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

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 2 Course program Introduction Outline of macroeconom(etr)ic models Macroeconomic framework Econometric methodology Applied econometrics with EViews Lesotho case studies

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 3 Course program Introduction Outline of macroeconom(etr)ic models Macroeconomic framework Econometric methodology Applied econometrics with EViews Lesotho case studies

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 4 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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DIMMoL Macro-Econom(etr)ic Modelling Course 1 5 Recommended literature Blanchard, O.: Macroeconomics, 3 rd ed., 2003. Wooldridge, J. M.: Introductory Econometrics – A Modern Approach, 3 rd ed., 2006. Enders, W.: Applied Economic Time Series, 2 nd 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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DIMMoL Macro-Econom(etr)ic Modelling Course 1 6 My contact data here in Maseru cell: 5847.0578 email: skooths@diw.de in Berlin DIW Berlin, German Institute for Economic Research Koenigin-Luise-Strasse 5 14195 Berlin fon: +49 30 89789-248 fax: +49 30 89789-102 email: skooths@diw.de

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 7 Introduction of participants Who are you? Where are you from? What are your specific questions and modelling needs?

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 8 Course program Introduction Outline of macroeconom(etr)ic models Macroeconomic framework Econometric methodology Applied econometrics with EViews Lesotho case studies

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 9 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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DIMMoL Macro-Econom(etr)ic Modelling Course 1 10 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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DIMMoL Macro-Econom(etr)ic Modelling Course 1 11 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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DIMMoL Macro-Econom(etr)ic Modelling Course 1 12 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 income interconnection of markets: direct vs. indirect effects

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 13 Sector interactions via markets: circular sectoral flow chart

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Consultancy to Develop and Implement a Macroeconomic Model for Lesotho (DIMMoL) Macro-Econom(etr)ic Modelling Part 2 Dr. Stefan Kooths DIW Berlin – Macro Analysis and Forecasting

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 15 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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DIMMoL Macro-Econom(etr)ic Modelling Course 1 16 The IS-identity: goods and capital market intertwined

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 17 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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DIMMoL Macro-Econom(etr)ic Modelling Course 1 18 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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DIMMoL Macro-Econom(etr)ic Modelling Course 1 19 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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DIMMoL Macro-Econom(etr)ic Modelling Course 1 20 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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DIMMoL Macro-Econom(etr)ic Modelling Course 1 21 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 play output 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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DIMMoL Macro-Econom(etr)ic Modelling Course 1 22 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) final domestic demand

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 23 Private consumption Important factors -real disposable income: current or permanent? -wealth -real interest rates Sub-components -durables -non-durables -services

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 24 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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DIMMoL Macro-Econom(etr)ic Modelling Course 1 25 Government expenditure Consumption Investment Expenditure 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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DIMMoL Macro-Econom(etr)ic Modelling Course 1 26 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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DIMMoL Macro-Econom(etr)ic Modelling Course 1 27 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)

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Consultancy to Develop and Implement a Macroeconomic Model for Lesotho (DIMMoL) Macro-Econom(etr)ic Modelling Part 3 Dr. Stefan Kooths DIW Berlin – Macro Analysis and Forecasting

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 29 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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DIMMoL Macro-Econom(etr)ic Modelling Course 1 30 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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DIMMoL Macro-Econom(etr)ic Modelling Course 1 31 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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DIMMoL Macro-Econom(etr)ic Modelling Course 1 32 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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DIMMoL Macro-Econom(etr)ic Modelling Course 1 33 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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DIMMoL Macro-Econom(etr)ic Modelling Course 1 34 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 endogenous Adjustment via current account and real exchange rate channel

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 35 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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DIMMoL Macro-Econom(etr)ic Modelling Course 1 36 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 rates appreciation = revaluation depreciation = devaluation

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 37 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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DIMMoL Macro-Econom(etr)ic Modelling Course 1 38 Putting it all together

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 39 A walk through the model

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 40 Course program Introduction Outline of macroeconom(etr)ic models Macroeconomic framework Econometric methodology Applied econometrics with EViews Lesotho case studies

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 41 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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DIMMoL Macro-Econom(etr)ic Modelling Course 1 42 Income-expenditure model: Closed economy

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 43 Income-expenditure model: Open economy

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Consultancy to Develop and Implement a Macroeconomic Model for Lesotho (DIMMoL) Macro-Econom(etr)ic Modelling Part 4 Dr. Stefan Kooths DIW Berlin – Macro Analysis and Forecasting

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 45 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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DIMMoL Macro-Econom(etr)ic Modelling Course 1 46 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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DIMMoL Macro-Econom(etr)ic Modelling Course 1 47 IS-curve: Construction

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 48 IS-curve: Response to fiscal policy

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 49 IS-curve: Response to price movements

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 50 LM-curve: Discussion

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 51 LM-curve: Response to monetary policy

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 52 IS-LM: Simultaneous equilibrium

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 53 IS-LM: Dynamics within a currency union 1

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 54 IS-LM: Dynamics within a currency union 2 Starting point: Equilibrium (i = i CMA ) 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 = i CMA )

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Consultancy to Develop and Implement a Macroeconomic Model for Lesotho (DIMMoL) Macro-Econom(etr)ic Modelling Part 5 Dr. Stefan Kooths DIW Berlin – Macro Analysis and Forecasting

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 56 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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DIMMoL Macro-Econom(etr)ic Modelling Course 1 57 AD-curve: Construction

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 58 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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DIMMoL Macro-Econom(etr)ic Modelling Course 1 59 Wage setting: Expectation-augmented Phillips curve (in levels)

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 60 Production function and labor demand

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 61 AS-Curve: Construction

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 62 AD-AS

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 63 Inflation and real exchange rate: Condition for constant demand

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 64 DAD-DAS: Equilibrium and adjustment drivers

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 65 Course program Introduction Outline of macroeconom(etr)ic models Macroeconomic framework Econometric methodology Applied econometrics with EViews Lesotho case studies

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 66 Econometric methodology: Overview Fundamentals of probability Fundamentals of mathematical statistics Principles of regression analysis Time series regression models

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 67 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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DIMMoL Macro-Econom(etr)ic Modelling Course 1 68 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(df 1,df 2 ) tabulated

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 69 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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DIMMoL Macro-Econom(etr)ic Modelling Course 1 70 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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DIMMoL Macro-Econom(etr)ic Modelling Course 1 71 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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DIMMoL Macro-Econom(etr)ic Modelling Course 1 72 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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DIMMoL Macro-Econom(etr)ic Modelling Course 1 73 Hypothesis testing: Design Null hypothesis: H 0 (particular value of ) Alternative hypothesis: H 1 -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 H 1 Interpretation: Rejection vs. non-rejection of H 0 Strategy: Trying to reject H 0

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 74 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 H 0 holds true) -note: T might depend on H 0 ! Rejection rule (depending on H 1 ) that determines when H 0 is rejected in favor of H 1 critical value of t -H1: > 0 t > t c -H1: < 0 t < -t c -H1: 0 t > |t c |

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 75 Hypothesis testing: Graphical interpretation

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 76 Hypothesis testing: p-values (prob-values) What is the largest significance level at which we could carry out the test without rejecting H 0 ? What is the probability to observe a value of T as large as t when H 0 is true? small p-values are evidence against H 0 high p-values are weak evidence against H 0 Procedure -design H 0 and H 1 and choose a test statistic T (possible rejection rules: t > c, 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 H 0 if p-value < (small p-values lead to rejection)

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 77 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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DIMMoL Macro-Econom(etr)ic Modelling Course 1 78 Types of data structures Cross-sectional data random sampling Time series data chronological ordering of observations conveys potentially important information correlation across time (non-random sampling!) Pooled cross sections combining independent cross sections from different years Panel data pooling identical cross sections across time

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Consultancy to Develop and Implement a Macroeconomic Model for Lesotho (DIMMoL) Macro-Econom(etr)ic Modelling Part 6 Dr. Stefan Kooths DIW Berlin – Macro Analysis and Forecasting

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 80 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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DIMMoL Macro-Econom(etr)ic Modelling Course 1 81 Econometric methodology: Overview Fundamentals of probability Fundamentals of mathematical statistics Principles of regression analysis (cross sections) Time series regression models

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 82 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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DIMMoL Macro-Econom(etr)ic Modelling Course 1 83 Population model and regression functions population model („true“ model) population regression function sample regression function using OLS estimation

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 84 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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DIMMoL Macro-Econom(etr)ic Modelling Course 1 85 Gauss-Markov assumptions Linearity in parameters population model is characterized by a linear regression function and additive errors Random sampling random sample of n observations following the population model No perfect collinearity none of the independent variables is constant and no exact linear relationships among them Zero conditional mean error has an expected value of zero given any values of the independent variables Homoskedasticity error has the same variance given any value of the explanatory variables OLS estimators are unbiased OLS estimators are BLUE (Gauss-Markov Theorem)

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 86 Fitted values and residuals

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 87 OLS strategy Finding the -vector that minimizes the sum of squared residuals (SSR)

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 88 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 R 2 = SSE/SST = 1 SSR/SST (coefficient of determination) R 2 = square of the correlation coefficient between y and yhat

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 89 Goodness-of-fit: Interpretation R 2 is the proportion of the sample variation in the dependent variable explained by the independent variables R 2 never decreases when any variable is added to a regression makes it a poor tool for deciding whether a particular variable should be added to a model R 2 is no goddess of fit (especially in time series analysis)!

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 90 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

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 91 Interpreting the slope coefficients Simple (bivariate) regression Multiple (multivariate) regression multicollinearity partialling-out effect omitted-variable bias

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 92 Variance of the slope coefficients Simple regression Multiple regression Sources of variance (1) error variance (2) sample variance in x j (3) multicollinearity (4) small sample size

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 93 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

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 94 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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DIMMoL Macro-Econom(etr)ic Modelling Course 1 95 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 bound asymptotic properties of the OLS estimators (as in case 1)

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 96 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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DIMMoL Macro-Econom(etr)ic Modelling Course 1 97 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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DIMMoL Macro-Econom(etr)ic Modelling Course 1 98 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 = n k 1

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 99 The t-Test: Rejection rules Two-sided test (H 1 : hypothesized value) Reject H 0 if: |t| > t c One sided test (H 1 : hypothesized value) Reject H 0 if: t t c One sided test (H 1 : hypothesized value) Recect H 0 if: t t c Alternative: Looking at respective p-values

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 100 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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DIMMoL Macro-Econom(etr)ic Modelling Course 1 101 Confidence intervals Regression parameters are („asymptotically“) t- distributed with n k 1 degrees of freedom Example: 95% confidence interval c = 97,5 th percentile in a t n k 1 distribution Rule of thumb (df = n k 1 50): c = 2

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 102 The F-Test Testing q multiple linear restrictions simultaneously (joint statistical significance) -unrestricted model: contains all independent variables -restricted model: contains q independent variables less than the unrestricted model Example for k 2 -H 0 : 1 = 2 = 0 -H 1 : H 0 is not true Ratio of SSR r and SSR ur is F-distributed with df 1 = q and df 2 = n k 1

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 103 The F-Test: Rejection rule Reject H 0 if: F c c depends on -nominator degrees of freedom (df 1 ) -denominator degrees of freedom (df 2 ) -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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DIMMoL Macro-Econom(etr)ic Modelling Course 1 104 The LM-Test (Lagrange-Multiplier Test) Step1: Estimate the restricted model (with q restrictions) and save the residuals u r Step 2: Regress u r on all of the independent variables and obtain the R 2 as UR 2 Step 3: Compute LM = n UR 2 Step 4: LM follows a Chi-Square distribution with df = q; reject H 0 if LM > c (alternatively, look at p-values)

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 105 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 F p,n k 1 p distributed

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 106 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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DIMMoL Macro-Econom(etr)ic Modelling Course 1 107 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 = n ln(SSR) 2(k 1) -Schwartz Baysian Criterion (SBC) SBC = n ln(SSR) (k 1) ln(n) General rule: Parsimony is buitiful smaller value is prefered (different implementations exist)

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 108 Functional forms involving logarithms level-level model: regressing y on x y = j x j level-log model: regressing y on log(x) y = ( j /100)% x j log-level model: regressing log(y) on x % y = (100 j ) x j 100 j = semi-elasticity log-log model: regressing log(y) on log(x) % y = j % x j j = elasticity

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 109 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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DIMMoL Macro-Econom(etr)ic Modelling Course 1 110 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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DIMMoL Macro-Econom(etr)ic Modelling Course 1 111 Functional form involving dummy variables Capture qualitative information g different groups g 1 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 model each dummy must have an economically justified interpretation

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 112 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)-models nothing happens to the regression parameters if the units of measurement of the dependent variable are changed

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 113 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

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Consultancy to Develop and Implement a Macroeconomic Model for Lesotho (DIMMoL) Macro-Econom(etr)ic Modelling Part 7 Dr. Stefan Kooths DIW Berlin – Macro Analysis and Forecasting

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 115 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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DIMMoL Macro-Econom(etr)ic Modelling Course 1 116 Econometric methodology: Overview Fundamentals of probability Fundamentals of mathematical statistics Principles of regression analysis (cross sections) Time series regression models

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 117 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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DIMMoL Macro-Econom(etr)ic Modelling Course 1 118 General strategy random sampling conditions that restrict temporal correlation in time series

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 119 Using OLS in time series analysis Case 1: Gauss-Markov-Assumptions -strictly exogenous regressors OLS 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 series OLS is super-consistent, inference methods apply error-correction model representation (trend-) stationary processes non-stationary processes

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 120 (Trend-) Stationarity A process y is stationary if it is identically distributed over time -constant mean y -constant variance Var(y) -constant autocovariance Cov(y t,y t-h ) Trend stationarity -stationarity after removing the trend -deviations from the trend are stationary

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 121 Gauss-Markov assumptions Linearity in parameters population model is characterized by a linear regression function and additive errors No perfect collinearity none 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 Homoskedasticity error has the same variance given any value of the explanatory variables for all time periods No serial correlation The errors in two different time periods are uncorrelated OLS estimators are unbiased OLS estimators are BLUE (Gauss-Markov Theorem)

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 122 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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DIMMoL Macro-Econom(etr)ic Modelling Course 1 123 Assymptotic Gauss-Markov assumptions Linearity and weak dependence population model is characterized by a linear regression function, additive errors, and weakly dependent processes No perfect collinearity none 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 Homoskedasticity error has the same variance given any contemporaneous value of the explanatory variables No serial correlation The errors in two different time periods are uncorrelated OLS estimators are consistent Asymptotic normality of OLS

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 124 Weak dependence A time series is weakly dependent, if -x t and x t+h are „almost independent“ as h increases without bound (autocorrelation dies out over time) -Cov(x t,x t+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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DIMMoL Macro-Econom(etr)ic Modelling Course 1 125 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)

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 126 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 valid otherwise: spurious regression problem, artificially high R 2

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 127 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

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 128 Estimating the first order autocorrelation Case 1: | | 1 (weakly dependent process) -regressing y t on y t-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 H 0 -Both cases might be not rejectable Power of unit root tests is rather poor

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 129 Transforming the AR(1) equation

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 130 (Augmented) Dickey-Fuller tests Three scenarios -Random walk: y t = y t-1 + e t -Random walk with drift: y t = a 0 + y t-1 + e t -Random walk with drift and trend: y t = a 0 + y t-1 + a 2 t + e t Scenarios may include lags of y (Augmented DF) -e.g. y t = y t-1 + 1 y t-1 + 1 y t-2 + e t Critical values t c (tabulated) depend on -scenario type -sample size Testing for = 0 (H 0 : existence of a unit root) Rejection rule: Reject H 0 if t < t c

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 131 Critical values for Dickey-Fuller test = a 0 = 0 = a 2 = 0 = a 0 = a 2 = 0 = 0

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 132 General-to-specific procedure for testing unit roots

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 133 Unit root processes in regression analysis Time series x t, y t 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 x t and y t is I(1) problem of spurious regression first differences as transformation method Case 2: Cointegration -an linear combination of x t and y t (cointegration vector) exists such that s t = y t – x t 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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DIMMoL Macro-Econom(etr)ic Modelling Course 1 134 Testing for cointegration: The Engle-Granger Methodology Step 1: Test x t and y t 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 H 0 : a 1 = 0 cannot be rejected

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 135 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 u t for white noise) -use residuals from step 3 for (y t-1 x t-1 ):

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 136 Critical values for Engle-Granger Cointegratoin test

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DIMMoL Macro-Econom(etr)ic Modelling Course 1 137 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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DIMMoL Macro-Econom(etr)ic Modelling Course 1 138 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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DIMMoL Macro-Econom(etr)ic Modelling Course 1 139 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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DIMMoL Macro-Econom(etr)ic Modelling Course 1 140 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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DIMMoL Macro-Econom(etr)ic Modelling Course 1 141 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

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Introduction to Regression Analysis, Chapter 13,

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