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Australian National University Faculty of Economics and Commerce School of Economics Advanced Econometric Methods EMET3011/EMET8014 Final Exam. Semester 1, 2002. Attempt all questions. Questions 1), 3), and 4) are worth 20 points. Question 2) is worth 10 points. Hypothesis tests should have size (or asymptotic size) equal to 5%. The 5% critical value from a chi- squared distribution with one degree of freedom is 3.84. 1) Consider the following three-equation model: (1.1) y1 = γ11 y2 + β11 x1 + β12 x2 + ε1 (1.2) y2 = γ23 y3 + β21 x1 + ε2 (1.3) y3 = β32 x2 + ε3 where the x’s are exogenous. The corresponding reduced form equations can be written as: (y1 y2 y3) = (x1 x2) Π + v π 11π 21π 31 where Π = . The data runs from i=1 to i=n, but the i indices have been dropped from the π 21π 22π 32 variables. a) Write the structural equations in the form yГ+xB = ε, where y = (y1 y2 y3) and x = (x1 x2). That is, give the forms of Г and B. b) Check the order conditions for identification. c) In the reduced form, we know that Π satisfies Π Г + B = 0. Use the equation Π Г + B = 0 to find γ23 in terms of the π ’s. Do the same for β21. d) Substitute the equation for y3 into that for y2 to obtain the reduced form equation for y2. What are the properties of the OLS estimator of this reduced form equation? (Assume the errors are homoskedastic and not serially correlated.) What are the properties of the OLS estimator of equation (1.3)? e) How do the estimators in part d) relate to the terms in part c)? 2) In the standard linear regression model, y=Xβ+ ε, define the residual vector, eIV = y-XbIV, where bIV is the IV estimator of β, bIV = (Z’X)-1 Z’y. a) Show that Z’eIV = 0. b) Suppose that the first element of β is the intercept and the first column of Z is a column of ones. What does the first element of Z’eIV imply about the residuals? 3) Consider the heteroskedastic regression model, yi = xiβ + εi , i = 1,…,n, where the εi have mean zero, var(εi) = σ i2 , and are not serially correlated. For simplicity, k=1. a) Define the vector of errors ε , where ε ' = (ε 1 ...ε n ) , and the covariance matrix Ω =E[ ε ε ' ]. Give the form of Ω. n ε i2 b) Show that ε’ Ω-1 ε = ∑ σ 2 . [Hint: the inverse of a diagonal matrix, i.e., a matrix with all i =1 i numbers off the main diagonal equal to zero, is a diagonal matrix in which the elements on the diagonal are the inverses of the elements of the original matrix.] c) Consider the transformed model, yi* = xi*β + εi*, where εi* = εi/σi , yi* = yi/σi, and xi* = xi/σi. Show that the εi* have mean zero and are homoskedastic. d) Give the form of the OLS estimator of β in the transformed model in c). Then give the form of this estimator in terms of yi, xi and σi. n e2 e) Consider the weighted sum of squares, ∑ i2 , where ei = yi - xiα. (Thus, α is any value for i =1 σi the parameter β.) Find the estimator of β obtained by minimizing the weighted sum of squares with respect to α. Show that the estimator is the same as that obtained in d). 4) Consider the linear regression model with serially correlated errors, (4.1) zt = β1 + β2xt + εt (4.2) εt = ρεt-1 + ut, where t=1,…T and ut is homoskedastic white noise; and the alternative model, (4.3) zt = γ0 + γ1zt-1 + γ2xt + γ3xt-1+ ut . a) Derive the restriction that must be imposed on the parameters in (4.3) to obtain (4.1)-(4.2). b) Suppose that you estimate the model (4.1) by OLS on a set of 150 observations and obtain the results in table 1 below. What does the value of the Durbin-Watson statistic say about the value of ρ? c) Regressing the residuals from OLS on (4.1) on an intercept, the lagged residuals, and xt gives the results in table 2. Using these results, or otherwise, test the null hypothesis that ρ = 0 against the alternative hypothesis that ρ ≠ 0. d) Suppose that you decide to estimate the model (4.1) – (4.2) using MLE. The results are shown in table 3. Use the results to test the null hypothesis that ρ = 0 against the alternative hypothesis that ρ ≠ 0. What is the difference from the test in c)? e) Suppose that you estimate the model (4.3) by OLS. The results are shown in table 4. Is (4.3) to be preferred to (4.1)-(4.2)? Why or why not? What does you answer imply about the cause of the serial correlation in the residuals from estimating (4.1) by OLS? Table 1. Estimation of (4.1) Dependent Variable: Z Method: Least Squares Sample: 1 150 Included observations: 150 Variable Coefficient Std. Error t-Statistic Prob. C 2.932922 0.530331 5.530356 0.0000 X 0.861947 0.312221 2.760691 0.0065 R-squared 0.048661 Mean dependent var 3.479133 Adjusted R-squared 0.042277 S.D. dependent var 6.178326 S.E. of regression 6.046317 Akaike info criterion 6.449932 Sum squared resid 5447.134 Schwarz criterion 6.489896 Log likelihood -484.9699 F-statistic 7.621414 Durbin-Watson stat 0.912215 Prob(F-statistic) 0.006493 Table 2. Residuals on Lagged Residuals Dependent Variable: RES_4_1 Method: Least Squares Sample (adjusted): 2 150 Included observations: 149 after adjusting endpoints Variable Coefficient Std. Error t-Statistic Prob. C 0.006936 0.448019 0.015481 0.9877 RES_4_1_LAG 0.541791 0.069307 7.817290 0.0000 X -0.107554 0.263625 -0.407981 0.6839 R-squared 0.293648 Mean dependent var -0.042263 Adjusted R-squared 0.284038 S.D. dependent var 6.023822 S.E. of regression 5.097030 Akaike info criterion 6.114990 Sum squared resid 3819.017 Schwarz criterion 6.175203 Log likelihood -455.6243 F-statistic 30.55578 Durbin-Watson stat 1.992793 Prob(F-statistic) 0.000000 Table 3. Estimation of (4.1) – (4.2) Dependent Variable: Z Method: MLE Sample (adjusted): 2 150 Included observations: 149 Convergence achieved after 8 iterations Variable Coefficient Std. Error t-Statistic Prob. C 2.694732 0.944007 2.854567 0.0049 X 1.240580 0.396397 3.129638 0.0021 RHO 0.542569 0.069070 7.855318 0.0000 R-squared 0.322214 Mean dependent var 3.479133 Adjusted R-squared 0.313055 S.D. dependent var 6.178326 S.E. of regression 5.120731 Akaike info criterion 6.124139 Sum squared resid 3880.839 Schwarz criterion 6.184085 Log likelihood -459.3725 F-statistic 35.17899 Durbin-Watson stat 1.912475 Prob(F-statistic) 0.000000 Table 4. Estimation of (4.3) Dependent Variable: Z Method: Least Squares Sample (adjusted): 2 150 Included observations: 149 Variable Coefficient Std. Error t-Statistic Prob. C 1.610133 0.488199 3.298107 0.0012 Z_LAG 0.522461 0.068548 7.621831 0.0000 X 1.600170 0.418872 3.820193 0.0002 X_LAG -1.507101 0.418091 -3.604724 0.0004 R-squared 0.347113 Mean dependent var 3.479133 Adjusted R-squared 0.333789 S.D. dependent var 6.178326 S.E. of regression 5.042860 Akaike info criterion 6.099957 Sum squared resid 3738.274 Schwarz criterion 6.179885 Log likelihood -456.5467 F-statistic 26.05125 Durbin-Watson stat 1.876878 Prob(F-statistic) 0.000000