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Transcript of Consultancy to Develop and Implement a Macroeconomic Model for Lesotho (DIMMoL) Macro-Econom(etr)ic...
Consultancy to Develop and Implement a Macroeconomic Model for Lesotho (DIMMoL)
Macro-Econom(etr)ic Modelling
Part 1
Dr. Stefan KoothsDIW Berlin – Macro Analysis and Forecasting
2DIMMoLMacro-
Econom(etr)icModelling Course 1
Course program
Introduction Outline of macroeconom(etr)ic models Macroeconomic framework Econometric methodology Applied econometrics with EViews Lesotho case studies
3DIMMoLMacro-
Econom(etr)icModelling Course 1
Course program
Introduction Outline of macroeconom(etr)ic models Macroeconomic framework Econometric methodology Applied econometrics with EViews Lesotho case studies
4DIMMoLMacro-
Econom(etr)icModelling Course 1
Fields of interest
Macroeconomics(model building)
Econometrics(applied mathematical statistics)
National accounting(data sources)
Country-specific knowledge(institutions, industries, policy regimes)
5DIMMoLMacro-
Econom(etr)icModelling Course 1
Recommended literature
Blanchard, O.: Macroeconomics, 3rd ed., 2003.
Wooldridge, J. M.: Introductory Econometrics – A Modern Approach, 3rd ed., 2006.
Enders, W.: Applied Economic Time Series, 2nd ed., 2004.
Matlanyane, R. A.: A Macroeconometric Model for the Economy of Lesotho: Policy analysis and Implications, 2004. (http://upetd.up.ac.za/thesis/available/etd-04182005-091509/)
6DIMMoLMacro-
Econom(etr)icModelling Course 1
My contact data
here in Maserucell: 5847.0578email: [email protected]
in BerlinDIW Berlin, German Institute for Economic ResearchKoenigin-Luise-Strasse 514195 Berlinfon: +49 30 89789-248fax: +49 30 89789-102email: [email protected]
7DIMMoLMacro-
Econom(etr)icModelling Course 1
Introduction of participants
Who are you? Where are you from? What are your specific questions and
modelling needs?
8DIMMoLMacro-
Econom(etr)icModelling Course 1
Course program
Introduction Outline of macroeconom(etr)ic models Macroeconomic framework Econometric methodology Applied econometrics with EViews Lesotho case studies
9DIMMoLMacro-
Econom(etr)icModelling Course 1
Scope of macroeconometric models
Forecasting- short-term behaviour of key macroeconomic
variables- long-term trends
Policy analysis- simulating the potential impact of alternative
policy measures- basis for long-term planning
Note:both aims are not allways harmonic“one size fits all” doesn’t apply
(different questions, different models)generally: „small is beautiful“
(robustness more important than detailed precision)
10DIMMoLMacro-
Econom(etr)icModelling Course 1
Building blocks, fundamental characteristics Institutional sectors (actors, agents) Markets (intermediaries) and regulations Time horizon and dynamics
- Equilibrium- Adjustment processes
Expectation formation
11DIMMoLMacro-
Econom(etr)icModelling Course 1
Institutional sectors
Private households- including non-profit organizations
Enterprises- independent of ownership
Public sector- government- social security systems
Rest of the world (external sector)
Financial sector
12DIMMoLMacro-
Econom(etr)icModelling Course 1
Markets and regulations
Goods market(s)- sectoral disaggregation
Labor market(s)- disaggregation by skills
Financial markets- capital market (implicit)- money market- foreign exchange
market
Income Redistribution
production = primary income
price formation(inflation rate)
(nominal) wage setting
(nominal) interest rates (nominal) exchange
rates,foreign reserves
disposable incomeinterconnection of markets:
direct vs. indirect effects
13DIMMoLMacro-
Econom(etr)icModelling Course 1
Sector interactions via markets:circular sectoral flow chart
Consultancy to Develop and Implement a Macroeconomic Model for Lesotho (DIMMoL)
Macro-Econom(etr)ic Modelling
Part 2
Dr. Stefan KoothsDIW Berlin – Macro Analysis and Forecasting
15DIMMoLMacro-
Econom(etr)icModelling Course 1
Course program
Introduction Outline of macroeconom(etr)ic models
(cont.) Macroeconomic framework Econometric methodology Applied econometrics with EViews Lesotho case studies
16DIMMoLMacro-
Econom(etr)icModelling Course 1
The IS-identity:goods and capital market intertwined
17DIMMoLMacro-
Econom(etr)icModelling Course 1
Time frames
The short run (a few years)- output driven primarily by demand- no significant price/wage movements- analytical framework: IS-LM
The medium run (up to a decade)- output determined by supply factors- adjustment via price and wage movements- fixed stock of capital, labor, technology- analytical framework: AD-AS
The long run (more than a decade)- accumulation effects of (physical and human)
capital, technological progress- analytical framework: growth-models
18DIMMoLMacro-
Econom(etr)icModelling Course 1
Types of variables
By status- endogenous- exogenous
third-party sources autoregressive forecasts (outside the model)
Most important/interesting variables- output- income- (un)employment- inflation
19DIMMoLMacro-
Econom(etr)icModelling Course 1
Types of equations
Assumption-based equations- Behavioural (e.g. consumption function)- Technological (e.g. production function)- Institutional (e.g. tax revenues)
Simple identities- e.g. disposable income
Equilibrium conditions- e.g. market clearing condition
Closed system of equations for capturing interactions and feed-backs
20DIMMoLMacro-
Econom(etr)icModelling Course 1
Supply, demand and market prices
What drives demand and supply?- components/inputs of both market sides- behavioural equations (assumptions) for all
involved sectors
What happens when demand and supply don’t match?- (temporal) disequilibriums- adjustment process (quantities, prices)
short run medium run
21DIMMoLMacro-
Econom(etr)icModelling Course 1
Goods market (income and price block)
Final demand meets production Price formation Short-run vs. long-run
- long-run: income creation (economic growth) is supply-side-driven
- short-run: level of final demand comes into playoutput gaps: actual GDP vs. potential GDP
(changing capacity utilization, business cylces) Potential GDP
filter-approach (HP-filter) production function + input factor stock approach
22DIMMoLMacro-
Econom(etr)icModelling Course 1
Goods market: demand side
Final demand: C + I + G + NX Private consumption (C) Private Investment (I) Government expenditure (G) Foreigen trade: Net exports (NX)
- Exports (X)- minus: Imports (IM)
finaldomestic demand
23DIMMoLMacro-
Econom(etr)icModelling Course 1
Private consumption
Important factors- real disposable income: current or permanent?- wealth- real interest rates
Sub-components- durables- non-durables- services
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Econom(etr)icModelling Course 1
Private investment
Private non-residential investment- [expected] output or output-gap (rate of capacity
utilization)- user cost of capital (influenced by real interest
rate) Private residential investment
- real disposable income (again: current or permanent)
- real interest rate
25DIMMoLMacro-
Econom(etr)icModelling Course 1
Government expenditure
Consumption InvestmentExpenditure for goods and services only!
Both usually (but not necessarily) exogenous- bound by budgetary rules- counter-cyclical use of fiscal policy
Distinction between consumption and investment matters in the long run!
26DIMMoLMacro-
Econom(etr)icModelling Course 1
Exports (= final foreign demand)
GDP of main trading partners Relative export prices
(international competitiveness)- domestic production costs- foreign prices (in main trading partners)- nominal exchange rates
Trade agreements, tariffs
real effective exchange rate
27DIMMoLMacro-
Econom(etr)icModelling Course 1
Imports (= foreign production)
Domestic final demand or production Relative import prices (see previous slide)
- domestic production costs- foreign prices- nominal exchange rates
Trade agreements, tariffs Special case: non-substitutional goods (oil,
raw materials)
Consultancy to Develop and Implement a Macroeconomic Model for Lesotho (DIMMoL)
Macro-Econom(etr)ic Modelling
Part 3
Dr. Stefan KoothsDIW Berlin – Macro Analysis and Forecasting
29DIMMoLMacro-
Econom(etr)icModelling Course 1
Course program
Introduction Outline of macroeconom(etr)ic models
(cont.) Macroeconomic framework Econometric methodology Applied econometrics with EViews Lesotho case studies
30DIMMoLMacro-
Econom(etr)icModelling Course 1
Goods market: supply side
Production of goods (and services) and generation of (domestic) income = use of (domestic) input factors
Production function for capturing production possibilities
Input factors- labor- (physical) capital stock- land usage- [Technology]
Most common: Cobb-Douglas production function and time trend for technological progress
Extensions: human capital, role of health, etc.
Disaggregation by industries
31DIMMoLMacro-
Econom(etr)icModelling Course 1
Goods market: inflation
Cost-push inflation- unit labor cost: wages and productivity- exchange rate/external prices (oil, etc.)
Regulatory influences- taxes- administrated prices- Import regulations
Demand-driven inflation- output-gap
What inflation?- GDP-deflator- Consumer price index (CPI)
example: oil price increase
32DIMMoLMacro-
Econom(etr)icModelling Course 1
Labor market (wage block)
Supply of labor- fix or real-wage dependent- long-run: population-dependent
(aging, health, behavior (e.g., participation rates)) Demand for labor
- derived from production function Disaggregation by skills Nominal wage setting
- unemployment rate (relative bargaining power)- inflation expectations- minimum expectation-augmented Phillips-curve
33DIMMoLMacro-
Econom(etr)icModelling Course 1
Money market 1 (interest block)
Demand for money- income-dependent via velocity of circulation
(income as a proxy for transaction volume)- interest-sensitive
Money supply- monetary base controlled by central-bank- money creation via lending of commercial banks
BUT: special case of Lesotho!
34DIMMoLMacro-
Econom(etr)icModelling Course 1
Money market 2 (The Lesotho case)
Common Monetary Area: fixed exchange rate with CMA partners
Small country within the CMA: exogenous exchange rate fluctuations (independent of domestic current account balance)
Money supply no longer exogenous Interest rate no longer endogenousAdjustment via current account and
real exchange rate channel
35DIMMoLMacro-
Econom(etr)icModelling Course 1
Foreign exchange market (external block) Demand-side
- imports of goods and services- exports of capital (portfolio or direct foreign
investment) Supply-side
- exports of goods and services- special treatment of income transfers from SA- imports of capital (portfolio or direct foreign
investment) Central bank interventions
- Linkages with monetary block- Sterilization policy?
36DIMMoLMacro-
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Terminology: Exchange rates
Exchange rate = price of foreign currency(foreign currency in terms of domestic currency)example: Euro-exchange rate: 9 [M/€]
Appreciation= decrease of exchange rate (example: 8 [M/€])
Depreciation= increase of exchange rate (example: 10 [M/€])
Case of fixed exchange ratesappreciation = revaluationdepreciation = devaluation
37DIMMoLMacro-
Econom(etr)icModelling Course 1
Government activities (fiscal block)
Public revenue- taxes (including customs receipts)- social contributions- interest payments from public assets
Public expenditure- goods and services- social transfers- interest payments on public debt
Budget surplus/deficit Just a model add-on in the short run
- except for existence of budgetary rules- debt/asset dynamics relevant in the medium and
long run
38DIMMoLMacro-
Econom(etr)icModelling Course 1
Putting it all together
39DIMMoLMacro-
Econom(etr)icModelling Course 1
A walk through the model
40DIMMoLMacro-
Econom(etr)icModelling Course 1
Course program
Introduction Outline of macroeconom(etr)ic models Macroeconomic framework Econometric methodology Applied econometrics with EViews Lesotho case studies
41DIMMoLMacro-
Econom(etr)icModelling Course 1
The IS-LM/AD-AS framework: Overview
Income-expenditure model (Keynesian multiplier)
IS-curve LM-curve IS-LM model IS-LM mechanics within a monetary union AD-curve AS-curve AD-AS model AD-AS dynamics Inflation: DAD-DAS
42DIMMoLMacro-
Econom(etr)icModelling Course 1
Income-expenditure model:Closed economy
43DIMMoLMacro-
Econom(etr)icModelling Course 1
Income-expenditure model:Open economy
Consultancy to Develop and Implement a Macroeconomic Model for Lesotho (DIMMoL)
Macro-Econom(etr)ic Modelling
Part 4
Dr. Stefan KoothsDIW Berlin – Macro Analysis and Forecasting
45DIMMoLMacro-
Econom(etr)icModelling Course 1
Course program
Introduction Outline of macroeconom(etr)ic models Macroeconomic framework (cont.) Econometric methodology Applied econometrics with EViews Lesotho case studies
46DIMMoLMacro-
Econom(etr)icModelling Course 1
Income-expenditure model:Main points Production follows demand
(no limiting supply-side factors) Exogenous prices
- goods market- interest rate- wage rate- exchange rate
Multiplier effect depends on- marginal propensity to consume (+)- marginal tax rate (-)- marginal import rate (-)
Income expansion reduces trade surplus
47DIMMoLMacro-
Econom(etr)icModelling Course 1
IS-curve:Construction
48DIMMoLMacro-
Econom(etr)icModelling Course 1
IS-curve:Response to fiscal policy
49DIMMoLMacro-
Econom(etr)icModelling Course 1
IS-curve:Response to price movements
50DIMMoLMacro-
Econom(etr)icModelling Course 1
LM-curve:Discussion
51DIMMoLMacro-
Econom(etr)icModelling Course 1
LM-curve:Response to monetary policy
52DIMMoLMacro-
Econom(etr)icModelling Course 1
IS-LM:Simultaneous equilibrium
53DIMMoLMacro-
Econom(etr)icModelling Course 1
IS-LM:Dynamics within a currency union 1
54DIMMoLMacro-
Econom(etr)icModelling Course 1
IS-LM:Dynamics within a currency union 2
Starting point: Equilibrium (i = iCMA) Increase in public spending (∆G > 0) Output expansion (multiplier process starts) Tendency for the interest rate to increase Arbitrage induces financial capital inflows Money supply increases according to
inflowing capital Higher quantity of money keeps interest rate
near to the initial level (i = iCMA)
Consultancy to Develop and Implement a Macroeconomic Model for Lesotho (DIMMoL)
Macro-Econom(etr)ic Modelling
Part 5
Dr. Stefan KoothsDIW Berlin – Macro Analysis and Forecasting
56DIMMoLMacro-
Econom(etr)icModelling Course 1
Course program
Introduction Outline of macroeconom(etr)ic models Macroeconomic framework (cont.) Econometric methodology Applied econometrics with EViews Lesotho case studies
57DIMMoLMacro-
Econom(etr)icModelling Course 1
AD-curve:Construction
58DIMMoLMacro-
Econom(etr)icModelling Course 1
AS-curve:Components Production function
- short-/medium-run: labor as only variable input factor
Quantity supplied (neocl.) | Price setting (keynes.)- real wage rate | unit labor cost - marginal productivity | rate of capacity
utilization- profit maximazition | mark-up pricing
Labor market model (wage setting equation)- rate of unemployment- inflation expectations
59DIMMoLMacro-
Econom(etr)icModelling Course 1
Wage setting:Expectation-augmented Phillips curve (in levels)
60DIMMoLMacro-
Econom(etr)icModelling Course 1
Production function and labor demand
61DIMMoLMacro-
Econom(etr)icModelling Course 1
AS-Curve:Construction
62DIMMoLMacro-
Econom(etr)icModelling Course 1
AD-AS
63DIMMoLMacro-
Econom(etr)icModelling Course 1
Inflation and real exchange rate:Condition for constant demand
64DIMMoLMacro-
Econom(etr)icModelling Course 1
DAD-DAS:Equilibrium and adjustment drivers
65DIMMoLMacro-
Econom(etr)icModelling Course 1
Course program
Introduction Outline of macroeconom(etr)ic models Macroeconomic framework Econometric methodology Applied econometrics with EViews Lesotho case studies
66DIMMoLMacro-
Econom(etr)icModelling Course 1
Econometric methodology: Overview
Fundamentals of probability Fundamentals of mathematical statistics Principles of regression analysis Time series regression models
67DIMMoLMacro-
Econom(etr)icModelling Course 1
Fundamentals of probability
Random variables Information about probability of possible
outcomes- Probability density function- Cumulative distribution function
Moments of the probability distribution- Measure of central tendency: Expected Value
(Mean)- Measures of variability: Variance and Standard
Deviation Measures of association ( causation):
- Covariance- Correlation
linear relationships only
68DIMMoLMacro-
Econom(etr)icModelling Course 1
Important probability distributions
Normal: X ~ Normal(mean, variance) Standard Normal: X ~ Normal(0, 1) Chi-Square: X ~ (df) t: X ~ t(df) F: X ~ F(df1,df2)
tabulated
69DIMMoLMacro-
Econom(etr)icModelling Course 1
Populations, parameters and sampling
Statistical inference = learning something about a well-defined
group by means of representatives of this group
well-defined group = population (unknown) something = parameters representatives = sample (observed) learning = estimation and hypothesis testing
70DIMMoLMacro-
Econom(etr)icModelling Course 1
Estimators and estimates
Estimator of a parameter = rule, that assigns each possible outcome of the sample a value of (which is then the concrete sample specific estimate)
Sampling variance of estimators Finite sample properties
- Unbiasedness- Efficiency
Asymptotic (= large sample) properties- Consistency, Law of Large Numbers (LLN)arbitrarily exact population mean by sufficiently
large sample Asymptotic normality, Central Limit Theorem (CLT)mean from a random sample of any population
has an asymptotic standard normal distribution
71DIMMoLMacro-
Econom(etr)icModelling Course 1
Using the sampling distribution of estimators Point estimate
best crisp guess at the population value (ignoring the sampling distribution)
Confidence intervals information about the estimate accuracy of the
estimate Hypothesis testing
answering concrete questions on a population value
72DIMMoLMacro-
Econom(etr)icModelling Course 1
Confidence intervals (CI)
Construction- point estimate- sampling distribution of the point estimate
sampling standard deviation functional form (large samples CLT)
- confidence level (usually 95 %) Interpretation
„There is a 95 % chance that the CI contains (before the sample is drawn).“
Rules of thumb (Standard Normal Distribution)- point estimate +/– 1 S.D. 66 % confidence
interval- point estimate +/– 2 S.D. 95 % confidence
interval
73DIMMoLMacro-
Econom(etr)icModelling Course 1
Hypothesis testing: Design
Null hypothesis: H0 (particular value of ) Alternative hypothesis: H1
- two-sided (one-tailed test)- one-sided (two-tailed test)
Errors types- Type 1 error (rejecting the null when it is in fact
true)- Type 2 error (failing to reject the null when it is
actually false) Significance level () = probability of a type 1
error- Given the power of the test is maximized- very small significance levels immunize against H1
Interpretation: Rejection vs. non-rejection of H0
Strategy: Trying to reject H0
74DIMMoLMacro-
Econom(etr)icModelling Course 1
Hypothesis testing: Test statistic
Test statistic T (particular outcome denoted t)- function of the random sample- usually: how many standard deviations is the
estimate for away from its assumed population mean (if H0 holds true)
- note: T might depend on H0!
Rejection rule (depending on H1) that determines when H0 is rejected in favor of H1 critical value of t- H1: > 0 t > tc
- H1: < 0 t < -tc
- H1: 0 t > |tc|
75DIMMoLMacro-
Econom(etr)icModelling Course 1
Hypothesis testing: Graphical interpretation
76DIMMoLMacro-
Econom(etr)icModelling Course 1
Hypothesis testing: p-values (prob-values) What is the largest significance level at
which we could carry out the test without rejecting H0?
What is the probability to observe a value of T as large as t when H0 is true?
small p-values are evidence against H0
high p-values are weak evidence against H0 Procedure
- design H0 and H1 and choose a test statistic T(possible rejection rules: t > c, t < -c, or |t| > c)
- use the observed value of t as the critical value and compute the corresponding significance level of the test
- given a significance level , reject H0 if p-value < (small p-values lead to rejection)
77DIMMoLMacro-
Econom(etr)icModelling Course 1
Inference: Final remarks
Confidence intervals and hypothesis testing are two sides of the same coin
Consistency- confidence intervals- hypothesis tests
Practical versus statistical significance: Magnitudes matter!
78DIMMoLMacro-
Econom(etr)icModelling Course 1
Types of data structures
Cross-sectional data random sampling
Time series datachronological ordering of observations conveys
potentially important informationcorrelation across time (non-random sampling!)
Pooled cross sectionscombining independent cross sections from
different years Panel data
pooling identical cross sections across time
Consultancy to Develop and Implement a Macroeconomic Model for Lesotho (DIMMoL)
Macro-Econom(etr)ic Modelling
Part 6
Dr. Stefan KoothsDIW Berlin – Macro Analysis and Forecasting
80DIMMoLMacro-
Econom(etr)icModelling Course 1
Course program
Introduction Outline of macroeconom(etr)ic models Macroeconomic framework Econometric methodology (cont.) Applied econometrics with EViews Lesotho case studies
81DIMMoLMacro-
Econom(etr)icModelling Course 1
Econometric methodology: Overview
Fundamentals of probability Fundamentals of mathematical statistics Principles of regression analysis (cross
sections) Time series regression models
82DIMMoLMacro-
Econom(etr)icModelling Course 1
Principles of regression analysis
Population regression model Properties of OLS estimates Functional forms and data scaling Confidence intervals and hypothesis testing OLS asymptotics Goodness-of-fit and selection of regressors Specification and data problems
83DIMMoLMacro-
Econom(etr)icModelling Course 1
Population model and regression functions
y = 0 + 1x1 + 2x2 + … + kxk + u
population model („true“ model)
population regression function
y = 0 + 1x1 + 2x2 + … + kxk
sample regression function
using OLS estimation
y = 0 + 1x1 + 2x2 + … + kxk + u
j j
84DIMMoLMacro-
Econom(etr)icModelling Course 1
Terminology
Dependent variable (y)- explained variable- response variable- predicted variable- regressand
Independent variables (x)- explanatory variables- control variables- predictor variables- regressors
Fitted value (y, speak: „y hat“) Error (u)
- disturbance- „unobserved“ variables
Residual (u)^
^
85DIMMoLMacro-
Econom(etr)icModelling Course 1
Gauss-Markov assumptions
Linearity in parameterspopulation model is characterized by a linear regression function and additive errors
Random samplingrandom sample of n observations following the population model
No perfect collinearitynone of the independent variables is constant and no exact linear relationships among them
Zero conditional meanerror has an expected value of zero given any values of the independent variables
Homoskedasticityerror has the same variance given any value of the explanatory variables
OLS
est
imato
rs a
re u
nb
iase
d
OLS
est
imato
rs a
re B
LUE
(Gau
ss-M
ark
ov T
heore
m)
86DIMMoLMacro-
Econom(etr)icModelling Course 1
Fitted values and residuals
87DIMMoLMacro-
Econom(etr)icModelling Course 1
OLS strategy
Finding the -vector that minimizes the sum of squared residuals (SSR)
i=1
n
(y i yi)2 =
i=1
n
u2i min!
88DIMMoLMacro-
Econom(etr)icModelling Course 1
Goodness-of-fit: Mechanics
Total sum of squares: SST( squared deviations of y from the sample mean)
Explained sum of squares: SSE( squared deviations of yhat from the sample mean)
Residual sum of squares: SSR( squared residuals), minimized by OLS
SST = SSE + SSR
R2 = SSE/SST = 1 SSR/SST(coefficient of determination)
R2 = square of the correlation coefficient
between y and yhat
89DIMMoLMacro-
Econom(etr)icModelling Course 1
Goodness-of-fit: Interpretation
R2 is the proportion of the sample variation in the dependent variable explained by the independent variables
R2 never decreases when any variable is added to a regressionmakes it a poor tool for deciding whether a
particular variable should be added to a modelR2 is no goddess of fit (especially in time series
analysis)!
90DIMMoLMacro-
Econom(etr)icModelling Course 1
Adjusted R-squared (corrected R-squared)
Penalizes the number of regressors (= loss of degrees of freedom)
Increases when t-statistic (F-statistic) of a single (group of) variable(s) is greater than 1
R_
2 = 1
SSRn k 1
SSTn 1
= R_
2 = 1 SSRSST
n 1n k 1
91DIMMoLMacro-
Econom(etr)icModelling Course 1
Interpreting the slope coefficients
Simple (bivariate) regression
Multiple (multivariate) regression
1 = Cov(x,y)Var(x) = 1
i=1
n
(xi x)ui
i=1
n
(xi x)2
= 1 Cov(x,u)Var(x)
j =
i=1
n
rijyi
i=1
n
rij2
= Cov(rj,y)Var(rj)
multicollinearity partialling-out
effectomitted-variable
bias
92DIMMoLMacro-
Econom(etr)icModelling Course 1
Variance of the slope coefficients
Simple regression
Multiple regression
Var(1) = 2
i=1
n
(xi x)2
Var( j) = 2
i=1
n
(xij xj)2(1 Rj
2)
Sources of variance
(1) error variance(2) sample variance
in xj
(3) multicollinearity(4) small sample
size
93DIMMoLMacro-
Econom(etr)icModelling Course 1
Estimating the error variance
Estimated error variance
- k = number of regressors- n k 1 = degrees of freedom
Standard error of the regression (SER)
- root squared error- standard error of the estimate
= 2
2 = 1
n k 1 i=1
n
u2i =
SSR n k 1
94DIMMoLMacro-
Econom(etr)icModelling Course 1
Misspecification
Overspecifying the model(including an irrelevant variable)- no effect on unbiasedness of OLS- multicollinearity increases the variances of the
remaining OLS estimators- consumes degrees of freedom
Underspecifying the model(excluding a relevant variable)- causes OLS to be biased if linearily correlated with
the remaining independent variables- multicollinearity might decrease the variances of
the remaining OLS estimators (bias vs. variability tradeoff)
95DIMMoLMacro-
Econom(etr)icModelling Course 1
Inference
Hypothesis testing and confidence intervals depend on the variances of OLS estimators
Error variance affects the variances of the OLS estimators
Case 1: Classical Linear Model- Gauss-Markov + Normality assumption- Normality assumption: population error is normally
distributed with zero mean and (constant!) variance 2
exact sampling distributions of the OLS estimators Case 2: OLS asymptotics
Gauss-Markov + large sample size properties emerge as the sample size grows
without boundasymptotic properties of the OLS estimators (as in
case 1)
96DIMMoLMacro-
Econom(etr)icModelling Course 1
CLM: Pro and cons
Pro- Central Limit Theorem: many unobserved
variables, each having a minor effect on the dependent variable have an aggregated average effect that is normally distributed
Cons- CLM captures additive errors only- discrete values cannot be normally distributed- many economic variables are non-negative (but:
often [logarithmic] transformations might restore normality)
97DIMMoLMacro-
Econom(etr)icModelling Course 1
Tests (overview)
t-Test (and confidence intervals)- single population parameter
F-Test- group of population parameters
LM-Test- group of population parameters (asymptotic
analysis) RESET Test
- functional form Davidson-MacKinnon test
- functional form for nonnested alternatives
98DIMMoLMacro-
Econom(etr)icModelling Course 1
The t-Test
Testing hypotheses about a single population parameter (usually testing for = 0)
General setting (t statistic or t ratio)
How many standard deviations is the estimated value away from the assumed (= tested) value?
Regression parameters are („asymptotically“)t-distributed with df = nk1
t = estimate hypothesized value
standard error
t j =
j j
se( j) ~ tn k 1
99DIMMoLMacro-
Econom(etr)icModelling Course 1
The t-Test: Rejection rules
Two-sided test (H1: hypothesized value)
Reject H0 if: |t| > tc
One sided test (H1: hypothesized value)
Reject H0 if: t tc
One sided test (H1: hypothesized value)
Recect H0 if: t tc
Alternative: Looking at respective p-values
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Practical guidelines
Check for statistical significance Check statisticially significant values for
practical significance (magnitudes of the estimates); be careful about functional form and units of measurement
Non-statistically significant values (at usual levels up to 10 %) might remain in the model if their economic influence in well-founded and if their magnitudes are important; p-values as large as 20 % might be acceptable in such cases
Statistically insignificant variables whose parameters have the „wrong“ sign can be ignored
Statistically significant variables with „wrong“ signs and a practically large effect indicate misspecification
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Confidence intervals
Regression parameters are („asymptotically“) t-distributed with nk1 degrees of freedom
Example: 95% confidence interval
c = 97,5th percentile in a tn k1 distribution
Rule of thumb (df = n k1 50): c = 2
j j
se( j) ~ tn k 1
j cse( j)
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The F-Test
Testing q multiple linear restrictionssimultaneously (joint statistical significance)- unrestricted model: contains all independent
variables- restricted model: contains q independent
variables less than the unrestricted model Example for k 2
- H0 : 1 = 2 = 0
- H1 : H0 is not true
Ratio of SSRr and SSRur is F-distributed with df1 = q and df2 = nk1
F = (SSRr SSRur)/qSSRur/(n k 1) =
(R2ur R2
r)/q
(1 R2ur)/n k 1
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The F-Test: Rejection rule
Reject H0 if: F c
c depends on- nominator degrees of freedom (df1)
- denominator degrees of freedom (df2)- signficance level
Alternative: Looking at p-value Remarks
- Note: F-Test tests for joint statistical significance, i.e. at least one (but not necessarily all) of the restricted variables is (are) statistically significant
- F-test for a single variable is equivalent to a two-sided t-test
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The LM-Test (Lagrange-Multiplier Test)
Step1:Estimate the restricted model (with q restrictions) and save the residuals ur
Step 2:Regress ur on all of the independent variables and obtain the R2 as UR2
Step 3:Compute LM = nUR2
Step 4:LM follows a Chi-Square distribution with df = q; reject H0 if LM > c (alternatively, look at p-values)
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The RESET Test
RESET = regression specification error test Tests for functional form misspecification
- not a general test for misspecification (i.e. linearly dependent omitted variables cannot be detected)
- if functional form is properly specified, heteroscedasticity is not detected
Strategy:- Add p polynomials in the OLS fitted values to the
original (= tested) estimation equation (here: p = 2):
- F-test for signficance of the -parameters; test statistic is Fp,nk1p distributed
y = 0 + 1x1 + 2x2 + … + kxk + 1y2 + 2y3 + e
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Tests against nonnested alternatives
Strategy 1: Comprehensive model approach- construct a comprehensive model that contains
each model as a special case- testing the restrictions that lead to each of the
models via F-tests Strategy 2: Davidson-MacKinnon test
- estimate each model seperately- check, whether the fitted values of alternative 1 are
significant when added as a regressor in alternative 2 and v.v.
Problems- a clear winner need not emerge (if none of the
special models can be rejected, use adjusted R-squared as creterion)
- only relative performance is tested, none of the alternatives needs to be the correct model
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Model selection criteria
Nested models- t-Tests for significance of a single variable- F-Tests for joint significance of a group of variables
Nonnested models- Davidson-MacKinnon + adjusted R-squared
(BUT: not to be used for functional form of the dependent variable!)
- Akaike Information Criterion (AIC)AIC = nln(SSR) 2(k1)
- Schwartz Baysian Criterion (SBC)SBC = nln(SSR) (k1) ln(n)
General rule: Parsimony is buitiful
smaller value is prefered (different implementations exist)
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Functional forms involving logarithms
level-level model: regressing y on xy = jxj
level-log model: regressing y on log(x)y = (j/100)%xj
log-level model: regressing log(y) on x%y = (100j)xj 100j = semi-elasticity
log-log model: regressing log(y) on log(x) %y = j%xj j = elasticity
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Rules of thumb for using logarithms
Strictly positive variables often tend to be heteroskedastic or skewed taking logs often mitigates/eliminates these problems
Taking logs narrows the range of the variable makes them less sensitive to outlying observations
Taking logs works for strictly positive variables only zero observations in y log(1+y) may work
Positive dollar amount or large integers try logs
Variables that are measures in years try levels
Variables that are proportions try rather levels
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Functional form involving quadratic terms Can capture increasing or diminishing
marginal effects ... ... but might also indicate functional form
misspecification (e.g. levels instead of logs or vice versa)
Note: Marginal effects are no longer constant, i.e. they depend on the value of the respective variable
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Functional form involving dummy variables Capture qualitative information g different groups g1 dummies Stand-alone dummies for group-specific
intercepts Interaction terms for group-specific slope
parameters BUT: Each observation is somewhat unique
- risk of over-dummying the modeleach dummy must have an economically justified
interpretation
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Units of measurements
No effect- on significance of parameters- on goodness-of-fit
Reflected in the magnitudes of the regression parameters
Special case: log(y)-modelsnothing happens to the regression parameters if
the units of measurement of the dependent variable are changed
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Heteroskedasticity
Does not cause bias or inconsistency in OLS estimators
BUT: The usual standard errors and test statistics are no longer valid (OLS estimators are no longer BLUE)
Tests: Regressing the squared OLS residuals ...- ... on the independent variables (Breusch-Pagan)- ... on the independent variables plus their squares
and all cross products (White)- ... on the fitted and squared fitted values (special
White) Solution
- Weighted least squares- constructing heteroskedasticity-robust statistics
Consultancy to Develop and Implement a Macroeconomic Model for Lesotho (DIMMoL)
Macro-Econom(etr)ic Modelling
Part 7
Dr. Stefan KoothsDIW Berlin – Macro Analysis and Forecasting
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Course program
Introduction Outline of macroeconom(etr)ic models Macroeconomic framework Econometric methodology (cont.) Applied econometrics with EViews Lesotho case studies Follow-up work
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Econometric methodology: Overview
Fundamentals of probability Fundamentals of mathematical statistics Principles of regression analysis (cross
sections) Time series regression models
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Conceptional differences to cross sections Sequence of random variables indexed by
time- time series process- stochastic process
Sample = one possible outcome (realization) of the stochastic process
Sample size = number of time periods observed
Temporal ordering The past can affect the present (and the
future) Randomness = different historic conditions
would have generated a different realization of the observed process
Population = set of all possible realization of the stochastic process
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General strategy
random sampling
conditions that restrict temporal correlation in time series
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Using OLS in time series analysis
Case 1: Gauss-Markov-Assumptions- strictly exogenous regressorsOLS estimators are BLUE
Case 2: Asymptotic Gauss-Markov-Assumptions contemporaneously exogenous regressors weakly dependent time series
(asymptotically uncorrelated)OLS is consistent, inference methods are
asymptotically valid Case 3: Cointegration analysis
strictly exogenous regressors (via leads and lags)
highly persistent, cointegrated time seriesOLS is super-consistent, inference methods
applyerror-correction model representation
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(Trend-) Stationarity
A process y is stationary if it is identically distributed over time- constant mean y
- constant variance Var(y)- constant autocovariance Cov(yt,yt-h)
Trend stationarity- stationarity after removing the trend- deviations from the trend are stationary
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Gauss-Markov assumptions
Linearity in parameterspopulation model is characterized by a linear regression function and additive errors
No perfect collinearitynone of the independent variables is constant nor a perfect linear combination of the others
Zero conditional mean (strict exogenity)for each t, the expected value of the error, given the regressors for all time periods, is zero
Homoskedasticityerror has the same variance given any value of the explanatory variables for all time periods
No serial correlationThe errors in two different time periods are uncorrelated
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Why strict exogenity might fail
Omitted variables Measurement errors in some of the
regressors feedback from the dependent variable on
future values of a regressor (policy response)
Lagged dependent variable as regressor
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Assymptotic Gauss-Markov assumptions
Linearity and weak dependencepopulation model is characterized by a linear regression function, additive errors, and weakly dependent processes
No perfect collinearitynone of the independent variables is constant nor a perfect linear combination of the others
Zero conditional mean(contemporaneous exogenity)for each t, the expected value of the error, given the regressors in the same period, is zero
Homoskedasticityerror has the same variance given any contemporaneous value of the explanatory variables
No serial correlationThe errors in two different time periods are uncorrelated
OLS
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LS
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Weak dependence
A time series is weakly dependent, if- xt and xt+h are „almost independent“ as h
increases without bound (autocorrelation dies out over time)
- Cov(xt,xt+h) 0 as h
Replaces the assumption of random sampling, making use of the - Law of Large Numbers and the - Central Limit Theorem
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Static and distributed lag Models
Static models (contemporaneous relationship)
Distributed lag models
- finite distributed lag models- infinite distributed lag models impact propensity (or: impact multiplier) long-run propensity (or: long-run multiplier)
yt = 0 + 1xt + ut
yt = 0 + 1xt + 2xt-1 + 3xt-2 +ut
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Deterministic trends and seasonality
Trends- linear- quadratic, cubic (BUT: parsimony condition!)- exponential
Seasonality- quarterly: 3 Dummies- monthly: 11 Dummies
Using trending/seasonal variables in regressions- including trend and/or seasonal component or- removing trends (detrending) and seasonality
(seasonal adjustment)usual inference procedures are asymptotically
validotherwise: spurious regression problem, artificially
high R2
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AR(1) processes
AR(1) = autoregressive process of order 1
Crucial assumption- 1 weakly dependent process
1 integrated of order zero: I(0)- 1 highly persistent (unit root) process
(Random walk) 1 integrated of order one: I(1)
Policy implication- weakly dependence: policy interventions have
temporary effects only- high persistence: policy interventions have
permanent effects
yt = 1yt-1 + et
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Estimating the first order autocorrelation Case 1: || 1 (weakly dependent process)
- regressing yt on yt-1
- consistent (but biased) estimator (LLN needed) Case 2: | | = 1 (unit root process)
- t-distribution no longer valid- Dickey-Fuller tests (based on Monte Carlo
Experiments) Problem
- Distribution of the test statistic depends on H0
- Both cases might be not rejectablePower of unit root tests is rather poor
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Transforming the AR(1) equation
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(Augmented) Dickey-Fuller tests
Three scenarios- Random walk: yt = yt-1 + et
- Random walk with drift: yt = a0 + yt-1 + et
- Random walk with drift and trend: yt = a0 + yt-1 + a2t + et
Scenarios may include lags of y (Augmented DF)- e.g. yt = yt-1 + 1yt-1 + 1yt-2 + et
Critical values tc (tabulated) depend on- scenario type - sample size
Testing for = 0 (H0: existence of a unit root)
Rejection rule: Reject H0 if t < tc
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Critical values for Dickey-Fuller test
= a0 = 0
= a2 = 0
= a0 = a2 = 0
= 0
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General-to-specific procedure for testing unit roots
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Unit root processes in regression analysis Time series xt, yt are I(1) processes
- also applies to higher identical orders of integration and more than two variables
Case 1: No cointegration- any linear combination of xt and yt is I(1) problem of spurious regression first differences as transformation method
Case 2: Cointegration an linear combination of xt and yt (cointegration vector)
exists such that st = yt – xt is I(0) OLS estimators show long-run equilibrium relationship error-correction model for short-run adjustment
dynamics(Granger representation theorem)
Test for cointegration: Engle-Granger cointegration test
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Testing for cointegration:The Engle-Granger Methodology
Step 1: Test xt and yt for integration- use Dickey-Fuller test- EXIT if both series are stationary or integrated of
different orders (= no cointegration) Step 2: Estimate long-run equilibrium
relationship
Step 3: Check residuals for stationarity
- special critical values apply- EXIT if H0: a1 = 0 cannot be rejected
yt = 0 + 1xt + st
st = a1 st-1 + et
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Testing for cointegration:The Engle-Granger Methodology (cont.)
Step 4: Estimate the error-correction model
- all variables are I(0), therefore OLS is valid- further lags of y and x may apply
(check ut for white noise)
- use residuals from step 3 for (yt-1 xt-1):
yt = 0 + 1yt-1 + 0xt + 1xt-1 + (yt-1 1xt-1) + ut
yt = 0 + 1yt-1 + 0xt + 1xt-1 + st-1 + ut
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Critical values for Engle-Granger Cointegratoin test
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Other topics in time series analysis
Serial correlation, Autokorrelationsfunktion ARMA (Box-Jenkins) and ARIMA models ARCH processes Vector autoregressive models (VAR),
interventions and impulse-response analysis Structural change Non-linear time series models
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Course program
Introduction Outline of macroeconom(etr)ic models Macroeconomic framework Econometric methodology (cont.) Applied econometrics with EViews Lesotho case studies Follow-up work
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Course program
Introduction Outline of macroeconom(etr)ic models Macroeconomic framework Econometric methodology (cont.) Applied econometrics with EViews Lesotho case studies Follow-up work
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Work groups
Private domestic demand- private consumption- private investment- income
Fiscal affairs- public consumption- public investment- taxation - subsidies- budgets and MTEF
External relations and monetary issues- trade flows- capital and transfer
flows- real effective exch.
rate- interest rate forecasts
and money demand Production and
Pricing- production function- labor demand and
wage setting- capital accumulation
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General tasks (all groups)
Economic theory and literature review Model formulation in African economies Functional form specification Data base checks Preliminary estimation of equations
Regular work group meetings Collective macro level discussions Remote assistence from DIW Berlin