Econometrics-I-14

Ε →-1-1-1-1 x x x x ω ω x x ω ω to the same

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Unformatted text preview: ε →-1-1-1-1 X' X - X' X Ω Ω X' - X' Ω Ω to the same random vector. Part 14: Generalized Regression Two Step FGLS VVIR (Theorem 9.6) To achieve full efficiency, we do not need an efficient estimate of the parameters in , only a consistent one. ˜˜˜˜™ ™ 36/44 Part 14: Generalized Regression Harvey’s Model Examine Harvey’s model once again. Methods of estimation: Two step FGLS: Use the least squares residuals to estimate , then use Full maximum likelihood estimation. Estimate all parameters simultaneously. A handy result due to Oberhofer and Kmenta - the “zig-zag” approach. Iterate back and forth between and . ˜˜˜˜˜™ ™ 37/44 ( 29 { } ( 29 1 1 1 ˆ ˆ ˆ ˆ--- ′ ′ X X X y β = Ω θ Ω θ Part 14: Generalized Regression Harvey’s Model for Groupwise Heteroscedasticity Groupwise sample, yig, xig,… N groups, each with Ng observations. Var[εig] = σg2 Let dig = 1 if observation i,g is in group g, 0 else. = group dummy variable. Var[εig] = σg2 exp(θ2d2 + … θGdG) Var1 = σg2 , Var2 = σg2 exp(θ2) and so on. ˜˜˜˜˜™ ™ 38/44 Part 14: Generalized Regression Estimating Variance Components p OLS is still consistent: p Est.Var1 = e1’e1/N1 estimates σg2 p Est.Var2 = e2’e2/N2 estimates σg2 exp(θ2) p Estimator of θ2 is ln[(e2’e2/N2)/(e1’e1/N1)] p (1) Now use FGLS – weighted least squares p Recompute residuals using WLS slopes p (2) Recompute variance estimators p Iterate to a solution… between (1) and (2) ˜˜˜˜˜™ ™ 39/44 Part 14: Generalized Regression Baltagi and Griffin’s Gasoline Data World Gasoline Demand Data, 18 OECD Countries, 19 years Variables in the file are COUNTRY = name of country YEAR = year, 1960-1978 LGASPCAR = log of consumption per car LINCOMEP = log of per capita income LRPMG = log of real price of gasoline LCARPCAP = log of per capita number of cars See Baltagi (2001, p. 24) for analysis of these data. The article on which the analysis is based is Baltagi, B. and Griffin, J., "Gasolne Demand in the OECD: An Application of Pooling and Testing Procedures," European Economic Review, 22, 1983, pp. 117-137. The data were downloaded from the website for Baltagi's text. ˜˜˜˜˜ ™ 40/44 Part 14: Generalized Regression Analysis of Variance ˜˜˜˜˜ ˜™ 41/44 Part 14: Generalized Regression Least Squares First Step---------------------------------------------------------------------- Multiplicative Heteroskedastic Regression Model... Ordinary least squares regression ............ LHS=LGASPCAR Mean = 4.29624 Standard deviation = .54891 Number of observs. = 342 Model size Parameters = 4 Degrees of freedom = 338 Residuals Sum of squares = 14.90436 Wald statistic [17 d.f.] = 699.43 (.0000) (Large) B/P LM statistic [17 d.f.] = 111.55 (.0000) (Large) Cov matrix for b is sigma^2*inv(X'X)(X'WX)inv(X'X)...
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ε →-1-1-1-1 X X X X Ω Ω X X Ω Ω to the same random...

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