Økonomisk Kandidateksamen 2004 (I) Econometrics 2

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1 Økonomisk Kandidateksamen 2004 (I) Econometrics 2 This is a closed-book exam (uden hjælpemidler). Answer all questions! The group of questions 1 to 4 have equal weight. Within each group, part (a) represents very basic questions, part (b) questions which require somewhat more detailed knowledge of the curriculum, and part (c) questions which require a deeper understanding, for example they may be technically demanding or they may require a good understanding for how to combine different theoretical results. A correct answer of all part (a) questions is sufficient for passing the exam. The answers can be in Danish or English. Question 1. (a) For the linear model with a fixed effect: y it = α + βx it + α i + u it explain informally why an OLS pooled regression of y it on x it may lead to inconsistent estimates of β. Suggest an alternative estimation procedure that does give consistent estimates. (b) Explain what we mean by a difference-in-differences estimator. (c) Three different investigators have the same panel data set and the same linear model. They use different estimation methods: OLS, random effects and first differencing. The three sets of parameter estimates are quite different and each investigator claims that their estimates are the correct ones. How would you devise an objective test between them? Question 2. We consider the linear model, y t = β 0 x t + u t, x 0 t = [1,x 1,t,x 2,t ]. The variables are defined by y t = UnR t, the unemployment rate (the number of unemployed per labour force), x 1,t = LrC t, the log of real aggregate consumption, x 2,t = Rb t, the 10 year bond rate, and t = 1983:2-2003:1. The output from an OLS regression analysis, inclusive various misspecification tests, is reported in Appendix, Part A. (a) Under which conditions is the OLS estimate ˆβ unbiased and efficient? Based on the misspecification tests reported in Appendix, Part A do you 1

2 think the conditions are satisfied. Motivate your answer. Assume that you know from previous experience that LrC t = a 0 +a 1 UnR t +...+e t. How would this affect your evaluation of the OLS properties of the present model? (b) Three different estimates of the standard errors of regression estimates (SE, HACSE, HCSE) are reported in Appendix, Part A. Explain briefly the difference between the three estimates. Based on the reported misspecification tests discuss which of the three is the most appropriate. Does any of the three standard errors of estimates correct for simultaneity bias? Motivate briefly! (c) A test for ARCH errors is reported in Appendix A. Formulate the H 0 and the H 1 hypothesis. Give the conditions under which this test is valid. Are they satisfied in this case? Motivate briefly! Question 3. (a) Appendix, Part B reports the output of a dynamic version of the above static OLS model y t = a 1 y t 1 + b 01 x 1,t + b 11 x 1,t 1 + b 02 x 2,t + b 12 x 2,t 1 + a 0 + ε t t = 1983:2 2003:1. Discuss whether you think the choice of one lag is consistent with the reported misspecification tests in Appendix, Part A. Compare with the same misspecification tests in Part B and discuss whether the OLS assumptions are now satisfied? Explain the difference between a residual correlogram, a residual autoregression and a partial autocorrelation function. Explain why R 2 and R 2 relative to difference give two very different results. (b) Derive the static long-run solution, y = β 0 + β 1 x 1 + β 2 x 2, of the estimated dynamic regression model based on the analysis of lag structure in Appendix, Part B (only approximate coefficients are needed). Interpret the estimated long-run coefficient of real consumption and compare it to the static regression estimate in Part A. Does the long-run estimate fall within the 95% confidence interval of the static OLS regression coefficient? Which of the two results do you think is more reliable? (c) Assume that inflation rate is an important omitted explanatory variable to the unemployment rate. Under which conditions would the estimates of the static regression coefficients in Part A and the estimates of the solved long-run coefficients in Part B remain unchanged? Discuss the role of the ceteris paribus assumptionsinaneconomicmodelandtheomittedvariables problem in an empirical model. 2

3 Question 4. (a) In the linear model the design matrix (X 0 T X T ) plays an important role, where X T =[x 1,.., x k ]andx i is (T 1). IntheOLSmodelweoften assume that the x variables are fixedorgiven. Whenthisassumptionisinappropriate, asymptotic theory is often used to derive properties of estimators and test procedures. Under which condition will 1 T (X0 T X T ) T M, where M is a matrix with constant parameters? Explain how the Dickey-Fuller test ofaunitrootisformulatedandspecifytheh 0 and the H 1 hypothesis of a unitrootinthevariablex t when the latter contains a linear trend. Based on the enclosed output would you say there is a unit root in unemployment rate? Asymptotic critical 5% values for unit root tests: τ nc = 1.94, τ c = 2.86, τ ct = 3.41, where nc stands for no constant, c for constant and ct for constant and trend in the model. (b) Specify the H 0 and the H 1 hypothesis of the common factor hypothesis in the dynamic regression model in Question 3. Based on the result of the COMFAC test reported in Part B would it be appropriate to estimate the static regression model in Question 2 with Generalized Least Squares? Motivate your answer. (c) Discuss a Wald procedure for testing the following hypotheses in the above dynamic regression model. H 0 : β 1 = b 01+b 11 a 1 =0, against the alternative H 1 : β 1 = b 01+b 11 a 1 6=0. 3

4 Appendix Part A: EQ( 1) Modelling UnR by OLS (using consdemo.in7) The estimation sample is: 1983 (2) to 2003 (1) Estimates of standard errors of regression coefficients based on 3 different methods: SE = OLS standard errors, HACSE = heteroscedasticity and autocorrelation consistent standard errors, HCSE = heteroscedasticity consistent standard errors. Coefficients SE HACSE HCSE Constant LrC Rb Coefficients t-se t-hacse t-hcse Constant LrC Rb Model misspecification tests: 1. Residual correlogram (ACF) from lag 1 to 5: Partial autocorrelation function (PACF): Testing for error autocorrelation from lags 1 to 3 Chi^2(3) = [0.0000]** and F-form F(3,74) = [0.0000]** Error autocorrelation coefficients in auxiliary regression: Testing for ARCH errors from lags 1 to 2 ARCH 1-2 test: F(2,73) = [0.0000]** ARCH coefficients: Testing for heteroscedasticity using squares and cross products Chi^2(5) = [0.0028]** and F-form F(5,71) = [0.0023]** Heteroscedasticity coefficients:

5 Coefficient Std.Error t-value LrC Rb LrC^ Rb^ LrC*Rb Part B. EQ( 2) Modelling UnR by OLS (using consdemo.in7) The estimation sample is: 1983 (2) to 2003 (1) Coefficient Std.Error HACSE t-hacse t-prob Part.R^2 UnR_ Constant LrC Rb Rb_ LrC_ sigma RSS R^ F(5,74) = 1543 [0.000]** DW no. of observations 80 no. of parameters 6 R^2 relative to difference = Analysis of lag structure, coefficients: Lag 0 Lag 1 Sum UnR Constant LrC Rb Unit-root t-test of each variable based on the assumption that there is a constant but no linear trend in the variables. Variable D-F t-test UnR LrC Rb COMFAC Wald test: COMFAC F(2,74) = [0.0007] ** Misspecification tests: Residual autocorrelograms: 1. Residual correlogram (ACF) from lag 1 to 5: Partial autocorrelation function (PACF): Testing for error autocorrelation from lags 1 to 2 Chi^2(2) = [0.0000]** and F-form F(2,72) = [0.0000]** Error autocorrelation coefficients in auxiliary regression:

6 Testing for ARCH errors from lags 1 to 2 ARCH 1-2 test: F(2,70) = [0.0049]** ARCH coefficients: