Consultancy to Develop and Implement a Macroeconomic Model for Lesotho (DIMMoL) Macro-Econom(etr)ic Modelling Part 1 Dr. Stefan Kooths DIW Berlin – Macro

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

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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

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Course program


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Fields of interest

Macroeconomics(model building)

Econometrics(applied mathematical statistics)

National accounting(data sources)

Country-specific knowledge(institutions, industries, policy regimes)

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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/)

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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]

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Introduction of participants

Who are you? Where are you from? What are your specific questions and

modelling needs?

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Course program


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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)

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Building blocks, fundamental characteristics Institutional sectors (actors, agents) Markets (intermediaries) and regulations Time horizon and dynamics

- Equilibrium- Adjustment processes

Expectation formation

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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

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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

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Sector interactions via markets:circular sectoral flow chart



Part 2


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Course program

Introduction Outline of macroeconom(etr)ic models

(cont.) Macroeconomic framework Econometric methodology Applied econometrics with EViews Lesotho case studies

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The IS-identity:goods and capital market intertwined

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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

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Types of variables

By status- endogenous- exogenous

third-party sources autoregressive forecasts (outside the model)

Most important/interesting variables- output- income- (un)employment- inflation

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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

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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

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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

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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

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Private consumption

Important factors- real disposable income: current or permanent?- wealth- real interest rates

Sub-components- durables- non-durables- services

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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

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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!

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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

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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)



Part 3


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Course program

Introduction Outline of macroeconom(etr)ic models

(cont.) Macroeconomic framework Econometric methodology Applied econometrics with EViews Lesotho case studies

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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

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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

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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

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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!

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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

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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?

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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

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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

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Putting it all together

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A walk through the model

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Course program


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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

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Income-expenditure model:Closed economy

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Income-expenditure model:Open economy



Part 4


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Course program

Introduction Outline of macroeconom(etr)ic models Macroeconomic framework (cont.) Econometric methodology Applied econometrics with EViews Lesotho case studies

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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

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IS-curve:Construction

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IS-curve:Response to fiscal policy

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IS-curve:Response to price movements

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LM-curve:Discussion

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LM-curve:Response to monetary policy

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IS-LM:Simultaneous equilibrium

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IS-LM:Dynamics within a currency union 1

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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)



Part 5


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Course program

Introduction Outline of macroeconom(etr)ic models Macroeconomic framework (cont.) Econometric methodology Applied econometrics with EViews Lesotho case studies

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AD-curve:Construction

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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

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Wage setting:Expectation-augmented Phillips curve (in levels)

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Production function and labor demand

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AS-Curve:Construction

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AD-AS

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Inflation and real exchange rate:Condition for constant demand

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DAD-DAS:Equilibrium and adjustment drivers

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Course program


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Econometric methodology: Overview

Fundamentals of probability Fundamentals of mathematical statistics Principles of regression analysis Time series regression models

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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

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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

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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

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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

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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

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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

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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

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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|

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Hypothesis testing: Graphical interpretation

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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)

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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!

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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



Part 6


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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

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Fundamentals of probability Fundamentals of mathematical statistics Principles of regression analysis (cross

sections) Time series regression models

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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

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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

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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)^

^

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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)

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Fitted values and residuals

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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!

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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

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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)!

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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

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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

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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

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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

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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)

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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)

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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)

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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

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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

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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



Part 7


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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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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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es

non

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pro

cess

es

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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

OLS

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iase

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OLS

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LUE

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m)

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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

est

imato

rs a

re c

on

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en

t

Asy

mp

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f O

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

137DIMMoLMacro-


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


139DIMMoLMacro-


Course program


140DIMMoLMacro-


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

Documents

Consultancy to Develop and Implement a Macroeconomic Model for Lesotho (DIMMoL) Macro-Econom(etr)ic Modelling Part 1 Dr. Stefan Kooths DIW Berlin – Macro