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Econometrics-I-18

Estimators of the asymptotic covariance asymptotic

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Estimators of the Asymptotic Covariance Asymptotic Covariance Matrix ™  21/47 ( 29 1 1 1 1 1 1 1 Conventional Estimator - Inverse of the information matrix "Usual" "Berndt, Hall, Hall, Hausman" (BHHH) [( ) ][( ) ] n i i i i n n i i i i i i i i i i y y - - = - - = = λ = = - λ - λ x x XΛX g g x x 1 2 1 ( ) n i i i i i y - = = - λ x x
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Part 18: Maximum Likelihood Estimation Robust Estimation p Sandwich Estimator p H-1 (GG) H-1 p Is this appropriate? Why do we do this? ™  22/47 ( 29 ( 29 1 1 2 1 ( ) n i i i i i y - - = - λ XΛX x x X ΛX
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Part 18: Maximum Likelihood Estimation Newton’s Method ™  23/47
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Part 18: Maximum Likelihood Estimation Newton’s Method for Poisson Regression ™  24/47
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Part 18: Maximum Likelihood Estimation Poisson Regression Iterations Poisson ; lhs = doctor ; rhs = one,female,hhninc,educ;mar;output=3$ Method=Newton; Maximum iterations=100 Convergence criteria: gtHg .1000D-05 chg.F .0000D+00 max|db| .0000D+00 Start values: .00000D+00 .00000D+00 .00000D+00 .00000D+00 1st derivs. -.13214D+06 -.61899D+05 -.43338D+05 -.14596D+07 Parameters: .28002D+01 .72374D-01 -.65451D+00 -.47608D-01 Itr 2 F= -.1587D+06 gtHg= .2832D+03 chg.F= .1587D+06 max|db|= .1346D+01 1st derivs. -.33055D+05 -.14401D+05 -.10804D+05 -.36592D+06 Parameters: .21404D+01 .16980D+00 -.60181D+00 -.48527D-01 Itr 3 F= -.1115D+06 gtHg= .9725D+02 chg.F= .4716D+05 max|db|= .6348D+00 1st derivs. -.42953D+04 -.15074D+04 -.13927D+04 -.47823D+05 Parameters: .17997D+01 .27758D+00 -.54519D+00 -.49513D-01 Itr 4 F= -.1063D+06 gtHg= .1545D+02 chg.F= .5162D+04 max|db|= .1437D+00 1st derivs. -.11692D+03 -.22248D+02 -.37525D+02 -.13159D+04 Parameters: .17276D+01 .31746D+00 -.52565D+00 -.49852D-01 Itr 5 F= -.1062D+06 gtHg= .5006D+00 chg.F= .1218D+03 max|db|= .6542D-02 1st derivs. -.12522D+00 -.54690D-02 -.40254D-01 -.14232D+01 Parameters: .17249D+01 .31954D+00 -.52476D+00 -.49867D-01 Itr 6 F= -.1062D+06 gtHg= .6215D-03 chg.F= .1254D+00 max|db|= .9678D-05 1st derivs. -.19317D-06 -.94936D-09 -.62872D-07 -.22029D-05 Parameters: .17249D+01 .31954D+00 -.52476D+00 -.49867D-01 Itr 7 F= -.1062D+06 gtHg= .9957D-09 chg.F= .1941D-06 max|db|= .1602D-10 * Converged ™  25/47
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Part 18: Maximum Likelihood Estimation Regression and Partial Effects +--------+--------------+----------------+--------+--------+----------+ |Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]| Mean of X| +--------+--------------+----------------+--------+--------+----------+ Constant| 1.72492985 .02000568 86.222 .0000 FEMALE | .31954440 .00696870 45.854 .0000 .47877479 HHNINC | -.52475878 .02197021 -23.885 .0000 .35208362 EDUC | -.04986696 .00172872 -28.846 .0000 11.3206310 +-------------------------------------------+ | Partial derivatives of expected val. with | | respect to the vector of characteristics. | | Effects are averaged over individuals. | | Observations used for means are All Obs. | | Conditional Mean at Sample Point 3.1835 | | Scale Factor for Marginal Effects 3.1835 | +-------------------------------------------+ +--------+--------------+----------------+--------+--------+----------+ |Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]| Mean of X| +--------+--------------+----------------+--------+--------+----------+ Constant| 5.49135704 .07890083 69.598 .0000 FEMALE | 1.01727755 .02427607 41.905 .0000 .47877479 HHNINC | -1.67058263 .07312900 -22.844 .0000 .35208362 EDUC | -.15875271 .00579668 -27.387 .0000 11.3206310 ™  26/47
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Part 18: Maximum Likelihood Estimation Comparison of Standard Errors Negative Inverse of Second Derivatives +--------+--------------+----------------+--------+--------+----------+ |Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]| Mean of X| +--------+--------------+----------------+--------+--------+----------+ Constant| 1.72492985 .02000568 86.222 .0000 FEMALE | .31954440 .00696870 45.854 .0000 .47877479 HHNINC | -.52475878 .02197021 -23.885 .0000 .35208362 EDUC | -.04986696 .00172872 -28.846 .0000 11.3206310 BHHH +--------+--------------+----------------+--------+--------+ |Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]| +--------+--------------+----------------+--------+--------+ Constant| 1.72492985 .00677787 254.495 .0000 FEMALE | .31954440 .00217499 146.918 .0000 HHNINC | -.52475878 .00733328 -71.559 .0000 EDUC | -.04986696 .00062283 -80.065 .0000 Why are they so different?  Model failure. This is a panel. There is autocorrelation. ™  27/47
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Part 18: Maximum Likelihood Estimation MLE vs. Nonlinear LS ™  28/47
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Part 18: Maximum Likelihood Estimation Testing Hypotheses – A Trinity of Tests The likelihood ratio test: Based on the proposition (Greene’s) that restrictions always “make life worse” Is the reduction in the criterion (log-likelihood) large?
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