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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.

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

1 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

2 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

3 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

4 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)

5 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/)

6 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

7 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?

8 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

9 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)

10 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

11 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

12 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

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

14 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

15 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

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

17 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

18 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

19 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

20 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

21 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

22 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

23 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

24 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

25 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!

26 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

27 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)

28 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

29 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

30 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

31 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

32 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

33 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!

34 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

35 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?

36 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

37 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

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

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

40 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

41 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

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

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

44 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

45 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

46 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

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

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

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

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

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

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

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

54 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 )

55 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

56 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

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

58 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

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

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

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

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

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

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

65 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

66 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

67 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

68 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

69 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

70 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

71 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

72 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

73 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

74 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 |

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

76 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)

77 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!

78 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

79 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

80 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

81 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

82 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

83 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

84 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) ^ ^

85 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)

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

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

88 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

89 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)!

90 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

91 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

92 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

93 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

94 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)

95 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)

96 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)

97 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

98 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

99 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

100 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

101 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

102 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

103 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

104 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)

105 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

106 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

107 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)

108 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

109 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

110 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

111 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

112 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

113 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

114 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

115 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

116 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

117 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

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

119 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

120 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

121 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)

122 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

123 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

124 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

125 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)

126 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

127 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

128 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

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

130 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

131 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

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

133 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

134 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

135 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 ):

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

137 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

138 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

139 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

140 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

141 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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