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Econometrics-I-14

2 1 2 3 1 1 1 1 1 ρ ρ ρ ρ ρ ρ σ σ ρ ρ ρ ρ

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2 1 2 3 1 1 1 1 1 - - - - - - ρ ρ ρ ρ ρ ρ σ σ = ρ ρ ρ ÷ ρ ρ ρ Ω L L L M M M O M L T T u T T T T
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Part 14: Generalized Regression Estimated AR(1) Model ---------------------------------------------------------------------- AR(1) Model: e(t) = rho * e(t-1) + u(t) Initial value of rho = .87566 Maximum iterations = 1 Method = Prais - Winsten Iter= 1, SS= .022, Log-L= 127.593 Final value of Rho = .959411 Std. Deviation: e(t) = .076512 Std. Deviation: u(t) = .021577 Autocorrelation: u(t) = .253173 N[0,1] used for significance levels --------+------------------------------------------------------------- Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] Mean of X --------+------------------------------------------------------------------------- Constant| -20.3373*** .69623 -29.211 .0000 FGLS LP| -.11379*** .03296 -3.453 .0006 3.72930 LY| .87040*** .08827 9.860 .0000 9.67215 LPNC| .05426 .12392 .438 .6615 4.38037 LPUC| -.04028 .06193 -.650 .5154 4.10545 RHO| .95941*** .03949 24.295 .0000 --------+------------------------------------------------------------------------- Constant| -21.2111*** .75322 -28.160 .0000 OLS LP| -.02121 .04377 -.485 .6303 3.72930 LY| 1.09587*** .07771 14.102 .0000 9.67215 LPNC| -.37361** .15707 -2.379 .0215 4.38037 LPUC| .02003 .10330 .194 .8471 4.10545 ™  33/44
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Part 14: Generalized Regression Two Step Estimation The general result for estimation when is estimated. GLS uses [ X-1X ] X -1 y which converges in probability to . We seek a vector which converges to the same thing that this does. Call it “Feasible GLS” or FGLS, based on [ X X ] X y The object is to find a set of parameters such that [ X X ] X y - [ X -1 X ] X -1 y 0 ™  34/44 ˆ -1 Ω ˆ -1 Ω ˆ -1 Ω ˆ -1 Ω
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Part 14: Generalized Regression Feasible GLS ™  35/44 For FGLS estimation, we do not seek an estimator of  such that ˆ                      ˆ This makes no sense, since   is nxn and does not "converge" to anything.  We seek a matrix  such that            Ω - 0 Ω Ω Ω   Ω ˆ  (1/n) (1/n) For the asymptotic properties, we will require that ˆ             (1/ n) (1/n) Note in this case, these are two random vectors, which we require to converge ε ε → -1 -1 -1 -1 X' X -   X' 0 Ω Ω X'  -   X'   0 Ω Ω  to the same random vector.
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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
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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
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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.
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