Question 3 (20 marks) The following regressions were run using 2,108 observations on females living in the US. The respondents in the sample are aged between 27 and 31, with many working and raising young families. WORKING = a binary variable equal to 1 if the respondent was working LEDU= natural logarithm of years of schooling of respondent ABIL = respondent's score on a cognitive ability test CHILDREN=number of children in the household MARRIED= a binary variable equal to 1 if the respondent was married ETHBLACK = a binary variable equal to 1 if the respondent was black ETHHISP = a binary variable equal to 1 if the respondent was Hispanic Regression 1 - Linear Probability Model reg WORKING LEDU ABIL CHILDREN MARRIED STHBLACK ETHHISP IT MALE--1) NS Model Residual 25.3798341 313.337433 6 2,101 1.22997231 .149137284 Number of o F16, 2101) Prob > R-squared Adj R-squared Root MSE 2,108 28.36 0.0050 0.0749 0.0723 . 38618 Total 338.717263 2,107 . 160759077 WORKTNCE Coat. Str. Fr. 1 P> 1954 Cant. Intervall -1.51 LEDU ABIG CHILDREN MARRIED ETHBLACK ETHNISE cons .3707367 .0407132 -.0071797 -.0650202 -.0080311 .0319012 -.1296264 .0487348 . 0112965 . 104 2015 .0175727 .0246139 .0267975 .1307303 0.000 D.GDU 1.12 0.000 0.745 0.234 0.322 -2751632 -0185597 -.01639948 --1024818 - .0563601 -.0208513 - 3860008 .4663103 .0628668 .0020204 -.0325586 .010258 .00 14536 .126740 -0.33 1.19 -0.99 THBLACK ETHHISE cons -.06DOODG . 2965293 -4.104034 . 1607939 .176193 .8351403 -0.37 1.06 -4.91 0.709 0.290 0.000 -.3751607 -.1588027 -5.740839 .2551594 .5318613 -2.467199 Regression 2 - Logit . Legit WORKING LEDU ABIL CHILDREN MARRIED ET BLACK ETAHISP iT MALE- Iteration : Iteration 1: Iteration 2: Iteration 3: Iteration 4 log 1. kelihood --1058.1669 log 1. kelihood --302.96666 log likelihood --980.10918 log 1. kelihood = -980.1038 log 1: kelihood = -980.1038 Logistic regresaian 2,108 Number of oba LR ch12 (6) Prob > chiz I'seudo R2 Log likelihood - 980.1038 WORKING Coef. Sid. Err. P12 (956 Conf. Intervall 7.12 3.61 -1.45 LEDU ABIL CHILCREN MARRIED ETHBLACK ETHHISP 2.254523 - 2686009 --0432632 --1835237 -.0600008 -1865293 -4.104034 .316596 .0743098 .0297784 1193119 .2607909 .176193 .8351403 0.000 0.000 0.146 0.000 0.709 0.290 0.000 1.634000 .1229564 .1016278 -.7173707 -.3751607 -. 1568027 -5.740979 2.875039 .4142454 .0151013 ..2496767 .2551594 .5316613 -2.463109 -0.37 1.06 -4.91 Regression 3 - Probit . probit WORKING LEDU ABIL CHILDREN MARRIED ET BLACK ETHHISP I HALE--0 Iteration 0: Iteration 1: Iteration 2: Iteration 3: Iteration 4 log likelihood - -1050.1669 log likelihood = -980.83399 Leg 11 kelihood --980.226799 Log likelihood - -980.2266 log likelihood - -980.2266 Probit regression 2,108 Number of obes IR ch12 (6) Probabil Iseudo R2 Log likelihood - 980.2266 WORKING Cuef. Std. Err. 21 21 (958 Cunt. Intervall TECC ABIL CHILDREN MARRIED ETHBLACK ETHHISP cons 1.312301 .1495463 -.025507 -.2718763 -.0502482 .0933996 2.310967 .1822144 .042753 .017219 .0677103 .0917524 -100238 .4842125 7.14 3.50 -1.48 -4.02 -0.55 0.000 0.000 0.135 0.000 0.584 9447754 .065752 -.0592557 -.404588 -.2300796 -.70301632 -3.290026 1.655827 .2333406 .0082417 -.1391665 - 1295832 - 2898625 -1.391948 -1.83 0.000 Regression 4 - Logit (Intercept only) Logit WORKING 1 MALEO Iteration : Iteration 11 Log likelihood - - 1038.1669 Log likelihood - 1058.1669 Logistic regression 2,108 0.00 Number of obs LR chi2 (0) Prob > chi2 Pseudu 12 Log likelihoud - -1058.1669 0.0000 WORKING Coef. Std. Er 2 P (95% Conf. Intervall con 1.379194 -0543352 25.38 0.000 1.222699 1.453689 Regression 5 - Probit (Intercept only) probit WORKING I MALEO Ileration 0: Iteration 1: loy likelihood - -1056.1669 log likelihood -1056.1669 Penbit regression Number of on LA chi 2 (D) Prob > chiz Paeudo R2 2,108 0.00 Log likelihood - -1050.1669 0000 WORKING Coef Std. Err. Z P>z! [95% Conf. Intervall _cons .375615 -0.310791 26.95 0.000 .7266425 .8944754 Regression 6 - Logit (WORKING on LEDU, ABIL and MARRIED only) Logit WORKING LEDU ADIL MARRIED if MALE--0 Iteration 0: Iteration 1: Iteration 2: Iteration 31 Iteration 1 : log likelihood = -1058.2669 log likelihood -994.60193 10g likellhood --981.87099 log likelihood - 981.86702 log likelihood - -981.86702 Logistic regression Number of cbs LR ch12 (2) Prob > ch12 Pseudo R2 2,108 -52.60 0. DOOD 0.0721 Log likelihood - -981.86702 WORKING Coef. Std. Err. P12 [959 Conf. Intervall LEDU ABIL MARRICS cons 2.256193 .2823305 -.1658482 -4.256176 .3097983 .0693063 . 1185693 .6045809 7.28 1.09 -4.00 -5.29 0.000 0.IDO 0.000 0.000 1.648998 .146569 -. 6943192 -5.833125 2.863387 .4176915 -.2373766 -2.679226 Summary Statistics sum WORKING LEDU ASIL CHILDREN MARRIED ETHBLACK ETHHISP MALE--D Variable Obs Mean Std. Dev. Min Max WORKING LEDO ABIL CHILDREN MARRIED 2,108 2,108 2,108 2,108 2,100 .7988615 2.64054 .0128978 3.462998 .420770 . 4009465 .2135372 .970222 1.858818 -4930011 D 1.791759 -2.912152 1 D 1 2.995732 2.640667 12 1 D ETHBLACK FITHHTSP 2,108 2,108 .1783681 . 1233397 .3829132 ..3289047 1 a. For each of regressions 1-3 above, write the underlying models. That is, express the dependent (latent) variable as a function of the regressors and the disturbance term. (1 mark) b. For each of regressions 1-3, write or derive the population regression function (PRF). That is, E(WORKINGX), where X = LEDU, ABIL etc. You may denote WORKING as Y and the regressors as X; if you want. (3 marks) c. Interpret the estimated slope coefficients (,) for each of regressions 1-3. Comment on their statistical significance. Any other comments on the estimated slope coefficients for these three regressions? (2 marks) d. Write expressions for the partial effect of education on the probability of working for each of regressions 1-3. Do a rough comparison of this partial effect across these three regressions by using the rule-of-thumb scale factors (the standard normal pdf at O is approximately 0.4, while the logistic pdf at 0 is approximately 0.25). Comment on your results. (3 marks) e. Repeat part (d) for the partial effect of ability on the probability of working. (1 mark) Average Partial Effects (APE) - Logit margina, dyrx (LED ABIL CHILDREN MARRIED ETHICK FTHATSP) Number of obs Average marginal effects Model VCE : DIM 2,108 Expression Er(WORKING), predicto dy/ax w.r.t. LEDU ABIL CHILDREN MARRIED ETHBLACK ETHHISP Delta-method dy/dx Std. Err. P> [95% Conf. Intervall LEDU ABIL CHILDREN MARRIED ETHBLACK ETOLISP .3337695 .0397649 -.0064049 -.071583 -.008882 .0276118 .0453811 .0109222 .0044020 .0175162 .0238036 .0260648 3.61 -1.45 -4.09 -0.3% 1.06 0.000 0.000 0.146 0.000 .2446241 .018 3577 -.0150342 -. 105914 -.0955 -.0231715 .4227149 .061172 .0022244 -.0372519 .0377714 .0787007 0.289 Partial Effects at the Average (PEA) - Logit .sargins, dydx (LEDU ABIL CHILDREN MARRIED ET BLACK ETHHISP) atmeans Number of obs 2,103 Conditional marginal effects Model VCE ! OIN Expression : Pr (WORKING), predict() dy/dxw.rit. LEDU ABIL CHILDREN MARRIED ETHBLACK ETHHISP at : LEDU 2.64054 mean ABIL .0128978 (nean) CHILDREN 3.462999 (en) MARRIED -420778 can ET BLACK .1783681 (nean ETHNTSP . 1233397 nean) Delta-mechod dy/dx Std. T. 7. P> Z (95% Cont. Interval LEDU ABIL CHILDREN MARRIED ETHBLACK THHIS -3337757 .0397656 - 006405 -.0715643 -.0088629 0276151 .9461592 .0109318 .004406 .0174073 .023038 -026079 7.23 3.64 -1.45 -4.09 0.000 0.000 0.146 0.000 0.209 0.290 .2433054 .0183397 -.0159405 -.1058538 -. 0555376 -.02.14943 .424246 .0611915 .0022305 -.0373090 .0377212 -0787296 Average Partial Effects (APE) - Probit .margins, dydx LEDU ABIL CHILDREN MARRIED ET BLACK ETAHISP) Number of oba Average marginal effects Model VCE 2.100 : DIM Expression : Pr (WORKING), predict() dy/dx w... LEDU ABIL CHILDREN MARRIED ETHBLACK ETHHISP P>21 [955 Conf. Intervall LEDU ABIL CHILDREN MARRIED ETHBLACK ETHHISE Delta method dy/dx std. Err. . 3382806 .0160984 .0386456 .0110151 -.0066256 .0044671 -.0706215 .0174593 -.0130523 -0238259 .0242611 .026031 7.34 3.52 -1.48 -4.04 -0.55 0.93 0.000 0.000 0.138 0.000 0.534 0.351 .2479254 .0171976 -.0153811 -.1046421 -.0597502 -.0267536 1286319 .0601936 .0021299 -.0364009 .0336456 .0752809 2,109 Partial Effects at the Average (PEA) - Probit margins, dydx (LEDU ABIS CHILDREN MARRIED ETHBLACK ETHH15) atmeans Conditional marginal effects Number of ube Model VCE : OIN Exprenninn : Pr(WORKING), predict dy/dx w.rit. LEDU ABIL CHILDREN MARRIED ETHBLACK ETHHISP at LEDU 2.64054 (mean) ABIL .C128978 (mean) CHILDREN 3.462990 (mean) MARRIED .420778 (mean) ETHBLACK .1783681 (mean) ETHHISP -1233397 (mean) 2 Delta-method dy/dx Std. Err. P> 21 (958 Conf. Intervall .3466644 .0483296 7.17 0.000 -2519399 .441389 ABIL .0398083 .011362 3.50 0.000 .0175393 .0620273 CHILDREN -.0067898 0045823 -1.48 0.138 -.015771 -0021914 MARRIED -.0723718 .0179502 -1.03 0.000 -.1075535 -.0371901 ETABLACK -.0233756 .0244231 -0.55 0.594 -.0612343 .0344620 ETHHISP .0248624 .0266884 0.93 0.352 -.0274459 .0771700 f. The tables above give the estimated Average Partial Effects (APE) and the Partial Effects at the Average (PEA) for both the logit and probit regressions. How do these compare with the rule-of-thumb rough estimates you calculated in parts (d) and (c) above? (2 marks) g. Using the likelihood ratio statistic, test the overall statistical significance of the probit model given in regression 3. (3 marks) h. Compute McFadden's measure of the pseudo R-squared for the logit regression (regression 2). (2 marks) i. Using the likelihood ratio statistic, test the null that the coefficients for CHILDREN, ETHBLACK and ETHHISP are jointly equal to zero in the logit regression. (3 marks) Question 3 (20 marks) The following regressions were run using 2,108 observations on females living in the US. The respondents in the sample are aged between 27 and 31, with many working and raising young families. WORKING = a binary variable equal to 1 if the respondent was working LEDU= natural logarithm of years of schooling of respondent ABIL = respondent's score on a cognitive ability test CHILDREN=number of children in the household MARRIED= a binary variable equal to 1 if the respondent was married ETHBLACK = a binary variable equal to 1 if the respondent was black ETHHISP = a binary variable equal to 1 if the respondent was Hispanic Regression 1 - Linear Probability Model reg WORKING LEDU ABIL CHILDREN MARRIED STHBLACK ETHHISP IT MALE--1) NS Model Residual 25.3798341 313.337433 6 2,101 1.22997231 .149137284 Number of o F16, 2101) Prob > R-squared Adj R-squared Root MSE 2,108 28.36 0.0050 0.0749 0.0723 . 38618 Total 338.717263 2,107 . 160759077 WORKTNCE Coat. Str. Fr. 1 P> 1954 Cant. Intervall -1.51 LEDU ABIG CHILDREN MARRIED ETHBLACK ETHNISE cons .3707367 .0407132 -.0071797 -.0650202 -.0080311 .0319012 -.1296264 .0487348 . 0112965 . 104 2015 .0175727 .0246139 .0267975 .1307303 0.000 D.GDU 1.12 0.000 0.745 0.234 0.322 -2751632 -0185597 -.01639948 --1024818 - .0563601 -.0208513 - 3860008 .4663103 .0628668 .0020204 -.0325586 .010258 .00 14536 .126740 -0.33 1.19 -0.99 THBLACK ETHHISE cons -.06DOODG . 2965293 -4.104034 . 1607939 .176193 .8351403 -0.37 1.06 -4.91 0.709 0.290 0.000 -.3751607 -.1588027 -5.740839 .2551594 .5318613 -2.467199 Regression 2 - Logit . Legit WORKING LEDU ABIL CHILDREN MARRIED ET BLACK ETAHISP iT MALE- Iteration : Iteration 1: Iteration 2: Iteration 3: Iteration 4 log 1. kelihood --1058.1669 log 1. kelihood --302.96666 log likelihood --980.10918 log 1. kelihood = -980.1038 log 1: kelihood = -980.1038 Logistic regresaian 2,108 Number of oba LR ch12 (6) Prob > chiz I'seudo R2 Log likelihood - 980.1038 WORKING Coef. Sid. Err. P12 (956 Conf. Intervall 7.12 3.61 -1.45 LEDU ABIL CHILCREN MARRIED ETHBLACK ETHHISP 2.254523 - 2686009 --0432632 --1835237 -.0600008 -1865293 -4.104034 .316596 .0743098 .0297784 1193119 .2607909 .176193 .8351403 0.000 0.000 0.146 0.000 0.709 0.290 0.000 1.634000 .1229564 .1016278 -.7173707 -.3751607 -. 1568027 -5.740979 2.875039 .4142454 .0151013 ..2496767 .2551594 .5316613 -2.463109 -0.37 1.06 -4.91 Regression 3 - Probit . probit WORKING LEDU ABIL CHILDREN MARRIED ET BLACK ETHHISP I HALE--0 Iteration 0: Iteration 1: Iteration 2: Iteration 3: Iteration 4 log likelihood - -1050.1669 log likelihood = -980.83399 Leg 11 kelihood --980.226799 Log likelihood - -980.2266 log likelihood - -980.2266 Probit regression 2,108 Number of obes IR ch12 (6) Probabil Iseudo R2 Log likelihood - 980.2266 WORKING Cuef. Std. Err. 21 21 (958 Cunt. Intervall TECC ABIL CHILDREN MARRIED ETHBLACK ETHHISP cons 1.312301 .1495463 -.025507 -.2718763 -.0502482 .0933996 2.310967 .1822144 .042753 .017219 .0677103 .0917524 -100238 .4842125 7.14 3.50 -1.48 -4.02 -0.55 0.000 0.000 0.135 0.000 0.584 9447754 .065752 -.0592557 -.404588 -.2300796 -.70301632 -3.290026 1.655827 .2333406 .0082417 -.1391665 - 1295832 - 2898625 -1.391948 -1.83 0.000 Regression 4 - Logit (Intercept only) Logit WORKING 1 MALEO Iteration : Iteration 11 Log likelihood - - 1038.1669 Log likelihood - 1058.1669 Logistic regression 2,108 0.00 Number of obs LR chi2 (0) Prob > chi2 Pseudu 12 Log likelihoud - -1058.1669 0.0000 WORKING Coef. Std. Er 2 P (95% Conf. Intervall con 1.379194 -0543352 25.38 0.000 1.222699 1.453689 Regression 5 - Probit (Intercept only) probit WORKING I MALEO Ileration 0: Iteration 1: loy likelihood - -1056.1669 log likelihood -1056.1669 Penbit regression Number of on LA chi 2 (D) Prob > chiz Paeudo R2 2,108 0.00 Log likelihood - -1050.1669 0000 WORKING Coef Std. Err. Z P>z! [95% Conf. Intervall _cons .375615 -0.310791 26.95 0.000 .7266425 .8944754 Regression 6 - Logit (WORKING on LEDU, ABIL and MARRIED only) Logit WORKING LEDU ADIL MARRIED if MALE--0 Iteration 0: Iteration 1: Iteration 2: Iteration 31 Iteration 1 : log likelihood = -1058.2669 log likelihood -994.60193 10g likellhood --981.87099 log likelihood - 981.86702 log likelihood - -981.86702 Logistic regression Number of cbs LR ch12 (2) Prob > ch12 Pseudo R2 2,108 -52.60 0. DOOD 0.0721 Log likelihood - -981.86702 WORKING Coef. Std. Err. P12 [959 Conf. Intervall LEDU ABIL MARRICS cons 2.256193 .2823305 -.1658482 -4.256176 .3097983 .0693063 . 1185693 .6045809 7.28 1.09 -4.00 -5.29 0.000 0.IDO 0.000 0.000 1.648998 .146569 -. 6943192 -5.833125 2.863387 .4176915 -.2373766 -2.679226 Summary Statistics sum WORKING LEDU ASIL CHILDREN MARRIED ETHBLACK ETHHISP MALE--D Variable Obs Mean Std. Dev. Min Max WORKING LEDO ABIL CHILDREN MARRIED 2,108 2,108 2,108 2,108 2,100 .7988615 2.64054 .0128978 3.462998 .420770 . 4009465 .2135372 .970222 1.858818 -4930011 D 1.791759 -2.912152 1 D 1 2.995732 2.640667 12 1 D ETHBLACK FITHHTSP 2,108 2,108 .1783681 . 1233397 .3829132 ..3289047 1 a. For each of regressions 1-3 above, write the underlying models. That is, express the dependent (latent) variable as a function of the regressors and the disturbance term. (1 mark) b. For each of regressions 1-3, write or derive the population regression function (PRF). That is, E(WORKINGX), where X = LEDU, ABIL etc. You may denote WORKING as Y and the regressors as X; if you want. (3 marks) c. Interpret the estimated slope coefficients (,) for each of regressions 1-3. Comment on their statistical significance. Any other comments on the estimated slope coefficients for these three regressions? (2 marks) d. Write expressions for the partial effect of education on the probability of working for each of regressions 1-3. Do a rough comparison of this partial effect across these three regressions by using the rule-of-thumb scale factors (the standard normal pdf at O is approximately 0.4, while the logistic pdf at 0 is approximately 0.25). Comment on your results. (3 marks) e. Repeat part (d) for the partial effect of ability on the probability of working. (1 mark) Average Partial Effects (APE) - Logit margina, dyrx (LED ABIL CHILDREN MARRIED ETHICK FTHATSP) Number of obs Average marginal effects Model VCE : DIM 2,108 Expression Er(WORKING), predicto dy/ax w.r.t. LEDU ABIL CHILDREN MARRIED ETHBLACK ETHHISP Delta-method dy/dx Std. Err. P> [95% Conf. Intervall LEDU ABIL CHILDREN MARRIED ETHBLACK ETOLISP .3337695 .0397649 -.0064049 -.071583 -.008882 .0276118 .0453811 .0109222 .0044020 .0175162 .0238036 .0260648 3.61 -1.45 -4.09 -0.3% 1.06 0.000 0.000 0.146 0.000 .2446241 .018 3577 -.0150342 -. 105914 -.0955 -.0231715 .4227149 .061172 .0022244 -.0372519 .0377714 .0787007 0.289 Partial Effects at the Average (PEA) - Logit .sargins, dydx (LEDU ABIL CHILDREN MARRIED ET BLACK ETHHISP) atmeans Number of obs 2,103 Conditional marginal effects Model VCE ! OIN Expression : Pr (WORKING), predict() dy/dxw.rit. LEDU ABIL CHILDREN MARRIED ETHBLACK ETHHISP at : LEDU 2.64054 mean ABIL .0128978 (nean) CHILDREN 3.462999 (en) MARRIED -420778 can ET BLACK .1783681 (nean ETHNTSP . 1233397 nean) Delta-mechod dy/dx Std. T. 7. P> Z (95% Cont. Interval LEDU ABIL CHILDREN MARRIED ETHBLACK THHIS -3337757 .0397656 - 006405 -.0715643 -.0088629 0276151 .9461592 .0109318 .004406 .0174073 .023038 -026079 7.23 3.64 -1.45 -4.09 0.000 0.000 0.146 0.000 0.209 0.290 .2433054 .0183397 -.0159405 -.1058538 -. 0555376 -.02.14943 .424246 .0611915 .0022305 -.0373090 .0377212 -0787296 Average Partial Effects (APE) - Probit .margins, dydx LEDU ABIL CHILDREN MARRIED ET BLACK ETAHISP) Number of oba Average marginal effects Model VCE 2.100 : DIM Expression : Pr (WORKING), predict() dy/dx w... LEDU ABIL CHILDREN MARRIED ETHBLACK ETHHISP P>21 [955 Conf. Intervall LEDU ABIL CHILDREN MARRIED ETHBLACK ETHHISE Delta method dy/dx std. Err. . 3382806 .0160984 .0386456 .0110151 -.0066256 .0044671 -.0706215 .0174593 -.0130523 -0238259 .0242611 .026031 7.34 3.52 -1.48 -4.04 -0.55 0.93 0.000 0.000 0.138 0.000 0.534 0.351 .2479254 .0171976 -.0153811 -.1046421 -.0597502 -.0267536 1286319 .0601936 .0021299 -.0364009 .0336456 .0752809 2,109 Partial Effects at the Average (PEA) - Probit margins, dydx (LEDU ABIS CHILDREN MARRIED ETHBLACK ETHH15) atmeans Conditional marginal effects Number of ube Model VCE : OIN Exprenninn : Pr(WORKING), predict dy/dx w.rit. LEDU ABIL CHILDREN MARRIED ETHBLACK ETHHISP at LEDU 2.64054 (mean) ABIL .C128978 (mean) CHILDREN 3.462990 (mean) MARRIED .420778 (mean) ETHBLACK .1783681 (mean) ETHHISP -1233397 (mean) 2 Delta-method dy/dx Std. Err. P> 21 (958 Conf. Intervall .3466644 .0483296 7.17 0.000 -2519399 .441389 ABIL .0398083 .011362 3.50 0.000 .0175393 .0620273 CHILDREN -.0067898 0045823 -1.48 0.138 -.015771 -0021914 MARRIED -.0723718 .0179502 -1.03 0.000 -.1075535 -.0371901 ETABLACK -.0233756 .0244231 -0.55 0.594 -.0612343 .0344620 ETHHISP .0248624 .0266884 0.93 0.352 -.0274459 .0771700 f. The tables above give the estimated Average Partial Effects (APE) and the Partial Effects at the Average (PEA) for both the logit and probit regressions. How do these compare with the rule-of-thumb rough estimates you calculated in parts (d) and (c) above? (2 marks) g. Using the likelihood ratio statistic, test the overall statistical significance of the probit model given in regression 3. (3 marks) h. Compute McFadden's measure of the pseudo R-squared for the logit regression (regression 2). (2 marks) i. Using the likelihood ratio statistic, test the null that the coefficients for CHILDREN, ETHBLACK and ETHHISP are jointly equal to zero in the logit regression