Time Series Econometrics: Asst. Prof. Dr. Mete Feridun Department of Banking and Finance Faculty of...

Time Series Time Series Econometrics:Econometrics:

Asst. Prof. Dr. Mete Feridun Asst. Prof. Dr. Mete Feridun Department of Banking and Finance Department of Banking and Finance Faculty of Business and Economics Faculty of Business and Economics Eastern Mediterranean University Eastern Mediterranean University

What is a time series?What is a time series?A time series is any series of data that A time series is any series of data that

varies over time. For examplevaries over time. For example• Monthly Tourist Arrivals from KoreaMonthly Tourist Arrivals from Korea• Quarterly GDP of LaosQuarterly GDP of Laos• Hourly price of stocks and sharesHourly price of stocks and shares• Weekly quantity of beer sold in a pubWeekly quantity of beer sold in a pubBecause of widespread availability of Because of widespread availability of

time series databases most empirical time series databases most empirical studies use time series data.studies use time series data.

Caveats in Using Time Series Caveats in Using Time Series Data in Applied Econometric Data in Applied Econometric ModelingModeling• Data Should be StationaryData Should be Stationary

• Presence of AutocorrelationPresence of Autocorrelation

• Guard Against Spurious RegressionsGuard Against Spurious Regressions

• Establish CointegrationEstablish Cointegration

• Reconcile SR with LR Behavior via ECM Reconcile SR with LR Behavior via ECM

• Implications to ForecastingImplications to Forecasting

• Possibility of Volatility ClusteringPossibility of Volatility Clustering

What is a Stationary Time What is a Stationary Time Series?Series?

• A Stationary Series is a Variable with A Stationary Series is a Variable with constant Mean across timeconstant Mean across time

• A Stationary Series is a Variable with A Stationary Series is a Variable with constant Variance across timeconstant Variance across time

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These are Examples of These are Examples of Non-Stationary Time SeriesNon-Stationary Time Series

These are Examples of These are Examples of Stationary Time SeriesStationary Time Series

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What is a “Unit Root”?What is a “Unit Root”?

If a Non-Stationary Time If a Non-Stationary Time Series Series YYtt has to be has to be “differenced” d times to make “differenced” d times to make it stationary, then it stationary, then YYtt is said is said to contain d “Unit Roots”. It is to contain d “Unit Roots”. It is customary to denote customary to denote YYtt ~ I(d) ~ I(d) which reads “which reads “YYtt is integrated is integrated of order d”of order d”

Establishment of Stationarity Establishment of Stationarity Using Differencing of Using Differencing of Integrated SeriesIntegrated Series

• If If YYtt ~ I(1), then ~ I(1), then ZZtt = = YYt – t – YYt-1t-1 is Stationary is Stationary

• If If YYtt ~ I(2), then ~ I(2), then ZZtt = = YYt – t – YYt-1t-1 – ( – (YYt – t – YYt-2t-2 )is )is

StationaryStationary

Unit Root Testing: Formal Tests Unit Root Testing: Formal Tests to Establish Stationarity of to Establish Stationarity of Time SeriesTime Series• Dickey-Fuller (DF) TestDickey-Fuller (DF) Test• Augmented Dickey-Augmented Dickey-

Fuller (ADF) TestFuller (ADF) Test• Phillips-Perron (PP) Phillips-Perron (PP)

Unit Root TestUnit Root Test• Dickey-Pantula Unit Dickey-Pantula Unit

Root TestRoot Test• GLS Transformed GLS Transformed

Dickey-Fuller TestDickey-Fuller Test• ERS Point Optimal TestERS Point Optimal Test• KPSS Unit Root TestKPSS Unit Root Test• Ng and Perron TestNg and Perron Test

What is a Spurious What is a Spurious Regression?Regression?A Spurious or Nonsensical relationship A Spurious or Nonsensical relationship

may result when one Non-stationary may result when one Non-stationary time series is regressed against one time series is regressed against one or more Non-stationary time seriesor more Non-stationary time series

The best way to guard against The best way to guard against Spurious Regressions is to check for Spurious Regressions is to check for “Cointegration” of the variables used “Cointegration” of the variables used in time series modelingin time series modeling

Symptoms of Likely Presence Symptoms of Likely Presence of Spurious Regressionof Spurious Regression

• If the If the RR2 2 of the regression is greater of the regression is greater than the Durbin-Watson Statisticthan the Durbin-Watson Statistic

• If the residual series of the regression If the residual series of the regression has a Unit Root has a Unit Root

CointegrationCointegration• Is the existence of a long run Is the existence of a long run

equilibrium relationship among time equilibrium relationship among time series variablesseries variables

• Is a property of two or more variables Is a property of two or more variables moving together through time, and moving together through time, and despite following their own individual despite following their own individual trends will not drift too far apart since trends will not drift too far apart since they are linked together in some sensethey are linked together in some sense

Two Cointegrated Time Two Cointegrated Time SeriesSeries

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Cointegration Analysis:Cointegration Analysis: Formal Tests Formal Tests

• Cointegrating Regression Durbin-Cointegrating Regression Durbin-Watson (CRDW) TestWatson (CRDW) Test

• Augmented Engle-Granger (AEG) TestAugmented Engle-Granger (AEG) Test

• Johansen Multivariate Cointegration Johansen Multivariate Cointegration Tests or the Johansen MethodTests or the Johansen Method

Error Correction Mechanism Error Correction Mechanism (ECM)(ECM)

• Reconciles the Static LR Equilibrium Reconciles the Static LR Equilibrium relationship of Cointegrated Time relationship of Cointegrated Time Series with its Dynamic SR Series with its Dynamic SR disequilibriumdisequilibrium

• Based on the Granger Based on the Granger Representation Theorem which Representation Theorem which states that “If variables are states that “If variables are cointegrated, the relationship among cointegrated, the relationship among them can be expressed as ECM”.them can be expressed as ECM”.

Forecasting: Main Forecasting: Main MotivationMotivation• Judicious planning Judicious planning

requires reliable requires reliable forecasts of decision forecasts of decision variablesvariables

• How can effective How can effective forecasting be forecasting be undertaken in the light undertaken in the light of non-stationary of non-stationary nature of most nature of most economic variables?economic variables?

• Featured techniques: Featured techniques: Box-Jenkins Approach Box-Jenkins Approach and Vector Auto and Vector Auto regression (VAR)regression (VAR)

Approaches to Economic Approaches to Economic ForecastingForecastingThe Box-Jenkins ApproachThe Box-Jenkins Approach

• One of most widely used methodologies for One of most widely used methodologies for the analysis of time-series datathe analysis of time-series data

• Forecasts based on a statistical analysis of Forecasts based on a statistical analysis of the past data. Differs from conventional the past data. Differs from conventional regression methods in that the mutual regression methods in that the mutual dependence of the observations is of primary dependence of the observations is of primary interestinterest

• Also known as the autoregressive integrated Also known as the autoregressive integrated moving average (ARIMA) modelmoving average (ARIMA) model

Advantages• Derived from solid mathematical statistics foundations

• ARIMA models are a family of models and the BJ approach is a strategy of choosing the best model out of this family

• It can be shown that an appropriate ARIMA model can produce optimal univariate forecasts

Disadvantages

• Requires large number of observations for model identification

• Hard to explain and interpret to unsophisticated users

• Estimation and selection an art form


Differencing the series Differencing the series to achieve stationarityto achieve stationarity

Identify model to be Identify model to be tentatively entertainedtentatively entertained

Estimate the parameters Estimate the parameters of the tentative modelof the tentative model

Diagnostic checking. Is Diagnostic checking. Is the model adequate?the model adequate?

NoNo

YesYesUse the model for Use the model for forecasting and forecasting and

controlcontrol


Approaches to Economic Approaches to Economic ForecastingForecastingThe Box-Jenkins Approach-Identification ToolsThe Box-Jenkins Approach-Identification Tools

• Autocorrelation function (ACF)- ratio of sample covariance (at lag k) to sample variance

• Partial autocorrelation function (PACF) – – measures correlation between (time series) observations that are k time periods apart after controlling for correlations at intermediate lags (i.e., lags less than k). In other words, it is the correlation between Yt and Yt-k after removing the effects of intermediate Y’s.

• Correlogram – – graph showing the ACF and the PACF at different lags.

Approaches to Economic Approaches to Economic ForecastingForecastingThe Box-Jenkins Approach-IdentificationThe Box-Jenkins Approach-Identification

Type of Type of ModelModel

Typical Pattern Typical Pattern of ACFof ACF

Typical Typical Pattern of Pattern of

PACFPACF

AR (AR (pp)) Decays Decays exponentially or exponentially or

with damped sine with damped sine wave pattern or wave pattern or

bothboth

Significant Significant spikes through spikes through

lags lags pp

MA (MA (qq)) Significant spikes Significant spikes through lags through lags qq

Declines Declines exponentiallyexponentially

ARMA ARMA ((p,qp,q))

Exponential decayExponential decay Exponential Exponential decaydecay

Theoretical Patterns of ACF and PACFTheoretical Patterns of ACF and PACF

Approaches to Economic Approaches to Economic ForecastingForecastingThe Box-Jenkins Approach-Diagnostic CheckingThe Box-Jenkins Approach-Diagnostic Checking

How do we know that the model we estimated is a reasonable fit to the data?

Check residualsRule of thumb: None of the ACF and the PACF are

individually statistically significant

Formal test:

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2 Box-Pierce Q

mkn

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Ljung-Box LB

Approaches to Economic Approaches to Economic ForecastingForecastingSome issues in the Box-Jenkins modelingSome issues in the Box-Jenkins modeling

Judgmental decisions• on the choice of degree of order

• on the choice of lags Data mining

• can be avoided if we confine to AR processes only

• fit versus parsimony

Seasonality

• observations, for example, in any month are often affected by some seasonal tendencies peculiar to that month.

• the differencing operation – considered as main limitation for a series that exhibit moving seasonal and moving trend.

Vector Autoregression (VAR)Vector Autoregression (VAR)IntroductionIntroduction

• VAR resembles a SEM modeling – we consider several endogenous variables together. Each endogenous variables is explained by its lagged values and the lagged values of all other endogenous variables in the model.

• In the SEM model, some variables are treated as endogenous and some are exogenous (predetermined). In estimating SEM, we have to make sure that the equation in the system are identified – this is achieved by assuming that some of the predetermined variables are present only in some equation (which is very subjective) – and criticized by Christopher Sims.

• If there is simultaneity among set of variables, they should all be treated on equal footing, i.e., there should not be any a priori distinction between endogenous and exogenous variables.

Vector Autoregression (VAR)Vector Autoregression (VAR)Its UsesIts Uses

Forecasting

VAR forecasts extrapolate expected values of current and future values of each of the variables using observed lagged values of all variables, assuming no further shocks

Impulse Response Function (IRFs)

IRFs trace out the expected responses of current and future values of each of the variables to a shock in one of the VAR equations

Vector Autoregression (VAR)Vector Autoregression (VAR)Its UsesIts Uses

Forecast Error Decomposition of Variance (FEDVs)

FEDVs provide the percentage of the variance of the error made in forecasting a variable at a given horizon due to specific shock. Thus, the FEDV is like a (partial) R2 for the forecast error

Granger Causality Tests

Granger-causality requires that lagged values of variable A are related to subsequent values in variable B, keeping constant the lagged values of variable B and any other explanatory variables

Vector Autoregression (VAR)Vector Autoregression (VAR)Mathematical DefinitionMathematical Definition

[Y]t = [A][Y]t-1 + … + [A’][Y]t-k + [e]t or

where: p = the number of variables be considered in the systemk = the number of lags be considered in the system[Y]t, [Y]t-1, …[Y]t-k = the 1x p vector of variables

[A], … and [A'] = the p x p matrices of coefficients to be estimated[e]t = a 1 x p vector of innovations that may be contemporaneously

correlated but are uncorrelated with their own lagged values and uncorrelated with all of the right-hand side variables.

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Vector Autoregression (VAR)Vector Autoregression (VAR)ExampleExample

Consider a case in which the number of variables n is 2, the number of lags p is 1 and the constant term is suppressed. For concreteness, let the two variables be called money, mt and output, yt .

The structural equation will be:

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Then, the reduced form is

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Among the statistics computed from VARs are:

Granger causality tests – which have been interpreted as testing, for example, the validity of the monetarist proposition that autonomous variations in the money supply have been a cause of output fluctuations.

Variance decomposition – which have been interpreted as indicating, for example, the fraction of the variance of output that is due to monetary versus that due to real factors.

Impulse response functions – which have been interpreted as tracing, for example, how output responds to shocks to money (is the return fast or slow?).

Vector Autoregression (VAR)Vector Autoregression (VAR)Granger CausalityGranger Causality

In a regression analysis, we deal with the dependence of one variable on other variables, but it does not necessarily imply causation. In other words, the existence of a relationship between variables does not prove causality or direction of influence.

In our GDP and M example, the often asked question is whether GDP M or M GDP. Since we have two variables, we are dealing with bilateral causality.

Given the previous GDP and M VAR equations:

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mttttt

ymmy

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Vector Autoregression (VAR)Vector Autoregression (VAR)Granger CausalityGranger Causality

We can distinguish four cases:

Unidirectional causality from M to GDP Unidirectional causality from GDP to M Feedback or bilateral causality Independence

Assumptions: Stationary variables for GDP and M Number of lag terms Error terms are uncorrelated – if it is, appropriate

transformation is necessary

Vector Autoregression (VAR)Vector Autoregression (VAR)Granger Causality – Estimation (t-test)Granger Causality – Estimation (t-test)

A variable, say mt is said to fail to Granger cause another variable, say yt, relative to an information set consisting of past m’s and y’s if: E[ yt | yt-1, mt-1, yt-2, mt-2, …] = E [yt | yt-1, yt-2, …].

mt does not Granger cause yt relative to an information set consisting of past m’s and y’s iff 21 = 0.yt does not Granger cause mt relative to an information set consisting of past m’s and y’s iff 12 = 0. In a bivariate case, as in our example, a t-test can be used to test

the null hypothesis that one variable does not Granger cause another variable. In higher order systems, an F-test is used.

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1. Regress current GDP on all lagged GDP terms but do not include the lagged M variable (restricted regression). From this, obtain the restricted residual sum of squares, RSSR.

2. Run the regression including the lagged M terms (unrestricted regression). Also get the residual sum of squares, RSSUR.

3. The null hypothesis is Ho: i = 0, that is, the lagged M terms do not belong in the regression.

5. If the computed F > critical F value at a chosen level of significance, we reject the null, in which case the lagged m belong in the regression. This is another way of saying that m causes y.

Vector Autoregression (VAR)Vector Autoregression (VAR)Granger Causality – Estimation (F-test)Granger Causality – Estimation (F-test)

)/(

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Vector Autoregression (VAR)Vector Autoregression (VAR)Variance DecompositionVariance Decomposition

Our aim here is to decompose the variance of each element of [Yt] into components due to each of the elements of the error term and to do so for various horizon. We wish to see how much of the variance of each element of [Yt] is due to the first error term, the second error term and so on.

Again, in our example:

The conditional variance of, say mt+j, can be broken down into a fraction due to monetary shock, mt and a fraction due to the output shock, yt .

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Vector Autoregression (VAR)Vector Autoregression (VAR)Impulse Response FunctionsImpulse Response Functions

Here, our aim is to trace out the dynamic response of each element of the [Yt] to a shock to each of the elements of the error term. Since there are n elements of the [Yt], there are n2

responses in all.

From our GDP and money supply example:

We have four impulse response functions:

yttttt

mttttt

ymmy

ymym

1221212

1121111

mtjtm /ytjtm /

mtjty / ytjty /

Vector Autoregression (VAR)Vector Autoregression (VAR)Pros and ConsPros and Cons

Advantages

The method is simple; one does not have to worry about determining which variables are endogenous and which ones exogenous. All variables in VAR are endogenous

Estimation is simple; the usual OLS method can be applied to each equation separately

The forecasts obtained by this method are in many cases better than those obtained from the more complex simultaneous-equation models.


Some Problems with VAR modeling

• A VAR model is a-theoretic because it uses less prior information. Recall that in simultaneous equation models exclusion or inclusion of certain variables plays a crucial role in the identification of the model.

• Because of its emphasis on forecasting, VAR models are less suited for policy analysis.

• Suppose you have a three-variable VAR model and you decide to include eight lags of each variable in each equation. You will have 24 lagged parameters in each equation plus the constant term, for a total of 25 parameters. Unless the sample size is large, estimating that many parameters will consume a lot of degree of freedom with all the problems associated with that.


• Strictly speaking, in an m-variable VAR model, all the m variables should be (joint) stationary. If they are not stationary, we have to transform (e.g., by first-differencing) the data appropriately. If some of the variables are non-stationary, and the model contains a mix of I(0) and I(1), then the transforming of data will not be easy.

• Since the individual coefficients in the estimated VAR models are often difficult to interpret, the practitioners of this technique often estimate the so-called impulse response function. The impulse response function traces out the response of the dependent variable in the VAR system to shocks in the error terms, and traces out the impact of such shocks for several periods in the future.

Time Series Econometrics: Asst. Prof. Dr. Mete Feridun Department of Banking and Finance Faculty of...

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