Econometrics-I-19 - Econometrics I Professor William Greene...

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Part 19: MLE Applications Econometrics I Professor William Greene Stern School of Business Department of Economics
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Part 19: MLE Applications Econometrics I Part 19 –MLE Applications  and a Two Step Estimator ™  1/29
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Part 19: MLE Applications Model for a Binary Dependent Variable p Binary outcome. n Event occurs or doesn’t (e.g., the democrat wins, the person enters the labor force,… n Model the probability of the event. P( x )=Prob(y=1| x ) n Probability responds to independent variables p Requirements n 0 < Probability < 1 n P( x ) should be monotonic in x – it’s a CDF ™  2/29
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Part 19: MLE Applications Two Standard Models p Based on the normal distribution: n Prob[y=1| x ] = ( β’x ) = CDF of normal distribution n The “probit” model p Based on the logistic distribution n Prob[y=1|x] = exp( β’x )/[1+ exp( β’x )] n The “logit” model p Log likelihood n P(y|x) = (1-F)(1-y) Fy where F = the cdf n LogL = Σi (1-yi)log(1-Fi) + yilogFi = Σi F[(2yi-1) β’x ] since F(-t)=1-F(t) for both. ™  3/29
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Part 19: MLE Applications Coefficients in the Binary Choice Models E[y|x] = 0*(1-Fi) + 1*Fi = P(y=1|x) = F( β’x ) The coefficients are not the slopes, as usual in a nonlinear model ∂E[y|x]/∂x= f( β’x ) β These will look similar for probit and logit ™  4/29
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Part 19: MLE Applications Application: Female Labor Supply 1975 Survey Data: Mroz (Econometrica) 753 Observations Descriptive Statistics Variable Mean Std.Dev. Minimum Maximum Cases Missing ============================================================================= = All observations in current sample -------- +--------------------------------------------------------------------- LFP | .568393 .495630 .000000 1.00000 753 0 WHRS | 740.576 871.314 .000000 4950.00 753 0 KL6 | .237716 .523959 .000000 3.00000 753 0 K618 | 1.35325 1.31987 .000000 8.00000 753 0 WA | 42.5378 8.07257 30.0000 60.0000 753 0 WE | 12.2869 2.28025 5.00000 17.0000 753 0 WW | 2.37457 3.24183 .000000 25.0000 753 0 RPWG | 1.84973 2.41989 .000000 9.98000 753 0 HHRS | 2267.27 595.567 175.000 5010.00 753 0 HA | 45.1208 8.05879 30.0000 60.0000 753 0 HE | 12.4914 3.02080 3.00000 17.0000 753 0 HW | 7.48218 4.23056 .412100 40.5090 753 0 FAMINC | 23080.6 12190.2 1500.00 96000.0 753 0 KIDS | .695883 .460338 .000000 1.00000 753 0 ™  5/29
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Part 19: MLE Applications ---------------------------------------------------------------------- Binomial Probit Model Dependent variable LFP Log likelihood function -488.26476 (Probit) Log likelihood function -488.17640 (Logit) --------+------------------------------------------------------------- Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] Mean of X --------+------------------------------------------------------------- |Index function for probability Constant| .77143 .52381 1.473 .1408 WA| -.02008 .01305 -1.538 .1241 42.5378 WE| .13881*** .02710 5.122 .0000 12.2869 HHRS| -.00019** .801461D-04 -2.359 .0183 2267.27 HA| -.00526 .01285 -.410 .6821 45.1208 HE| -.06136*** .02058 -2.982 .0029 12.4914 FAMINC| .00997** .00435 2.289 .0221 23.0806 KIDS| -.34017*** .12556 -2.709 .0067 .69588 --------+------------------------------------------------------------- Binary Logit Model for Binary Choice --------+------------------------------------------------------------- |Characteristics in numerator of Prob[Y = 1] Constant| 1.24556 .84987 1.466 .1428 WA| -.03289 .02134 -1.542 .1232 42.5378 WE| .22584*** .04504 5.014 .0000 12.2869 HHRS| -.00030** .00013 -2.326 .0200 2267.27 HA| -.00856 .02098 -.408 .6834 45.1208 HE| -.10096*** .03381 -2.986 .0028 12.4914 FAMINC| .01727** .00752 2.298 .0215 23.0806 KIDS| -.54990*** .20416 -2.693 .0071 .69588 --------+------------------------------------------------------------- Estimated Choice Models for Labor Force Participation ™  6/29
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Part 19: MLE Applications Partial Effects ---------------------------------------------------------------------- Partial derivatives of probabilities with respect to the vector of characteristics. They are computed at the means of the Xs. Observations used are All Obs. --------+------------------------------------------------------------- Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] Elasticity --------+------------------------------------------------------------- |PROBIT: Index function for probability WA| -.00788 .00512 -1.538 .1240 -.58479 WE| .05445*** .01062 5.127 .0000 1.16790 HHRS|-.74164D-04** .314375D-04 -2.359 .0183 -.29353 HA| -.00206 .00504 -.410 .6821 -.16263 HE| -.02407*** .00807 -2.983 .0029 -.52488 FAMINC| .00391** .00171 2.289 .0221 .15753 |Marginal effect for dummy variable is P|1 - P|0.
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