Econometrics-I-19

# 26341 01568 16804 0000 75869 hhkids 06512 01399 4657

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-.26341*** .01568 -16.804 .0000 .75869 HHKIDS| .06512*** .01399 4.657 .0000 .40272 FEMALE| -.00542 .01234 -.439 .6603 .47881 --------+------------------------------------------------------------- Note: ***, **, * = Significance at 1%, 5%, 10% level. ---------------------------------------------------------------------- ™  13/29

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Part 19: MLE Applications Variance Estimators histogram;rhs=hhninc\$ reject ; hhninc=0\$ namelist ; X = one, age,educ,married,hhkids,female \$ loglinear ; lhs=hhninc ; rhs = x ; model=exp \$ create ; thetai = exp(b'x) \$ create ; gi = (hhninc/thetai - 1) ; gi2 = gi^2 \$\$ create ; hi = (hhninc/thetai) \$ matrix ; Expected = <X'X> ; Stat(b,Expected,X)\$ matrix ; Actual = <X'[hi]X> ; Stat(b,Actual,X) \$ matrix ; BHHH = <X'[gi2]X> ; Stat(b,BHHH,X) \$ matrix ; Robust = Actual * X'[gi2]X * Actual ; Stat(b,Robust,X) \$ ™  14/29
Part 19: MLE Applications Estimates --------+-------------------------------------------------- Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] --------+-------------------------------------------------- --> matrix ; Expected = <X'X> ; Stat(b,Expected,X)\$ Constant| 1.77430*** .04548 39.010 .0000 AGE| .00205*** .00061 3.361 .0008 EDUC| -.05572*** .00269 -20.739 .0000 MARRIED| -.26341*** .01558 -16.902 .0000 HHKIDS| .06512*** .01425 4.571 .0000 FEMALE| -.00542 .01235 -.439 .6605 --> matrix ; Actual = <X'[hi]X> ; Stat(b,Actual,X) \$ Constant| 1.77430*** .11922 14.883 .0000 AGE| .00205 .00181 1.137 .2553 EDUC| -.05572*** .00631 -8.837 .0000 MARRIED| -.26341*** .04954 -5.318 .0000 HHKIDS| .06512* .03920 1.661 .0967 FEMALE| -.00542 .03471 -.156 .8759 --> matrix ; BHHH = <X'[gi2]X> ; Stat(b,BHHH,X) \$ Constant| 1.77430*** .05409 32.802 .0000 AGE| .00205*** .00069 2.973 .0029 EDUC| -.05572*** .00331 -16.815 .0000 MARRIED| -.26341*** .01737 -15.165 .0000 HHKIDS| .06512*** .01637 3.978 .0001 FEMALE| -.00542 .01410 -.385 .7004 --> matrix ; Robust = Actual * X'[gi2]X * Actual \$ Constant| 1.77430*** .28500 6.226 .0000 AGE| .00205 .00481 .427 .6691 EDUC| -.05572*** .01306 -4.268 .0000 MARRIED| -.26341* .14581 -1.806 .0708 HHKIDS| .06512 .09459 .689 .4911 FEMALE| -.00542 .08580 -.063 .9496 ™  15/29

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Part 19: MLE Applications GARCH Models: A Model for Time Series with Latent Heteroscedasticity         Bollerslev/Ghysel, 1974 ™  16/29
Part 19: MLE Applications ARCH Model ™  17/29

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Part 19: MLE Applications GARCH Model ™  18/29
Part 19: MLE Applications Estimated GARCH Model ---------------------------------------------------------------------- GARCH MODEL Dependent variable Y Log likelihood function -1106.60788 Restricted log likelihood -1311.09637 Chi squared [ 2 d.f.] 408.97699 Significance level .00000 McFadden Pseudo R-squared .1559676 Estimation based on N = 1974, K = 4 GARCH Model, P = 1, Q = 1 Wald statistic for GARCH = 3727.503 --------+------------------------------------------------------------- Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] Mean of X --------+------------------------------------------------------------- |Regression parameters Constant| -.00619 .00873 -.709 .4783 |Unconditional Variance Alpha(0)| .01076*** .00312 3.445 .0006 |Lagged Variance Terms Delta(1)| .80597*** .03015 26.731 .0000 |Lagged Squared Disturbance Terms Alpha(1)| .15313*** .02732 5.605 .0000 |Equilibrium variance, a0/[1-D(1)-A(1)] EquilVar| .26316 .59402 .443 .6577 --------+------------------------------------------------------------- ™  19/29

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Part 19: MLE Applications 2 Step Estimation (Murphy-Topel) Setting, fitting a model which contains parameter estimates from another model. Typical application, inserting a prediction from one model into another. A. Procedures: How it's done. B. Asymptotic results: 1. Consistency 2. Getting an appropriate estimator of the asymptotic covariance matrix The Murphy - Topel result Application: Equation 1: Number of children Equation 2: Labor force participation ™  20/29
Part 19: MLE Applications Setting p Two equation model: n Model for y1 = f(y1 | x 1, θ 1) n Model for y2 = f(y2 | x 2, θ 2, x 1, θ 1) n (Note, not ‘simultaneous’ or even ‘recursive.’) p Procedure: n Estimate θ 1 by ML, with covariance matrix (1/n) V 1 n Estimate θ 2 by ML treating θ 1 as if it were known. n Correct the estimated asymptotic covariance matrix, (1/n) V 2 for the estimator of θ 2 ™  21/29

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Part 19: MLE Applications Murphy and Topel (1984,2002) Results Both MLEs are consistent ™  22/29 [ ] = + - - = - = - ÷ ÷ ÷
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