

15:1

15:3

16:1

16:3

17:1

17:3

CPI

CPI_SA

0

2,000

4,000

6,000

8,000

10,000

12,000

14,000

16,000

00

:10

0:3

01

:10

1:3

02

:10

2:3

03

:10

3:3

04

:10

4:3

05

:10

5:3

06

:10

6:3

07

:10

7:3

08

:10

8:3

09

:10

9:3

10

:11

0:3

11

:11

1:3

12

:11

2:3

13

:11

3:3

14

:11

4:3

15

:11

5:3

16

:11

6:3

17

:11

7:3

M2

M2_SA

Based on the evidence of the presence of seasonality effects, going forward,

core_cpi, GDP, M2 and oil_prices are adjusted for seasonal effects, but for

simplicity, the suffix_sa is dropped.

4.2 Deriving the Output GAP

Another important variable transformation which is common in monetary

policy making circles is the output gap. Though may not be used here because

we are not analyzing a gap model, it is important to understand how it is

derived. It is the deviation of actual output from trend, where trend output is

generated in EViews using an in-built Hodrick-Prescott (HP) filter routine,


35 |

from the GDP series available to the user. A positive output gap indicates

‘excess demand’ position, which implies that output is above its trend level and

the reverse is true when it is negative. Output gap is an important premise of

the Taylor principle in the wider standard neo-Keynesian macroeconomic

models, which are used by central banks for policy analysis and forecasting. In

here, we want to describe how the output gap is generated in EViews using the

inbuilt HP filter routine.

Turning to Demo page of EViews COMESA2018 work file (WF) that we have

used all along, there, among other series, is GDP_sa. The first thing we want to

do, also a standard practice in applied time series modelling, is to transform the

GDP_sa series to natural logarithms. To do this, in the command window, type

a command of the form genr lgdp = log(gdp_sa), then press the enter button on

your key board. This adds a new series, lgdp onto the list of variables in the

work space window.

Now double click on lgdp to open the series in an excel-like worksheet. Whilst

in the lgdp open spread sheet, click on Proc, and in the drop-down window, is

Hodrick-Prescott Filter…, as shown in the screen shot.

Clicking on Hodrick-Prescott Filter…, - brings Hodrick-Prescott Filter

window - also printed here. This specifies the Smoothed series as hptrend01

but is no more than GDP trend series (in this case). The Cycle series, here, is

the output gap, and is computed behind the scene as the difference between

the actual GDP (entered as raw data) and the software’s computed GDP trend

series. Here, I have intentionally declared its series name upfront as

‘output_gap’ and so upon execution; it will be reflected among the other series


|36

in the work space window. Else, if not declared upfront, it will not be

computed and the user would have to compute it through the command

window, with the following command: genr output_gap=lgdp-hptrend01, which

inevitably is a longer route. The value of the smoothing parameter, lambda (λ)

varies with data frequency:

𝜆 =

100: 𝑖𝑓 𝑎𝑛𝑛𝑢𝑎𝑙 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦1600: 𝑖𝑓 𝑞𝑢𝑎𝑟𝑡𝑒𝑟𝑙𝑦 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦

14400: 𝑖𝑓 𝑚𝑜𝑛𝑡ℎ𝑙𝑦 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦.

Check OK for the output in Figure 4.

Figure 4: Output gap: Hodrick-Prescott Filter (λ=1600)

-.06

-.04

-.02

.00

.02

.04

.06

8.4

8.6

8.8

9.0

9.2

9.4

9.6

9.8

00:1

00:3

01:1

01:3

02:1

02:3

03:1

03:3

04:1

04:3

05:1

05:3

06:1

06:3

07:1

07:3

08:1

08:3

09:1

09:3

10:1

10:3

11:1

11:3

12:1

12:3

13:1

13:3

14:1

14:3

15:1

15:3

16:1

16:3

17:1

17:3

LGDP (secondary axix)

Trend (secondary axis)

Cycle

It is the cycle or output gap that is of interest for policy analysis and

forecasting, and this, in our EViews work space, is appeared as output_gap.

Note however, for emphasis that we are not analyzing a gap model and

therefore, while generating the output gap is a useful exercise for our

knowledge, the variable is not of use here.

The next thing we do, except for interest rates, which are percentages, is to

express the rest of the variables (duly tested for seasonal effects) in natural

logarithms to stabilize the second moment, the variance.

In the active COMESA2018 work file, supply, for demonstration purposes, in

the command window, commands of the following form:


37 |

Genr lcpi = log(cpi_sa) Genr lexr=log(exr) Genr lm2=log(m2_sa) Genr loil=log(oil_sa) Genr lpsc=log(psc)

Each of these commands

is executed on pressing

the enter button on your

key board. Resulting from

this, a whole host of new

corresponding series, lcpi,

lexr, lgdp, lm2, loil and lpsc

are created and are

automatically added onto

the list of variables in the

work space, as shown in the screen print.

All the data are from the bank of Uganda data bases. But prior to any empirical

analysis of this kind of data and models, it is always a good idea to take some

time to simply examine the data. This often, as a precursor, requires the

modeller to have a graphical exposition of the level and first difference of the

series to, among others, unearth important data features.

The series in level and first difference are given in Figure 5. Plotting the series

is important, not least because it gives us the opportunity to get familiar with

the data’s important characteristics - features that make modelling even far

more exciting. If we do not know what the features of our data are, we are not

going to be able to develop a particularly good model of them. It is in general

vital that we know our data, especially in time series modelling.


|38

Figure 5: Variables in levels and first differences

4.0

4.2

4.4

4.6

4.8

5.0

5.2

00:1

00:3

01:1

01:3

02:1

02:3

03:1

03:3

04:1

04:3

05:1

05:3

06:1

06:3

07:1

07:3

08:1

08:3

09:1

09:3

10:1

10:3

11:1

11:3

12:1

12:3

13:1

13:3

14:1

14:3

15:1

15:3

16:1

16:3

17:1

17:3

lcpi

-.01

.00

.01

.02

.03

.04

.05

.06

.07

.08

00:1

00:3

01:1

01:3

02:1

02:3

03:1

03:3

04:1

04:3

05:1

05:3

06:1

06:3

07:1

07:3

08:1

08:3

09:1

09:3

10:1

10:3

11:1

11:3

12:1

12:3

13:1

13:3

14:1

14:3

15:1

15:3

16:1

16:3

17:1

17:3

dlcpi

7.3

7.4

7.5

7.6

7.7

7.8

7.9

8.0

8.1

8.2

00:1

00:3

01:1

01:3

02:1

02:3

03:1

03:3

04:1

04:3

05:1

05:3

06:1

06:3

07:1

07:3

08:1

08:3

09:1

09:3

10:1

10:3

11:1

11:3

12:1

12:3

13:1

13:3

14:1

14:3

15:1

15:3

16:1

16:3

17:1

17:3

lexr

-.10

-.05

.00

.05

.10

.15

.20

00:1

00:3

01:1

01:3

02:1

02:3

03:1

03:3

04:1

04:3

05:1

05:3

06:1

06:3

07:1

07:3

08:1

08:3

09:1

09:3

10:1

10:3

11:1

11:3

12:1

12:3

13:1

13:3

14:1

14:3

15:1

15:3

16:1

16:3

17:1

17:3

dlexr


39 |

8.6

8.8

9.0

9.2

9.4

9.6

9.8

00:1

00:3

01:1

01:3

02:1

02:3

03:1

03:3

04:1

04:3

05:1

05:3

06:1

06:3

07:1

07:3

08:1

08:3

09:1

09:3

10:1

10:3

11:1

11:3

12:1

12:3

13:1

13:3

14:1

14:3

15:1

15:3

16:1

16:3

17:1

17:3

lgdp

-.08

-.06

-.04

-.02

.00

.02

.04

.06

.08

00:1

00:3

01:1

01:3

02:1

02:3

03:1

03:3

04:1

04:3

05:1

05:3

06:1

06:3

07:1

07:3

08:1

08:3

09:1

09:3

10:1

10:3

11:1

11:3

12:1

12:3

13:1

13:3

14:1

14:3

15:1

15:3

16:1

16:3

17:1

17:3

dlgdp

6.8

7.2

7.6

8.0

8.4

8.8

9.2

9.6

00:1

00:3

01:1

01:3

02:1

02:3

03:1

03:3

04:1

04:3

05:1

05:3

06:1

06:3

07:1

07:3

08:1

08:3

09:1

09:3

10:1

10:3

11:1

11:3

12:1

12:3

13:1

13:3

14:1

14:3

15:1

15:3

16:1

16:3

17:1

17:3

lM2

-.04

-.02

.00

.02

.04

.06

.08

.10

.12

00:1

00:3

01:1

01:3

02:1

02:3

03:1

03:3

04:1

04:3

05:1

05:3

06:1

06:3

07:1

07:3

08:1

08:3

09:1

09:3

10:1

10:3

11:1

11:3

12:1

12:3

13:1

13:3

14:1

14:3

15:1

15:3

16:1

16:3

17:1

17:3

dlM2


|40

2.8

3.2

3.6

4.0

4.4

4.8

5.2

00:

10

0:3

01:

10

1:3

02:

10

2:3

03:

10

3:3

04:

10

4:3

05:

10

5:3

06:

10

6:3

07:

10

7:3

08:

10

8:3

09:

10

9:3

10:

11

0:3

11:

11

1:3

12:

11

2:3

13:

11

3:3

14:

11

4:3

15:

11

5:3

16:

11

6:3

17:

11

7:3

loil

-.8

-.6

-.4

-.2

.0

.2

.4

00:1

00:3

01:1

01:3

02:1

02:3

03:1

03:3

04:1

04:3

05:1

05:3

06:1

06:3

07:1

07:3

08:1

08:3

09:1

09:3

10:1

10:3

11:1

11:3

12:1

12:3

13:1

13:3

14:1

14:3

15:1

15:3

16:1

16:3

17:1

17:3

dloil

6.0

6.5

7.0

7.5

8.0

8.5

9.0

9.5

00:1

00:3

01:1

01:3

02:1

02:3

03:1

03:3

04:1

04:3

05:1

05:3

06:1

06:3

07:1

07:3

08:1

08:3

09:1

09:3

10:1

10:3

11:1

11:3

12:1

12:3

13:1

13:3

14:1

14:3

15:1

15:3

16:1

16:3

17:1

17:3

lpsc

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

00:1

00:3

01:1

01:3

02:1

02:3

03:1

03:3

04:1

04:3

05:1

05:3

06:1

06:3

07:1

07:3

08:1

08:3

09:1

09:3

10:1

10:3

11:1

11:3

12:1

12:3

13:1

13:3

14:1

14:3

15:1

15:3

16:1

16:3

17:1

17:3

dlpsc


41 |

16

18

20

22

24

26

28

00:1

00:3

01:1

01:3

02:1

02:3

03:1

03:3

04:1

04:3

05:1

05:3

06:1

06:3

07:1

07:3

08:1

08:3

09:1

09:3

10:1

10:3

11:1

11:3

12:1

12:3

13:1

13:3

14:1

14:3

15:1

15:3

16:1

16:3

17:1

17:3

lrate

0

4

8

12

16

20

24

00:1

00:3

01:1

01:3

02:1

02:3

03:1

03:3

04:1

04:3

05:1

05:3

06:1

06:3

07:1

07:3

08:1

08:3

09:1

09:3

10:1

10:3

11:1

11:3

12:1

12:3

13:1

13:3

14:1

14:3

15:1

15:3

16:1

16:3

17:1

17:3

tb91

16

18

20

22

24

26

28

00:1

00:3

01:1

01:3

02:1

02:3

03:1

03:3

04:1

04:3

05:1

05:3

06:1

06:3

07:1

07:3

08:1

08:3

09:1

09:3

10:1

10:3

11:1

11:3

12:1

12:3

13:1

13:3

14:1

14:3

15:1

15:3

16:1

16:3

17:1

17:3

lrate (%)

0

4

8

12

16

20

24

00:1

00:3

01:1

01:3

02:1

02:3

03:1

03:3

04:1

04:3

05:1

05:3

06:1

06:3

07:1

07:3

08:1

08:3

09:1

09:3

10:1

10:3

11:1

11:3

12:1

12:3

13:1

13:3

14:1

14:3

15:1

15:3

16:1

16:3

17:1

17:3

tb91 (%)


|42

Save for tb91 and to a slight extent lrate whose mean reversion behaviour is

somewhat unclear, the remainder of the series in levels clearly exhibit trend like

behaviour over time (i.e. trending) as they are not mean-reverting but appears

stationary around trend (i.e. trend-stationary) in first difference. The first

difference plots also seem to point to a deep in GDP and oil_prices around

2008 and a spike in inflation around 2011. These correspond to events that we

all can pin down quite easily, namely the 2008 global financial crisis and the

increases in world commodity prices beginning the third quarter of 2011,

amplified by unprecedented exchange rate depreciation and rapid expansion in

private sector credit in Uganda in 2011. We might want to think of shift

dummies to capture these shocks, but only if they can be found to improve the

analysis.

4.3 Unit Root Testing

After visual inspection of the data, the series are formally tested for the order

of integration or non-stationarity and the degree of differencing required to

induce stationarity using the commonly used Augmented Dickey Fuller (ADF)

unit root test (Dickey and Fuller, 1979). But before deploying the test static,

some explanatory notes are in order.

Economic time series are commonly characterised by strong ‘trend-like’

behaviour. Orthodox methods of estimation and hypothesis testing assume

that all variables are stationary (loosely speaking, ‘trend-free’). If no account is

made of this trend-like behaviour then the OLS estimator can give rise to

highly misleading results (spurious regression - Granger and Newbold (1974)) -

often characterised by significant t-ratios and a high explanatory power, even

though the regressors are economically unrelated to the variable being

explained.

In this section, we show how the univariate tests for non-stationarity can be

used to test for the existence of real (as opposed to ‘spurious’) relationships

between data series. Because (most) economic time series have strong trend

components we often find that many different regressions will apparently

‘explain’ the same variable of interest. Each competing regression may have

significant t-ratios and high explanatory power, even though the regressors are

economically unrelated to the variable being explained. This arises due to the

presence of trends in the variables, nothing more. Any trending variable will be

highly correlated with any other, even though the reason for their trends is


43 |

separate (in general, correlation does not mean causation). Moreover, where

trends are present in the data, test statistics (t and F) are not distributed

according to their usual distributions and thus standard critical values are

almost always incorrect.

Although the presence of trend-like behaviour is apparent in the series in levels

simply by plotting the data (as in Figure 5 above), the behaviour can be

generated in two distinct ways. Thus, in order to be able to remove the trend in

the data, it is essential that we know what type of mechanism is creating the

trend in the first place. To go about this, consider the autoregressive [AR]

model,

ttt ztccz 121 ; 2,0~ nidt (8)

This class of process embodies a wide range of processes, some of which

exhibit trend-like behaviour, and some that do not.

If we assume that 021 cc , then ttz is devoid of any ‘structure’

(i.e., tz is a completely random process), and thus clearly cannot have any

trend-like behaviour.

For 01 c , 02 c the process has a non-zero mean, but still no trend.

For 10 , the series has some structure (it is related to its previous value,

i.e. what happens today (and in the past) influences what happens tomorrow -

some dependence) but no trend. Such processes are called stationary or integrated

of order zero, denoted I(0).

If we now allow 02 c , ty has a deterministic (linear trend) and is said to be

stationary about trend or trend stationary. Note that series of this sort will always rise

( 02 c ) or fall ( 02 c ) albeit with some 'noise'. Such deterministic trend

behaviour approximates many processes we tend to find in the real world. And

are called non-stationary series and are often integrated of order one, denoted I(1).

They are said to contain a unit root because they occur when, in (8), 1

Besides I(1) processes exhibiting trend-like behaviour, the unit root could, in

addition, be due to the stochastic trend. Stochastic trend is generated through

the accumulation of the random process, t . It may seem counter-intuitive, but


|44

the accumulation of a white noise process such as t produces a series that has

long swings of trend-like behaviour, that we describe as stochastic trend. Over

(arbitrarily) long periods of time, the ‘trend’ may be positive and then become

negative. The point at which the ‘trend’ switches, the magnitude of the ‘trend’

and the duration of the period in which a ‘trend’ is apparent is completely

random, depending simply on the run of disturbances that make up the

stochastic process t (the inverted commas are used above since the behaviour

is not actually due to the presence of a (deterministic linear) trend, it simply

looks like it). Asymptotically, (i.e. as t ) the clear distinction between

stochastic and deterministic trend is obvious, however even in samples of 100

observations, they can be easily confused, even though the mechanisms

generating the behaviours are poles apart (one is deterministic and the other is

stochastic).

Random Walk process:

The idea of stochastic trends is fundamental to modern time series analysis, not least

because they seem to mimic rather well the real-world series that we observe in

economics. To fix the idea of the stochastic trend, consider the following analogy.

Imagine we are tracing the journey of a man who takes a step forward to the left if the

toss of a coin is tails and a step forward to the right when the coin comes up heads.

Each step forward (whether to the left or the right) on the path of his randomly

selected footsteps will take him away from his starting point. Successive 'tails' will

mean she walks forward in a leftwards direction.

Occasional 'heads' on this part of his journey will cause his path to deviate but she will

still be walking in an approximately leftward direction. Only when there are a number

of successive heads will she appear to change course and head off in the rightwards

direction. Changes in direction are akin to changes in trend and because they are

caused by chance (successive flips of the coin that come out as heads or tails), a time

series that contains stochastic trends is often called a random walk. So, although each

step is random to the right or left, this does not mean that she stays put; far from it.

Because her journey represents the accumulation of her steps, some in the leftwards

direction and some in a rightwards direction, she meanders further away from her

starting point the more steps there are to her journey.

It may strike you as odd, but this random walk behaviour mimics rather well the sort

of ‘trend-like’ behaviour we observe in economic time series, in that sometimes,

economic time series (say, GDP) are trending upward, sometimes downward and

sometimes they appear to move without any trend-like behaviour at all. In contrast, if

a true (linear) deterministic trend were driving the process (say, GDP), the trend


45 |

would relentlessly take the variable in one direction only. Deviations from upward (or

downward) trend would only be transient. It is not plausible to observe the series

display a mixture of upward, downward (or indeed no) trend if the series has a

deterministic trend driving it (except if the trend is varying across sub-periods). Such

deterministic behaviour of relentless growth or decline, while possible in economics, is

not really plausible particularly over long-time scales.

As we will be showing, it is important to be able to distinguish between

deterministic and stochastic trends. Testing for the existence of trend is called

testing for non-stationarity (or testing for a unit root) and this is now a

standard pre-test that is conducted prior to all regression analysis involving

time series data. Note that if the ‘trend’ is deterministic, the process is

sometimes called trend-stationary whereas if the ‘trend’ is stochastic, the process

is said to be difference-stationary. These additional labels simply reflect that ‘de-

trending’ removes a deterministic trend whereas differencing removes a

stochastic trend. Strictly speaking, taking the first difference will remove both

types of trend-like behaviour. Subtracting a linear trend on the other hand will

not remove a stochastic trend, but only a deterministic trend. The downside is

that while this removes the trends in the data, it throws away valuable

information about the ‘long-run’ behaviour of the variables (about which

economic theory is informative) leaving only the ‘short-run’ behaviour (the

deviation about trend)

about which economic

theory has little to say.

As we show later, the

modern estimation

techniques for the non-

stationary environment

allows us to describe

both short and long run

behaviours yet avoid

‘spurious regression’,

that is so common with

I(1) data. In the

literature, the terms

difference stationary,

I(1), unit root and non-

stationary all denote a non-stationary process and are used interchangeably.


|46

Given the discussion above, having a graphical visual inspection (as in Figure

5), complemented by a look at the level and difference autocorrelation plots (a

process we describe below), is a really good starting point to categorising the

series.

Taking an example of lgdp, that is now

familiar; to generate the autocorrelation,

double click on the lgdp series in the

COMESA2018 EViews file to open the

series. In the window containing the data

spreadsheet, click on View icon at the

extreme left-hand side corner and select

Correlogram…, a route shown in the

screen print.

In the small window that opens from the above route (also shown here),

ensure, as shown, under Correlogram of, choose Level (if level data) and 1st

difference (if in first difference). Now we choose Level and 1st difference

thereafter. Click OK to generate the figure in the left-hand side panel of Figure

6. Now go through the same process, but this time, choose 1st difference to

generate the figure on the right-hand side panel of Figure 6.

Figure 6: Level, first difference autocorrelations

Autocorrelation Partial Correlation

Autocorrelation Partial Correlation

The autocorrelations of a non-stationary series tend to decay very slowly to

zero as t while the decay is rapid for a stationary series. The

autocorrelations behaviour in levels and first difference mimics non-stationarity


47 |

and stationarity, respectively. Nonetheless, appropriate categorization requires a

formal test.

In theory, a vector tz is said to be integrated of order d (i.e., )(~ dIzt ) if

variables in tz can be differenced d times to induce stationarity. A commonly

used test of non-stationarity is the Augmented Dickey-Fuller (ADF) test

(Dickey and Fuller, 1979). This test is designed to distinguish between

stationary (about mean or trend) and non-stationary processes.

Rather than estimating equation (8) directly, Dickey and Fuller proposed

subtracting from both sides1tz , a manipulation that gives:

ttt ztccz 121 (9)

Where 1 - since testing for the presence of a unit root is testing that

1 . To allow for more general autoregressive [AR(p)] processes, the ADF

regression takes the general form given by

1

120

i

tititt zztccz

(10)

Where, 0c is the intercept term, 2c and are coefficients of time trend and level

of lagged dependent variable, respectively, is the first difference operator and

t are white noise residuals. is the number of lagged ∆𝐳𝑡 terms required to

achieve white noise residuals in the ADF regression of tz.

The appropriate order of is an

empirical issue, thus in practice

is set high enough to include

the ‘true’ lag length for the

variable and is selected on the

basis of Akaike (AIC), Schwartz

(SC) and Hannan-Quinn (HQ)

information criteria, as shown in

the screen shot, in the drop-down

menu under Test type. EViews

default here is Augmented

Dickey-Fuller (or ADF for short).


|48

These criteria have the same basic formulation, i.e., derive from the log

likelihood ratio (LR) function but penalise for the loss of degrees of freedom

due to extra lags to different degrees (see Juselius 2006, 70-1), for detailed

exposition of these frameworks). In practice, we choose the model which

minimizes each model selection criterion. The marginal cost (or penalty) of

adding redundant regressors is greater with the Schwarz Bayesian Criterion

(SBC) than the AIC and hence they need not all select the same preferred

model. Although this lack of unanimity is frustrating, unless the unit root

inference changes between models of differing lag length; the choice of lag

length is, from a practical perspective, irrelevant. Remember that lagged

dependent variables are introduced to remove autocorrelation, so if the model

does not have any, you do not need to correct for it. Overall, SBC has superior

large sample properties, (it is consistent) but because it is prone to select an

under-parameterised (and hence potentially serially correlated) model in small

(i.e. usual) samples, the AIC is often the preferred test.

In eqn. 10, if 0 , the sequence tz contains a unit root, else it is stationary.

Therefore, the ADF test is conducted under the null hypothesis that 0 , so

is rejected if the

calculated t-statistic is

larger than the critical

value reported by

Dickey and Fuller

(1981) - the statistic

(see ADF table in

attachment or Table A

in Enders, 2010: 488).

The t-ratio on is

called the ADF test

statistic, and is

constructed to account

for the fact that critical

values of the t-statistic

do depend on whether

an intercept (0c )


49 |

and/or time trend ( t ) and intercept (0c ) and /or none of this is included in

the regression equation and on the sample size (Enders 2010: 206).

In an estimated ADF equation, an extensive exercise we turn to shortly, this is

given in the upper results window, shown here for ADF test on lrate equation

of eqn. 10.

Based on the same equation, we also evaluate whether the data generating

process (DGP) is characterized by non-stationarity with or without a linear

deterministic trend and a drift, and non-stationarity with or without a linear

deterministic trend. This involves testing joint hypotheses on the coefficients

of interest, i.e. 0,c and

2c . However, under non-stationarity, the computed

ADF- test statistic does not follow a standard t-distribution, but rather a dickey

Fuller (DF) distribution and so the critical values for these joint tests are also

non-standard. They follow the non-standard F-statistics denoted by 2 and

3

statistics which are constructed in the same way as ordinary F-tests (adopted

from Enders, 2010: 207), i.e.

KTedunrestrictSSR

redunrestrictSSRrestrictedSSRi

(11)

Where SSR (restricted) and SSR (unrestricted) are the sums of the squared residuals

from the restricted and unrestricted models, while r is the number of

restrictions, T is the number of usable observations and k is the number of

parameters estimated in the restricted model.

The joint hypothesis 020 cc , i.e. the significance or otherwise of a

constant term, time trend and non-stationarity is tested using the 2 -statistic.

The null hypothesis is then that the data are generated by the restricted model

and the alternative hypothesis is that the data are generated by the unrestricted

model. Thus, if 2 (calculated)

is smaller than 2 (critical)

, we accept the

restricted model and conclude that the restriction is not binding.

Similarly, the joint hypothesis 02 c , i.e. the sequence tz contains a

unit root and no linear deterministic trend is tested using the 3 -statistic, and is

evaluated on exactly the same grounds as the 2 -statistic. That is, the restricted

model is accepted, and the restriction is not binding if 3 (calculated) is smaller


|50

than 3 (critical)

. Critical values for the i - statistics are obtained from Table

B in Enders (2010: 489).

The ADF test is known to have low power if the series has undergone a

(permanent) regime shift during the period under consideration (Harris and

Sollis, 2005: 57) or if there are outliers in regression residuals. Such economic

behaviour needs to be included in the deterministic part of the model and is

likely to bias estimates and result in invalid inference if ignored (Juselius, 2003).

More precisely, Perron (1989) argues that when there are structural breaks, the

various Dickey-Fuller test statistics are biased towards the non-rejection of a

unit root, when in reality the series could simply be trend-stationary but

characterized by a structural break, which the test would fail to take into

account. Thus, because a break in the deterministic time trend introduces a

spurious unit root, one econometric procedure to test for unit roots in the

presence of a structural break involves splitting the sample into ‘q’- number of

regimes arising from the structural breaks and subjecting each part of the

sample period to the ADF test. Taking for example, the simplest case where

q=2, that is, one structural break, the sample series period would have to be

split into two sub-regimes, that is, t1 and Tt 1 with being the

proportion of the way through the sample where the break occurs.

However, this application is limited on two grounds: First, the degrees of

freedom for each of the resulting regressions are greatly diminished; generating

the likelihood that the probability estimates and associated critical values may

be unreliable for inference and may lack power in small samples (Mackinnon,

1996). Second, you may not know when the break point actually occurs (only

known to be possible with CATS procedure). It is therefore preferable to have

a single test based on the full sample (Perron, 1989). The test is based on the

estimates of the following equation;

∆𝑧𝑡 = 𝛼0 + 𝛼1𝑧𝑡−1 + 𝛼2𝑡 + 𝜇2𝐷𝐿 + ∑ 𝛽𝑖𝑘𝑖=1 ∆𝑧𝑡−𝑖 + 휀𝑡 (12)

Where, t denotes the time trend and, LD , is in general, a dummy that allows for

a one time change in the drift or a onetime change in both the mean and the

drift such that oDL for ,...,1t and 1LD for Tt ,...,1 . The L

subscript indicates the level of the dummy changes. This equation is the formal

Phillips-Perron (P-P) procedure for testing for unit roots in the presence of


51 |

structural change at time period 1t based on the full sample and test the

:0H 11 a , that is, tz is a unit root process, allowing for any structural break.

Note that unlike the ADF test for which we can evaluate the significance of

21 ,cc and coefficients from the critical values, the P-P test only generates

critical values for testing the significance of 1a coefficient, also the P-P statistic.

4.3.1 Demonstration of unit root testing using ADF test

Given the exposition above, in

the following, we conduct unit

root test for all the variables in

Figure 5 but using the most

popular ADF test. To proceed,

make active our COMESA2018

EViews work file, in which we

want to test for a unit root (s) in

lcpi variable, for demonstration

purposes.

Double click on lcpi. This opens

a spread sheet of lcpi raw data. In

this spread sheet window, click

on View (this is appeared at the

extreme left-hand side corner of

the window) and in the drop-down menu, there lies Unit Root Test…entry, as

highlighted here in the screen print.

Choosing Unit Root

Test…brings forth Unit Root

Test dialogue box, also

printed here.

As can be seen, under the Test

Type, if you check the drop-

down key, we find all the

available unit root tests, of

which we have discussed only

ADF and P-P, in both of


|52

which the null hypothesis is that the series contains a unit root(s). Please note

that you will need to fully understand each of these other tests before you can

ably apply them – the null hypothesis of non-stationarity is not always true for

all these tests! As mentioned earlier, I have limited myself in this demonstration

to the ADF unit root test, as appeared in the Test Type box. The next thing the

user will have to decide on is whether to test unit root in level, 1st difference or

2nd difference.

EViews understands that rarely will you get a time series integrated of order

more than two. The practice is that we first test for unit root in level and

depending on our judgement of its order of integration, i.e. the number of

times we will need to difference it to make it stationary, decide on whether it is

I(1) or I(2), where the former is 1st difference while the latter is 2nd difference.

We first conduct unit root on lcpi in level. We also have to decide on the

deterministic terms to include in the ADF equation, given by eqn. 10 in the

text. EViews provides all possible combinations of deterministic terms under

Includes in test equation. An appropriate specification of eqn. 10

corresponds to option 2 under here, as indicated. However, after the equation

has been estimated, we do a post mortem to evaluate the statistical significance

of each of the deterministic terms (intercept and trend) in the ADF model,

which then informs a true specification of the data generating process (d.g.p)

for the series at hand. This is not withstanding the fact that statistical

significance is not necessarily economic significance so some modeller’s

judgement may be exercised. Should there be need to drop any of these

deterministic terms or even both, the model is re-estimated for true

characterization of the unit root process.

Lastly, we must decide on how to determine the lag-length or the size of in

eqn. 10. EViews again provides all the available known lag-length criteria in the

Lag length drop-down key, some of which we have discussed. Here, as argued

above and as shown in the screen print, I adopt the Modified Akaike and in the

interest of preserving the degrees of freedom, set Maximum lags to 5 informed

by the rule of thumb of (𝑓 + 1), where 𝑓 is the data frequency. With these

entries, click OK, for the test results in Table 1.


53 |

Table 1: Unit test in CPI (levels)

Null Hypothesis: LCPI has a unit rootExogenous: Constant, Linear TrendLag Length: 3 (Automatic - based on Modified AIC, maxlag=5)

t-Statistic Prob.*

Augmented Dickey-Fuller test statistic -2.585515 0.2881Test critical values: 1% level -4.100935

5% level -3.47830510% level -3.166788

*MacKinnon (1996) one-sided p-values.

Augmented Dickey-Fuller Test EquationDependent Variable: D(LCPI)Method: Least SquaresDate: 09/11/18 Time: 09:48Sample (adjusted): 2001Q1 2017Q3Included observations: 67 after adjustments

Variable Coefficien... Std. Error t-Statistic Prob.

LCPI(-1) -0.064924 0.025111 -2.585515 0.0121D(LCPI(-1)) 0.606732 0.117779 5.151461 0.0000D(LCPI(-2)) 0.109973 0.141216 0.778753 0.4391D(LCPI(-3)) -0.211118 0.117674 -1.794095 0.0778

C 0.258458 0.097636 2.647169 0.0103@TREND("2000Q1") 0.001191 0.000444 2.680725 0.0094

R-squared 0.483579 Mean dependent var 0.015174Adjusted R-squared 0.441250 S.D. dependent var 0.012906S.E. of regression 0.009647 Akaike info criterion -6.359048Sum squared resid 0.005677 Schwarz criterion -6.161612Log likelihood 219.0281 Hannan-Quinn criter. -6.280922F-statistic 11.42415 Durbin-Watson stat 1.737229Prob(F-statistic) 0.000000

The results show that we needed no more than 3 lags to appropriately fit the

lcore_cpi equation with no traces of any remaining serial correlation.

The ADF statistic is -2.586 and

is greater than the 5% critical

value of -3.478, which shows the

series has a unit root.

In the lower panel, both the

intercept C and Trend or 0c and

t , respectively in eqn.9 are

statistically significant, with t-

statistic values of 2.647 and

2.681, respectively. This shows

the model as specified in EViews, is the true d.g.p.

After the test in level, we test the same series in first difference. While in the

same unit root output window, click on View and in the drop-down menu choose

Unit Root Test…, as in the above case to see the now familiar Unit Root Test


|54

dialogue box. As in level, the test type is the ADF, but this time, we’re testing

for unit root in 1st difference. In practice, for some reason (which we explore

later), when testing for unit in differences, we choose None for the

deterministic terms, but this is not always the case.

The ADF statistic is sensitive to this choice. Here I chose Intercept,

maintained Modified Akaike under lag-length and provided for the true lag

length of 3 as determined earlier. With these entries, click OK, for the test results

in Table 2.

Clearly, the ADF statistic is now smaller (-4.023) relative to the 5% critical

value of -2.904. Thus, the series, in first difference, is stationary.

Table 2: Unit root test in LCPI (difference)

Null Hypothesis: D(LCORE_CPI) has a unit root

Exogenous: Constant

Lag Length: 0 (Automatic - based on Modified AIC, maxlag=3)

t-Statistic Prob.*

Augmented Dickey-Fuller test statistic -4.029994 0.0023

Test critical values: 1% level -3.528515

5% level -2.904198

10% level -2.589562


Augmented Dickey-Fuller Test Equation

Dependent Variable: D(LCORE_CPI,2)

Method: Least Squares

Date: 04/17/18 Time: 15:16

Sample (adjusted): 2000Q3 2017Q3

Included observations: 69 after adjustments

Variable Coefficient Std. Error t-Statistic Prob.

D(LCORE_CPI(-1)) -0.393687 0.097689 -4.029994 0.0001

C 0.006047 0.001956 3.091753 0.0029

R-squared 0.195107 Mean dependent var -4.31E-05

Adjusted R-squared 0.183093 S.D. dependent var 0.011411

S.E. of regression 0.010313 Akaike info criterion -6.282233

Sum squared resid 0.007126 Schwarz criterion -6.217476

Log likelihood 218.7370 Hannan-Quinn criter. -6.256542

F-statistic 16.24086 Durbin-Watson stat 1.807840

Prob(F-statistic) 0.000145

Replicating these steps yields the following results for the rest of the series in

Figure 5, both in level and differences.


55 |

Table 3: Unit root test in exr

Null Hypothesis: LEXR has a unit root

Exogenous: Constant, Linear Trend


t-Statistic Prob.*



5% level -3.478305

10% level -3.166788



Dependent Variable: D(LEXR)


Date: 04/17/18 Time: 15:34




LEXR(-1) -0.079718 0.051789 -1.539293 0.1289

D(LEXR(-1)) 0.223365 0.124018 1.801068 0.0766

D(LEXR(-2)) -0.069814 0.124810 -0.559360 0.5780

D(LEXR(-3)) -0.129550 0.123666 -1.047582 0.2990

C 0.577234 0.377574 1.528796 0.1315

@TREND("2000Q1") 0.001242 0.000600 2.070689 0.0426

R-squared 0.155751 Mean dependent var 0.010189







Null Hypothesis: D(LEXR) has a unit root

Exogenous: None


t-Statistic Prob.*



5% level -1.945596

10% level -1.613719



Dependent Variable: D(LEXR,2)


Date: 04/17/18 Time: 15:35




D(LEXR(-1)) -0.714894 0.115986 -6.163611 0.0000

R-squared 0.358399 Mean dependent var -0.000387





Durbin-Watson stat 1.926933


|56

Table 4: Unit root test in lgdp

Null Hypothesis: LGDP has a unit root

Exogenous: Constant


t-Statistic Prob.*



5% level -2.904848

10% level -2.589907



Dependent Variable: D(LGDP)


Date: 04/17/18 Time: 15:40




LGDP(-1) -0.010365 0.008783 -1.180061 0.2423

D(LGDP(-1)) -0.430965 0.119319 -3.611863 0.0006

D(LGDP(-2)) -0.231227 0.119291 -1.938348 0.0570

C 0.119063 0.080656 1.476173 0.1448








Null Hypothesis: D(LGDP) has a unit root

Exogenous: Constant


t-Statistic Prob.*



5% level -2.904198

10% level -2.589562



Dependent Variable: D(LGDP,2)


Date: 04/17/18 Time: 15:41




D(LGDP(-1)) -1.356456 0.113847 -11.91474 0.0000

C 0.019423 0.003066 6.334388 0.0000








Notes: The trend was not significant, so the ADF equation assumes the intercept only. Critical values therefore differ from the ADF under intercept and trend assumptions.


57 |

Table 5: Unit root test in lM2

Null Hypothesis: LM2 has a unit root

Exogenous: Constant


t-Statistic Prob.*



5% level -2.903566

10% level -2.589227



Dependent Variable: D(LM2)


Date: 04/17/18 Time: 15:55




LM2(-1) -0.005858 0.004227 -1.386003 0.1703

C 0.086515 0.035314 2.449884 0.0169








Null Hypothesis: D(LM2) has a unit root

Exogenous: Constant


t-Statistic Prob.*



5% level -2.904198

10% level -2.589562



Dependent Variable: D(LM2,2)


Date: 04/17/18 Time: 15:56




D(LM2(-1)) -0.934056 0.122259 -7.639987 0.0000

C 0.035477 0.005865 6.049077 0.0000










|58

Table 6: Unit root test in loil_price

Null Hypothesis: LOIL_PRICE has a unit root

Exogenous: Constant


t-Statistic Prob.*



5% level -2.904198

10% level -2.589562



Dependent Variable: D(LOIL_PRICE)


Date: 04/17/18 Time: 16:20




LOIL_PRICE(-1) -0.060237 0.031932 -1.886398 0.0636

D(LOIL_PRICE(-1)) 0.270850 0.116427 2.326341 0.0231

C 0.250947 0.130140 1.928292 0.0581








Null Hypothesis: D(LOIL_PRICE) has a unit root

Exogenous: Constant


t-Statistic Prob.*



5% level -2.904198

10% level -2.589562



Dependent Variable: D(LOIL_PRICE,2)


Date: 04/17/18 Time: 16:22




D(LOIL_PRICE(-1)) -0.749102 0.118139 -6.340849 0.0000

C 0.007505 0.017113 0.438567 0.6624








Notes: The trend was not significant, so the ADF equation assumes the intercept only. Critical

values therefore differ from the ADF under intercept and trend assumptions.


59 |

Table 7: Unit root test in lpsc

Null Hypothesis: LPSC has a unit root

Exogenous: Constant, Linear Trend


t-Statistic Prob.*



5% level -3.478305

10% level -3.166788



Dependent Variable: D(LPSC)


Date: 04/17/18 Time: 16:26




LPSC(-1) -0.171435 0.090588 -1.892457 0.0632

D(LPSC(-1)) 0.106514 0.124477 0.855693 0.3955

D(LPSC(-2)) 0.134873 0.125092 1.078185 0.2852

D(LPSC(-3)) -0.280375 0.125619 -2.231950 0.0293

C 1.139737 0.560926 2.031887 0.0465

@TREND("2000Q1") 0.007888 0.004534 1.739677 0.0870








Null Hypothesis: D(LPSC) has a unit root

Exogenous: None


t-Statistic Prob.*



5% level -1.945669

10% level -1.613677



Dependent Variable: D(LPSC,2)


Date: 04/17/18 Time: 16:27




D(LPSC(-1)) -0.762826 0.161806 -4.714456 0.0000

D(LPSC(-1),2) -0.138716 0.121621 -1.140561 0.2582








|60

Table 8: Unit root test in lrate

Null Hypothesis: LRATE has a unit root

Exogenous: Constant


t-Statistic Prob.*



5% level -2.903566

10% level -2.589227



Dependent Variable: D(LRATE)


Date: 04/17/18 Time: 17:02




LRATE(-1) -0.148448 0.063577 -2.334926 0.0225

C 3.170276 1.365202 2.322203 0.0232



S.E. of regression 1.185875 Akaike info criterion 3.206994

Sum squared resid 95.62833 Schwarz criterion 3.271236

Log likelihood -110.2448 Hannan-Quinn criter. 3.232512



Null Hypothesis: D(LRATE) has a unit root

Exogenous: None


t-Statistic Prob.*



5% level -1.945596

10% level -1.613719



Dependent Variable: D(LRATE,2)


Date: 04/17/18 Time: 17:04




D(LRATE(-1)) -1.003975 0.121024 -8.295674 0.0000









61 |

Table 9: Unit root test in tb91

Null Hypothesis: TB91 has a unit root

Exogenous: Constant


t-Statistic Prob.*



5% level -2.903566

10% level -2.589227



Dependent Variable: D(TB91)


Date: 04/17/18 Time: 17:07




TB91(-1) -0.222967 0.076327 -2.921201 0.0047

C 2.436678 0.897054 2.716311 0.0084








However, with seven endogenous variables for only 71 observations there is a

limit on how much a VAR can bear due to diminishing degrees of freedom

(d.o.f), defined as 𝑁 − 𝐾 where 𝑁 is the number of observations and 𝐾 is the

number of coefficients to be estimated, including the constant, defined as

[𝑘(𝑘 𝑥 𝜌 ) + 𝒏]. 𝑘 is the number of variables in the VAR system, 𝜌 is the VAR

lag length, and 𝒏 is a vector of constants for the n equations. For small samples

therefore, it is pertinent that we carefully consider modelling as few variables as

possible while at the same time minimise on the number of lags – ensuring they

are enough to remove any remaining residual serial correlation. Given this, it is

not possible that we can analyse all the possible channels of MTM embedded

in our data in one system at the same time.

For purposes of demonstration and brevity, we consider a five variable VAR

model for the two non-policy variables (GDP and CPI) and three policy

variables (tb91, M2 and exr), capturing the interest rate, money and exchange

rate channels, respectively. We also add a sixth variable, oil price, but as

indicated earlier, enter the model exogenously. Central to VAR specification is

the determination of the order of VAR or lag-length that describes the true

d.g.p.


|62

4.4 Setting up a MTM VAR model

A key consideration before any VAR estimation can be attempted is the form

the variables must have when they enter the VAR: whether in levels, gaps or

first differences. The answer is simply ‘it depends’ on what the VAR is going to

be used for. If for forecasting purposes, we must avoid potential spurious

regressions that may result in spurious forecasts. For the purpose of identifying

shocks, we have to care about the statistical properties of the residuals, the

stability of the VAR and the reliability of the impulse response functions. For a

given choice and form of variables, setting up a VAR starts with the

appropriate specification, i.e., lag length specification.

4.4.1 Determination of lag length

In practice, when choosing the lag-length, we want to reduce the number of

lags as much as possible to get as simple a model as is possible, but at the same

time we want enough lags to remove autocorrelation of the VAR residuals. The

appropriate lag-length ( ) of the VAR (in eqn. 6) is chosen using the

minimum of the information criteria: the Akaike (AIC), Schwarz (SC) and

Hannan-Quinn (HQ) information criteria. These criteria have the same basic

formulation, i.e., derive from the log likelihood ratio (LR) function but penalise

for the loss of degrees of freedom due to extra 𝜌 lags to different degrees,

hence, in practice, need not to select the same preferred model and often they

do not. Juselius (2006, 70-1) gives a very detailed exposition of these

frameworks. AIC asymptotically over estimates the order with positive

probability, HQ estimates the order consistently (i.e. ppp ˆlim ) and SC is

even more strongly asymptotically consistent (i.e. pp ˆ ) (Lütkepohl and

Krätzig, 2004), i.e., it selects a shorter lag than the other criteria.

It is further shown that even in small samples of fixed size 16n , the

following relation AICpQHpSCp ˆˆˆ hold, hence the reason why, in

applied work SC is usually favoured in choosing the appropriate order of VAR.

This notwithstanding, it is very important to note that the residuals from the

estimated VAR should be well behaved, i.e., there should be no problems with

autocorrelation.


63 |

As stated earlier, SC usually favours a shorter lag, which in a great majority of

cases results in an under-parameterised (and hence potentially serially

correlated) model especially in small samples. Given this, the most crucial

assumption in the VAR environment is that of time independence of the

residuals (Juselius, 2006). Thus, whilst the AIC, SC or HQ may be good

starting points for

determining the lag-length

of the VAR, it is crucial to

check for residual

autocorrelation – a practical

process we now describe.

To recap, we have

considered a five variable

VAR model (lGDP, lcpi,

tb91, lM2 and lexr). With

this in mind, make active

our work-horse EViews

work file, COMESA2018.

Check and highlight the

five series above (click on for example lcpi, and while holding down, on your

key board, the control button, click on the rest of the series but one at a time –

and clearly, all the five will appear highlighted). Point the cursor in any of the

highlighted variables (this assumes you have not let loose the control button on

your key board) and right click. Follow the Open command and in the drop-

down menu, select as VAR… as shown

in the screen print.

Selecting this gives VAR specification

window, herewith.

As seen, this estimates Unrestricted

VAR for the estimation sample,

corresponding to our data points. All

the five variables are endogenous. The

lag interval for now is left as in

EViews default, where the first entry (1) is the system and the second (2) is the

initial value of 𝜌 in our equations. And under exogenous dialogue box, in


|64

addition to c (default), I have input loil as an exogenous variable. Click OK to

generate the output in Table 10

Table 10: VAR(2) Estimates

Vector Autoregression Estimates

Date: 04/21/18 Time: 14:07



Standard errors in ( ) & t-statistics in [ ]

LCPI LEXR LGDP LM2 TB91

LCPI(-1) 1.358467 -0.616656 -0.297740 -0.608976 108.2125

(0.12836) (0.49656) (0.27758) (0.34306) (31.4928)

[ 10.5836] [-1.24185] [-1.07264] [-1.77512] [ 3.43611]

LCPI(-2) -0.437665 0.603780 0.178212 0.637168 -108.9942

(0.11656) (0.45092) (0.25206) (0.31153) (28.5980)

[-3.75492] [ 1.33900] [ 0.70701] [ 2.04530] [-3.81125]

LEXR(-1) 0.021762 0.847628 0.008170 -0.134891 14.13248

(0.03576) (0.13835) (0.07734) (0.09558) (8.77456)

[ 0.60850] [ 6.12656] [ 0.10564] [-1.41122] [ 1.61062]

LEXR(-2) -0.018084 -0.204031 -0.056458 -0.047848 -6.988324

(0.03350) (0.12961) (0.07245) (0.08954) (8.22007)

[-0.53978] [-1.57420] [-0.77925] [-0.53435] [-0.85015]

LGDP(-1) 0.006963 -0.433617 0.420086 0.019165 -3.544776

(0.06061) (0.23446) (0.13106) (0.16198) (14.8699)

[ 0.11488] [-1.84942] [ 3.20522] [ 0.11831] [-0.23839]

LGDP(-2) 0.003725 -0.129905 0.234311 0.047588 -10.74061

(0.06135) (0.23735) (0.13268) (0.16398) (15.0533)

[ 0.06072] [-0.54731] [ 1.76599] [ 0.29020] [-0.71351]

LM2(-1) 0.081943 0.290661 -0.051731 0.768638 0.676937

(0.05227) (0.20222) (0.11304) (0.13971) (12.8248)

[ 1.56768] [ 1.43739] [-0.45764] [ 5.50186] [ 0.05278]

LM2(-2) -0.053812 0.047443 0.244084 0.247399 3.315820

(0.05552) (0.21478) (0.12006) (0.14839) (13.6218)

[-0.96927] [ 0.22089] [ 2.03298] [ 1.66726] [ 0.24342]

TB91(-1) 0.000930 0.002967 -0.000547 -0.000764 0.679838

(0.00049) (0.00191) (0.00107) (0.00132) (0.12127)

[ 1.88188] [ 1.55178] [-0.51160] [-0.57842] [ 5.60606]

TB91(-2) -0.000570 -0.000523 0.000907 -0.000297 -0.080569

(0.00050) (0.00192) (0.00107) (0.00133) (0.12176)

[-1.14774] [-0.27259] [ 0.84480] [-0.22386] [-0.66172]

C -0.013058 5.455254 2.561532 0.703407 49.46761

(0.37736) (1.45988) (0.81607) (1.00859) (92.5876)

[-0.03460] [ 3.73679] [ 3.13887] [ 0.69742] [ 0.53428]

LOIL 0.003467 -0.081395 -0.016110 -0.025377 -0.211361

(0.00523) (0.02023) (0.01131) (0.01398) (1.28316)

[ 0.66287] [-4.02303] [-1.42446] [-1.81551] [-0.16472]

R-squared 0.999374 0.980472 0.995834 0.999153 0.756174

Adj. R-squared 0.999254 0.976704 0.995030 0.998989 0.709119

Sum sq. resids 0.005302 0.079346 0.024794 0.037872 319.1515

S.E. equation 0.009644 0.037310 0.020856 0.025776 2.366252

F-statistic 8277.875 260.1746 1238.768 6110.235 16.07026

Log likelihood 228.9412 135.5909 175.7220 161.1071 -150.7456

Akaike AIC -6.288151 -3.582345 -4.745565 -4.321945 4.717262

Schwarz SC -5.899611 -3.193805 -4.357025 -3.933404 5.105803

Mean dependent 4.548391 7.693571 9.162380 8.372681 10.89884

S.D. dependent 0.353023 0.244445 0.295854 0.810731 4.387363

Determinant resid covariance (dof adj.) 1.08E-13

Determinant resid covariance 4.15E-14

Log likelihood 573.4909

Akaike information criterion -14.88380

Schwarz criterion -12.94109


65 |

Each column in the Table corresponds to the equation for each of the

endogenous variable in the VAR. For each right-hand side variable, EViews

reports the coefficient point estimate, the estimated coefficient standard error

(in round brackets) and the t-statistic (in square brackets)

As pointed out earlier, this traditional VAR output is not economically

intuitive, in part because it lacks economic structure that would aid

interpretation of estimated coefficients, but also and indeed as can be deduced,

is spurious (very high R-squared and on average strong t-ratios) due to non-

stationarity of the data. Instead the result from this implementation is a step to

the specification of an appropriate VAR.

What we really want is to unravel the true lag structure of the VAR, at which

the VAR residuals

are independent of

each other across

time (i.e., serially

uncorrelated).

To achieve this,

within this very

results window,

click on View (at

the extreme left-

hand side of the

results window),

and in the drop-down

menu, lies Lag

Structure, and

following the highlights in the screen print, we choose Lag Length

Criteria….

This gives the Lag Specification window, where

the only modification I have made is changing

the lags to include from the default of 6 to 5, for

the reasons given earlier. Click OK, to generate

the results in Table 11.


|66

Table 11: VAR Lag order selection criteria

VAR Lag Order Selection Criteria

Endogenous variables: LCPI LEXR LGDP LM2 TB91

Exogenous variables: C LOIL

Date: 04/21/18 Time: 14:12

Sample: 2000Q1 2017Q3

Included observations: 66

Lag LogL LR FPE AIC SC HQ

0 124.1461 NA 2.16e-08 -3.458974 -3.127208 -3.327877

1 521.4094 710.2585 2.74e-13 -14.73968 -13.57850* -14.28084

2 557.6360 59.28000* 1.98e-13* -15.07988 -13.08928 -14.29330*

3 582.8379 37.42098 2.05e-13 -15.08600* -12.26599 -13.97168

4 600.4145 23.43541 2.76e-13 -14.86104 -11.21162 -13.41898

5 628.0872 32.70420 2.88e-13 -14.94204 -10.46320 -13.17224

* indicates lag order selected by the criterion

LR: sequential modified LR test statistic (each test at 5% level)

FPE: Final prediction error

AIC: Akaike information criterion

SC: Schwarz information criterion

HQ: Hannan-Quinn information criterion

As expected, there are variations in lag length choices by the AIC, H-Q and SC.

AIC selects 3 lags, H-Q favours 2 lags, while SC suggests 1 lag, which, as

discussed earlier is not entirely surprising.

In as much as in applied work, it is the lag structure as chosen by SC that is

often preferred. However, as already stated, the fact that SC often favours a

shorter lag, such a short lag is likely to under-parameterise the model, resulting

in a potentially serially correlated model, more so in small samples. Given this,

it is highly recommended that we test for serial correlation in the VAR

residuals. This is achieved using Autocorrelation LM Test – one of the default

tests in a battery of EViews Residual Tests. Note that in this exercise, it is the

Residual Serial Correlation LM Test that we use to pin down the actual lag-

structure that we consider. Implementing this is also straight forward.

In the open VAR results window, click on View, and in the drop-down menu,

select Residual Tests, and following the arrow,

are a battery of residual misspecification tests,

Autocorrelation LM Test… inclusive. Clicking

on this brings forth Lag Specification dialogue

box, in this case with 3 lags to include (always

better to adjust this to accommodate choices of

all the information criteria: AIC, HQ and SC).

Click OK to generate VAR Residual Serial

Correlation LM Test output in Table 12.


67 |

Table 12: Residual Serial Correlation LM test

VAR Residual Serial Correlation LM T...

Null Hypothesis: no serial correlation a...

Date: 04/21/18 Time: 14:28

Sample: 2000Q1 2017Q3


Lags LM-Stat Prob

1 24.48707 0.4914

2 29.87300 0.2291

3 26.61641 0.3753

Probs from chi-square with 25 df.

All lags, 1 – 3 appear to guarantee time independence of the VAR residuals.

Note that the guiding principle when choosing the lag-length is that we want to

reduce the number of lags as much as possible to get as simple a model as is

possible, but at the same time, we want enough lags to remove autocorrelation

of the VAR residuals. Any of the three lags is accommodative, but for purposes

of uncovering the dynamics embedded in the data, while preserving the degrees

of freedom, our exercise will estimate VAR(2) model.

4.4.2 VAR(2) residual statistical properties

It is also important, upon establishing the true order of VAR, to assess the

suitability of the choice model in terms of a battery of residual misspecification

tests (see inter alia Godfrey, 1988). This comprises: the residuals plots;

normality; autocorrelation and ARCH effects.

The idea with residual plots is to see if there are outlier observations and /or

change in behaviour over time. Such features, if detected, would require the

modeller to take appropriate modelling actions, to account for such outlier

observations or mean shifts – usually, use of dummies has been found to be

very helpful particularly in the interest of preserving the degrees of freedom.


|68

To generate the residual plots, re-estimate the VAR(2) model. You may not

want to go through the entire process we have described above, but instead,

while in the above VAR Residual Serial Correlation LM Test window, click on

Estimate in the result window menu bar. This should give us a VAR(2)

Specification window we have seen before, with 2 lags in the Lag Intervals for

Endogenous comb box. Then click OK to estimate the traditional VAR like we

have done before, noting that these are still spurious results and with no

economic structure so is not economically intuitive.

To generate residual plots from this VAR(2) choice model output, click on

View in the menu bar of this results window. In the drop-down menu, select

Residuals and following the arrow, select Graphs – a road map given in the

screen print. The resulting residual plots are given in Figure 7, and look on

average, to be well behaved.


69 |

Figure 7: Residual plots

-.03

-.02

-.01

.00

.01

.02

.03

.04

00:3

01:1

01:3

02:1

02:3

03:1

03:3

04:1

04:3

05:1

05:3

06:1

06:3

07:1

07:3

08:1

08:3

09:1

09:3

10:1

10:3

11:1

11:3

12:1

12:3

13:1

13:3

14:1

14:3

15:1

15:3

16:1

16:3

17:1

17:3

LCPI Residuals

-.10

-.05

.00

.05

.10

.15

00:3

01:1

01:3

02:1

02:3

03:1

03:3

04:1

04:3

05:1

05:3

06:1

06:3

07:1

07:3

08:1

08:3

09:1

09:3

10:1

10:3

11:1

11:3

12:1

12:3

13:1

13:3

14:1

14:3

15:1

15:3

16:1

16:3

17:1

17:3

LEXR Residuals

-.06

-.04

-.02

.00

.02

.04

.06

.08

00:3

01:1

01:3

02:1

02:3

03:1

03:3

04:1

04:3

05:1

05:3

06:1

06:3

07:1

07:3

08:1

08:3

09:1

09:3

10:1

10:3

11:1

11:3

12:1

12:3

13:1

13:3

14:1

14:3

15:1

15:3

16:1

16:3

17:1

17:3

LGDP Residuals

-.08

-.06

-.04

-.02

.00

.02

.04

.06

00:3

01:1

01:3

02:1

02:3

03:1

03:3

04:1

04:3

05:1

05:3

06:1

06:3

07:1

07:3

08:1

08:3

09:1

09:3

10:1

10:3

11:1

11:3

12:1

12:3

13:1

13:3

14:1

14:3

15:1

15:3

16:1

16:3

17:1

17:3

LM2 Residuals

-8

-4

0

4

8

00:3

01:1

01:3

02:1

02:3

03:1

03:3

04:1

04:3

05:1

05:3

06:1

06:3

07:1

07:3

08:1

08:3

09:1

09:3

10:1

10:3

11:1

11:3

12:1

12:3

13:1

13:3

14:1

14:3

15:1

15:3

16:1

16:3

17:1

17:3

TB91 Residuals

Following the same route, we can undertake several of Residual Tests,

including normality and heteroskedasticity tests.


|70

Table 13: VAR Residual Normality Tests

VAR Residual Normality Tests

Orthogonalization: Cholesky (Lutkepohl)

Null Hypothesis: residuals are multivariate normal

Date: 04/21/18 Time: 14:49

Sample: 2000Q1 2017Q3


Component Skewness Chi-sq df Prob.

1 0.677531 5.279061 1 0.0216

2 -0.046807 0.025195 1 0.8739

3 0.818639 7.706948 1 0.0055

4 -0.374700 1.614602 1 0.2038

5 -0.809223 7.530684 1 0.0061

Joint 22.15649 5 0.0005

Component Kurtosis Chi-sq df Prob.

1 5.413742 16.75018 1 0.0000

2 3.103978 0.031083 1 0.8601

3 4.175388 3.971919 1 0.0463

4 2.810353 0.103402 1 0.7478

5 4.863281 9.981467 1 0.0016

Joint 30.83805 5 0.0000

Component Jarque-Bera df Prob.

1 22.02924 2 0.0000

2 0.056277 2 0.9723

3 11.67887 2 0.0029

4 1.718004 2 0.4236

5 17.51215 2 0.0002

Joint 52.99454 10 0.0000

We know the standard normal distribution has Skewness of 0 and kurtosis of 3.

Looking at the results, normality of residuals is rejected at the conventional 5

percent level of significance for the first, third and fifth residuals. Consistent

with all joint tests, the overall Jarque-Bera test is not significantly different from

zero, indicating that the assumption of multivariate normality is not supported.

The non-normality of the residuals results from a relatively large degree of

skewness – which is usually due to large outliers – seen in CPI (2011), GDP

(2008) and tb91 (2000), and excess kurtosis (fat tails).

Rather than EViews’ default setting of Cholesky of covariance (Lütkepohl) as

the Orthogonalisation method, some authors prefer to use Square root of

correlation (Doornik-Hendry). This is because we must choose a factorisation

of the residuals for the multivariate normality test, such that residuals are

orthogonal to each other. The approach due to Doornik and Hansen (2008)

has two advantages over the one in Lütkepohl (2005, p. 174-181). First,

Lütkepohl’s test uses the inverse of the lower triangular Cholesky factor of the

residual covariance matrix, resulting in a test which is not invariant to a re-


71 |

ordering of the dependent variables. Second, Doornik and Hansen perform a

small-sample correction to the transformed residuals before computing their

statistics. This problem on non-normality notwithstanding, the good news is that estimates of

the VAR model are robust to deviations from normality provided residuals are not

autocorrelated (Juselius 2006).

The results of multivariate test for ARCH effects are given in Table 14 and

reveal moderate ARCH effects in the system (p-value = 0.0324). Rahbek et al.

(2002) cited in Juselius (2006) and Dennis (2006) show that the rank tests are

robust to moderate ARCH effects, so this should not be a problem here.

Table 14: VAR Residual Heteroskedasticity Test

VAR Residual Heteroskedasticity Tests: No Cross Terms (only levels and squares)

Date: 04/21/18 Time: 14:57

Sample: 2000Q1 2017Q3


Joint test:

Chi-sq df Prob.

379.0245 330 0.0324

4.4.3 Stability of VAR(2)

The last thing we would want to do is to check the stability of the VAR. If the

VAR is not stable, certain results (such as impulse response standard errors) are

not valid. In doing this procedure, there will be (n × p) roots overall, where n is

the number of endogenous variables (i.e., 5) and p is the particular lag length

(i.e., 2). It is easy to check for stability in EViews. Go to View, Lag Structure

and click on AR Roots Table. You should get the results, as in the table 15.


|72

Table 15: VAR (2) Roots of the characteristic polynomial

Roots of Characteristic Polynomial

Endogenous variables: LCPI LEXR LGDP LM2 TB91

Exogenous variables: C LOIL

Lag specification: 1 2

Date: 04/21/18 Time: 15:00

Root Modulus

0.990389 0.990389

0.852970 0.852970

0.750850 - 0.307318i 0.811308

0.750850 + 0.307318i 0.811308

0.684853 0.684853

0.345303 - 0.209800i 0.404043

0.345303 + 0.209800i 0.404043

-0.354578 0.354578

-0.145642 - 0.162338i 0.218095

-0.145642 + 0.162338i 0.218095

No root lies outside the unit circle.

VAR satisfies the stability condition.

The VAR is stable as none of the roots lie outside the unit circle (EViews very

kindly tells you this at the bottom of the table): all the moduli of the roots of

the characteristic polynomial are less than one in magnitude. In case one or

more roots fall outside the unit circle, adding a time trend (denoted @trend in

EViews) as an exogenous variable can help, but this does not always correct

VAR instability.

To circumvent the short comings associated with unrestricted VAR

frameworks discussed above, models that identify a set of behavioral

relationships often between endogenous variables - that together explain the

overall working of the economy, have been advocated for – birthing Structural

VAR (SVAR) models, i.e., where economic theory is used to inform the

construction of the model. SVARS models are discussed in detail in the next

section.

Chapter 5

Structural Vector Autoregressive Models (SVARs)

5.1 Motivation

Structural econometric models may be used to explain and predict the effect of

policies on key macroeconomic aggregates. This could for instance include how

a change in monetary policy that is achieved through a reduction/increase in

the central bank policy interest rate propagates to the rest of the economy.

While in the previous section, it is plausible to construct impulse response

functions based on reduced-form VARs, the framework relies on an unrealistic

assumption, whereby the interpretation of the effect of each shock largely

assumes that all other shocks are held constant. In other words, as shown in

eqn. 7, shocks are orthogonal or uncorrelated. However, from the perspective

of Wold representation, we know that shocks are not always orthogonal due to

the contemporaneous correlation between variables. Consequently, we need to

impose a structure on the evolution of the shock processes in a VAR to

achieve contemporaneity, where the value of a variable depends on the

contemporaneous values of other variables, as opposed to a variable depending

solely on the past values of other variables. SVARs also ensure that the shocks

have economic meaning (technology, demand, labour supply, monetary policy

etc.).

5.2 VAR Identification

When Sims (1980a) first advocated for the use of VARs in economics, it was in

response to the prevailing orthodoxy at the time that all economic models

should be structural models, i.e., that they should include identifying

restrictions. Instead, he argued for the use of an unrestricted VAR wherein, no

distinction is made between endogenous and exogenous variables. The aim was

to free-up econometric modelling from the constraints applied by economic


|74

theory and, in effect, to ‘let the data speak. However, as pointed out earlier,

unrestricted VARs have been criticised for their lack of economic structure

(Lucas, 1972), which forms a basis for structural analysis.

To undertake the structural analysis, a reduced-form VAR is first fitted to

summarise the data and then a structural VAR is proposed whose structural

equation errors are taken to be the structural, primitive or economic shocks or

innovations. The parameters of these structural equations are then ‘identified’

(estimated) by utilising the information in the estimated reduced-form VAR. In

other words, the VAR in a reduced-form model summarises the data, while the

SVAR provides an interpretation of the data.

It follows then that when estimating a SVAR model, the starting point is the

estimation of a reduced form VAR, given by eqn. 6, here compressed to take

the form:

𝑦𝑡 = 𝐹𝑦𝑡−1 + 휀𝑡 , 휀𝑡~𝑁(0 , Ω) (13)

A corresponding underlying structural system of equations in EViews is of the

form:

𝐴𝑦𝑡 = 𝐶(𝐿)𝑦𝑡−1 + 𝐵𝑢𝑡 ,

𝑦𝑡 = 𝐴−1𝐶(𝐿)𝑦𝑡−1 + 𝐴

−1𝐵𝑢𝑡, 𝑢𝑡~𝑁(0 , 𝐼) (14)

Where, 𝑢𝑡 is a vector of normally distributed structural shocks, i.e.,

𝑢𝑡~𝑁(0 , Σ), where the Σ is specified as a diagonal matrix - essentially because

the structural shocks are assumed to originate from independent sources, i.e.,

are purely exogenous and mutually uncorrelated. Moreover, Σ is frequently

normalised such that 𝐸(𝑢𝑡𝑢𝑡′) = Σ = 𝐼𝑛 such that 𝑢𝑡~𝑁(0 , 𝐼). In other words,

the assumptions underlying 𝑢𝑡 are that there are as many structural shocks as

there are variables in the model and that these shocks are by definition

mutually uncorrelated, which implies that Σ is diagonal.

The matrices A, B and the Ci’s (i = 1, 2, …, p) are not separately observable

from the estimated variance-covariance matrix, 𝐸(휀𝑡휀𝑡′) = Ω, of the reduced-

form shocks, 휀𝑡 in eqn. 7, which is a diagonal matrix. The vector of non-policy

variables and policy variables are contained in 𝑦𝑡, while the contemporaneous

relations among the variables, i.e., recovered structural shocks from the

reduced form shocks are contained in matrix 𝐴, and 𝐵 is a matrix of reduced-


75 |

form shocks. However, eqn. 14 cannot be estimated directly due to

identification issues and therefore must be recovered from eqn.13. Therefore,

the major task here is how to recover eqn. 14 from eqn. 13. This process is

achieved by imposing economic theory informed restrictions on unrestricted

VAR (i.e., eqn. 13) to identify an underlying structure embedded in the data

(i.e., eqn. 14).

The process to finding “enough” restrictions is arguably the most difficult part

in the estimation of SVAR models. However, as a general rule, this process is

largely guided by our knowledge of economic theory. Some of the most

popular approaches that are found in the literature include: use of recursive and

non-recursive ordering of shock processes; imposing parametric restrictions on

the 𝐴 matrix; and imposing parametric restrictions on the shocks in 𝑢𝑡. i.e. the

impulse responses.

While there are many types of restrictions that can be used to identify a SVAR,

EViews allows two different types of restrictions. One type imposes

restrictions on the short-run behaviour of the system, while the other,

introduced by Blanchard and Quah (1989) imposes restrictions on the long-

run. Blanchard and Quah restriction scheme consider the vector moving-

average representation and thus its impulse responses, concentrating on the

impact that shocks have on the long-run of variables. However, in the current

application of identification of monetary policy shocks, long run restrictions are

often not admissible because it is now a stylized fact that monetary policy

shocks have a zero long-run effect- the so called long-run monetary policy

neutrality (Christiano et al.,1999). Given this, in this User Guide, we limit our

application to the imposition of short-run restrictions. Nonetheless, where

permissible, starting with EViews 7.1, it is possible to impose both long- and

short-run restrictions at the same time. Moreover, it is also possible to impose

sign restrictions to identify SVARs, subject to having access to MATLAB

software on the same computer for it to work – a real embarrassment of riches

when it comes to imposing sign restrictions in EViews.

5.3 Imposing short-run identifications

To impose short-run restrictions in EViews, we use eqn. 14, in which we

estimate the random stochastic residual, 𝐴−1𝐵𝑢𝑡 from the residuals, 휀𝑡 of the

estimated VAR, given in Table 10. Hence in view of eqns. 13 and 14, 휀𝑡

becomes:


|76

휀𝑡 = 𝐴−1𝐵𝑢𝑡 (15)

Or equivalently, as in EViews 𝐴휀𝑡 = 𝐵𝑢𝑡 (15*)

In requiring that restrictions or identifying schemes are of the form given by

eqn. 15*, EViews follows what is known as the AB model, where A and B are

interpreted as above.

From eqn. 14, we can show that:

휀𝑡휀𝑡′= 𝐴−1𝐵𝑢𝑡𝑢𝑡

′𝐵′(𝐴−1)′. (16)

Applying the expectation operator on both sides, we get

𝐸(휀𝑡휀𝑡′) = 𝐸(𝐴−1𝐵𝑢𝑡𝑢𝑡

′𝐵′(𝐴−1)′) = 𝐴−1𝐵𝐸(𝑢𝑡𝑢𝑡′)𝐵′(𝐴−1)′ )

= 𝐴−1𝐵𝐵′(𝐴−1)′ ; 𝐸(𝑢𝑡𝑢𝑡′) = 𝐼𝑛. (17)

Clearly relating coefficients of the structural and reduced form equations

Under the assumption that 𝐸(𝑢𝑡𝑢𝑡′) = Σ = 𝐼𝑛 and also that 𝐸(휀𝑡휀𝑡

′) = Ω, then

from eqn. 17:

𝐸(휀𝑡휀𝑡′) = Ω = 𝐴−1𝐵𝐵′(𝐴−1)′ ) (18)

Where Ω is a symmetric matrix that has 𝑘(𝑘 + 1)/2 different elements. This

matrix plays a key role in the identification scheme.

The key question is whether we can identify all the elements in A and B from

Ω. A necessary condition in the identification scheme of eqn.18 is the

fulfilment of the requirement that the number of equations in the system

should be equal to the number of the unknown variables. A sufficient

condition to facilitate the identification process is that the equations in eqn. 18

should not be a linear combination of each another. When the VAR model has

𝑘 endogenous variables, the symmetry property of the variance-covariance

matrix 𝐸(휀𝑡휀𝑡′) = Ω in eqn. 18, implies that we would need to impose 𝑘(𝑘 +

1)/2 (identity) restrictions on the 2𝑘2 unknown elements in 𝐴 and 𝐵. Since the

two matrices 𝐴 𝑎𝑛𝑑 𝐵 have 𝑘2 elements each, we would need to impose a total

of 2𝑘2 −𝑘(𝑘+1)

2= (3𝑘2 − 𝑘)/2 restrictions in the two matrices to achieve

identification, either recursively or non-recursively. In so doing, we impose

restrictions on the SVAR given by eqn. 14.

5.3.1 Imposing a recursive Identification Scheme


77 |

This approach involves imposing restrictions on the 𝐴 matrix so that it is a

lower triangular matrix, and the structural shocks are uncorrelated. In practical

terms, the approach makes use of Cholesky decomposition for estimation.

In this case, and using the above AB model, we have a VAR model with k = 5

endogenous variables. This would require us to impose at least (3(52) −

5)/2 = 35 restrictions. Where, the resultant Cholesky decomposition scheme

may be represented as:

𝐴 =

[ 1 0 0𝑎21𝑎31

1𝑎32

01

𝑎41𝑎51

𝑎42𝑎52

𝑎43𝑎53

0001𝑎54

00001]

, 𝐵 =

[ 𝑏11 0 0

00

𝑏220

0𝑏33

00

00

00

000𝑏440

0000𝑏55]

(19)

The 35 restrictions constitute 10 zero restrictions in matrix 𝐴, 20 zero

restrictions in matrix 𝑩 and 5 normalisation restrictions from the diagonal of

matrix 𝐴. Here the Cholesky decomposition assumes that shocks are

propagated in the order of output, inflation, money, interest rate and exchange

rate. And is intended to ensure that non-policy variables, output and inflation,

have a slow response to changes in policy variables (money, interest rate and

exchange rate), as monetary policy would not be expected to affect output over

a one period horizon.

These five equations tell a nice story about initial or short-run responses of the

endogenous variables. The set-up is such that a given endogenous variable is

contemporaneously determined by those variables “above it” in the system but

not by those “below it”, which affect it with a lag. For example, in our ordering

scheme, output only responds to lags of itself and to lags of the other four

variables, i.e., in the short-run, output does not respond to any variable

contemporaneously. Inflation, on the other hand reacts to the output

contemporaneously, with the size and magnitude of this contemporaneous

response given by the size of coefficient, 𝑎21. Putting it all together, we readily

see that a variable ordered first does not react to variables ordered below it

contemporaneously but does affect all the ones ordered below it

contemporaneously, i.e., every variable ordered above the other is affected by

variables below it only with a lag or not on impact. Similarly, a variable ordered

second responds contemporaneously to the variable above it but has a delayed

response from the variable ordered third in the ordering scheme, and so on.

And finally, the variable ordered last responds contemporaneously to all the


|78

variables ordered above it or, alternatively, it does not affect the variables

above it in the current period.

Put simply, if we take 𝑢1𝑡 to be a demand shock, 𝑢2𝑡 to be a supply or cost-

push shock, 𝑢3𝑡 to be a money shock, 𝑢4𝑡 to be a monetary or interest-rate

shock, and 𝑢5𝑡 to be an exchange rate shock, then: the cost push shock does

not contemporaneously affect output, i.e., output is not affected on impact by a

cost push shock. Both output and inflation are not affected on impact by a

money shock, i.e., the money shock does affect inflation and output with a lag.

Output, inflation and money are not affected on impact by the interest rate

shock and so on. In other words, according to the interest rate rule, the impact

of monetary policy on money, inflation and output is felt with a lag.

In EViews, these restrictions can be imposed either in Text or Matrix form,

following from an estimated unrestricted VAR, given in Table 10. Unlike Text

form, imposing restrictions in Matrix form is relatively straight forward. We

will, as such, while still fresh enough, start this implementation with an

illustration of the Text form restrictions.


79 |

5.3.1.1 Text form restrictions

In terms of implementation in EViews, we will need to navigate through the

route leading us to VAR (2) results in Table 10, and here I proceed on this very

assumption. However, unlike before, in this round, variables are ordered in the

order consistent with that in eqn. 18, i.e., lgdp, lcpi, lm2, tb91, lexr.

With VAR(2) results, as in Table 10, to impose the 35 restrictions given in eqn.

18, and in Text format, select Proc, and from the drop-down menu, select

Estimate Structural Factorization, as shown in the screen print here.

This brings to the fore SVAR Options dialogue box (shown here). The SVAR

Options window consists of Identifying restrictions and Optimization

Control options. We will deal with Identifying restrictions option for now,

which allows us to impose short run and long run restrictions, under

Identifying restrictions (𝐀𝐞 = 𝐁𝐮 where 𝐄(𝐮′𝐮) = 𝐈) box – a

representation given by eqn. 15* in the text. Select Text. This option

presupposes that each endogenous variable has a specific variable number, for

example in, k=5 variable SVAR;

@e1 for LGDP residuals @e2 for LCPI residuals @e3 for LM2 residuals @e4 for TB91 residuals @e5 for LEXR residuals

The identifying restrictions are imposed

in terms of the 휀’s, which are the

residuals from the reduced-form VAR

estimates, and the 𝑢’s, which are the

structural, fundamental or ‘primitive’

random (stochastic) errors in the

structural system, formulated for

estimation according to the relationship in eqn. 15*.

At this point, we enter the following in the text box (it is easiest to simply copy

and paste the suggested short-run factorisation example in the Identifying

restrictions box into the empty Identifying Restrictions box at the bottom),

as in the above screen print:

@e1 = C(1)*@u1


|80

@e2 = C(2)*@e1 + C(3)*@u2 @e3 = C(4)*@e1 + C(5)*@e2 + C(6)*@u3 @e4 = C(7)*@e1 + C(8)*@e2 + C(9)*@e3 + C(10)*@u4 @e5 = C(11)*@e1 + C(12)*@e2 + C(13)*@e3 + C(14)*@e4 + C(15)*@u5

The way to interpret these restrictions is that they represent the entries in the

𝐴−1𝐵 matrix linking 휀𝑡 (@e) and 𝑢𝑡 (@u) via eqn. 15, i.e., 휀𝑡 = 𝐴−1𝐵𝑢𝑡.

Given matrices A and B (eqn. 19), the matrix representation of 휀𝑡 = 𝐴−1𝐵𝑢𝑡 is

no more than:

(

휀1𝑡휀2𝑡휀3𝑡휀4𝑡휀5𝑡)

=

(

1 0 0𝑎21𝑎31

1𝑎32

01

𝑎41𝑎51

𝑎42𝑎52

𝑎43𝑎53

0001𝑎54

00001)

′

[ 𝑏11 0 0

00

𝑏220

0𝑏33

00

00

00

000𝑏440

0000𝑏55]

(

𝑢1𝑡𝑢2𝑡𝑢3𝑡𝑢4𝑡𝑢5𝑡)

(20)

Exploring the fact that the inverse of a lower (upper) triangular matrix is also a

lower (upper) triangular matrix readily facilitates the computations of the

underlying matrix algebra that results in the above set of EViews restrictions –

though frankly speaking, the matrix algebra involved is quite tedious. Check

OK and the resultant output is given in Table 16.


81 |

Table 16: Just-Identified SVAR estimates (short-run text form)

Structural VAR Estimates Date: 04/29/18 Time: 11:48 Sample (adjusted): 2000Q3 2017Q3 Included observations: 69 after adjustments Estimation method: method of scoring (analytic derivatives) Convergence achieved after 15 iterations Structural VAR is just-identified

Model: Ae = Bu where E[uu']=IRestriction Type: short-run text form@e1 = C(1)*@u1@e2 = C(2)*@e1 + C(3)*@u2@e3 = C(4)*@e1 + C(5)*@e2 + C(6)*@u3@e4 = C(7)*@e1 + C(8)*@e2 + C(9)*@e3 + C(10)*@u4@e5 = C(11)*@e1 + C(12)*@e2 + C(13)*@e3 + C(14)*@e4 + C(15)*@u5where@e1 represents LGDP residuals@e2 represents LCPI residuals@e3 represents LM2 residuals@e4 represents TB91 residuals@e5 represents LEXR residuals

Coefficient Std. Error z-Statistic Prob.

C(2) -0.124607 0.053609 -2.324360 0.0201C(4) 0.241603 0.151700 1.592638 0.1112C(5) 0.200320 0.328061 0.610617 0.5415C(7) -4.083252 13.96435 -0.292405 0.7700C(8) -3.757790 29.73856 -0.126361 0.8994C(9) -22.57606 10.88356 -2.074328 0.0380C(11) 0.287316 0.178884 1.606159 0.1082C(12) 2.183418 0.380760 5.734368 0.0000C(13) 0.305309 0.143611 2.125945 0.0335C(14) 0.003931 0.001541 2.550563 0.0108C(1) 0.020856 0.001775 11.74734 0.0000C(3) 0.009287 0.000791 11.74734 0.0000C(6) 0.025309 0.002154 11.74734 0.0000C(10) 2.288076 0.194774 11.74734 0.0000C(15) 0.029292 0.002494 11.74734 0.0000


Estimated A matrix: 1.000000 0.000000 0.000000 0.000000 0.000000 0.124607 1.000000 0.000000 0.000000 0.000000-0.241603 -0.200320 1.000000 0.000000 0.000000 4.083252 3.757790 22.57606 1.000000 0.000000-0.287316 -2.183418 -0.305309 -0.003931 1.000000

Estimated B matrix: 0.020856 0.000000 0.000000 0.000000 0.000000 0.000000 0.009287 0.000000 0.000000 0.000000 0.000000 0.000000 0.025309 0.000000 0.000000 0.000000 0.000000 0.000000 2.288076 0.000000 0.000000 0.000000 0.000000 0.000000 0.029292

The correspondence between the estimated residuals, 휀𝑡 (denoted by @e), and

the structural shocks, 𝑢𝑡 (denoted by @u) in EViews should be obvious.

And so is correspondence between:

C(1) = 𝑏11 = 0.021∗∗∗

C(2) = 𝑎21 = 0.125∗∗


|82

C(3) = 𝑏22 = 0.009∗∗∗

C(4) = 𝑎31 = 0.242

C(5) = 𝑎32 = 0.200

C(6) = 𝑏33 = 0.025∗∗∗

C(7) = 𝑎41 = 4.083

C(8) = 𝑎42 = 3.758

C(9) = 𝑎43 = 22.576∗∗

C(10) = 𝑏44 = 2.288∗∗∗

C(11) = 𝑎51 = 0.287

C(12) = 𝑎52 = 2.183∗∗∗

C(13) = 𝑎53 = 0.305∗∗

C(14) = 𝑎54 = 0.004∗∗

C(15) = 𝑏55 = 0.029∗∗∗

Asterisks ***, **, and * indicate 1%. 5% and 10% levels of significance, respectively. All numbers are in absolute values.

Structural VAR proponents try to avoid over identifying the VAR structure

and propose just enough restrictions to identify the parameters uniquely, which

is what we have just done with Table 16 – where in EViews output, it is

explicitly mentioned of the fact that the structural VAR is just identified.

Accordingly, in almost most cases, providing the identification scheme is

appropriate, SVAR models will often be just identified. It is always a good idea

to consider this recursive solution first, which then serve as a benchmark for

later analysis. Based on this, we could assess if there is anything unreasonable

about the recursive solution and think about how the system could be

modified. In our output in Table 16, five freely estimated coefficients in the A

matrix [C(4), C(5), C(7), C(8) and C(11)], are not statistically different from

zero or are insignificant. In other words, the recursive output suggests we could

impose five additional zero (over identifying) restrictions on the 𝐀 matrix in

eqn. 18 so that the resultant Cholesky decomposition scheme now takes the

form:

𝐴 =

[ 1 0 0𝑎210

10

01

00

0𝑎52

𝑎43𝑎53

0001𝑎54

00001]

, 𝐵 =

[ 𝑏11 0 0

00

𝑏220

0𝑏33

00

00

00

000𝑏440

0000𝑏55]

(21)

We could test this empirical finding further and assess the impact of setting

C(4), C(5), C(7), C(8) and C(11) equal to zero. In the text form, this over

identifying restrictions becomes:


83 |

@e1 = C(1)*@u1 @e2 = C(2)*@e1 + C(3)*@u2 @e3 = C(6)*@u3 @e4 = C(9)*@e3 + C(10)*@u4 @e5 = C(12)*@e2 + C(13)*@e3 + C(14)*@e4 + C(15)*@u5

Note the absence of C(4) and C(5) in the equations for @e3 and C(7) and C(8)

in the equation for @e4 and C(11) in the equation for @e5.).

Re-estimating the model with these five additional restrictions yield the results

in Table 17.

Table 17: Over-identified SVAR estimates (short-run text form)

Structural VAR Estimates

Date: 04/30/18 Time: 08:01



Estimation method: method of scoring (analytic derivatives)

Convergence achieved after 10 iterations

Structural VAR is over-identified (5 degrees of freedom)

Model: Ae = Bu where E[uu']=I

Restriction Type: short-run text form

@e1 = C(1)*@u1

@e2 = C(2)*@e1 + C(3)*@u2

@e3 = C(6)*@u3

@e4 = C(9)*@e3 + C(10)*@u4

@e5 = C(12)*@e2 + C(13)*@e3 + C(14)*@e4 + C(15)*@u5

where

@e1 represents LGDP residuals

@e2 represents LCPI residuals

@e3 represents LM2 residuals

@e4 represents TB91 residuals

@e5 represents LEXR residuals


C(2) -0.124607 0.053609 -2.324360 0.0201

C(9) -23.18650 10.69300 -2.168381 0.0301

C(12) 2.013359 0.372419 5.406169 0.0000

C(13) 0.345457 0.144009 2.398854 0.0164

C(14) 0.003844 0.001569 2.450259 0.0143

C(1) 0.020856 0.001775 11.74734 0.0000

C(3) 0.009287 0.000791 11.74734 0.0000

C(6) 0.025776 0.002194 11.74734 0.0000

C(10) 2.289530 0.194898 11.74734 0.0000

C(15) 0.029835 0.002540 11.74734 0.0000


LR test for over-identification:

Chi-square(5) 5.145566 Probability 0.3984

Estimated A matrix:

1.000000 0.000000 0.000000 0.000000 0.000000

0.124607 1.000000 0.000000 0.000000 0.000000

0.000000 0.000000 1.000000 0.000000 0.000000

0.000000 0.000000 23.18650 1.000000 0.000000

0.000000 -2.013359 -0.345457 -0.003844 1.000000

Estimated B matrix:

0.020856 0.000000 0.000000 0.000000 0.000000

0.000000 0.009287 0.000000 0.000000 0.000000

0.000000 0.000000 0.025776 0.000000 0.000000

0.000000 0.000000 0.000000 2.289530 0.000000

0.000000 0.000000 0.000000 0.000000 0.029835

By imposing five additional (zero) restrictions on C(4), C(5), C(7), C(8) and

C(11), we have more restrictions than are necessary to identify the SVAR. We

needed to impose a minimum of 35 restrictions to achieve identification.

Counting restrictions in the A and B matrices, we find that we now have 15


|84

zero restrictions in matrix A, 20 zero restrictions in matrix B and another five

normalisation restrictions on the diagonal of matrix A, giving us a total of 40

restrictions. The five extra restrictions are reflected in a new entry in Table 17,

which is a likelihood ratio (LR) test for over-identification, with a p-value of

0.398. The fact that the LR test of the over-identification is not significant,

indicates that we cannot reject the null hypothesis that the five elements, C(4),

C(5), C(7), C(8) and C(11), are equal to zero. We could therefore, actually, set

them equal to zero.

Further to that, the value of the test statistic, 5.146, is actually twice the

difference between the log-likelihood values of the unrestricted (540.534) and

the restricted model (537.961). In essence, these five restrictions mean that the

re-estimated relationship between the reduced-form errors, 휀𝑡, and the

structural errors, 𝑢𝑡, mean that the estimated reduced-form shocks are two

different linear combinations of the structural shocks (@e1 and @e3

equations). In particular, they are no longer recursive, which is why additional

restrictions are very rarely tested and almost never imposed. We also note that

the new estimates of C(2), C(9), C(12), C(13) and C(14) are qualitatively little

changed from their earlier values. The same is true for the entries of the B

matrix containing the variances of the reduced-form shocks. In short, unless

we have a (very) good reason for doing so, we should be wary of over-

identifying SVARs.

5.3.1.2 Matrix form restrictions

An alternative approach to inputting identifying restrictions would be to use

the matrix option. Under this option, you would create the two matrices, 𝐀

and 𝐁 with the following entries:

𝐴 =

[ 1 0 0𝑁𝐴𝑁𝐴

1𝑁𝐴

01

𝑁𝐴𝑁𝐴

𝑁𝐴𝑁𝐴

𝑁𝐴𝑁𝐴

0001𝑁𝐴

00001]

, 𝐵 =

[ 𝑁𝐴 0 000

𝑁𝐴0

0𝑁𝐴

00

00

00

000𝑁𝐴0

0000𝑁𝐴]

(22)

Unlike in the text form above, matrices are created by going to object at the

main menu of EViews work file window, then New Object, as shown in the

screen print. This opens New Object window of the form:


85 |

From the list of available possibilities, under Type of object, choose the

Matrix-Vector-Coef option. Under Name for object, make sure to give the

matrix an appropriate

name – here

"𝑚𝑎𝑡𝑟𝑖𝑥_𝑎" and then

later, 𝑚𝑎𝑡𝑟𝑖𝑥_𝑏. Click

OK.

A new window for

New Matrix that pops

up, as shown in the

EViews default, the

new matrix Type is

Matrix. It also provides

for the Dimension of

the matrix, where I

have input 5 rows and 5 columns

(the size of matrix A in the text) as

shown in the screen print. Click OK.

With this execution, in the matrix

that comes up, all entries are zero.

This must be edited to reflect the

structure in eqn. 22 above before it

can be used in the estimation of

SVAR. To do this, select Edit +/-

option in the open matrix window, to edit the

individual cells in the matrix we have created.

We then enter the individual elements of the

first matrix, i.e., matrix A, as in eqn. 22,

consisting of ones, zeros and NAs, where,

NA is used to represent those elements in the

matrix that are unknown values of

coefficients and are to be estimated. Once we are done, click on Edit +/-

again, and then close the window.


|86

Repeat the procedure for the second matrix, matrix B. The said input,

consistent with entries in Matrices A and B in eqn. 22, are shown here.

Once the two matrices have been created, both matrix_a and matrix_b will

appear among the variables listed in EViews work file. The next procedure is to

return to the estimated VAR(2) model in Table 10, but as with the text form,

ordering variables in the order: lgdp, lcpi, lm2, tb91, lexr.

Go to Proc, and in the drop-down

menu, navigate through to Estimate

Structural Factorization, as

described before, and select Matrix

and then short-run pattern,

providing for the names of matrices

A (matrix_a) and B (matrix_b) as

appropriate, as shown in the screen

print. Click OK.


87 |

Table 18: SVAR estimates (short-run pattern matrix option)


Date: 04/30/18 Time: 11:32





Structural VAR is just-identified


Restriction Type: short-run pattern matrix

A =

1 0 0 0 0

C(1) 1 0 0 0

C(2) C(5) 1 0 0

C(3) C(6) C(8) 1 0

C(4) C(7) C(9) C(10) 1

B =

C(11) 0 0 0 0

0 C(12) 0 0 0

0 0 C(13) 0 0

0 0 0 C(14) 0

0 0 0 0 C(15)


C(1) 0.124607 0.053609 2.324360 0.0201

C(2) -0.241603 0.151700 -1.592638 0.1112

C(3) 4.083252 13.96435 0.292405 0.7700

C(4) -0.287316 0.178884 -1.606159 0.1082

C(5) -0.200320 0.328061 -0.610617 0.5415

C(6) 3.757790 29.73856 0.126361 0.8994

C(7) -2.183418 0.380760 -5.734368 0.0000

C(8) 22.57606 10.88356 2.074328 0.0380

C(9) -0.305309 0.143611 -2.125945 0.0335

C(10) -0.003931 0.001541 -2.550563 0.0108

C(11) 0.020856 0.001775 11.74734 0.0000

C(12) 0.009287 0.000791 11.74734 0.0000

C(13) 0.025309 0.002154 11.74734 0.0000

C(14) 2.288076 0.194774 11.74734 0.0000

C(15) 0.029292 0.002494 11.74734 0.0000


Estimated A matrix:

1.000000 0.000000 0.000000 0.000000 0.000000

0.124607 1.000000 0.000000 0.000000 0.000000

-0.241603 -0.200320 1.000000 0.000000 0.000000

4.083252 3.757790 22.57606 1.000000 0.000000

-0.287316 -2.183418 -0.305309 -0.003931 1.000000

Estimated B matrix:

0.020856 0.000000 0.000000 0.000000 0.000000

0.000000 0.009287 0.000000 0.000000 0.000000

0.000000 0.000000 0.025309 0.000000 0.000000

0.000000 0.000000 0.000000 2.288076 0.000000

0.000000 0.000000 0.000000 0.000000 0.029292


|88

The resulting output, given here in Table 18, is identical to the one obtained in

Table 16, obtained using the text form.

5.4 Generating Impulse Response Functions and Forecast Error Variance Decomposition

Two very useful outputs from the above VARs modelling process are the

impulse response function (IRF) and the forecast error variance decomposition

(FEVD). An IRF traces the effect of a shock to one of the innovations of the

VAR on current and future values of the endogenous variables. As such, a

shock to the i-th variable directly affects the i-th variable itself

contemporaneously, and is also transmitted to all of the endogenous variables

with a lag through the dynamic structure of the VAR.

In the case of identified SVARs, impulse responses show how the different

variables in the system respond to identified structural shocks, i.e., they show

the dynamic interactions between the endogenous variables in the VAR(p)

process. Since we have ‘identified’ the SVAR, the impulse responses will be

depicting the responses to the structural shocks to interest rates, here the

interest rate shock. We do this because the results of the impulse response

analysis are often more informative than the parameter estimates of the SVAR

coefficients themselves.

The same is true for forecast error variance decompositions, which are also

popular tools for interpreting VAR models. While IRF trace the effect of a

shock to one endogenous variable onto the other variables in the system,

forecast error variance decompositions (or variance decompositions for short)

separate the variation in an endogenous variable into the contributions

explained by the component shocks in the VAR. In other words, the variance

decomposition tells us the proportion of the movements in a variable due to its

‘own’ shock versus shocks to the other variables. Thus, the variance

decomposition provides information about the relative importance of each

(structural) shock in affecting the variables in the VAR. In much empirical

work, it is typical for a variable to explain almost all of its own forecast error

variance at short horizons and smaller proportions at longer horizons. Such a

delayed effect of the other endogenous variables is not unexpected, as the

effects from the other variables are propagated through the reduced-form VAR

with lags. In what follows, we attempt to generate IRFs using the Cholesky

decomposition following a shock to tb91.


89 |

To do this, in the open

SVAR output, click on

the Impulse button at

the top of the estimated

SVAR output box. This

opens the impulse

responses window,

where we select the

multiple graphs

option under the

Display Format, then

the Analytic

(asymptotic option) for Response Standard Errors. We enter (under

Display Information) the variable for which we want to shock (Impulses) -

here tb91 and the variables for which we want to observe the responses

(Responses) – here lgdp lcpi lm2 tb91 lexr – noting that these appear in the

order in which the SVAR has been estimated. Alternatively, one may simply

enter the numbers corresponding to the ordering of the variables. As shown in

the screen print, we type tb91 (or 4) in the Impulses box and leave all the five

variables as they are in the Responses box. This option will show the impulse

response of each variable to a structural shock to tb91 (or 𝑢4𝑡). We plot

impulse responses over 20 Periods (some five years horizon). The two

standard error bands of the impulse response functions are based on

analytical (or asymptotic, i.e., large-sample) results, as in the screen print.

We also need to define the nature of impulse responses we are interested in

estimating. This is done in the Impulse Definition option. The default option

in EViews is the Cholesky-dof adjusted, which is a choice option for

unrestricted VAR, estimated in Table 10. However, as our model is structural,

we need to choose the Structural Decomposition option which makes use of

the identification scheme that we had earlier specified in estimating the SVAR,

either the text or the matrix form. Then click on OK, which yields Figure 8.

Figure 8 consists of five charts of impulse response functions, corresponding

to the five endogenous variables we have modelled. Specifically, it shows the

impact of a one standard deviation shock, defined as an exogenous, one-time

positive shock to short-term interest rate (i.e. monetary policy shock). The solid

line in each graph is the estimated response while the dashed lines denote a two


|90

standard error confidence band around the estimate. The short-term interest

rate obviously increases as a result of a one-time positive shock to itself (the

increase is equal to 2.3, a value we have come across before as C(14) in Table

18, or C(10) in Table 16, i.e., the standard deviation of the structural monetary

policy shock), but the effect of the monetary shock decays to zero as 𝑡 → ∞,

which is consistent with Christiano et al. (1999) stylized fact of long-run

monetary policy neutrality.

In response to a positive one-standard deviation structural shock to tb91, gdp

first falls for some two periods before rising for some two periods, but there is

a zero long-run effect of the monetary policy shock. The aggregate price level

responds quite strongly – but with the opposite sign, contrary to what theory

would predict. While this is a manifestation of the so-called ‘inflation puzzle’ –

since we would not expect inflation to increase with policy rate tightening, it is

also true that inflation responds to policy tightening with a lag due to sticky

prices. M2 falls as expected, while the exchange rate depreciates (another

puzzle). Once we add the plus and minus two standard error bands, we can see

how significant these effects are. The negative response of GDP is insignificant

throughout and so is inflation and M2. The positive shock of tb91 to itself is

significant and persists for some four periods before becoming insignificant

while exr shows a significant response to tb91 for the immediate three periods.


91 |

Figure 8: IRF to one standard-deviation structural shock in tb91 (recursive structural factorisation)

-.012

-.008

-.004

.000

.004

.008

2 4 6 8 10 12 14 16 18 20

Response of LGDP to Shock4

-.010

-.005

.000

.005

.010

2 4 6 8 10 12 14 16 18 20

Response of LCPI to Shock4

-.03

-.02

-.01

.00

.01

2 4 6 8 10 12 14 16 18 20

Response of LM2 to Shock4

-1

0

1

2

3

2 4 6 8 10 12 14 16 18 20

Response of TB91 to Shock4

-.02

-.01

.00

.01

.02

.03

2 4 6 8 10 12 14 16 18 20

Response of LEXR to Shock4

Response to Structural One S.D. Innovations ± 2 S.E.

We now demonstrate how to generate forecast error variance decompositions

for the recursively identified SVAR, assuming we can still navigate through the

process to the results in either Table 16 or Table 18.


|92

While in the SVAR results window, select View and Variance

Decomposition… and in the VAR Variance Decomposition dialogue box

that pops up (in the screen print here), select, under the Display Format,

Table option and 12 periods, and Structural Decomposition under

Factorization. EViews default is Cholesky Decomposition, an option we

would pick if we were generating variance decomposition from the unrestricted

VAR, given in Table 10. Click OK for output in Table 19.


93 |

Table 19: Variance decomposition

Variance Decomposition of LGDP:

Period S.E. Shock1 Shock2 Shock3 Shock4 Shock5

1 0.020856 100.0000 2.25E-30 2.27E-28 2.76E-27 5.41E-31

2 0.023083 98.23935 1.319503 0.170100 0.260295 0.010749

3 0.025898 93.32557 3.897530 2.186899 0.217606 0.372391

4 0.028073 87.52252 7.435605 3.585088 0.366931 1.089853

5 0.030473 80.68891 10.75726 5.506875 0.864001 2.182953

6 0.032984 74.09178 13.92871 6.822371 1.764045 3.393101

7 0.035614 68.31438 16.62026 7.584801 2.932796 4.547764

8 0.038248 63.67140 18.78895 7.846986 4.164553 5.528115

9 0.040792 60.17446 20.43094 7.805573 5.291171 6.297856

10 0.043171 57.66623 21.62332 7.614181 6.225563 6.870701

11 0.045342 55.92972 22.46042 7.377514 6.948161 7.284192

12 0.047294 54.75151 23.03587 7.151855 7.480612 7.580151

Variance Decomposition of LCPI:


1 0.009644 7.261370 92.73863 1.33E-31 0.000000 0.000000

2 0.016862 5.883174 91.10335 0.971154 1.899416 0.142909

3 0.022703 5.327304 89.35431 1.883251 3.185051 0.250080

4 0.027045 5.083353 87.59703 3.366052 3.728604 0.224960

5 0.030023 4.868406 85.85633 5.323202 3.766082 0.185980

6 0.031967 4.583894 83.99350 7.611276 3.522957 0.288369

7 0.033242 4.271190 81.80704 10.00898 3.259347 0.653448

8 0.034170 4.081406 79.14278 12.25134 3.196361 1.328113

9 0.034993 4.208826 75.96740 14.10455 3.450010 2.269207

10 0.035861 4.800899 72.39206 15.43595 4.006707 3.364390

11 0.036836 5.895774 68.62550 16.23871 4.758733 4.481283

12 0.037925 7.421278 64.89101 16.60111 5.571908 5.514690

Variance Decomposition of LM2:


1 0.025776 3.072632 0.520949 96.40642 4.80E-29 2.49E-32

2 0.033667 4.427342 4.552485 88.86896 0.773812 1.377400

3 0.042345 6.270284 9.073304 75.66169 4.106420 4.888305

4 0.050956 8.543194 12.87681 62.38385 7.858896 8.337249

5 0.059205 11.46141 15.19725 51.55981 11.01162 10.76992

6 0.066749 14.66089 16.48401 43.41708 13.22545 12.21258

7 0.073476 17.89582 17.12614 37.45861 14.55706 12.96236

8 0.079378 20.94977 17.40697 33.12309 15.22674 13.29344

9 0.084538 23.69892 17.50004 29.94770 15.45788 13.39545

10 0.089083 26.08451 17.51274 27.58868 15.42920 13.38487

11 0.093157 28.09800 17.50815 25.80202 15.26426 13.32757

12 0.096895 29.76242 17.52230 24.41610 15.04103 13.25816

Variance Decomposition of TB91:


1 2.366252 0.562070 0.105622 5.830751 93.50156 3.02E-33

2 3.232146 2.314144 14.87812 3.951660 77.21564 1.640431

3 3.838388 4.813424 22.00545 3.172673 67.19508 2.813371

4 4.156673 6.675522 23.52914 4.007825 62.67250 3.115019

5 4.295984 8.118661 23.15342 5.465784 60.24070 3.021438

6 4.355723 8.875209 22.54899 6.865976 58.75413 2.955692

7 4.394796 9.038630 22.35688 7.744007 57.77082 3.089663

8 4.436156 8.898150 22.58065 8.067919 57.08877 3.364506

9 4.480784 8.746168 22.98271 8.044305 56.58043 3.646386

10 4.522175 8.726422 23.35126 7.909328 56.16557 3.847414

11 4.554874 8.839252 23.59371 7.802269 55.81333 3.951439

12 4.577234 9.017644 23.71177 7.760710 55.52457 3.985312

Variance Decomposition of LEXR:


1 0.037310 0.071858 30.31975 2.158130 5.811360 61.63891

2 0.050002 1.388463 22.64017 5.450584 11.54432 58.97646

3 0.056614 4.860304 18.95208 9.688222 13.79740 52.70200

4 0.060763 8.709735 16.93905 13.84737 13.64499 46.85885

5 0.063471 11.38523 15.79756 17.10348 12.74365 42.97009

6 0.065198 12.68156 15.13367 19.25494 12.09297 40.83686

7 0.066297 13.00616 14.72476 20.46982 11.96752 39.83174

8 0.067031 12.87898 14.45275 21.02417 12.23125 39.41286

9 0.067563 12.67738 14.25879 21.19181 12.63849 39.23354

10 0.067978 12.58546 14.11501 21.17524 13.00522 39.11907

11 0.068319 12.64150 14.00704 21.09330 13.25427 39.00388

12 0.068608 12.81189 13.92312 21.00285 13.38703 38.87510

Factorization: Structural


|94

Table 19 gives the variance decomposition of the five variables in the VAR to

the identified structural shocks. As you may have realized during

implementation, variance decompositions in Table 19 are given without

standard errors.

Of some interest is the effect of nominal on real variables, as it is sometimes

found that monetary policy shocks explain only a small part of the variance of

output. As such, the short-term interest rate predicts only a small percentage of

the variance of the output, equal to some 7.5% after 12 periods. The variance

decomposition of output due to a shock to itself is as high as 55%, with a

combined contribution of policy variables (money, interest and exchange rate)

of about 22% at the end of the observation period. The picture is the same for

CPI and tb91, where the percentage of the variance explained by the other

variables amounts to 35% and 44.5%, respectively, with more than ½ of the

variance of CPI and tb91 explained by themselves, respectively, in period 12.

5.5 Non-recursive Identification Scheme

In non-recursive identification scheme, we are not worried about a particular

ordering of the endogenous variables but rather our interest is in ensuring that

the short-run restrictions make economic sense. Recall that previously in our

recursive ordering scheme, we had assumed that the non- policy variables come

first and the policy variables come last, now under a non-recursive scheme we

could start with policy variables and end with non-policy variables, while

ensuring all the contemporaneous relationships are identified. Recall that the

main requirement of just-identification is to ensure that we can uniquely

recover all the parameters in the A and B matrices from the variance-

covariance matrix Ω of the estimated residuals, which we achieve by imposing

the necessary (3𝑘2 − 𝑘)/2 additional restrictions.

We can therefore impose identifying structures different from the recursive

one, but for the variables in order exr, GDP, CPI, tb91 and M2. Accordingly, as

in eqn. 19, we postulate the following non-recursive system for matrix A:

𝐴 =

[ 1 𝑎12 𝑎1300

1𝑎32

01

00

𝑎42𝑎52

𝑎43𝑎53

𝑎140010

𝑎1500𝑎451 ]

, 𝐵 =

[ 𝑏11 0 0

00

𝑏220

0𝑏33

00

00

00

000𝑏440

0000𝑏55]

(23)


95 |

Where, the first relationship now postulates a contemporaneous effect of exr

on the rest of the variables in the system, while in the second row, GDP has a

contemporaneous relationship with itself. In the third relationship, there is a

contemporaneous effect of GDP on CPI, and a contemporaneous effect of

GDP, CPI and M2 on interest rates in the fourth relation. Finally, in the fifth

relationship, we capture a contemporaneous effect of CP and GDP on money.

Note that while we have the same number of zeros as in eqn. 19, we have

located them in different places, and the basic structures of matrices A and B

remain unchanged: matrix A still has ones on the main diagonal (this is a

normalisation).

These restrictions are most easily imposed using the matrix form in EViews. As

before, we need to create these two matrices A and B by the procedure

described earlier, but for purposes of clarity, here as follows:

𝐴 =

[ 1 𝑁𝐴 𝑁𝐴00

1𝑁𝐴

01

00

𝑁𝐴𝑁𝐴

𝑁𝐴𝑁𝐴

𝑁𝐴0010

𝑁𝐴00𝑁𝐴1 ]

, 𝐵 =

[ 𝑁𝐴 0 000

𝑁𝐴0

0𝑁𝐴

00

00

00

000𝑁𝐴0

0000𝑁𝐴]

(24)

I have named these matrix_a2 and matrix_b2, respectively, in the EViews work

file. Once you have created the two new matrices, as I have, estimate VAR(2)

as before, noting the ordering of variables for this round (lexr, lgdp, lcpi, tb91,

lM2). In the estimated VAR(2) results window, select Proc, Estimate

Structural Factorisation..., select Matrix, Short-run pattern and enter the

two matrices (matrix_a2 and matrix_b2) in the appropriate places. Click on OK

to estimate the elements of the two matrices, as shown in Table 20.


|96

Table 20: Just-identified Non-recursive SVAR estimates (matrix form)


Date: 05/02/18 Time: 09:53





Structural VAR is just-identified


Restriction Type: short-run pattern matrix

A =

1 C(1) C(5) C(8) C(9)

0 1 0 0 0

0 C(2) 1 0 0

0 C(3) C(6) 1 C(10)

0 C(4) C(7) 0 1

B =

C(11) 0 0 0 0

0 C(12) 0 0 0

0 0 C(13) 0 0

0 0 0 C(14) 0

0 0 0 0 C(15)


C(1) -0.287316 0.178884 -1.606159 0.1082

C(2) 0.124607 0.053609 2.324360 0.0201

C(3) 4.083252 13.96435 0.292405 0.7700

C(4) -0.241603 0.151700 -1.592638 0.1112

C(5) -2.183418 0.380760 -5.734368 0.0000

C(6) 3.757790 29.73856 0.126361 0.8994

C(7) -0.200320 0.328061 -0.610617 0.5415

C(8) -0.003931 0.001541 -2.550563 0.0108

C(9) -0.305309 0.143611 -2.125945 0.0335

C(10) 22.57606 10.88356 2.074328 0.0380

C(11) 0.029292 0.002494 11.74734 0.0000

C(12) 0.020856 0.001775 11.74734 0.0000

C(13) 0.009287 0.000791 11.74734 0.0000

C(14) 2.288076 0.194774 11.74734 0.0000

C(15) 0.025309 0.002154 11.74734 0.0000


Estimated A matrix:

1.000000 -0.287316 -2.183418 -0.003931 -0.305309

0.000000 1.000000 0.000000 0.000000 0.000000

0.000000 0.124607 1.000000 0.000000 0.000000

0.000000 4.083252 3.757790 1.000000 22.57606

0.000000 -0.241603 -0.200320 0.000000 1.000000

Estimated B matrix:

0.029292 0.000000 0.000000 0.000000 0.000000

0.000000 0.020856 0.000000 0.000000 0.000000

0.000000 0.000000 0.009287 0.000000 0.000000

0.000000 0.000000 0.000000 2.288076 0.000000

0.000000 0.000000 0.000000 0.000000 0.025309


97 |

5.6 Comparing Recursively and Non-recursively Identified SVAR

One natural thing to think about is which of the two, i.e., recursive and non-

recursive SVAR is better, something we evaluate using the maximised value of

the log of the likelihood function. A model with a lower log-likelihood is

generally better. In both the non-recursive (Table 20) and recursive (Table 18),

the log-likelihood value is 540.5339, and is exactly the same in both cases. This

means that the two are observationally equivalent. In other words, we cannot

choose between them based on the fit of the data alone, and other criteria may

be required. One possibility in this regard is the visual inspection of the IRFs.

Since we have described the process of generating IRFs before, leading to

charts in Figure 8, there is no need of replicating the same story here. I will

therefore proceed, following strictly the same procedure to produce impulse

response charts from the non-recursive scheme in Figure 9.

Figure 9: IRF to one standard-deviation structural shock in tb91 (non-recursive structural factorisation)

-.02

-.01

.00

.01

.02

.03

2 4 6 8 10 12 14 16 18 20


-.012

-.008

-.004

.000

.004

.008

2 4 6 8 10 12 14 16 18 20


-.010

-.005

.000

.005

.010

2 4 6 8 10 12 14 16 18 20


-1

0

1

2

3

2 4 6 8 10 12 14 16 18 20


-.03

-.02

-.01

.00

.01

2 4 6 8 10 12 14 16 18 20




|98

-.02

-.01

.00

.01

.02

.03

2 4 6 8 10 12 14 16 18 20


-.012

-.008

-.004

.000

.004

.008

2 4 6 8 10 12 14 16 18 20


-.010

-.005

.000

.005

.010

2 4 6 8 10 12 14 16 18 20


-1

0

1

2

3

2 4 6 8 10 12 14 16 18 20


-.03

-.02

-.01

.00

.01

2 4 6 8 10 12 14 16 18 20



These new impulses are broadly well aligned to those in Figures 8, and notably,

the price and exchange rate puzzles persist following a contractionary monetary

policy shock. Therefore, until now, we are unable to judge superiority of one

ordering over the other!

Nonetheless, some caution about non-recursive ordering scheme is in order.

While there may be arguments favouring non-recursive ordering, there is also a

substantial cost. A broader set of economic relations must be identified to

make sense of the non-recursive structure, and it turns out, in practice, that it is

not enough to just have the same number of restrictions in the A matrix.

Assume for example, as in eqn.19, the following non-recursive system.

𝐴 =

[ 1 𝑎12 00𝑎31

10

𝑎231

𝑎41𝑎51

0𝑎52

𝑎43𝑎53

0𝑎24010

000𝑎451 ]

, 𝐵 =

[ 𝑏11 0 0

00

𝑏220

0𝑏33

00

00

00

000𝑏440

0000𝑏55]

(25)

Such that there is a contemporaneous relationship between gdp and cpi (row

1), a contemporaneous effect of cpi on M2 and interest rates (row 2), and a

contemporaneous effect of GDP on M2 (row 3). There is also a

contemporaneous effect of GDP, M2 and exr on the interest rate (row 4) and

finally a contemporaneous effect of GDP, CPI and M2 on exr (row 5). Again,

we have the same number of zeros as in eqn. 18, except that we have located

them in different places of the A matrix. Furthermore, the basic structures of

matrices A and B remain unchanged, with matrix A having ones on the main

diagonal.

As with the earlier cases, these restrictions are easily imposed using the matrix

form in EViews. For clarity again, we will need to create these two matrices A


99 |

and B, (matrix_a3 and matrix_b3) by the same procedures described earlier.

The matrices take the form:

𝐴 =

[ 1 𝑁𝐴 00𝑁𝐴

10

𝑁𝐴1

𝑁𝐴𝑁𝐴

0𝑁𝐴

𝑁𝐴𝑁𝐴

0𝑁𝐴010

000𝑁𝐴1 ]

, 𝐵 =

[ 𝑁𝐴 0 000

𝑁𝐴0

0𝑁𝐴

00

00

00

000𝑁𝐴0

0000𝑁𝐴]

(26)

Once these have been created, as I have, estimate VAR(2) as before, noting the

ordering of variables for this round (lgdp, lcpi, lM2, tb91, lexr). In the estimated

VAR(2) results window, select Proc, Estimate Structural Factorisation...,

select Matrix, Short-run pattern and enter the two matrices (matrix_a3 and

matrix_b3) in the appropriate places. Click on OK to estimate the elements of

the two matrices, but other than the usual neat output, this time, we receive an

error message!

Note that even with the right number of necessary restrictions for

identification (35), such that the order condition for identification is fulfilled,

we encounter a problem with the estimation, in particular the fact that the

Hessian matrix is near singular at final iteration parameter values.

Although it has been shown that this problem can be resolved by going to the

Optimisation Control tab and changing the option for Starting Values from

Fixed to Draw from Uniform (0,1), as in the screen print,


|100

sadly, this time around, the problem has no quick fix, as we have run afoul of

the rank condition of identification, which is described in, inter alia, Hamilton

(1994, p. 332). Taking this into account makes coming up with a good

economic rationale for non-recursive restrictions even harder.

Chapter 6

VAR and Vector Error Correction Model (VECM)

6.1 Deriving the VECM Framework

On the basis of unit root unit testing, we treat all variables, save for tb91 which

is boarder line stationary so, as unit root non-stationary, i.e., I(1). I(1) behaviour

allows (potentially) for cointegration, i.e., a linear combination of I(1) variables

that is I(0) – which is a statistical analogy of economic equilibrium. The logic

for this is straight forward. If the trend in one variable is cancelled out by the

trend in one (or a linear combination) of other variable(s), then there must exist

forces ensuring that these variable(s) ‘do not move too far apart’ (i.e. they share

a common trend). In economics, such forces constitute the ‘equilibrium’

relationships posited by economic theory. If there were no relationship tying

one variable to another (or set of other variables) then they would drift

arbitrarily far apart, independent of one another.

The link between VAR in eqn. 6 and cointegration is accommodated in a

vector error correction model (VECM) (Johansen 1988), a link that we now

turn to. Providing the data for the k system series are I(1), it is possible to

rewrite VAR(p) in eqn. 6 as an VECM of the form:

∆Z𝑡 = A0 + Γ1∆Z𝑡−1 + Γ2∆Z𝑡−2 +⋯+ Γ𝑘−1∆Z𝑡−𝑝−1

+ΠZ𝑡−𝑝 + ε𝑡 (27)

Where

Γ𝑖 = −(I − A1 − A2 −⋯− A𝑘−1); 𝑖 = (1,2, … , 𝑘 − 1), and

Π = −(I − A1 − A2 −⋯− A𝑘)

Making the derivation simpler, consider the VAR(2) model:


|102

Z𝑡 = A0 + A1Z𝑡−1 + A2Z𝑡−2 + ε𝑡 (28)

Subtract 𝐙𝒕−𝟏 from both sides to give

∆Z𝑡 = (A1 − I)Z𝑡−1 + A2Z𝑡−2 + ε𝑡 (29)

From RHS subtract and add (A1 − I)Z𝑡−2, to give

∆Z𝑡 = A0 + (A1 − I)Z𝑡−1 + −(A1 − I)Z𝑡−2⏟ (A1−I)∆Z𝑡−1

+ (A1 − I)Z𝑡−2 +

A2Z𝑡−2 + ε𝑡 (30)

∆Z𝑡 = A0 + (A1 − I)∆Z𝑡−1 + (A1 + A2 − I)Z𝑡−2 + ε𝑡 (31)

∆Z𝑡 = A0 + Γ1∆Z𝑡−1 +ΠZ𝑡−2 + ε𝑡 (32)

Where Γ1 = −(I − A1) and Π = −(I − A1 − A2)

In general VAR(p) in VECM is of the form:

∆Z𝑡 = A0 + ΠZ𝑡−𝑝 + ∑ Γ𝑖∆Z𝑡−𝑖𝑝−1𝑖=1 + ε𝑡. (33)

Where 𝒊 = 𝟏,… , 𝒑 − 𝟏 is the lag-length, which is sufficient to remove any

remaining serial correlation in the model - and is statistical metric. We readily

see from this expression that the VECM expresses the VAR in I(0) space, i.e.,

the dependent variable is now ∆𝐙𝒕~𝐈(𝟎) and ∆𝐙𝒕−𝑖~𝐈(𝟎), which facilitates

standard hypothesis testing.

The advantage of this parametrization is in the economic interpretation of the

coefficients. The short run dynamics are isolated in 𝚪𝒊, which is also useful in

examining the concept of Granger causality (Appendix 1). The 𝚷 matrix

contains the long-run (cointegrating) relationships and the nature of long-run

causality, which is the idea behind Granger’s Representation Theorem that for any

set of cointegrated variables, there is error correction representation.

Johansen (1988) shows that we can write the long-run matrix, 𝚷 as,

Π = α𝛽′

Where 𝛂 and 𝛃 are both matrices of dimensions (k x r).

Under this decomposition, eqn. 32 becomes:

VAR and Vector Error Correction Model

103 |

∆Z𝑡 = A0 + α𝛽′Z𝑡−𝑝 + ∑ Γ𝑖∆Z𝑡−𝑖

𝑝−1𝑖=1 + ε𝑡. (36)

The 𝒓 columns of 𝛃 represent the co-integrating vectors that quantify the

‘long-run’ (or equilibrium) relation(s) between the variables in the system, and

as we have suggested, this could be the statistical analogue of economic

equilibrium. The 𝒓 columns of error correction coefficients 𝛂 load deviations

from equilibrium into ∆𝐙𝒕 for correction, thereby ensuring that the equilibrium

is maintained.

Note that:

If 𝐙𝒕~𝐈(𝟏), 𝒓 can be at most 𝒌 − 𝟏. In other words, where 𝒌 = 𝟐, there can at

most be one equilibrium relationship between the two variables (i.e., 𝒓 = 𝟏).

If 𝒓 = 𝟎, there is no cointegration, and the VECM collapses to a VAR in first

differences, that is,

∆Z𝑡 = A0 + ∑ Γ𝑖∆Z𝑡−𝑖𝑝−1𝑖=1 + ε𝑡 (37)

If 𝒓 = 𝟏, there is a single cointegrating relationship - an equilibrium which is a

‘long run’ property of the data.

In addition, some interpretation is in order. For purposes of clarity, we

continue with VAR(2) model in eqn. 32, but for k =2 variables, z1 and z2.

which when unpacked, ignoring the 𝐀𝟎 term, is:

[∆𝑍1𝑡∆𝑍2𝑡

] = [𝛼11𝛼21

] [𝛽11 𝛽21] [𝑍1𝑡−2𝑍2𝑡−2

] + Γ1 [∆𝑍1𝑡−1∆𝑍2𝑡−1

] + [휀1𝑡휀2𝑡] (38)

Ignoring the 𝚪𝟏 term for the moment, we will focus on the decomposition of

the long-run matrix:

Π = αβ′ ⇒ [𝛼11𝛼21

] [𝛽11 𝛽21] [𝑍1𝑡−2𝑍2𝑡−2

] (39)

𝛃𝒋𝒊 describes the cointegrating or ‘long-run relationship while the error

correction coefficients, 𝜶𝒊𝒋, quantifies the speed at which deviation from each

equilibrium is corrected.

Literally, cointegration implies that in equilibrium:

β11𝑧1 + β21𝑧2 = 0. (40)


|104

Normalizing on say (𝒛𝟏), this relation implies

𝑧1 = β𝑧2, (41)

Where β = −β21 −β11⁄

This is the cointegrating (equilibrium) relationship and 𝛃 is called the

cointegrating or long-run parameter – noting that equilibrium normalization is

arbitrary.

Deviations from equilibrium are not zero, i.e.,

β11𝑧1𝑡−2 + β21𝑧2𝑡−2 ≠ 0 (42)

And serves to measure the extent to which variables are out of their

equilibrium, with ‘disequilibrium:

𝑒 = (𝑧1 − β𝑧2). (43)

Because of cointegration, 𝒆 is I(0). Incorporating the notion of 𝒆, eqn. 38 can

be rewritten as:


] = Γ1 [∆𝑍1𝑡−1∆𝑍2𝑡−1

] + [𝛼11𝛼21

] e𝑡−2 + [휀1𝑡휀2𝑡] (44)

The error correction coefficients (𝜶𝒊𝒋) measure how fast each variable adjusts

to deviations from equilibrium (𝒆), if at all. Moreover, if indeed the system

variables are cointegrated, then at least one of the 𝜶𝒊𝒋 ≠ 𝟎. This is essentially

the Granger (Nobel Prize) Representation Theorem, i.e., that cointegration ⇔

error correction. This error correction is the means by which equilibrium is

maintained and whether one or all are significant can be tested.

Where 𝛂𝟏𝟏 = 𝟎, then 𝒛𝟏 does not adjust to disequilibrium, and in this case, it is

‘weakly exogenous’ for the long-run, i.e., 𝒛𝟏 drives the long-run relationship

but itself is not driven by other variables in the long-run relationship – this will

constitute part of the sign restrictions to be imposed in practical applications.

Lastly, we examine a related concept of ‘Granger Causality’ in the VECM

which relates to the short run parameters contained in 𝚪𝟏matrix, which for

exposition, we unpack to:


] = [Γ11 Γ12Γ21 Γ22

] [∆𝑍1𝑡−1∆𝑍2𝑡−1

] + [𝛼11𝛼21

] e𝑡−2 + [휀1𝑡휀2𝑡] (45)


105 |

If 𝚪𝟏𝟐 = 𝟎, then 𝒁𝟐𝒕is not a Granger cause of 𝒛𝟏𝒕. Thus, if, from eqn. 44,

𝒛𝟏𝒕 is weakly exogenous and 𝒛𝟐𝒕 does not Granger-cause it, then 𝒛𝟏𝒕 is also

‘strongly exogenous’.

I want to stress for emphasis that this extension of VAR to VECM, in the sort

of exercise at hand, i.e., MTM is only necessary for hypothesis testing, and

specifically, testing for weak exogeneity, but only where we find ourselves in a

mix of theory regarding the ordering of variables in the SVAR. We limit

VECM estimation here to hypothesis testing largely because, as we may know,

and as indeed has been established from IRFs, monetary policy has a zero long-

run effect on real variables – the so called long-run monetary policy neutrality.

Implementing this useful extension exercise demands of us to cautiously think

about;

i. The deterministic terms that enter into the cointegrating equation;

ii. Determining that cointegration exists

iii. Determining that all or subset of variables are important for the

existing long-run equilibrium, i.e., long-run exclusion tests.

iv. Testing for weak exogeneity; and

v. Performing, probably Granger non-causality tests.

6.2 Demonstration Determination of Cointegration and Estimation of VECM

We have already established that four of the five variables - lgdp, lcpi, lM2 and

lexr are unit root non-stationary, i.e., I(1), while tb91 is boarder line stationary.

Arguably therefore, a linear combination involving lgdp, lcpi, lexr, lM2 – all I(1)

and tb91 - the only I(0) variable could be cointegrated. Note that in a

multivariate model, a combination of I(1) and I(0) data is permissible, but the

number of cointegrating relations increases correspondingly with every

stationary variable included in the system (Harris and Sollis, 2005 p. 112).

In what follows, we evaluate the existence of equilibrium relation among the

five endogenous variables using the Johansen’s (1988) trace statistic (given in

eqn. 46), as this is asymptotically more correct than the bottom-to-top

alternative of the Max-Eigen statistic (Juselius, 2006: 131-134).

𝜆𝑡𝑟𝑎𝑐𝑒 = −𝑇∑ 𝑙𝑜𝑔(1 − 𝜆𝑛𝑖=𝑟+1 ) (46)


|106

Where T is the effective sample and 𝜆 is the estimated Eigenvalues.

The Johansen’s (1988) trace test has however since been shown to have a finite

sample bias, with the implication that it often indicates too many cointegrating

relations, i.e., the test is over-sized (Juselius, 2006: 140-2; Cheung and Lai,

1993b; Reimers, 1992). Hence, for the sort of small samples we encounter quite

often, especially on developing COMESA countries, it could be helpful having

to adjust for finite-sample bias. This is achieved using the correcting procedure

suggested by Reimers (1992) as in eqn. 47 (Harris and Sollis, 2005: 122-24).

𝜆𝑡𝑟𝑎𝑐𝑒 = −(𝑇 − 𝑛𝑘)∑ 𝑙𝑜𝑔(1 − 𝜆𝑛𝑖=𝑟+1 ) (47)

Where in the adjustment equation, T is as defined before and n and k are the

number of variables in the system (5 in this case) and lag-length used when

testing for cointegration (2 in this case).

With the order of VAR beforehand, prior to implementation of the trace test,

is a choice of the deterministic components (linear trend and constant)

(Johansen, 1994) that must enter into the cointegrating space. Note that from

the visual inspection of the data in Fig. 2, it looks reasonable to supply the

standard unrestricted VECM model given by eqn. 33 with a restricted trend

and an unrestricted constant, as is, at least initially. This is because, whilst the

variables in levels appear

to be trending, we cannot

be sure these linear trends

will cancel out in the

cointegrating relation. We

therefore allow for the

possibility of a linear trend

in the data by restricting

the trend and unrestricting

the constant. Including

unrestricted constant

allows for linear trends in

both cointegrating space

and in the variables in

levels and produces a non-

zero mean in the

cointegrating relation and


107 |

avoids creation of quadratic trends in the levels, which would arise if both the

constant and trend are unrestricted (Juselius, 2006: 99-100). Accordingly, I

unrestricted the intercept, 𝐀𝟎 to lie in the cointegrating space but restricted the

trend to zero.

With a lag of 2 and unrestricted intercept and no trend, corresponding to

option 3 of the deterministic trend assumption of test in the Cointegration Test

Specification dialogue box of EViews, we implement the Johansen (1988) trace

statistic test for cointegration through the steps described here-under.

Turn to and make active our COMESA2018 EViews work file as we have done

before. Select or highlight the five series: lcpi, lexr, lgdp, lm2 and tb91. Click on

Quick at the Top most menu window and navigate through the drop-down

menu to Group Statistics. Follow the arrow and navigate through to

Johansen Cointegration Test, as shown

in the screen print.

Choosing this brings forth the Series List

box, highlighting the variables we have

selected and on which we want to

perform cointegration test (note that this

list can be adjusted). Click OK, for the

Johansen Cointegration Test dialogue

box.

As already alluded to, we allow

for the unrestricted intercept,

but restrict the trend in the

deterministic trend assumption

of the test, i.e., option 3 – which

incidentally is EViews default. I

have entered under Exog

variables*, loil, just as it was

with VAR, though it is of no

added value as the test critical

values assume no exogenous

series. The Lag Intervals default

of 2 is incidentally what has

been determined beforehand. Click OK, for the results in Table 21.


|108

Table 21: Johansen's trace test results

Date: 04/23/18 Time: 16:56 Sample (adjusted): 2000Q4 2017Q3 Included observations: 68 after adjustments Trend assumption: Linear deterministic trend Series: LCPI LEXR LGDP LM2 TB91 Exogenous series: LOIL Warning: Critical values assume no exogenous series Lags interval (in first differences): 1 to 2 Unrestricted Cointegration Rank Test (Trace)

Hypothesized No. of CE(s) None *

Eigenvalue 0.43181

Trace Statistic 98.57763

Trace Statistic# 84.08097

0.05 Critical Value 69.81889

Prob.** 0.0001

At most 1 * 0.40296 60.13727 51.29361 47.85613 0.0023

At most 2 0.183746 25.06482 21.37888 29.79707 0.1591

At most 3 0.133213 11.25883 9.60316 15.49471 0.1961

At most 4 0.022356 1.537441 1.31136 3.841466 0.215

Trace test indicates 2 cointegrating eqn(s) at the 0.05 level

* denotes rejection of the hypothesis at the 0.05 level

**MacKinnon-Haug-Michelis (1999) p-values # small sample correction

Looking at the results, we see that the values for both trace statistic and trace

statistic* in the rows for None and At most 1 are higher than the

corresponding 5% critical values, i.e., 98.578 (84.081) > 69.819, and 60.137

(51.294) > 47.856, respectively. This suggests the presence of two equilibrium

(stationary) relations, even when corrected for small sample bias among the

variables at the conventional 5% level of significance cannot be rejected. Even

further, EViews generously indicates with * in the output table, but also

explicitly indicates the same in the first line of the accompanying notes beneath

the table.

However, we know that tb91 variable is I(0) while the rest of the variables are

I(1). This implies that potentially, the I(0) variable may have added an

additional cointegrating relation in the model. Thus, adjusting the number of

cointegration equations for the one I(0) variable leaves only one long-run

relation.


109 |

6.3 Estimating a VECM

With a unique relationship as indicated in Table 21 among the five endogenous

variables, identification of the long-run relation becomes relatively direct. As

normalization is arbitrary, the only existing cointegration relation will here be

normalized on GDP, translating into the VECM estimates in Table 22. The

procedure for achieving this is detailed as follows.

As with VAR estimation procedure described earlier, check and highlight the

five series in the cointegrating relationship. Point the cursor in any of the

highlighted variables and right click. Follow the Open command and in the

drop-down menu, select as VAR…. The menu bar of the VAR Specification

dialogue box that pops up, as in the screen print, has 3 parts: i) Basics, ii)

Cointegration, and iii) VEC restrictions. We begin with the Basics window,

where under VAR Type, we select Vector Error Correction (the default is

Unrestricted VAR). The rest of the entries in the screen print are as before.


|110

Next check Cointegration in the menu bar, and input Number of

cointegrating rank, as may have been determined (i.e. 1 in our case) and we

are allowing for intercept (no trend in cointegrating Equation (CE) and

VAR), i.e. option 3) in EViews, as shown in the screen print.

Finally, we turn to VEC Restrictions window.

It is in this window that we have the leverage to impose: long-run

normalization, long-run exclusion and weak exogeneity restrictions, and some

detailed illustration is warranted.

Restrictions may be placed on the coefficients B(r, k) of the r-th cointegrating

relation:

B(r,1)*LCPI + B(r,2)*LEXR + B(r,3)*LGDP + B(r,4)*LM2 + B(r,5)*TB91 (48)

This equation is akin to eqn. 39 of the theoretical model and is the long-run

equilibrium. For r = 1 as determined in Table 21, imposing the restriction that

B(1, i)=1 for i = 1…5, is an equivalent of the normalization given by eqn. 41.

In the screen print, the restriction b(1,3)=1, implies that normalization is on

GDP or that we are estimating the GDP equation in a VECM framework.

Note that normalization is arbitrary and can be done on any one variable,

providing there is leaning on some economic intuition. In what follows, we

demonstrate the VECM output. In the screen print above, make active

Impose Restrictions box and provide in the box thereunder, the restriction

b(1,3)=1, i.e., normalize on GDP. Click OK for the VECM results in Table 22.


111 |

Table 22: Vector Error Correction Estimates

Vector Error Correction Estimates

Date: 04/24/18 Time: 15:05




Cointegration Restrictions:

B(1,3)=1

Convergence achieved after 1 iterations.

Restrictions identify all cointegrating vectors

Restrictions are not binding (LR test not available)

Cointegrating Eq: CointEq1

LCPI(-1) -0.104111

(0.26980)

[-0.38588]

LEXR(-1) 0.779756

(0.16835)

[ 4.63169]

LGDP(-1) 1.000000

LM2(-1) -0.612232

(0.12040)

[-5.08508]

TB91(-1) 0.005176

(0.00372)

[ 1.39244]

C -9.617435

Error Correction: D(LCPI) D(LEXR) D(LGDP) D(LM2) D(TB91)

CointEq1 0.004642 -0.354163 -0.018260 -0.141348 -9.499563

(0.02223) (0.07782) (0.04327) (0.06011) (5.37615)

[ 0.20883] [-4.55104] [-0.42197] [-2.35149] [-1.76698]

D(LCPI(-1)) 0.651879 -0.164099 -0.293812 -0.359738 164.7688

(0.15481) (0.54194) (0.30135) (0.41860) (37.4395)

[ 4.21074] [-0.30280] [-0.97498] [-0.85937] [ 4.40093]

D(LCPI(-2)) -0.218317 -0.865829 -0.055406 -0.106765 -114.4386

(0.15478) (0.54181) (0.30128) (0.41850) (37.4303)

[-1.41054] [-1.59804] [-0.18391] [-0.25511] [-3.05738]

D(LEXR(-1)) -0.014779 0.184653 0.046651 -0.088046 3.661002

(0.03445) (0.12060) (0.06706) (0.09315) (8.33130)

[-0.42899] [ 1.53117] [ 0.69568] [-0.94519] [ 0.43943]

D(LEXR(-2)) 0.002703 0.011326 -0.021026 -0.177891 1.229858

(0.03328) (0.11651) (0.06479) (0.08999) (8.04887)

[ 0.08122] [ 0.09721] [-0.32455] [-1.97671] [ 0.15280]

D(LGDP(-1)) 0.034705 -0.151297 -0.457682 0.157668 9.235799

(0.06493) (0.22730) (0.12639) (0.17557) (15.7030)

[ 0.53448] [-0.66562] [-3.62108] [ 0.89802] [ 0.58815]

D(LGDP(-2)) -0.017465 -0.506523 -0.307409 0.057205 -15.22198

(0.06291) (0.22024) (0.12247) (0.17012) (15.2151)

[-0.27760] [-2.29987] [-2.51015] [ 0.33627] [-1.00045]

D(LM2(-1)) 0.100083 -0.096778 -0.062131 -0.159615 0.597304

(0.05470) (0.19148) (0.10648) (0.14791) (13.2285)

[ 1.82967] [-0.50541] [-0.58352] [-1.07917] [ 0.04515]

D(LM2(-2)) 0.003324 -0.251315 0.331758 0.190495 -7.798826

(0.05418) (0.18966) (0.10546) (0.14650) (13.1026)

[ 0.06136] [-1.32508] [ 3.14574] [ 1.30033] [-0.59521]

D(TB91(-1)) 0.001116 0.003931 0.000297 0.000737 0.052385

(0.00049) (0.00173) (0.00096) (0.00134) (0.11957)

[ 2.25752] [ 2.27132] [ 0.30892] [ 0.55124] [ 0.43810]

D(TB91(-2)) 7.38E-05 0.003318 0.001898 0.000450 0.080294

(0.00047) (0.00164) (0.00091) (0.00127) (0.11340)

[ 0.15749] [ 2.02111] [ 2.07882] [ 0.35500] [ 0.70803]

C -0.020032 0.309989 0.042652 0.153174 6.871951

(0.02231) (0.07809) (0.04342) (0.06032) (5.39455)

[-0.89805] [ 3.96980] [ 0.98230] [ 2.53955] [ 1.27387]

LOIL 0.006054 -0.064604 -0.005462 -0.026815 -1.839598

(0.00527) (0.01843) (0.01025) (0.01424) (1.27328)

[ 1.14982] [-3.50524] [-0.53299] [-1.88357] [-1.44477]

R-squared 0.500961 0.489082 0.387289 0.301453 0.415049

Adj. R-squared 0.392080 0.377609 0.253606 0.149043 0.287423

Sum sq. resids 0.005599 0.068612 0.021215 0.040936 327.4575

S.E. equation 0.010090 0.035320 0.019640 0.027282 2.440035

F-statistic 4.600991 4.387442 2.897078 1.977904 3.252080

Log likelihood 223.2711 138.0715 177.9793 155.6310 -149.9307

Akaike AIC -6.184444 -3.678572 -4.852333 -4.195030 4.792080

Schwarz SC -5.760127 -3.254255 -4.428016 -3.770712 5.216398

Mean dependent 0.015397 0.011209 0.014714 0.037783 -0.101765

S.D. dependent 0.012940 0.044770 0.022733 0.029574 2.890549

Determinant resid covariance (dof adj.) 1.30E-13

Determinant resid covariance 4.48E-14


Akaike information criterion -14.48748

Schwarz criterion -12.20269

𝛃′

𝚪𝒊


|112

Each column in the table corresponds to the equation for one endogenous

variable in the VAR. For each right-hand side variable, EViews reports the

coefficient point estimate, the estimated coefficient standard error (in round

brackets) and the t-statistic (in square brackets)

6.4 Long-run Exclusion Tests

Based on the estimated VECM, zero restrictions on 𝛃 imply long-run

exclusion, i.e., test whether (or not) a corresponding variable can be excluded

from the estimated long-run relation. If accepted, the variable is redundant to

the long-run relation(s) (Juselius, 2006: 176) and so can at most have a short-

run impact. For purposes of demonstration, we may want to check the long-

held view of monetary policy neutrality, i.e. that monetary policy has a zero

long-run effect on real variables. To test this view, we impose a VEC

restriction that B(1,5)=0, while maintaining normalization .

To do this, while in the VECM result (Table 22) window (assuming this is still

live or else re-estimate), click on Estimate to get back to the VEC

Restrictions dialogue box above. Click VEC Restrictions in the menu bar of

the window and impose a restriction as shown in the screen print. Execute the

restriction to generate the output in Table 23.


113 |

Table 23: A test of monetary policy neutrality


Date: 04/24/18 Time: 15:44





B(1,3)=1, B(1,5)=0

Convergence achieved after 61 iterations.


LR test for binding restrictions (rank = 1):

Chi-square(1) 0.207866

Probability 0.648445


LCPI(-1) -0.182368

(0.24636)

[-0.74025]

LEXR(-1) 0.853532

(0.14676)

[ 5.81591]

LGDP(-1) 1.000000

LM2(-1) -0.595378

(0.10569)

[-5.63348]

TB91(-1) 0.000000

C -9.913518


CointEq1 0.004871 -0.410354 -0.020845 -0.144820 -4.414180

(0.02371) (0.08012) (0.04614) (0.06435) (5.86383)

[ 0.20544] [-5.12147] [-0.45179] [-2.25041] [-0.75278]

Clearly, we see that the p-value resulting from the restriction that B(1,5)=0 is

0.648. Based on this, long-run exclusion of tb91, in GDP normalization,

cannot be rejected; amounting to the finding that monetary policy has no long-

run impact but could at best have a short-run impact. This is not only

consistent with the long-held view of monetary policy neutrality, i.e., that

monetary policy has a zero long-run effect on real variables (Christiano et

al.,1999), but also with our IRFs estimates.

6.5 Long-run Weak Exogeneity Tests

This constitute restrictions on the adjustment coefficients, iα given as A(k, r)

where k= 1, …5 for k = 1 D(LCPI) equation; k = 2 D(LEXR) equation; k = 3

D(LGDP) equation; k = 4 D(LM2) equation and k = 5 D(TB91) equation, and

r = 1 in the VEC Restrictions window, and are accomplished by a zero row in


|114

α (Johansen, 1996). Each iα measure the speed at which the corresponding

variable in ∆𝐙𝒕 in eqn. 44 adjusts to deviations from the equilibrium.

Therefore, a zero coefficient implies that the variable impacts on the long-run

stochastic path of the other variables of the system, while at the same time has

not been influenced by them (Juselius, 2006: 193), and is as such, considered to

be weakly exogenous for the long-run parametersβ .

For purposes of demonstration, we may want to check for example if exchange

rate is weakly exogenous to the domestic variables, a test we accomplish by

imposing a VEC restriction that A(2,1)=0.

To do this, while in the VECM result window (assuming this is still live or else

re-estimate), click on Estimate to get back to the VEC Restrictions dialogue

box above. Click VEC Restrictions in the menu bar of the window and

impose a restriction as shown in the screen print. Execute the restriction to

generate the following output in Table 24.


115 |

Table 24: A test of weak exogeneity


Date: 04/25/18 Time: 07:12





B(1,3)=1, A(2,1)=0

Maximum iterations (500) reached.


LR test for binding restrictions (rank = 1):

Chi-square(1) 2.717220

Probability 0.099271


LCPI(-1) 1.843897

(1.38927)

[ 1.32724]

LEXR(-1) -0.878474

(0.86690)

[-1.01335]

LGDP(-1) 1.000000

LM2(-1) -1.028502

(0.61996)

[-1.65897]

TB91(-1) 0.099142

(0.01914)

[ 5.17959]

C -3.263896


CointEq1 0.000308 0.000000 -0.000692 -0.013772 -4.715715

(0.00415) (0.00000) (0.00880) (0.01219) (0.92670)

[ 0.07413] [NA] [-0.07865] [-1.12955] [-5.08870]


|116

The p-value resulting from the restriction that A(2,1)=0 is 0.099, which is

significantly different zero, even if boarder line so. This suggests, weak

exogeneity of exr can be rejected and amounts to the finding that it is

endogenous, i.e., are influenced by the other variables in the system.

6.6 Granger Non-causality Test

Determining as in the above that variables are cointegrated implies there must

be Granger causality in at least one direction. To illustrate this, we estimate

VEC Granger Causality in the VECM framework for the output in Table 25.

Assuming we can navigate through the process leading to the results in Table

22, once in results window, click on View and in the drop-down menu, choose

Lag Structure, and following the arrow, choose Granger Causality/ Block

Exogeneity Tests – a road map shown here in the screen print. Implement

this for the results in Table 25.


117 |

Table 25: VEC Granger Causality/Block Exogeneity test

VEC Granger Causality/Block Exogeneity Wald Tests

Date: 04/25/18 Time: 09:38

Sample: 2000Q1 2017Q3


Dependent variable: D(LCPI)

Excluded Chi-sq df Prob.

D(LEXR) 0.184041 2 0.9121

D(LGDP) 0.583467 2 0.7470

D(LM2) 3.412090 2 0.1816

D(TB91) 5.096852 2 0.0782

All 9.179960 8 0.3273

Dependent variable: D(LEXR)


D(LCPI) 4.998432 2 0.0821

D(LGDP) 5.383924 2 0.0677

D(LM2) 1.836494 2 0.3992

D(TB91) 8.722769 2 0.0128

All 18.07900 8 0.0206

Dependent variable: D(LGDP)


D(LCPI) 1.858613 2 0.3948

D(LEXR) 0.520750 2 0.7708

D(LM2) 11.18510 2 0.0037

D(TB91) 4.355405 2 0.1133

All 15.87207 8 0.0442

Dependent variable: D(LM2)


D(LCPI) 1.651997 2 0.4378

D(LEXR) 5.755381 2 0.0563

D(LGDP) 0.807757 2 0.6677

D(TB91) 0.407811 2 0.8155

All 11.09241 8 0.1965

Dependent variable: D(TB91)


D(LCPI) 19.66035 2 0.0001

D(LEXR) 0.252507 2 0.8814

D(LGDP) 2.203567 2 0.3323

D(LM2) 0.376372 2 0.8285

All 31.29667 8 0.0001

The null hypothesis of the test, in part, is that individually, variable i is

excludable from any of the five system equations, and that collectively, all

system variables are excludable from each of the five system equations.

The result table has 5 tranches of variable exclusion, corresponding to the five

equations in the system. The interpretation is readily facilitated by the p-values.


|118

In summary, we see a two-way causation, with influences running from CPI to

interest rates (d(tb91) block) and from interest rates to inflation (d(lcpi) block),

i.e., cpi ↔ tb91, which is intuitive more so in an environment of inflation

targeters. Dynamics in inflation very much influence the monetary policy

signals and also monetary policy signals are expected to anchor inflation

expectations. We also find unidirectional influences from cpi → exr; gdp →

exr; tb91 → exr; money → gdp; and exr → money, and not vice versa. All

these are economically intuitive.

Appendix 1

Granger Causality Test

Determining in the above that variables are cointegrated implies there must be

Granger non- causality in at least one direction. On this account, and following

Granger (1969), y is said to be Granger-caused by x if x helps in the prediction

of y or if the coefficients on the lagged x's are statistically significant in y and

vice versa. Thus, the Granger non-causality model is specified as follows:

n

i

n

i

tntntt xyy1 1

112111 (34)

n

i

n

i

tntntt xyx1 1

222212 (35)

Where n is the maximum lag-length; and t1 and

t2 are additive stochastic error

terms, which are by assumption normally distributed with a zero mean and a

constant variance. In light of equations (34) and (35), we can deduce two

testable hypotheses:

that 012 while 011 , i.e. x does not Granger-cause y (no causality

from x to y)

that 022 while 021 , i.e. y does not Granger-cause x (no causality

from y to x)

Acceptance of either hypotheses would suggest the existence of unidirectional

causality between x and y, and feedback between x and y may be understood to

exist if 02i and 01i . Alternatively, no causality between x and y

exists if 02i and 01i .

References Abuka, C., Alinda, R.K., Minoiu, C., Peydro, J-L and Presbitero, A.F (2015)

Monetary Policy in a Developing Country: Loan Applications and Real

Effects, IMF working paper, WP/15/270

Bernanke B. S. and Mark G. (1995) Inside the Black Box: The Credit Channel of

Monetary Policy Transmission‖, Journal of Economic Perspectives 9(4), 27-48.

Bwire, T. (2018) Modelling and forecasting volatility in Financial markets using

EViews, practical manual developed for COMESA central banks (forth

coming).

Blanchard, O.J and Quah, D.T. (1989) The dynamic effects of aggregate demand

and supply disturbances, American Economic Review 79 (4), 655 – 73.

Cecchetti, S. (1995) Distinguishing theories of monetary policy transmission

mechanism Federal Reserve Bank of St. Louis, 77, 83-97.

Cheung, Y-W., and Lai, K.S. (1993a) A Fractional Cointegration Analysis of

Purchasing Power Parity, Journal of Business and Economic Statistics 11: 103-

112

_____. (1993b) Finite-Sample Sizes of Johansen’s Likelihood Ratio Tests for

Cointegration, Oxford Bulletin of Economics and Statistics 55: 313-328

Christiano, L. J., Eichenbaum, M. and Evans, C. (1997) Sticky prices and limited

participation models of money: a comparison, European Economic Review,

41, 1201-49

______. (1999) Monetary policy shocks: what have we learned and to what end? in

Taylor, JB and Woodford, M (eds), Handbook of Macroeconomics, Vol. 1A,

Amsterdam, Elsevier Science, North-Holland, pp 65 – 148.

Clarida, R., Gal´ı, J. and M. Gertler, M. (1998) Monetary Policy Rules in Practice:

Some International Evidence, European Economic Review 42, 1033–

1067.

Davoodi, H.R, Dixit, S., Pinter, G. (2013) Monetary Transmission Mechanisms in

the East African Community: An Empirical Investigation, IMF working

paper, WP/13/39

Dennis, G. J. (2006) CATS in RATS Cointegration Analysis of Time Series, Version 2,

Estima, Evanston, Illinois, USA


121 |

Dickey, D.A., and W.A. Fuller (1979) Distribution of estimators for Autoregressive

time series with a unit root, Journal of American Statistical Association 74(366),

427-31.

______. (1981) Likelihood ratio statistics for Autogressive time series with a unit

root, Econometrica, Econometric society 49(4), 1057-1072

Doornik, J.A. and Hansen, H. (2008) An omnibus test for Univariate and

Multivariate Normality, Oxford Bulletin of Economics and Statistics 70, 927-939.

Enders, Walter (2010). Applied Time Series Econometrics, Wiley

Engel, F. Robert (1982) Autoregressive Conditional Heteroskedasticity with

Estimates of the Variance of United Kingdom Inflation, Econometrica 50(4),

987-1007

Engle R. and C.W. Granger (1987) Cointegration and Error Correction:

Representation and Testing, Econometrica 48, 1-48.

Godfrey, L.G. (1988) Misspecification Tests in Econometrics, Cambridge: Cambridge

University Press

Granger, C. and Newbold, P. (1974) Spurious Regressions in Econometrics, Journal

of Econometrics 2, 111-20

Hamilton, J.D. (1994) Time Series Analysis, Princeton University Press, Princeton.

Harris, R. and Sollis, R. (2005) Applied Time Series Modelling and Forecasting, John

Wiley & sons Ltd

Johansen, S. (1988) Statistical Analysis of Co-integrating Vectors, Journal of Economic

Dynamics and Control, 12, (1988): 231-254.

______. (1994) The role of Constant and Linear Terms in Cointegration Analysis

of Non-stationary variables, Econometric Reviews 13, 205-229.

Juselius, K. (2006) The Cointegrated VAR Model: Methodology and Applications, Advanced

Texts in Econometrics, Oxford University Press, Oxford

______. (2003) The Cointegrated VAR model: Econometric Methodology and

Macroeconomic Applications, manuscript.

Lucas, R. (1972) Expectations and the Neutrality of Money, Journal of Economic

Theory 4: 103-124

Lütkepohl, H. (2005) New Introduction to Multiple Time Series Analysis, Berlin,

Springer Verlag.


|122

Lütkepohl, H., and Krätzig, M. (2004) Applied Time Series Econometrics: Themes in

Modern Econometrics, Cambridge University Press.

Rahbek, A., Hansen, E., and Dennis, J.D. (2002) ARCH innovations and their

impact on cointegration rank testing, Working Paper, Department of Applied

Mathematics and Statistics, University of Copenhagen.

Reimers, H.-E. (1992) Comparison of tests for multivariate cointegration, Statistical

Papers 33: 335 359

Rotemberg, Julio R., and Michael Woodford (1997) An Optimization-Based

Econometric Framework for the Evaluation of Monetary Policy, In

NBER Macroeconomics Annual, 297-346. Cambridge. MA. MIT Press

Sims, C. (1980) Macroeconomics and Reality, Econometrica 48(1), 1-48.

Mackinnon, J.G. (1996) Numerical Distribution Functions for Unit Root and

Cointegration Tests, Journal of Applied Econometrics 11, 601 - 618

Mishkin, F. (1995) Symposium on the Monetary Transmission Mechanism, Journal

of Economic Perspectives 9(4) 3-10.

Montiel, P. (2013) The Monetary Transmission Mechanism in Uganda, S-43002-

UGA-1

Mugume, A. (2011) Monetary Policy Transmission Mechanisms in Uganda, Bank

of Uganda Working Paper Series, BOUWPS/01/2011, Bank of Uganda

Opolot, J. (2013) Bank lending channel of the Monetary Policy Transmission

Mechanism in Uganda: Evidence from Panel Data Analysis, Bank of

Uganda Working Paper, BOUWPS 01/2013.

Perron (1989) The Great Crash, the Oil Price Shock, and the unit root hypothesis,

Econometrica 57, 1361 -1401.

Taylor, John B. (1993). Discretion versus Policy Rules in Practice, Carnegie –

Rochester Conference Series on Public Policy 39, 195-214

Vector Autoregressions and Structural VARs for Monetary ...

Documents

Transcript of Vector Autoregressions and Structural VARs for Monetary ...