Springer Texts in Business and Economics
Limited Dependent Variables
|
13.1 The Linear Probability Model
|
a. Let к і = Pr[y; = 1], then y; = 1 when u; = 1 — x0" with probability к; as shown in the table above. Similarly, y; = 0 when u; = —x0" with probability 1 — к ;. Hence, E(u;) = к; (1 — x[") + (1 — к ;) (—x0").
For this to equal zero, we get, к; — к ;xi" + к ;xi" — x0" = 0 which gives к ; = xi" as required.
b. var(u;) = E(u2) = (1 — xi")2 к ; + (—x0")2 (1 — к ;)
1 — 2x0" + (x0")2 к; + (xi")2 (1 — к i)
= к ; — 2x0" к ; + (x0")2 = к ; — к 2 = к ;(1 — к ;) = x0" (1 — xi") using the fact that к ; = xi".
13.2 a. Since there are no slopes and only a constant, x0" = a and (13.16) becomes
n
log ' = J]{y; logF(a) + (1 — y;) log[1 — F(a)]} differentiating with respect
i=1
to a we get
9log' y; л (1 — y;) (
= £ щ •f(a)+£г—щ (-f(a».
n
Setting this equal to zero yields J2 (yi — F(a))f(a) = 0.
i=1
n
Therefore, F(a) = J2 Уі/п = y. This is the proportion of the sample with
i=1
Уі = 1
B. H. Baltagi, Solutions Manual for Econometrics, Springer Texts in Business and Economics, DOI 10.1007/978-3-642-54548-1_13, © Springer-Verlag Berlin Heidelberg 2015
b. Using F(a) = y, the value of the maximized likelihood, from (13.16), is
n
log'r =2>logy C (1 Уі)log(l-y)} = nylogy C (n—ny)log(l-y)
i=i
= n[y log y C (1 — y) log(1 — y)] as required.
c. For the empirical example in Sect. 13.9, we know that y = 218/595 = 0.366. Substituting in (13.33) we get, log'r = n[0.366 log0.366 C (1 — 0.366) log(1 — 0.366)] = —390.918.
13.3 Union participation example. See Tables 13.3-13.5. These were run using EViews.
a. OLS ESTIMATION
LS // Dependent Variable is UNION
Sample: 1 595
Included observations: 595
|
Variable |
Coefficient |
Std. Error |
t-Statistic |
Prob. |
|
|
C |
1.195872 |
0.227010 |
5.267922 |
0.0000 |
|
|
EX |
-0.001974 |
0.001726 |
-1.143270 |
0.2534 |
|
|
WKS |
-0.017809 |
0.003419 |
-5.209226 |
0.0000 |
|
|
OCC |
0.318118 |
0.046425 |
6.852287 |
0.0000 |
|
|
IND |
0.030048 |
0.038072 |
0.789229 |
0.4303 |
|
|
SOUTH |
-0.170130 |
0.039801 |
-4.274471 |
0.0000 |
|
|
SMSA |
0.084522 |
0.038464 |
2.197419 |
0.0284 |
|
|
MS |
0.098953 |
0.063781 |
1.551453 |
0.1213 |
|
|
FEM |
-0.108706 |
0.079266 |
-1.371398 |
0.1708 |
|
|
ED |
-0.016187 |
0.008592 |
-1.883924 |
0.0601 |
|
|
BLK |
0.050197 |
0.071130 |
0.705708 |
0.4807 |
|
|
R-squared |
0.233548 |
Mean dependent var |
0.366387 |
||
|
Adjusted R-squared |
0.220424 |
S. D. dependent var |
0.482222 |
||
|
S. E. of regression |
0.425771 |
Akaike info criterion |
-1.689391 |
||
|
Sum squared resid |
105.8682 |
Schwarz criterion |
-1.608258 |
||
|
Log likelihood |
-330.6745 |
F-statistic |
17.79528 |
||
|
Durbin-Watson stat |
1.900963 |
Prob(F-statistic) |
0.000000 |
LOGIT ESTIMATION
LOGIT // Dependent Variable is UNION
Sample: 1 595
Included observations: 595
Convergence achieved after 4 iterations
|
Variable |
Coefficient |
Std. Error |
t-Statistic |
Prob. |
|
C |
4.380828 |
1.338629 |
3.272624 |
0.0011 |
|
EX |
-0.011143 |
0.009691 |
-1.149750 |
0.2507 |
|
WKS |
-0.108126 |
0.021428 |
-5.046037 |
0.0000 |
|
OCC |
1.658222 |
0.264456 |
6.270325 |
0.0000 |
|
IND |
0.181818 |
0.205470 |
0.884888 |
0.3766 |
|
SOUTH |
-1.044332 |
0.241107 |
-4.331411 |
0.0000 |
|
SMSA |
0.448389 |
0.218289 |
2.054110 |
0.0404 |
|
MS |
0.604999 |
0.365043 |
1.657336 |
0.0980 |
|
FEM |
-0.772222 |
0.489665 |
-1.577040 |
0.1153 |
|
ED |
-0.090799 |
0.049227 |
-1.844501 |
0.0656 |
|
BLK |
0.355706 |
0.394794 |
0.900992 |
0.3680 |
|
Log likelihood |
-312.3367 |
|||
|
Obs with Dep=1 |
218 |
|||
|
Obs with Dep=0 |
377 |
|||
|
Variable |
Mean All |
Mean D=1 |
Mean D=0 |
|
|
C |
1.000000 |
1.000000 |
1.000000 |
|
|
EX |
22.85378 |
23.83028 |
22.28912 |
|
|
WKS |
46.45210 |
45.27982 |
47.12997 |
|
|
OCC |
0.512605 |
0.766055 |
0.366048 |
|
|
IND |
0.405042 |
0.513761 |
0.342175 |
|
|
SOUTH |
0.292437 |
0.197248 |
0.347480 |
|
|
SMSA |
0.642017 |
0.646789 |
0.639257 |
|
|
MS |
0.805042 |
0.866972 |
0.769231 |
|
|
FEM |
0.112605 |
0.059633 |
0.143236 |
|
|
ED |
12.84538 |
11.84862 |
13.42175 |
|
|
BLK |
0.072269 |
0.082569 |
0.066313 |
PROBIT ESTIMATION
PROBIT // Dependent Variable is UNION
Sample: 1 595
Included observations: 595
Convergence achieved after 3 iterations
|
Variable |
Coefficient |
Std. Error |
t-Statistic |
Prob. |
|
C |
2.516784 |
0.762606 |
3.300242 |
0.0010 |
|
EX |
-0.006932 |
0.005745 |
-1.206501 |
0.2281 |
|
WKS |
-0.060829 |
0.011785 |
-5.161707 |
0.0000 |
|
OCC |
0.955490 |
0.152136 |
6.280522 |
0.0000 |
|
IND |
0.092827 |
0.122773 |
0.756089 |
0.4499 |
|
SOUTH |
-0.592739 |
0.139100 |
-4.261243 |
0.0000 |
|
SMSA |
0.260701 |
0.128629 |
2.026756 |
0.0431 |
|
MS |
0.350520 |
0.216282 |
1.620664 |
0.1056 |
|
FEM |
-0.407026 |
0.277034 |
-1.469226 |
0.1423 |
|
ED |
-0.057382 |
0.028842 |
-1.989533 |
0.0471 |
|
BLK |
0.226482 |
0.228843 |
0.989683 |
0.3227 |
Log likelihood -313.3795 ObswithDep=1 218 Obs with Dep=0 377
d. Dropping the industry variable (IND).
OLS ESTIMATION
LS // Dependent Variable is UNION
Sample: 1 595
Included observations: 595
|
Variable |
Coefficient |
Std. Error |
t-Statistic |
Prob. |
|
C |
1.216753 |
0.225390 |
5.398425 |
0.0000 |
|
EX |
-0.001848 |
0.001718 |
-1.075209 |
0.2827 |
|
WKS |
-0.017874 |
0.003417 |
-5.231558 |
0.0000 |
|
OCC |
0.322215 |
0.046119 |
6.986568 |
0.0000 |
|
SOUTH |
-0.173339 |
0.039580 |
-4.379418 |
0.0000 |
|
SMSA |
0.085043 |
0.038446 |
2.212014 |
0.0274 |
|
MS |
0.100697 |
0.063722 |
1.580267 |
0.1146 |
|
FEM |
-0.114088 |
0.078947 |
-1.445122 |
0.1490 |
|
ED |
-0.017021 |
0.008524 |
-1.996684 |
0.0463 |
|
BLK |
0.048167 |
0.071061 |
0.677822 |
0.4982 |
|
R-squared |
0.232731 |
Mean dependent var |
0.366387 |
|
Adjusted R-squared |
0.220927 |
S. D. dependent var |
0.482222 |
|
S. E. of regression |
0.425634 |
Akaike info criterion |
-1.691687 |
|
Sum squared resid |
105.9811 |
Schwarz criterion |
-1.617929 |
|
Log likelihood |
-330.9916 |
F-statistic |
19.71604 |
|
Durbin-Watson stat |
1.907714 |
Prob(F-statistic) |
0.000000 |
LOGIT ESTIMATION
LOGIT // Dependent Variable is UNION
Sample: 1 595
Included observations: 595
Convergence achieved after 4 iterations
|
Variable |
Coefficient |
Std. Error |
t-Statistic |
Prob. |
|
C |
4.492957 |
1.333992 |
3.368053 |
0.0008 |
|
EX |
-0.010454 |
0.009649 |
-1.083430 |
0.2791 |
|
WKS |
-0.107912 |
0.021380 |
-5.047345 |
0.0000 |
|
OCC |
1.675169 |
0.263654 |
6.353652 |
0.0000 |
|
SOUTH |
-1.058953 |
0.240224 |
-4.408193 |
0.0000 |
|
SMSA |
0.449003 |
0.217955 |
2.060074 |
0.0398 |
|
MS |
0.618511 |
0.365637 |
1.691599 |
0.0913 |
|
FEM |
-0.795607 |
0.489820 |
-1.624285 |
0.1049 |
|
ED |
-0.096695 |
0.048806 |
-1.981194 |
0.0480 |
|
BLK |
0.339984 |
0.394027 |
0.862845 |
0.3886 |
|
Log likelihood |
-312.7267 |
|||
|
Obs with Dep=1 |
218 |
|||
|
Obs with Dep=0 |
377 |
|||
|
Variable |
Mean All |
Mean D=1 |
Mean D=0 |
|
|
C |
1.000000 |
1.000000 |
1.000000 |
|
|
EX |
22.85378 |
23.83028 |
22.28912 |
|
|
WKS |
46.45210 |
45.27982 |
47.12997 |
|
|
OCC |
0.512605 |
0.766055 |
0.366048 |
|
|
SOUTH |
0.292437 |
0.197248 |
0.347480 |
|
|
SMSA |
0.642017 |
0.646789 |
0.639257 |
|
|
MS |
0.805042 |
0.866972 |
0.769231 |
|
|
FEM |
0.112605 |
0.059633 |
0.143236 |
|
|
ED |
12.84538 |
11.84862 |
13.42175 |
|
|
BLK |
0.072269 |
0.082569 |
0.066313 |
PROBIT ESTIMATION
PROBIT // Dependent Variable is UNION
Sample: 1 595
Included observations: 595
Convergence achieved after 3 iterations
|
Variable |
Coefficient |
Std. Error |
t-Statistic |
Prob. |
|
C |
2.570491 |
0.759181 |
3.385875 |
0.0008 |
|
EX |
-0.006590 |
0.005723 |
-1.151333 |
0.2501 |
|
WKS |
-0.060795 |
0.011777 |
-5.162354 |
0.0000 |
|
OCC |
0.967972 |
0.151305 |
6.397481 |
0.0000 |
|
SOUTH |
-0.601050 |
0.138528 |
-4.338836 |
0.0000 |
|
SMSA |
0.261381 |
0.128465 |
2.034640 |
0.0423 |
|
MS |
0.357808 |
0.216057 |
1.656085 |
0.0982 |
|
FEM |
-0.417974 |
0.276501 |
-1.511657 |
0.1312 |
|
ED |
-0.060082 |
0.028625 |
-2.098957 |
0.0362 |
|
BLK |
0.220695 |
0.228363 |
0.966423 |
0.3342 |
Log likelihood -313.6647 ObswithDep = 1 218 Obs with Dep = 0 377
f. The restricted regressions omitting IND, FEM and BLK are given below:
LS // Dependent Variable is UNION
Sample: 1 595
Included observations: 595
|
Variable |
Coefficient |
Std. Error |
t-Statistic |
Prob. |
|
C |
1.153900 |
0.218771 |
5.274452 |
0.0000 |
|
EX |
-0.001840 |
0.001717 |
-1.071655 |
0.2843 |
|
WKS |
-0.017744 |
0.003412 |
-5.200421 |
0.0000 |
|
OCC |
0.326411 |
0.046051 |
7.088110 |
0.0000 |
|
SOUTH |
-0.171713 |
0.039295 |
-4.369868 |
0.0000 |
|
SMSA |
0.086076 |
0.038013 |
2.264382 |
0.0239 |
|
MS |
0.158303 |
0.045433 |
3.484351 |
0.0005 |
|
ED |
-0.017204 |
0.008507 |
-2.022449 |
0.0436 |
|
R-squared |
0.229543 |
Mean dependent var |
0.366387 |
|
|
Adjusted R-squared |
0.220355 |
S. D. dependent var |
0.482222 |
|
|
S. E. of regression |
0.425790 |
Akaike info criterion |
-1.694263 |
|
|
Sum squared resid |
106.4215 |
Schwarz criterion |
-1.635257 |
|
|
Log likelihood |
-332.2252 |
F-statistic |
24.98361 |
|
|
Durbin-Watson stat |
1.912059 |
Prob(F-statistic) |
0.000000 |
LOGIT// Dependent Variable is UNION Sample: 1 595 Included observations: 595 Convergence achieved after 4 iterations
|
Variable |
Coefficient |
Std. Error |
t-Statistic |
Prob. |
|
C |
4.152595 |
1.288390 |
3.223088 |
0.0013 |
|
EX |
-0.011018 |
0.009641 |
-1.142863 |
0.2536 |
|
WKS |
-0.107116 |
0.021215 |
-5.049031 |
0.0000 |
|
OCC |
1.684082 |
0.262193 |
6.423059 |
0.0000 |
|
SOUTH |
-1.043629 |
0.237769 |
-4.389255 |
0.0000 |
|
SMSA |
0.459707 |
0.215149 |
2.136687 |
0.0330 |
|
MS |
0.975711 |
0.272560 |
3.579800 |
0.0004 |
|
ED |
-0.100033 |
0.048507 |
-2.062229 |
0.0396 |
|
Log likelihood |
-314.2744 |
|||
|
Obs with Dep=1 |
218 |
|||
|
Obs with Dep=0 |
377 |
|||
|
Variable |
Mean All |
Mean D= |
1 Mean D= |
0 |
|
C |
1.000000 |
1.000000 |
1.000000 |
|
|
EX |
22.85378 |
23.83028 |
22.28912 |
|
|
WKS |
46.45210 |
45.27982 |
47.12997 |
|
|
OCC |
0.512605 |
0.766055 |
0.366048 |
|
|
SOUTH |
0.292437 |
0.197248 |
0.347480 |
|
|
SMSA |
0.642017 |
0.646789 |
0.639257 |
|
|
MS |
0.805042 |
0.866972 |
0.769231 |
|
|
ED |
12.84538 |
11.84862 |
13.42175 |
|
|
PROBIT // Dependent Variable is UNION |
||||
|
Sample: 1 595 |
||||
|
Included observations: 595 |
||||
|
Convergence achieved after 3 iterations |
||||
|
Variable |
Coefficient |
Std. Error |
t-Statistic |
Prob. |
|
C |
2.411706 |
0.741327 |
3.253228 |
0.0012 |
|
EX |
-0.006986 |
0.005715 |
-1.222444 |
0.2220 |
|
WKS |
-0.060491 |
0.011788 |
-5.131568 |
0.0000 |
|
OCC |
0.971984 |
0.150538 |
6.456745 |
0.0000 |
|
SOUTH |
-0.580959 |
0.136344 |
-4.260988 |
0.0000 |
|
SMSA |
0.273201 |
0.126988 |
2.151388 |
0.0319 |
|
MS |
0.545824 |
0.155812 |
3.503105 |
0.0005 |
|
ED |
-0.063196 |
0.028464 |
-2.220210 |
0.0268 |
|
Log likelihood |
-315.1770 |
|||
|
Obs with Dep=1 |
218 |
|||
|
Obs with Dep=0 |
377 |
13.4 Occupation regression.
a. OLS Estimation
LS // Dependent Variable is OCC
Sample: 1 595
Included observations: 595
|
Variable |
Coefficient |
Std. Error |
t-Statistic |
Prob. |
|
C |
2.111943 |
0.182340 |
11.58245 |
0.0000 |
|
ED |
-0.111499 |
0.006108 |
-18.25569 |
0.0000 |
|
WKS |
-0.001510 |
0.003044 |
-0.496158 |
0.6200 |
|
EX |
-0.002870 |
0.001533 |
-1.872517 |
0.0616 |
|
SOUTH |
-0.068631 |
0.035332 |
-1.942452 |
0.0526 |
|
SMSA |
-0.079735 |
0.034096 |
-2.338528 |
0.0197 |
|
IND |
0.091688 |
0.033693 |
2.721240 |
0.0067 |
|
MS |
0.006271 |
0.056801 |
0.110402 |
0.9121 |
|
FEM |
-0.064045 |
0.070543 |
-0.907893 |
0.3643 |
|
BLK |
0.068514 |
0.063283 |
1.082647 |
0.2794 |
|
R-squared |
0.434196 Mean dependent var |
0.512605 |
||
|
Adjusted R-squared |
0.425491 S. D. dependent var |
0.500262 |
||
|
S. E. of regression |
0.379180 Akaike info criterion |
-1.922824 |
||
|
Sum squared resid |
84.10987 Schwarz criterion |
-1.849067 |
||
|
Log likelihood |
-262.2283 F-statistic |
49.88075 |
||
|
Durbin-Watson stat |
1.876105 Prob(F-statistic) |
0.000000 |
LOGIT ESTIMATION
LOGIT // Dependent Variable is OCC
Sample: 1 595
Included observations: 595
Convergence achieved after 5 iterations
|
Variable |
Coefficient |
Std. Error |
t-Statistic |
Prob. |
|
C |
11.62962 |
1.581601 |
7.353069 |
0.0000 |
|
ED |
-0.806320 |
0.070068 |
-11.50773 |
0.0000 |
|
WKS |
-0.008424 |
0.023511 |
-0.358297 |
0.7203 |
|
EX |
-0.017610 |
0.011161 |
-1.577893 |
0.1151 |
|
SOUTH |
-0.349960 |
0.260761 |
-1.342073 |
0.1801 |
|
SMSA |
-0.601945 |
0.247206 |
-2.434995 |
0.0152 |
|
IND |
0.689620 |
0.241028 |
2.861157 |
0.0044 |
PROBIT ESTIMATION
PROBIT // Dependent Variable is OCC
Sample: 1 595
Included observations: 595
Convergence achieved after 4 iterations
|
Variable |
Coefficient |
Std. Error |
t-Statistic |
Prob. |
|
C |
6.416131 |
0.847427 |
7.571312 |
0.0000 |
|
ED |
-0.446740 |
0.034458 |
-12.96473 |
0.0000 |
|
WKS |
-0.003574 |
0.013258 |
-0.269581 |
0.7876 |
|
EX |
-0.010891 |
0.006336 |
-1.718878 |
0.0862 |
|
SOUTH |
-0.240756 |
0.147920 |
-1.627608 |
0.1041 |
|
SMSA |
-0.327948 |
0.139849 |
-2.345016 |
0.0194 |
|
IND |
0.371434 |
0.135825 |
2.734658 |
0.0064 |
|
MS |
-0.097665 |
0.245069 |
-0.398522 |
0.6904 |
|
FEM |
-0.358948 |
0.296971 |
-1.208697 |
0.2273 |
|
BLK |
0.215257 |
0.252219 |
0.853453 |
0.3938 |
Log likelihood -246.6581 Obs with Dep=1 305 Obs with Dep=0 290
13.5 Truncated Uniform Density.
|
1 ■ |
Ї1 1 |
1 |
"3" |
|
|
Pr |
x > — |
= - dx = |
||
|
2 |
У-1/2 2 |
2 |
_2_ |
|
= - .So that 4 |
|
f x/x > -- = |
|
f(x) |
1/2 2 1
і — Г, = —— = - tor----------------- < x <1.
2) Pr[x > - i] 3/4 3 2
var(x) = E(x2) - (E(x))2 = E(x2) = 3
E(x2/x >- д=/-1/2x2 - 2 - dx=2 - 3[x3]L1/2=2
Therefore, as expected, truncation reduces the variance.
13.6 Truncated Normal Density.
a. From the Appendix, Eq. (A.1), using c = 1, p = 1, ct2 = 1 and Ф(0) = 2, we get, f(x/x >1) = ¥-x-(0}) = 2¥(x - 1) for x >1
Similarly, using Eq. (A.2), for c = 1, p = 1 and ct2 = 1 with Ф(0) = 3 we getf(x/x < 1) = ¥ф(0)1) = 2¥(x - 1) forx < 1
b. The conditional mean is given in (A.3) and for this example we get
with c* = = = 0. Similarly, using (A.4) we get,
о 1
¥(c*) ф(0) 2
E(x/x < 1) = 1 - 1 • ) = 1 - = 1 - 2ф(0) = 1 -
c. From (A.5) we get, var(x/x >1) = 1(1 - 8(c*)) = 1 - 8(0) where
2 4 2
= 2ф(0)[2ф(0)] = 4ф2(0) = = = 0.64 for x >1
2
From (A.6), we get var(x/x >1) = 1 - 8(0) where
Both conditional truncated variances are less than the unconditional var(x) = 1 .
13.7 Censored Normal Distribution.
a. From the Appendix we get,
E(y) = Pr[y = c] E(y/y = c) C Pr[y > c] E(y/y > c)
= cФ(c*) C (1 - Ф(c*))E(y*/y* > c)
Ф(c*)
1 - Ф(c*)_
where E(y*/y* > c) is obtained from the mean of a truncated normal density, see (A.3).
b. Using the result on conditional variance given in Chap. 2 we get, var(y) = E(conditional variance) C var(conditional mean). But
E(conditional variance) = P[y = c] var(y/y = c)CP[y > c] var(y/y > c)
= Ф(е*) • 0 + (1 - Ф(о*Х)ст2(1 - 8(c*)) where var(y/y > c) is given by (A.5).
var(conditional mean) = P[y = c] • (c — E(y))2 + Pr(y > c)[E(y/y>c)-E(y)]2 = Ф(c*)(c — E(y))2+[1 — Ф(^)][E(y/y > c) — E(y)]2
where E(y) is given by (A.7) and E(y/y > c) is given by (A.3). This gives
var(conditional mean) = Ф(^) fc — cФ(c*) — (1 — Ф^*))
as required. Similarly, from part (b), using c* = —fi/o and Ф(— fi/o) =
13.8 Fixed vs. adjustable mortgage rates. This is based on Dhillon et al. (1987).
a. The OLS regression of Y on all variables in the data set is given below. This was done using EViews. The R2 = 0.434 and the F-statistic for the significance of all slopes is equal to 3.169. This is distributed as F(15,62) under the null hypothesis. This has a p-value of 0.0007. Therefore, we reject Ho and we conclude that this is a significant regression. As explained in Sect. 13.6, using BRMR this also rejects the insignificance of all slopes in the logit specification.
Unrestricted Least Squares
LS // Dependent Variable is Y
Sample: 1 78
Included observations: 78
|
Variable |
Coefficient |
Std. Error |
t-Statistic |
Prob. |
|
C |
1.272832 |
1.411806 |
0.901563 |
0.3708 |
|
BA |
0.000398 |
0.007307 |
0.054431 |
0.9568 |
|
BS |
0.017084 |
0.020365 |
0.838887 |
0.4048 |
|
NW |
-0.036932 |
0.025320 |
-1.458609 |
0.1497 |
|
FI |
-0.221726 |
0.092813 |
-2.388949 |
0.0200 |
|
PTS |
0.178963 |
0.091050 |
1.965544 |
0.0538 |
|
MAT |
0.214264 |
0.202497 |
1.058108 |
0.2941 |
|
MOB |
0.020963 |
0.009194 |
2.279984 |
0.0261 |
|
MC |
0.189973 |
0.150816 |
1.259635 |
0.2125 |
|
FTB |
-0.013857 |
0.136127 |
-0.101797 |
0.9192 |
|
SE |
0.188284 |
0.360196 |
0.522728 |
0.6030 |
|
YLD |
0.656227 |
0.366117 |
1.792399 |
0.0779 |
|
MARG |
0.129127 |
0.054840 |
2.354621 |
0.0217 |
|
CB |
0.172202 |
0.137827 |
1.249403 |
0.2162 |
|
STL |
-0.001599 |
0.005994 |
-0.266823 |
0.7905 |
|
LA |
-0.001761 |
0.007801 |
-0.225725 |
0.8222 |
|
R-squared |
0.433996 |
Mean dependent var |
0.589744 |
|
Adjusted R-squared |
0.297059 |
S. D. dependent var |
0.495064 |
|
S. E. of regression |
0.415069 |
Akaike info criter |
-1.577938 |
|
Sum squared resid |
10.68152 |
Schwarz criterion |
-1.094510 |
|
Log likelihood |
-33.13764 |
F-statistic |
3.169321 |
|
Durbin-Watson stat |
0.905968 |
Prob(F-statistic) |
0.000702 |
|
Plot of Y and YHAT |
b. The URSS from part (a) is 10.6815 while the RRSS by including only the cost variables is 14.0180 as shown in the enclosed output from EViews. The Chow-F statistic for insignificance of 10 personal characteristics variables is
F= (14.0180 - 10.6815)/10
10.6815/62 '
which is distributed as F(10,62) under the null hypothesis. This has a 5% critical value of 1.99. Hence, we cannot reject Ho. The principal agent theory suggests that personal characteristics are important in making this mortgage choice. Briefly, this theory suggests that information is asymmetric and the borrower knows things about himself or herself that the lending institution does not. Not rejecting Ho does not provide support for the principal agent theory.
TESTING THE EFFICIENT MARKET HYPOTHESIS WITH THE LINEAR PROBABILITY MODEL
Restricted Least Squares
LS // Dependent Variable is Y
Sample: 1 78
Included observations: 78
|
Variable |
Coefficient |
Std. Error |
t-Statistic |
Prob. |
|
FI |
-0.237228 |
0.078592 |
-3.018479 |
0.0035 |
|
MARG |
0.127029 |
0.051496 |
2.466784 |
0.0160 |
|
YLD |
0.889908 |
0.332037 |
2.680151 |
0.0091 |
|
PTS |
0.054879 |
0.072165 |
0.760465 |
0.4495 |
|
MAT |
0.069466 |
0.196727 |
0.353108 |
0.7250 |
|
C |
1.856435 |
1.289797 |
1.439324 |
0.1544 |
|
R-squared |
0.257199 |
Mean dependent var |
0.589744 |
|
Adjusted R-squared |
0.205616 |
S. D. dependent var |
0.495064 |
|
S. E. of regression |
0.441242 |
Akaike info criter |
-1.562522 |
|
Sum squared resid |
14.01798 |
Schwarz criterion |
-1.381236 |
|
Log likelihood |
-43.73886 |
F-statistic |
4.986087 |
|
Durbin-Watson stat |
0.509361 |
Prob(F-statistic) |
0.000562 |
c. The logit specification output using EViews is given below. The unrestricted log-likelihood is equal to —30.8963. The restricted specification output is also given showing a restricted log-likelihood of —41.4729. Therefore, the LR test statistic is given by LR = 2(41.4729 — 30.8963/ = 21.1532 which is distributed as x20 under the null hypothesis. This is significant given that the 5% critical value of x20 is 18.31. This means that the logit specification does not reject the principal agent theory as personal characteristics are not jointly insignificant.
TESTING THE EFFICIENT MARKET HYPOTHESIS WITH THE LOGIT MODEL Unrestricted Logit Model
LOGIT // Dependent Variable is Y Sample: 1 78 Included observations: 78 Convergence achieved after 5 iterations
|
Variable |
Coefficient |
Std. Error |
t-Statistic |
Prob. |
|
C |
4.238872 |
10.47875 |
0.404521 |
0.6872 |
|
BA |
0.010478 |
0.075692 |
0.138425 |
0.8904 |
|
BS |
0.198251 |
0.172444 |
1.149658 |
0.2547 |
|
NW |
-0.244064 |
0.185027 |
-1.319072 |
0.1920 |
|
FI |
-1.717497 |
0.727707 |
-2.360149 |
0.0214 |
|
PTS |
1.499799 |
0.719917 |
2.083294 |
0.0414 |
|
MAT |
2.057067 |
1.631100 |
1.261153 |
0.2120 |
|
MOB |
0.153078 |
0.097000 |
1.578129 |
0.1196 |
|
MC |
1.922943 |
1.182932 |
1.625575 |
0.1091 |
|
FTB |
-0.110924 |
0.983688 |
-0.112763 |
0.9106 |
|
SE |
2.208505 |
2.800907 |
0.788496 |
0.4334 |
|
YLD |
4.626702 |
2.919634 |
1.584686 |
0.1181 |
|
MARG |
1.189518 |
0.485433 |
2.450426 |
0.0171 |
|
CB |
1.759744 |
1.242104 |
1.416744 |
0.1616 |
|
STL |
-0.031563 |
0.051720 |
-0.610265 |
0.5439 |
|
LA |
-0.022067 |
0.061013 |
-0.361675 |
0.7188 |
Log likelihood -30.89597 Obs with Dep=1 46
Obs with Dep=0 32
|
Variable |
Mean All |
Mean D=1 |
Mean D=0 |
|
C |
1.000000 |
1.000000 |
1.000000 |
|
BA |
36.03846 |
35.52174 |
36.78125 |
|
BS |
16.44872 |
15.58696 |
17.68750 |
|
NW |
3.504013 |
2.075261 |
5.557844 |
|
FI |
13.24936 |
13.02348 |
13.57406 |
|
PTS |
1.497949 |
1.505217 |
1.487500 |
|
MAT |
1.058333 |
1.027609 |
1.102500 |
|
MOB |
4.205128 |
4.913043 |
3.187500 |
|
MC |
0.602564 |
0.695652 |
0.468750 |
|
FTB |
0.615385 |
0.521739 |
0.750000 |
|
SE |
0.102564 |
0.043478 |
0.187500 |
|
YLD |
1.606410 |
1.633261 |
1.567813 |
|
MARG |
2.291923 |
2.526304 |
1.955000 |
|
CB |
0.358974 |
0.478261 |
0.187500 |
|
STL |
13.42218 |
11.72304 |
15.86469 |
|
LA |
5.682692 |
4.792174 |
6.962812 |
|
Restricted Logit Model |
|||
|
LOGIT // Dependent Variable is Y |
|||
|
Sample: 1 78 |
|||
|
Included observations: 78 |
|||
|
Convergence achieved after 4 iterations |
|||
|
Variable |
Coefficient |
Std. Error t-Statistic Prob. |
|
|
FI |
-1.264608 |
0.454050 -2.785172 0.0068 |
|
|
MARG |
0.717847 |
0.313845 2.287265 0.0251 |
|
|
YLD |
4.827537 |
1.958833 2.464497 0.0161 |
|
|
PTS |
0.359033 |
0.423378 0.848019 0.3992 |
|
|
MAT |
0.550320 |
1.036613 0.530883 0.5971 |
|
|
C |
6.731755 |
7.059485 0.953576 0.3435 |
|
|
Log likelihood |
-41.47292 |
||
|
Obs with Dep=1 |
46 |
||
|
Obs with Dep=0 |
32 |
||
|
Variable |
Mean All |
Mean D=1 |
Mean D=0 |
|
FI |
13.24936 |
13.02348 |
13.57406 |
|
MARG |
2.291923 |
2.526304 |
1.955000 |
|
YLD |
1.606410 |
1.633261 |
1.567813 |
|
PTS |
1.497949 |
1.505217 |
1.487500 |
|
MAT |
1.058333 |
1.027609 |
1.102500 |
|
C |
1.000000 |
1.000000 |
1.000000 |
d. Similarly, the probit specification output using EViews is given below. The unrestricted log-likelihood is equal to —30.7294. The restricted log - likelihood is —41.7649. Therefore, the LR test statistic is given by LR = 2(41.7649 — 30.7294/ = 22.0710 which is distributed as x?0 under the null hypothesis. This is significant given that the 5% critical value of x20 is 18.31. This means that the probit specification does not reject the principal agent theory as personal characteristics are not jointly insignificant.
TESTING THE EFFICIENT MARKET HYPOTHESIS WITH THE PROBIT MODEL
Unrestricted Probit Model
PROBIT // Dependent Variable is Y Sample: 1 78 Included observations: 78 Convergence achieved after 5 iterations
|
Variable |
Coefficien |
Std. Error |
t-Statistic |
Prob. |
|
C |
3.107820 |
5.954673 |
0.521913 |
0.6036 |
|
BA |
0.003978 |
0.044546 |
0.089293 |
0.9291 |
|
BS |
0.108267 |
0.099172 |
1.091704 |
0.2792 |
|
NW |
-0.128775 |
0.103438 |
-1.244943 |
0.2178 |
|
FI |
-1.008080 |
0.418160 |
-2.410750 |
0.0189 |
|
PTS |
0.830273 |
0.379895 |
2.185533 |
0.0326 |
|
MAT |
1.164384 |
0.924018 |
1.260131 |
0.2123 |
|
MOB |
0.093034 |
0.056047 |
1.659924 |
0.1020 |
|
MC |
1.058577 |
0.653234 |
1.620518 |
0.1102 |
|
FTB |
-0.143447 |
0.550471 |
-0.260589 |
0.7953 |
|
SE |
1.127523 |
1.565488 |
0.720237 |
0.4741 |
|
YLD |
2.525122 |
1.590796 |
1.587332 |
0.1175 |
|
MARG |
0.705238 |
0.276340 |
2.552069 |
0.0132 |
|
CB |
1.066589 |
0.721403 |
1.478493 |
0.1443 |
|
STL |
-0.016130 |
0.029303 |
-0.550446 |
0.5840 |
|
LA |
-0.014615 |
0.035920 |
-0.406871 |
0.6855 |
Log likelihood -30.72937 Obs with Dep=1 46
Obs with Dep=0 32
Restricted Probit Model
PROBIT // Dependent Variable is Y Sample: 1 78 Included observations: 78 Convergence achieved after 3 iterations
|
Variable |
Coefficient |
Std. Error |
t-Statistic |
Prob. |
|
FI |
-0.693584 |
0.244631 |
-2.835225 |
0.0059 |
|
MARG |
0.419997 |
0.175012 |
2.399811 |
0.0190 |
|
YLD |
2.730187 |
1.099487 |
2.483146 |
0.0154 |
|
PTS |
0.235534 |
0.247390 |
0.952076 |
0.3442 |
|
MAT |
0.221568 |
0.610572 |
0.362886 |
0.7178 |
|
C |
3.536657 |
4.030251 |
0.877528 |
0.3831 |
Log likelihood -41.76443 Obs with Dep=1 46
Obs with Dep=0 32
13.13 Problem Drinking and Employment. The following Stata output replicates the OLS results given in Table 5 of Mullahy and Sindelar (1996, p. 428) for males. The first regression is for employment, given in column 1 of Table 5 of the paper, and the second regression is for unemployment, given in column 3 of Table 5 of the paper. Robust standard errors are reported.
. reg emp hvdrnk90 ue88 age agesq educ married famsize white hlstat1 hlstat2 hlstat3 hlstat4 region1 region2 region3 msa1 msa2 q1 q2 q3, robust
|
Regression with robust standard errors |
Number of obs |
= 9822 |
|
F (20, 9801) |
= 46.15 |
|
|
Prob > F |
= 0.0000 |
|
|
R-squared |
= 0.1563 |
|
|
Root MSE |
= .27807 |
|
Robust
|
|
educ |
.0078258 |
.0011271 |
6.94 |
0.000 |
.0056166 |
.0100351 |
|
married |
.0505682 |
.0098396 |
5.14 |
0.000 |
.0312805 |
.0698558 |
|
famsize |
.0020612 |
.0021796 |
0.95 |
0.344 |
-.0022113 |
.0063336 |
|
white |
.0773332 |
.0104289 |
7.42 |
0.000 |
.0568905 |
.097776 |
|
hlstat1 |
.5751898 |
.0306635 |
18.76 |
0.000 |
.5150831 |
.6352965 |
|
hlstat2 |
.5728 |
.0306427 |
18.69 |
0.000 |
.512734 |
.632866 |
|
hlstat3 |
.537617 |
.0308845 |
17.41 |
0.000 |
.4770769 |
.598157 |
|
hlstat4 |
.3947391 |
.0354291 |
11.14 |
0.000 |
.3252908 |
.4641874 |
|
region1 |
-.0013608 |
.0094193 |
-0.14 |
0.885 |
-.0198247 |
.017103 |
|
region2 |
.0050446 |
.0084215 |
0.60 |
0.549 |
-.0114633 |
.0215526 |
|
region3 |
.0254332 |
.0081999 |
3.10 |
0.002 |
.0093596 |
.0415067 |
|
msa1 |
-.0159492 |
.0083578 |
-1.91 |
0.056 |
-.0323322 |
.0004337 |
|
msa2 |
.0073081 |
.0072395 |
1.01 |
0.313 |
-.0068827 |
.0214989 |
|
q1 |
-.0155891 |
.0079415 |
-1.96 |
0.050 |
-.0311561 |
-.000022 |
|
q2 |
-.0068915 |
.0077786 |
-0.89 |
0.376 |
-.0221392 |
.0083561 |
|
q3 |
-.0035867 |
.0078474 |
-0.46 |
0.648 |
-.0189692 |
.0117957 |
|
_cons |
-.0957667 |
.0623045 |
-1.54 |
0.124 |
-.2178964 |
.0263631 |
. reg unemp hvdrnk90 ue88 age agesq educ married famsize white hlstatl hlstat2 hlstat3 hlstat4 region1 region2 region3 msa1 msa2 q1 q2 q3, robust
|
Regression with robust standard errors |
Number of obs |
= 9822 |
|
F(20, 9801) |
= 3.37 |
|
|
Prob > F |
= 0.0000 |
|
|
R-squared |
= 0.0099 |
|
|
Root MSE |
= .17577 |
|
1 emp | |
Coef. |
Robust Std. Err. |
t |
P>|t| |
[95% Conf. Interval] |
|
|
hvdrnk90 | |
.0100022 |
.0066807 |
1.50 |
0.134 |
-.0030934 |
.0230977 |
|
ue88 | |
.0045029 |
.0014666 |
3.07 |
0.002 |
.0016281 |
.0073776 |
|
age | |
-.0014753 |
.0017288 |
-0.85 |
0.393 |
-.0048641 |
.0019134 |
|
agesq | |
.0000123 |
.0000206 |
0.60 |
0.551 |
-.0000281 |
.0000527 |
|
educ | |
-.0028141 |
.0006307 |
-4.46 |
0.000 |
-.0040504 |
-.0015777 |
|
married | |
-.0092854 |
.0060161 |
-1.54 |
0.123 |
-.0210782 |
.0025073 |
|
famsize | |
.0003859 |
.0013719 |
0.28 |
0.778 |
-.0023033 |
.0030751 |
|
white | |
-.0246801 |
.0063618 |
-3.88 |
0.000 |
-.0371506 |
-.0122096 |
|
hlstat1 j |
.0150194 |
.0113968 |
1.32 |
0.188 |
-.0073206 |
.0373594 |
|
hlstat2 | |
.0178594 |
.0114626 |
1.56 |
0.119 |
-.0046097 |
.0403285 |
|
hlstat3 | |
.0225153 |
.0116518 |
1.93 |
0.053 |
-.0003245 |
.0453552 |
|
hlstat4 | |
.0178865 |
.0136228 |
1.31 |
0.189 |
-.0088171 |
.0445901 |
|
region1 | |
.0007911 |
.005861 |
0.13 |
0.893 |
-.0106977 |
.01228 |
|
region2 | |
-.0029056 |
.0053543 |
-0.54 |
0.587 |
-.0134011 |
.0075898 |
|
region3 | |
-.0065005 |
.005095 |
-1.28 |
0.202 |
-.0164877 |
.0034868 |
|
msa1 | |
-.0008801 |
.0052004 |
-0.17 |
0.866 |
-.011074 |
.0093139 |
|
msa2 I |
-.0055184 |
.0047189 |
-1.17 |
0.242 |
-.0147685 |
.0037317 |
|
q11 |
.0145704 |
.0051986 |
2.80 |
0.005 |
.00438 |
.0247607 |
|
q21 |
.0022831 |
.0047579 |
0.48 |
0.631 |
-.0070434 |
.0116096 |
|
q31 |
.000043 |
.0047504 |
0.01 |
0.993 |
-.0092687 |
.0093547 |
|
.cons j |
.0927746 |
.0364578 |
2.54 |
0.011 |
.0213098 |
.1642394 |
The following Stata output replicates the OLS results given in Table 6 of Mullahy and Sindelar (1996, p. 429) for females. The first regression is for employment, given in column 1 of Table 6 of the paper, and the second regression is for unemployment, given in column 3 of Table 6 of the paper. Robust standard errors are reported.
. reg emp hvdrnk90 ue88 age agesq educ married famsize white hlstatl hlstat2 hlstat3 hlstat4 region1 region2 region3 msa1 msa2 q1 q2 q3, robust
|
Regression with robust standard errors |
Number of obs |
= 12534 |
|
F(20,12513) |
= 117.99 |
|
|
Prob > F |
= 0.0000 |
|
|
R-squared |
= 0.1358 |
|
|
Root MSE |
= .42932 |
|
j emp j |
Coef. |
Robust Std. Err. |
t |
P>jtj |
[95% Conf. Interval] |
|
|
hvdrnk90 j |
.0059878 |
.0120102 |
0.50 |
0.618 |
-.017554 |
.0295296 |
|
ue88 j |
-.0168969 |
.002911 |
-5.80 |
0.000 |
-.0226028 |
-.011191 |
|
age j |
.04635 |
.0036794 |
12.60 |
0.000 |
.0391378 |
.0535622 |
|
agesq j |
-.0005898 |
.0000449 |
-13.13 |
0.000 |
-.0006778 |
-.0005018 |
|
educ j |
.0227162 |
.0015509 |
14.65 |
0.000 |
.0196762 |
.0257563 |
|
married j |
.0105416 |
.0111463 |
0.95 |
0.344 |
-.0113068 |
.0323901 |
|
famsize j |
-.0662794 |
.0030445 |
-21.77 |
0.000 |
-.072247 |
-.0603118 |
|
white j |
-.0077594 |
.0104111 |
-0.75 |
0.456 |
-.0281668 |
.012648 |
|
hvdrnk90 j |
.0059878 |
.0120102 |
0.50 |
0.618 |
-.017554 |
.0295296 |
|
ue88 j |
-.0168969 |
.002911 |
-5.80 |
0.000 |
-.0226028 |
-.011191 |
|
age j |
.04635 |
.0036794 |
12.60 |
0.000 |
.0391378 |
.0535622 |
|
agesq j |
-.0005898 |
.0000449 |
-13.13 |
0.000 |
-.0006778 |
-.0005018 |
|
educ j |
.0227162 |
.0015509 |
14.65 |
0.000 |
.0196762 |
.0257563 |
|
married j |
.0105416 |
.0111463 |
0.95 |
0.344 |
-.0113068 |
.0323901 |
|
famsize j |
-.0662794 |
.0030445 |
-21.77 |
0.000 |
-.072247 |
-.0603118 |
|
white j |
-.0077594 |
.0104111 |
-0.75 |
0.456 |
-.0281668 |
.012648 |
|
hlstat1 j |
.4601695 |
.0253797 |
18.13 |
0.000 |
.4104214 |
.5099177 |
|
hlstat2 j |
.4583823 |
.0252973 |
18.12 |
0.000 |
.4087957 |
.5079689 |
|
hlstat3 |
.4096243 |
.0251983 |
16.26 |
0.000 |
.3602317 |
.4590169 |
|
hlstat4 |
.2494427 |
.027846 |
8.96 |
0.000 |
.1948602 |
.3040251 |
|
region1 |
-.0180596 |
.0129489 |
-1.39 |
0.163 |
-.0434415 |
.0073223 |
|
region2 |
.0095951 |
.0114397 |
0.84 |
0.402 |
-.0128285 |
.0320186 |
|
region3 |
.0465464 |
.0108841 |
4.28 |
0.000 |
.0252119 |
.067881 |
|
msa1 |
-.0256183 |
.0109856 |
-2.33 |
0.020 |
-.0471518 |
-.0040848 |
|
msa2 |
.0051885 |
.0103385 |
0.50 |
0.616 |
-.0150765 |
.0254534 |
|
q1 |
-.0058134 |
.0107234 |
-0.54 |
0.588 |
-.0268329 |
.0152061 |
|
q2 |
-.0061301 |
.0109033 |
-0.56 |
0.574 |
-.0275022 |
.0152421 |
|
q3 |
-.0168673 |
.0109023 |
-1.55 |
0.122 |
-.0382376 |
.0045029 |
|
_cons |
-.5882924 |
.0782545 |
-7.52 |
0.000 |
-.7416831 |
-.4349017 |
. reg unemp hvdrnk90 ue88 age agesq educ married famsize white hlstatl hlstat2 hlstat3 hlstat4 region1 region2 region3 msa1 msa2 q1 q2 q3, robust
Regression with robust standard errors Number of obs = 12534
F(20, 12513) = 5.99
Prob > F = 0.0000
R-squared = 0.0141
Root MSE = .18409
|
1 emp | |
Coef. |
Robust Std. Err. |
t |
P>|t| |
[95% Conf. Interval] |
|
|
hvdrnk90 | |
.0149286 |
.0059782 |
2.50 |
0.013 |
.0032104 |
.0266468 |
|
ue88 | |
.0038119 |
.0013782 |
2.77 |
0.006 |
.0011105 |
.0065133 |
|
age | |
-.0013974 |
.0015439 |
-0.91 |
0.365 |
-.0044237 |
.0016289 |
|
agesq | |
4.43e-06 |
.0000181 |
0.24 |
0.807 |
-.0000311 |
.00004 |
|
educ | |
-.0011631 |
.0006751 |
-1.72 |
0.085 |
-.0024865 |
.0001602 |
|
married | |
-.0066296 |
.0058847 |
-1.13 |
0.260 |
-.0181645 |
.0049053 |
|
famsize | |
.0013304 |
.0013075 |
1.02 |
0.309 |
-.0012325 |
.0038933 |
|
white | |
-.0308826 |
.0051866 |
-5.95 |
0.000 |
-.0410493 |
-.020716 |
|
hlstat1 j |
.008861 |
.0092209 |
0.96 |
0.337 |
-.0092135 |
.0269354 |
|
hlstat2 | |
.0079536 |
.0091305 |
0.87 |
0.384 |
-.0099435 |
.0258507 |
|
hlstat3 | |
.0224927 |
.0093356 |
2.41 |
0.016 |
.0041934 |
.0407919 |
|
hlstat4 | |
.0193116 |
.0106953 |
1.81 |
0.071 |
-.0016528 |
.040276 |
|
region1 | |
.0020325 |
.0055618 |
0.37 |
0.715 |
-.0088694 |
.0129344 |
|
region2 | |
-.0005405 |
.0049211 |
-0.11 |
0.913 |
-.0101866 |
.0091057 |
|
region3 | |
-.0079708 |
.0046818 |
-1.70 |
0.089 |
-.0171479 |
.0012063 |
|
msa1 | |
-.002055 |
.0049721 |
-0.41 |
0.679 |
-.0118011 |
.007691 |
|
msa2 | |
-.0130041 |
.0041938 |
-3.10 |
0.002 |
-.0212246 |
-.0047835 |
|
q1 | |
.0025441 |
.0043698 |
0.58 |
0.560 |
-.0060214 |
.0111095 |
|
q21 |
.0080984 |
.0046198 |
1.75 |
0.080 |
-.0009571 |
.0171539 |
|
q31 |
.0102601 |
.0046839 |
2.19 |
0.029 |
.001079 |
.0194413 |
|
_cons | |
.0922081 |
.0350856 |
2.63 |
0.009 |
.023435 |
.1609813 |
The corresponding probit equation for employment for males is given by the following stata output (this replicates Table 13.6 in the text): . probit emp hvdrnk90 ue88 age agesq educ married famsize white hlstatl hlstat2 hlstat3 hlstat4 region1 region2 region3 msa1 msa2 q1 q2 q3, robust
|
Probit regression |
Number of obs |
= 9822 |
|
Wald chi2(20) |
= 928.34 |
|
|
Prob > chi2 |
= 0.0000 |
|
|
Log pseudolikelihood = -2698.1797 |
Pseudo R2 |
= 0.1651 |
|
1 emp | |
Coef. |
Robust Std. Err. |
z |
P>|z| |
[95% Conf. |
Interval] |
|
hvdrnk90 | |
-.1049465 |
.0589878 |
-1.78 |
0.075 |
-.2205606 |
.0106675 |
|
ue88 | |
-.0532774 |
.0142024 |
-3.75 |
0.000 |
-.0811135 |
-.0254413 |
|
age | |
.0996338 |
.0171184 |
5.82 |
0.000 |
.0660824 |
.1331853 |
|
agesq | |
-.0013043 |
.0002051 |
-6.36 |
0.000 |
-.0017062 |
-.0009023 |
|
educ | |
.0471834 |
.0066738 |
7.07 |
0.000 |
.034103 |
.0602638 |
|
married | |
.2952921 |
.0540855 |
5.46 |
0.000 |
.1892866 |
.4012976 |
|
famsize | |
.0188906 |
.0140462 |
1.34 |
0.179 |
-.0086395 |
.0464206 |
|
white | |
.3945226 |
.0483378 |
8.16 |
0.000 |
.2997822 |
.489263 |
|
hlstat1 | |
1.816306 |
.0983443 |
18.47 |
0.000 |
1.623554 |
2.009057 |
|
hlstat2 | |
1.778434 |
.0991528 |
17.94 |
0.000 |
1.584098 |
1.97277 |
|
hlstat3 | |
1.547836 |
.0982635 |
15.75 |
0.000 |
1.355244 |
1.740429 |
|
hlstat4 | |
1.043363 |
.1077276 |
9.69 |
0.000 |
.8322209 |
1.254505 |
|
region1 | |
.0343123 |
.0620016 |
0.55 |
0.580 |
-.0872085 |
.1558331 |
|
region2 | |
.0604907 |
.0537881 |
1.12 |
0.261 |
-.044932 |
.1659135 |
|
region3 | |
.1821206 |
.0542342 |
3.36 |
0.001 |
.0758236 |
.2884176 |
|
msa1 | |
-.0730529 |
.0518715 |
-1.41 |
0.159 |
-.1747192 |
.0286134 |
|
msa2 | |
.0759533 |
.0513087 |
1.48 |
0.139 |
-.02461 |
.1765166 |
|
q11 |
-.1054844 |
.0527723 |
-2.00 |
0.046 |
-.2089162 |
-.0020525 |
|
q21 |
-.0513229 |
.052818 |
-0.97 |
0.331 |
-.1548444 |
.0521985 |
|
q31 |
-.0293419 |
.0543746 |
-0.54 |
0.589 |
-.1359142 |
.0772303 |
|
_cons | |
-3.017454 |
.3592294 |
-8.40 |
0.000 |
-3.72153 |
-2.313377 |
We can see how the probit model fits by looking at its predictions.
. estat classification Probit model for emp
|
— True -- |
|||
|
Classified | |
D |
~D I |
Total |
|
+ I |
8743 |
826 | |
9569 |
|
- I |
79 |
174 | |
253 |
|
Total | |
8822 |
1000 | |
9822 |
Classified + if predicted Pr(D) >= .5 True D defined as emp!= 0
|
Sensitivity Specificity Positive predictive value Negative predictive value |
Pr(+| D) Pr(-| ~D) Pr(D| +) Pr(~ D| -) |
99.10% 17.40% 91.37% 68.77% |
|
False + rate for true ~D |
Pr(+| ~ D) |
82.60% |
|
False - rate for true D |
Pr(-| D) |
0.90% |
|
False + rate for classified + |
Pr(~D| +) |
8.63% |
|
False - rate for classified - |
Pr(D| -) |
31.23% |
|
Correctly classified |
90.79% |
We could have alternatively run a logit regression on employment for males. logit emp hvdrnk90 ue88 age agesq educ married famsize white hlstat1 hlstat2 hlstat3 hlstat4 region1 region2 region3 msa1 msa2 q1 q2 q3, robust
|
Logistic regression |
Number of obs |
= 9822 |
|
Wald chi2(20) |
= 900.15 |
|
|
Prob > chi2 |
= 0.0000 |
|
|
Log pseudolikelihood = -2700.0567 |
Pseudo R2 |
= 0.1646 |
|
| emp | |
Coef. |
Robust Std. Err. |
z |
P>|z| |
[95% Conf. Interval] |
|
|
hvdrnk90 | |
-.1960754 |
.1114946 |
-1.76 |
0.079 |
-.4146008 |
.02245 |
|
ue88 | |
-.1131074 |
.0273316 |
-4.14 |
0.000 |
-.1666764 |
-.0595384 |
|
age | |
.1884486 |
.0332284 |
5.67 |
0.000 |
.123322 |
.2535751 |
|
agesq | |
-.0024584 |
.0003965 |
-6.20 |
0.000 |
-.0032356 |
-.0016813 |
|
educ | |
.0913569 |
.0127978 |
7.14 |
0.000 |
.0662738 |
.1164401 |
|
married | |
.5534291 |
.1057963 |
5.23 |
0.000 |
.3460721 |
.760786 |
|
famsize | |
.0365059 |
.0276468 |
1.32 |
0.187 |
-.0176808 |
.0906927 |
|
white |
.7224036 |
.0912559 |
7.92 |
0.000 |
.5435454 |
.9012619 |
|
hlstat1 |
3.145481 |
.1721925 |
18.27 |
0.000 |
2.80799 |
3.482972 |
|
hlstat2 |
3.067279 |
.1741295 |
17.61 |
0.000 |
2.725992 |
3.408567 |
|
hlstat3 |
2.613691 |
.1707421 |
15.31 |
0.000 |
2.279042 |
2.948339 |
|
hlstat4 |
1.725571 |
.1844904 |
9.35 |
0.000 |
1.363976 |
2.087166 |
|
region1 |
.0493715 |
.1220065 |
0.40 |
0.686 |
-.1897568 |
.2884999 |
|
region2 |
.1146108 |
.105504 |
1.09 |
0.277 |
-.0921733 |
.3213948 |
|
region3 |
.3738274 |
.1066491 |
3.51 |
0.000 |
.1647991 |
.5828558 |
|
msa1 |
-.1690904 |
.1016459 |
-1.66 |
0.096 |
-.3683127 |
.0301319 |
|
msa2 |
.1345974 |
.1021625 |
1.32 |
0.188 |
-.0656374 |
.3348323 |
|
q1 |
-.1954528 |
.1034703 |
-1.89 |
0.059 |
-.3982508 |
.0073453 |
|
q2 |
-.1052494 |
.1033014 |
-1.02 |
0.308 |
-.3077163 |
.0972176 |
|
q3 |
-.0418287 |
.1074896 |
-0.39 |
0.697 |
-.2525045 |
.168847 |
|
.cons |
-5.538271 |
.6935383 |
-7.99 |
0.000 |
-6.897581 |
-4.178961 |
And the corresponding predictions for the logit model are given by
. estat classification Logistic model for emp
|
........ True - |
|||
|
Classified | |
D |
~D I |
Total |
|
+ I |
8740 |
822 | |
9562 |
|
- I |
82 |
178 | |
260 |
|
Total | |
8822 |
1000 | |
9822 |
Classified + if predicted Pr(D) >= .5 True D defined as emp!=0
|
Sensitivity Specificity Positive predictive value Negative predictive value |
Pr(+- D) Pr(—~D) Pr(D - +) Pr(~D - -) |
99.07% 17.80% 91.40% 68.46% |
|
False + rate for true ~D |
Pr(+| ~D) |
82.20% |
|
False - rate for true D |
Pr(--D) |
0.93% |
|
False + rate for classified + |
Pr(~D - +) |
8.60% |
|
False - rate for classified - |
Pr(D--) |
31.54% |
|
Correctly classified |
90.80% |
The marginal effects for the probit model can be obtained as follows:
.dprobit emp hvdrnk90 ue88 age agesq educ married famsize white hlstatl hlstat2 hlstat3 hlstat4 region1 region2 region3 msa1 msa2 q1 q2 q3, robust
Iteration 0: log pseudolikelihood =-3231.8973
Iteration 1: log pseudolikelihood =-2707.0435
Iteration 2: log pseudolikelihood =-2698.2015
Iteration 3: log pseudolikelihood =-2698.1797
|
Probit regression, reporting marginal effects |
Number of obs |
= 9822 |
|
Wald chi2(20) |
= 928.34 |
|
|
Prob > chi2 |
= 0.0000 |
|
|
Log pseudolikelihood = -2698.1797 |
Pseudo R2 |
= 0.1651 |
|
| emp | |
dF/dx |
Robust Std. Err. |
z |
P>|z| |
x-bar |
[95% Conf. Interval] |
|
|
hvdrnk90[7] | |
-.0161704 |
.0096242 |
-1.78 |
0.075 |
.099165 |
-.035034 |
.002693 |
|
ue88 | |
-.0077362 |
.0020463 |
-3.75 |
0.000 |
5.56921 |
-.011747 |
-.003725 |
|
age | |
.0144674 |
.0024796 |
5.82 |
0.000 |
39.1757 |
.009607 |
.019327 |
|
agesq | |
-.0001894 |
.0000297 |
-6.36 |
0.000 |
1627.61 |
-.000248 |
-.000131 |
|
educ | |
.0068513 |
.0009621 |
7.07 |
0.000 |
13.3096 |
.004966 |
.008737 |
|
married* | |
.0488911 |
.010088 |
5.46 |
0.000 |
.816432 |
.029119 |
.068663 |
|
famsize | |
.002743 |
.002039 |
1.34 |
0.179 |
2.7415 |
-.001253 |
.006739 |
|
white* | |
.069445 |
.0100697 |
8.16 |
0.000 |
.853085 |
.049709 |
.089181 |
|
hlstat1 * | |
.2460794 |
.0148411 |
18.47 |
0.000 |
.415903 |
.216991 |
.275167 |
|
hlstat2* | |
.1842432 |
.0099207 |
17.94 |
0.000 |
.301873 |
.164799 |
.203687 |
|
hlstat3* | |
.130786 |
.0066051 |
15.75 |
0.000 |
.205254 |
.11784 |
.143732 |
|
hlstat4* | |
.0779836 |
.0041542 |
9.69 |
0.000 |
.053451 |
.069841 |
.086126 |
|
region1 * | |
.0049107 |
.0087468 |
0.55 |
0.580 |
.203014 |
-.012233 |
.022054 |
|
region2* | |
.0086088 |
.0075003 |
1.12 |
0.261 |
.265628 |
-.006092 |
.023309 |
|
region3* | |
.0252543 |
.0071469 |
3.36 |
0.001 |
.318265 |
.011247 |
.039262 |
|
msa1 * | |
-.0107946 |
.0077889 |
-1.41 |
0.159 |
.333232 |
-.026061 |
.004471 |
|
msa2* | |
.0109542 |
.0073524 |
1.48 |
0.139 |
.434942 |
-.003456 |
.025365 |
|
q1* | |
-.0158927 |
.0082451 |
-2.00 |
0.046 |
.254632 |
-.032053 |
.000267 |
|
q2* | |
-.0075883 |
.0079484 |
-0.97 |
0.331 |
.252698 |
-.023167 |
.00799 |
|
q3* | |
-.0043066 |
.0080689 |
-0.54 |
0.589 |
.242822 |
-.020121 |
.011508 |
|
obs. P | |
.8981877 |
||||||
|
pred. P | |
.9224487 |
(at x-bar) |
13.15 Fertility and Female Labor Supply
a. Carrasco (2001, p. 391) Table 4, column 1, ran a fertility probit equation, which we replicate below using Stata:
probit f dsex ags26l educ_2 educ_3 age drace inc
|
Probit regression |
Number of obs |
= 5768 |
|
LR chi2 (7) |
= 964.31 |
|
|
Prob > chi2 |
= 0.0000 |
|
|
Log likelihood = -1561.1312 |
Pseudo R2 |
= 0.2360 |
|
f |
Coef. |
Std. Err. |
z |
P>|z| |
[95% Conf. Interval] |
|
|
dsex |
.3250503 |
.0602214 |
5.40 |
0.000 |
.2070184 |
.4430822 |
|
ags26l |
-2.135365 |
.1614783 |
-13.22 |
0.000 |
-2.451857 |
-1.818873 |
|
educ_2 |
.0278467 |
.1145118 |
0.24 |
0.808 |
-.1965922 |
.2522856 |
|
educ_3 |
.3071582 |
.1255317 |
2.45 |
0.014 |
.0611207 |
.5531958 |
|
age |
-.0808522 |
.0048563 |
-16.65 |
0.000 |
-.0903703 |
-.071334 |
|
drace |
-.0916409 |
.0629859 |
-1.45 |
0.146 |
-.215091 |
.0318093 |
|
inc |
.003161 |
.0029803 |
1.06 |
0.289 |
-.0026803 |
.0090022 |
|
_cons |
1.526893 |
.1856654 |
8.22 |
0.000 |
1.162996 |
1.890791 |
|
For part (b) the predicted probabilities are obtained as follows: |
. Istat
Probit model for f
|
True-
|
Classified + if predicted Pr(D) >= .5 True D defined as f!= 0
|
Sensitivity Specificity Positive predictive value Negative predictive value |
Pr(+| D) Pr(-| ~D) Pr(D| +) Pr(~D| -) |
0.30% 99.94% 40.00% 88.65% |
|
False + rate for true ~D |
Pr(+| ~D) |
0.06% |
|
False - rate for true D |
Pr(-| D) |
99.70% |
|
False + rate for classified + |
Pr(~D| +) |
60.00% |
|
False - rate for classified - |
Pr( Dj -) |
11.35% |
|
Correctly classified |
88.61% |
The estimates reveal that having children of the same sex has a significant and positive effect on the probability of having an additional child. The marginal effects are given by dprobit in Stata. dprobit f dsex ags26l educ_2 educ_3 age drace inc
|
Probit regression, reporting marginal effects |
Number of obs |
= 5768 |
|
LR chi2 (7) |
= 964.31 |
|
|
Prob > chi2 |
= 0.0000 |
|
|
Log likelihood = —1561.1312 |
Pseudo R2 |
= 0.2360 |
|
f |
dF/dx |
Std. Err. |
z |
P>|z| |
x-bar |
[95% C. I.] |
|
|
dsex* |
.0302835 |
.0069532 |
5.40 |
0.000 |
.256415 |
.016655 |
.043912 |
|
ags26l* |
-.1618148 |
.0066629 |
-13.22 |
0.000 |
.377601 |
-.174874 |
-.148756 |
|
educ_2* |
.0022157 |
.0090239 |
0.24 |
0.808 |
.717753 |
-.015471 |
.019902 |
|
educ_3* |
.0288636 |
.0140083 |
2.45 |
0.014 |
.223994 |
.001408 |
.056319 |
|
age |
-.0065031 |
.0007644 |
-16.65 |
0.000 |
32.8024 |
-.008001 |
-.005005 |
|
drace* |
-.0077119 |
.0055649 |
-1.45 |
0.146 |
.773232 |
-.018619 |
.003195 |
|
inc |
.0002542 |
.000241 |
1.06 |
0.289 |
12.8582 |
-.000218 |
.000727 |
|
obs. P |
.1137309 |
||||||
|
pred. P |
.0367557 |
(at x-bar) |
|
(*) dF/dx is for discrete change of dummy variable from 0 to 1 z and P> |z| correspond to the test of the underlying coefficient being 0 |
If we replace same sex by its components: same sex female and same sex male variables, the results do not change indicating that having both boys or girls does not matter, see Carrasco (2001,p.391) Table 4, column 2.
. probit f dsexm dsexf ags26l educ_2 educ_3 age drace inc
|
Probit regression |
Number of obs |
= 5768 |
|
LR chi2 (8) |
= 964.32 |
|
|
Prob > chi2 |
= 0.0000 |
|
|
Log likelihood = -1561.1284 |
Pseudo R2 |
= 0.2360 |
|
f |
Coef. |
Std. Err. |
z |
P>|z| |
[95% Conf. Interval] |
|
|
dsexm |
.328542 |
.0764336 |
4.30 |
0.000 |
.1787349 |
.4783491 |
|
dsexf |
.3209239 |
.0820417 |
3.91 |
0.000 |
.1601252 |
.4817226 |
|
ags26l |
-2.135421 |
.1614518 |
-13.23 |
0.000 |
-2.451861 |
-1.818981 |
|
educ_2 |
.027657 |
.1145384 |
0.24 |
0.809 |
-.1968342 |
.2521482 |
|
educ_3 |
.3068706 |
.1255904 |
2.44 |
0.015 |
.0607179 |
.5530233 |
|
age |
-.0808669 |
.0048605 |
-16.64 |
0.000 |
-.0903934 |
-.0713404 |
|
drace |
-.0918074 |
.0630233 |
-1.46 |
0.145 |
-.2153308 |
.031716 |
|
inc |
.0031709 |
.0029829 |
1.06 |
0.288 |
-.0026754 |
.0090173 |
|
_cons |
1.527551 |
.1858818 |
8.22 |
0.000 |
1.163229 |
1.891872 |
Probit model for f
True-
|
Classified |
D |
Total |
|
|
+ |
2 |
3 |
5 |
|
- |
654 |
5109 |
5763 |
|
Total |
656 |
5112 |
5768 |
Classified + if predicted Pr(D) >= .5 True D defined as f!= 0
|
Sensitivity Specificity Positive predictive value Negative predictive value |
Pr(+| D) Pr(-| ~D) Pr(D| +) Pr(-D| -) |
0.30% 99.94% 40.00% 88.65% |
|
False + rate for true —D |
Pr(+| ~D) |
0.06% |
|
False - rate for true D |
Pr(-| D) |
99.70% |
|
False + rate for classified + |
Pr(-D| +) |
60.00% |
|
False - rate for classified - |
Pr(D| -) |
11.35% |
|
Correctly classified |
88.61% |
. dprobit f dsexm dsexf ags26l educ_2 educ_3 age drace inc
|
Probit regression, reporting marginal effects |
Number of obs |
= 5768 |
|
LR chi2 (7) |
= 964.32 |
|
|
Prob > chi2 |
= 0.0000 |
|
|
Log likelihood = —1561.1284 |
Pseudo R2 |
= 0.2360 |
|
dF/dx |
Std. Err. |
z |
P>|z| |
x-bar |
[95% C. I.] |
||
|
dsexm[8] |
.0325965 |
.0095475 |
4.30 |
0.000 |
.145111 |
.013884 |
.051309 |
|
dsexf* |
.032261 |
.0103983 |
3.91 |
0.000 |
.111304 |
.011881 |
.052641 |
|
ags26l* |
-.16182 |
.0066634 |
-13.23 |
0.000 |
.377601 |
-.17488 |
-.14876 |
|
educ_2* |
.0022008 |
.0090273 |
0.24 |
0.809 |
.717753 |
-.015492 |
.019894 |
|
educ_3* |
.0288323 |
.01401 |
2.44 |
0.015 |
.223994 |
.001373 |
.056291 |
|
age |
-.0065042 |
.0007645 |
-16.64 |
0.000 |
32.8024 |
-.008003 |
-.005006 |
|
drace* |
-.0077266 |
.0055692 |
-1.46 |
0.145 |
.773232 |
-.018642 |
.003189 |
|
inc |
.000255 |
.0002412 |
1.06 |
0.288 |
12.8582 |
-.000218 |
.000728 |
|
obs. P |
.1137309 |
||||||
|
pred. P |
.0367556 |
(at x-bar) |
c. Carrasco (2001, p. 392) Table 5, column 4, ran a female labor force participation OLS equation, which we replicate below using Stata 10:
. reg dhw f ags26l fxag26l educ_2 educ_3 age drace inc dhwl
|
Number of obs |
= 5768 |
|
F(9, 5758) |
= 445.42 |
|
Prob > F |
= 0.0000 |
|
R-squared |
= 0.4104 |
|
Adj R-squared |
= 0.4095 |
|
Root MSE |
= .32361 |
|
dhw |
Coef. |
Std. Err. |
t |
P>|t| |
[95% Conf. Interval] |
|
|
f |
-.0888995 |
.0144912 |
-6.13 |
0.000 |
-.1173077 |
-.0604912 |
|
ags26l |
-.0194454 |
.0093334 |
-2.08 |
0.037 |
-.0377424 |
-.0011484 |
|
fxag26l |
-.0581458 |
.1629414 |
-0.36 |
0.721 |
-.3775723 |
.2612806 |
|
educ_2 |
.0491989 |
.0186018 |
2.64 |
0.008 |
.0127323 |
.0856655 |
|
educ_3 |
.0725501 |
.0207404 |
3.50 |
0.000 |
.0318912 |
.1132091 |
|
age |
.0014193 |
.0007854 |
1.81 |
0.071 |
-.0001203 |
.002959 |
|
drace |
-.0098333 |
.010379 |
-0.95 |
0.343 |
-.03018 |
.0105134 |
|
inc |
-.0018149 |
.0004887 |
-3.71 |
0.000 |
-.002773 |
-.0008568 |
|
dhwl |
.6253973 |
.0103188 |
60.61 |
0.000 |
.6051686 |
.645626 |
|
_cons |
.2373022 |
.032744 |
7.25 |
0.000 |
.1731117 |
.3014927 |
Carrasco (2001, p. 392) Table 5, column 1, ran a female labor force participation probit equation, which we replicate below using Stata:
. probit dhw f ags26l fxag26l educ_2 educ_3 age drace inc dhwl
Number of obs = 5768
LR chi2 (7) = 2153.17
Prob > chi2 = 0.0000
Pseudo R2 = 0.3458
|
Log likelihood = -2036.8086
|
Probit model for dhw
|
Classified |
D |
~D |
Total |
|
+ |
4073 |
378 |
4451 |
|
- |
366 |
951 |
1317 |
|
Total |
4439 |
1329 |
5768 |
Classified + if predicted Pr(D) >= .5 True D defined as dhw!= 0
|
Sensitivity Specificity Positive predictive value Negative predictive value |
Pr(+| D) Pr(-| ~D) Pr(D| +) Pr(~D| -) |
91.75% 71.56% 91.51% 72.21% |
|
False + rate for true ~D |
Pr(+| ~D) |
28.44% |
|
False - rate for true D |
Pr(- D) |
8.25% |
|
False + rate for classified + |
Pr(~D| +) |
8.49% |
|
False - rate for classified - |
Pr( Dj -) |
27.79% |
|
Correctly classified |
87.10% |
The marginal effects are given by dprobit in Stata:
. dprobit dhw f ags26l fxag26l educ_2 educ_3 age drace inc dhwl
Number of obs = 5768
LR chi2 (9) = 2153.17
Prob > chi2 = 0.0000
Pseudo R2 = 0.3458
|
dhw |
dF/dx |
Std. Err. |
z |
P>|z| |
x-bar |
[95% C. I.] |
|
|
f[9] |
-.1200392 |
.0224936 |
-5.94 |
0.000 |
.113731 |
-.164126 |
-.075953 |
|
ags26l* |
-.0275503 |
.0125892 |
-2.21 |
0.027 |
.377601 |
-.052225 |
-.002876 |
|
fxag26l* |
-.0524753 |
.2127127 |
-0.27 |
0.790 |
.000693 |
-.469385 |
.364434 |
|
educ_2* |
.0626367 |
.0239923 |
2.72 |
0.006 |
.717753 |
.015613 |
.109661 |
|
educ_3* |
.0870573 |
.0206089 |
3.77 |
0.000 |
.223994 |
.046665 |
.12745 |
|
age |
.0023327 |
.0010504 |
2.22 |
0.027 |
32.8024 |
.000274 |
.004391 |
|
drace* |
-.0145508 |
.0134701 |
-1.06 |
0.288 |
.773232 |
-.040952 |
.01185 |
|
inc |
-.0022631 |
.0006189 |
-3.65 |
0.000 |
12.8582 |
-.003476 |
-.00105 |
|
dhwl* |
.6249756 |
.0134883 |
41.80 |
0.000 |
.771671 |
.598539 |
.651412 |
|
obs. P |
.7695908 |
||||||
|
pred. P |
.8271351 |
(at x-bar) |
d. The 2sls estimates in Table 5, column 5, of Carrasco (2001, p. 392) using as instruments the same sex variables and their interactions with ags26l is given below, along with the over-identification test and the first stage diagnostics:
. ivregress 2sls dhw (f fxag26l =dsexm dsexf sexm_26l sexf_26l) ags26l educ_2 e > duc_3 age drace inc dhwl
|
Instrumental variables (2SLS) regression |
Number of obs = |
5768 |
|
Wald chi2(9) = |
3645.96 |
|
|
Prob > chi2 = |
0.0000 |
|
|
R-squared = |
0.3565 |
|
|
Root MSE = |
.3378 |
|
dhw |
Coef. |
Std. Err. |
z |
P>|z| |
[95% Conf. Interval] |
|
|
f |
-.2164685 |
.2246665 |
-0.96 |
0.335 |
-.6568067 |
.2238697 |
|
fxag26l |
-3.366305 |
3.512783 |
-0.96 |
0.338 |
-10.25123 |
3.518623 |
|
ags26l |
-.0385731 |
.0467522 |
-0.83 |
0.409 |
-.1302058 |
.0530596 |
|
educ_2 |
.0331807 |
.0288653 |
1.15 |
0.250 |
-.0233943 |
.0897557 |
|
educ_3 |
.064607 |
.0348694 |
1.85 |
0.064 |
-.0037357 |
.1329497 |
|
age |
-.0001934 |
.0030344 |
-0.06 |
0.949 |
-.0061407 |
.0057539 |
|
drace |
-.0163251 |
.012366 |
-1.32 |
0.187 |
-.0405621 |
.0079118 |
|
inc |
-.0017194 |
.0005162 |
-3.33 |
0.001 |
-.0027312 |
-.0007076 |
|
dhwl |
.6230639 |
.017256 |
36.11 |
0.000 |
.5892427 |
.6568851 |
|
_cons |
.3330965 |
.141537 |
2.35 |
0.019 |
.0556891 |
.610504 |
Instrumented: f fxag26l
Instruments: ags26l educ_2 educ_3 age drace inc dhwl dsexm dsexf sexm_26l sexf_26l
. estat overid
Tests of overidentifying restrictions:
Sargan (score) chi2(2) = .332468 (p = 0.8468) Basmann chi2(2) =.331796 (p = 0.8471)
. estat firststage
Shea’s partial R-squared
|
Shea's |
Shea's |
|
|
Variable |
Partial R-sq. |
Adj. Partial R-sq. |
|
f |
0.0045 |
0.0028 |
|
fxag26l |
0.0023 |
0.0006 |
Minimum eigenvalue statistic = 3.36217
Critical Values # of endogenous regressors: 2
Ho: Instruments are weak # of excluded instruments: 4
|
5% |
10% |
20% |
30% |
|
|
2SLS relative bias |
11.04 |
7.56 |
5.57 |
4.73 |
|
10% |
15% |
20% |
25% |
|
|
2SLS Size of nominal 5% Wald test |
16.87 |
9.93 |
7.54 |
6.28 |
|
LIML Size of nominal 5% Wald test |
4.72 |
3.39 |
2.99 |
2.79 |
e. So far, heterogeneity across the individuals is not taken into account. Carrasco (2001, p. 393) Table 7, column 4, ran a female labor force participation fixed effects equation with robust standard errors, which we replicate below using Stata:
. xtreg dhw f ags26l fxag26l dhwl, fe r
|
Fixed-effects (within) regression |
Number of obs |
= 5768 |
|
Group variable: ident |
Number of groups |
= 1442 |
|
R-sq: within = 0.0059 |
Obs per group: min |
=4 |
|
between = 0.6185 |
avg |
= 4.0 |
|
overall = 0.2046 |
max |
=4 |
|
F(4,4322) |
= 4.64 |
|
|
corr(u_i, Xb) = 0.4991 |
Prob > F |
= 0.0010 |
|
(Std. Err. adjusted for clustering on ident) |
|
dhw |
Coef. |
Robust Std. Err. |
t |
P>|t| |
[95% Conf. Interval] |
|
|
f |
-.0547777 |
.0155326 |
-3.53 |
0.000 |
-.0852296 |
-.0243257 |
|
ags26l |
.0012836 |
.0126213 |
0.10 |
0.919 |
-.0234607 |
.0260279 |
|
fxag26l |
-.2204885 |
.2013721 |
-1.09 |
0.274 |
-.615281 |
.1743041 |
|
dhwl |
.0356233 |
.0236582 |
1.51 |
0.132 |
-.0107588 |
.0820055 |
|
_cons |
.7479995 |
.0193259 |
38.70 |
0.000 |
.7101108 |
.7858881 |
|
sigma_u sigma_e rho |
.33260036 .27830212 .58818535 |
(fraction of variance due to u_i) |
Note that only fertility is significant in this equation.
Fixed effects 2sls using as instruments the same sex variables and their interactions with ags26l is given below: . xtivreg dhw (f fxag26l =dsexm dsexf sexm_26l sexf_26l)ags26l age inc dhwl, fe
|
Fixed-effects (within) IV regression |
Number of obs = |
5768 |
|
Group variable: ident |
Number of groups = |
1442 |
|
R-sq: within = . |
Obs per group: min = |
4 |
|
between = 0.1125 |
avg = |
4.0 |
|
overall = 0.0332 |
max = |
4 |
|
Wald chi2(6) = |
39710.29 |
|
|
corr(u_i, Xb) = 0.0882 |
Prob > chi2 = |
0.0000 |
|
dhw |
Coef. |
Std. Err. |
z |
P>|z| |
[95% Conf. Interval] |
|
|
f |
-.2970225 |
.156909 |
-1.89 |
0.058 |
-.6045584 |
.0105134 |
|
fxag26l |
-2.1887 |
2.433852 |
-0.90 |
0.369 |
-6.958963 |
2.581562 |
|
ags26l |
-.0584866 |
.0467667 |
-1.25 |
0.211 |
-.1501476 |
.0331744 |
|
age |
.000651 |
.0043265 |
0.15 |
0.880 |
-.0078287 |
.0091307 |
|
inc |
-.0011213 |
.0010482 |
-1.07 |
0.285 |
-.0031758 |
.0009331 |
|
dhwl |
.0362943 |
.0160524 |
2.26 |
0.024 |
.0048322 |
.0677565 |
|
_cons |
.7920305 |
.1623108 |
4.88 |
0.000 |
.4739071 |
1.110154 |
|
sigma_u |
.3293446 |
|||||
|
sigma_e |
.29336161 |
|||||
|
rho |
.55759255 |
(fraction of variance due to u. i) |
||||
|
F test that all u_ |
=0: |
F(1441,4320) |
= 2.16 |
Prob > F = 0.0000 |
Instrumented: f fxag26l
Instruments: ags26l age inc dhwl dsexm dsexf sexm_26l sexf_26l
13.16 multinomial logit model
a. Table II of Terza (2002, p. 399) columns 3,4, 9 and 10 are replicated below for the male data using Stata:
. mlogit y alc90th ue88 age agesq schooling married famsize white excellent verygood good fair northeast midwest south centercity othermsa q1 q2 q3, baseoutcome(1)
|
У |
Coef. |
Std. Err. |
z |
P>|z| |
[95% Conf. Interval] |
|
|
2 alc90th |
.1270931 |
.21395 |
0.59 |
0.552 |
-.2922412 |
.5464274 |
|
ue88 |
.0458099 |
.051355 |
0.89 |
0.372 |
-.0548441 |
.1464639 |
|
age |
.1617634 |
.0663205 |
2.44 |
0.015 |
.0317776 |
.2917492 |
|
agesq |
-.0024377 |
.0007991 |
-3.05 |
0.002 |
-.004004 |
-.0008714 |
|
schooling |
-.0092135 |
.0245172 |
-0.38 |
0.707 |
-.0572664 |
.0388393 |
|
married |
.4004928 |
.1927458 |
2.08 |
0.038 |
.022718 |
.7782677 |
|
famsize |
.0622453 |
.0503686 |
1.24 |
0.217 |
-.0364753 |
.1609659 |
|
white |
.0391309 |
.1705625 |
0.23 |
0.819 |
-.2951653 |
.3734272 |
|
excellent |
2.91833 |
.4486757 |
6.50 |
0.000 |
2.038942 |
3.797719 |
|
verygood |
2.978336 |
.4505932 |
6.61 |
0.000 |
2.09519 |
3.861483 |
|
good |
2.493939 |
.4446815 |
5.61 |
0.000 |
1.622379 |
3.365499 |
|
fair |
1.460263 |
.4817231 |
3.03 |
0.002 |
.5161027 |
2.404422 |
|
northeast |
.0849125 |
.2374365 |
0.36 |
0.721 |
-.3804545 |
.5502796 |
|
midwest |
.0158816 |
.2037486 |
0.08 |
0.938 |
-.3834583 |
.4152215 |
|
south |
.1750244 |
.2027444 |
0.86 |
0.388 |
-.2223474 |
.5723962 |
|
centercity |
-.2717445 |
.1911074 |
-1.42 |
0.155 |
-.6463081 |
.1028192 |
|
othermsa |
-.0921566 |
.1929076 |
-0.48 |
0.633 |
-.4702486 |
.2859354 |
|
q1 |
.422405 |
.1978767 |
2.13 |
0.033 |
.0345738 |
.8102362 |
|
q2 |
-.0219499 |
.2056751 |
-0.11 |
0.915 |
-.4250657 |
.3811659 |
|
q3 |
-.0365295 |
.2109049 |
-0.17 |
0.862 |
-.4498954 |
.3768364 |
|
_cons |
-6.113244 |
1.427325 |
-4.28 |
0.000 |
-8.910749 |
-3.315739 |
|
3 alc90th |
-.1534987 |
.1395003 |
-1.10 |
0.271 |
-.4269144 |
.1199169 |
|
ue88 |
-.0954848 |
.033631 |
-2.84 |
0.005 |
-.1614004 |
-.0295693 |
|
age |
.227164 |
.0409884 |
5.54 |
0.000 |
.1468282 |
.3074999 |
|
agesq |
-.0030796 |
.0004813 |
-6.40 |
0.000 |
-.0040228 |
-.0021363 |
|
schooling |
.0890537 |
.0152314 |
5.85 |
0.000 |
.0592008 |
.1189067 |
|
married |
.7085708 |
.1219565 |
5.81 |
0.000 |
.4695405 |
.9476012 |
|
famsize |
.0622447 |
.0332365 |
1.87 |
0.061 |
-.0028975 |
.127387 |
|
white |
.7380044 |
.1083131 |
6.81 |
0.000 |
.5257147 |
.9502941 |
|
excellent |
3.702792 |
.1852415 |
19.99 |
0.000 |
3.339725 |
4.065858 |
|
verygood |
3.653313 |
.1894137 |
19.29 |
0.000 |
3.282069 |
4.024557 |
|
good |
2.99946 |
.1786747 |
16.79 |
0.000 |
2.649264 |
3.349656 |
|
fair |
1.876172 |
.1885159 |
9.95 |
0.000 |
1.506688 |
2.245657 |
|
northeast |
.088966 |
.1491191 |
0.60 |
0.551 |
-.203302 |
.3812341 |
|
midwest |
.1230169 |
.1294376 |
0.95 |
0.342 |
-.130676 |
.3767099 |
|
south |
.4393047 |
.1298054 |
3.38 |
0.001 |
.1848908 |
.6937185 |
|
centercity |
-.2689532 |
.1231083 |
-2.18 |
0.029 |
-.510241 |
-.0276654 |
|
othermsa |
.0978701 |
.1257623 |
0.78 |
0.436 |
-.1486195 |
.3443598 |
|
q1 |
-.0274086 |
.1286695 |
-0.21 |
0.831 |
-.2795961 |
.224779 |
|
q2 |
-.110751 |
.126176 |
-0.88 |
0.380 |
-.3580514 |
.1365494 |
|
q3 |
-.0530835 |
.1296053 |
-0.41 |
0.682 |
-.3071052 |
.2009382 |
|
_cons |
-6.237275 |
.8886698 |
-7.02 |
0.000 |
-7.979036 |
-4.495515 |
|
(y==1 is the base outcome) |
**using bootstrap for the var-cov matrix
. mlogit y alc90th ue88 age agesq schooling married famsize white excellent ver > ygood good fair northeast midwest south centercity othermsa q1 q2 q3, baseout > come(1) vce(bootstrap)
(running mlogit on estimation sample)
Bootstrap replications (50)
----+---1 ---+--- 2 ---+--- 3 ---+--- 4 ---+--- 5 ............................ 50
|
Multinomial logistic regression |
Number of obs = |
9822 |
|
Replications = |
50 |
|
|
Wald chi2 (40) = |
7442.69 |
|
|
Prob > chi2 = |
0.0000 |
|
|
Log likelihood = -3217.481 |
Pseudo R2 = |
0.1655 |
|
Observed |
Bookstrap |
Normal-based |
|||
|
y |
Coef. |
Std. Err. |
z |
P>|z| |
[95% Conf. Interval] |
|
2
|
|
3
|
(y==1 is the base outcome)
**using robust for the var-cov matrix
. mlogit y alc90th ue88 age agesq schooling married famsize white excellent ver > ygood good fair northeast midwest south centercity othermsa q1 q2 q3, baseout > come(1) vce(robust)
Iteration 0: log pseudolikelihood = -3855.7148 Iteration 1: log pseudolikelihood = -3692.5753 Iteration 2: log pseudolikelihood = -3526.5092 Iteration 3: log pseudolikelihood = -3236.3918 Iteration 4: log pseudolikelihood = -3219.1826 Iteration 5: log pseudolikelihood = -3217.5569 Iteration 6: log pseudolikelihood = -3217.4813 Iteration 7: log pseudolikelihood = -3217.481
Multinomial logistic regression Number of obs = 9822
Wald chi2 (40) = 1075.69
Prob > chi2 = 0.0000
Log pseudolikelihood = —3217.481 Pseudo R2 = 0.1655
|
Robust
(y==1 is the base outcome) |
b. For the female data, the multinomial logit estimates yield:
. mlogit y alc90th ue88 age agesq schooling married famsize white excellent verygood good fair northeast midwest south centercity othermsa q1 q2 q3,
|
- |
Coef. |
Std. Err. |
z |
P>|z| |
[95% Conf. Interval] |
|
|
2 alc90th |
-.1241993 |
.2365754 |
-0.52 |
0.600 |
-.5878785 |
.3394799 |
|
ue88 |
-.001862 |
.0514214 |
-0.04 |
0.971 |
-.1026462 |
.0989221 |
|
age |
-.0392239 |
.0612728 |
-0.64 |
0.522 |
-.1593164 |
.0808687 |
|
agesq |
.0004834 |
.0007411 |
0.65 |
0.514 |
-.0009691 |
.0019359 |
|
schooling |
-.0121174 |
.0254645 |
-0.48 |
0.634 |
-.0620269 |
.037792 |
|
married |
.0117958 |
.2220045 |
0.05 |
0.958 |
-.423325 |
.4469167 |
|
famsize |
.0092434 |
.0495871 |
0.19 |
0.852 |
-.0879456 |
.1064324 |
|
white |
.2817941 |
.1935931 |
1.46 |
0.146 |
-.0976414 |
.6612296 |
|
excellent |
.0420423 |
.4579618 |
0.09 |
0.927 |
-.8555463 |
.939631 |
|
verygood |
.0449091 |
.4574373 |
0.10 |
0.922 |
-.8516516 |
.9414698 |
|
good |
.0182444 |
.4544742 |
0.04 |
0.968 |
-.8725086 |
.9089974 |
|
fair |
.2925131 |
.4839658 |
0.60 |
0.546 |
-.6560424 |
1.241069 |
|
northeast |
-.1721726 |
.2163151 |
-0.80 |
0.426 |
-.5961425 |
.2517973 |
|
midwest |
-.2643294 |
.1944624 |
-1.36 |
0.174 |
-.6454687 |
.11681 |
|
south |
-.0161982 |
.1814209 |
-0.09 |
0.929 |
-.3717766 |
.3393803 |
|
centercity |
-.0812978 |
.1869101 |
-0.43 |
0.664 |
-.447635 |
.2850393 |
|
othermsa |
.044578 |
.1738872 |
0.26 |
0.798 |
-.2962347 |
.3853908 |
|
q1 |
22.30515 |
1.328553 |
16.79 |
0.000 |
19.70123 |
24.90906 |
|
q2 |
22.24068 |
1.32893 |
16.74 |
0.000 |
19.63603 |
24.84534 |
|
q3 |
18.65596 |
1.360765 |
13.71 |
0.000 |
15.98891 |
21.32301 |
|
_cons |
-23.50938 |
|||||
|
3 alc90th |
.1288509 |
.0812475 |
1.59 |
0.113 |
-.0303912 |
.2880931 |
|
ue88 |
.0148758 |
.0188237 |
0.79 |
0.429 |
-.022018 |
.0517696 |
|
age |
.0175243 |
.0230613 |
0.76 |
0.447 |
-.0276751 |
.0627236 |
|
agesq |
-.0002381 |
.00028 |
-0.85 |
0.395 |
-.0007868 |
.0003106 |
|
schooling |
.0035127 |
.0095824 |
0.37 |
0.714 |
-.0152685 |
.0222939 |
|
married |
-.0997914 |
.078327 |
-1.27 |
0.203 |
-.2533095 |
.0537266 |
|
famsize |
.0027002 |
.0184619 |
0.15 |
0.884 |
-.0334844 |
.0388849 |
|
white |
-.0277798 |
.066196 |
-0.42 |
0.675 |
-.1575217 |
.1019621 |
|
excellent |
-.1178398 |
.1636194 |
-0.72 |
0.471 |
-.4385278 |
.2028483 |
|
verygood |
-.1170045 |
.1633395 |
-0.72 |
0.474 |
-.437144 |
.203135 |
|
good |
-.1144024 |
.1622966 |
-0.70 |
0.481 |
-.4324979 |
.2036931 |
|
fair |
-.0344312 |
.1775054 |
-0.19 |
0.846 |
-.3823353 |
.3134729 |
|
northeast |
-.0548967 |
.0819514 |
-0.67 |
0.503 |
-.2155184 |
.105725 |
|
midwest |
.0572296 |
.0720545 |
0.79 |
0.427 |
-.0839946 |
.1984538 |
(y==1 is the base outcome)
13.17 Tobit estimation of Married Women Labor Supply
a. A detailed summary of the hours of work show that mean hours of work is
741, the median is 288, the minimum is zero and the maximum is 4950.
. sum hours, detail
hours worked, 1975
|
Percentiles |
Smallest |
|||
|
1% |
0 |
0 |
||
|
5% |
0 |
0 |
||
|
10% |
0 |
0 |
Obs |
753 |
|
25% |
0 |
0 |
Sum of Wgt. |
753 |
|
50% |
288 |
Mean |
740.5764 |
|
|
Largest |
Std. Dev. |
871.3142 |
||
|
75% |
1516 |
3640 |
||
|
90% |
1984 |
3686 |
Variance |
759188.5 |
|
95% |
2100 |
4210 |
Skewness |
.9225315 |
|
99% |
3087 |
4950 |
Kurtosis |
3.193949 |
b. Using the notation of solution 11.31, OLS on this model yields
. reg hours nwifeinc kidslt6 kidsge6 ‘control’ ‘E’, r
Linear regression Number of obs = 753
F(7,745) = 45.81
Prob > F = 0.0000
R-squared = 0.2656
Root MSE = 750.18
|
hours |
Coef. |
Robust Std. Err. |
t |
P>|t| |
[95% Conf. Interval] |
|
|
nwifeinc |
-3.446636 |
2.240662 |
-1.54 |
0.124 |
-7.845398 |
.9521268 |
|
kidslt6 |
-442.0899 |
57.46384 |
-7.69 |
0.000 |
-554.9002 |
-329.2796 |
|
kidsge6 |
-32.77923 |
22.80238 |
-1.44 |
0.151 |
-77.5438 |
11.98535 |
|
age |
-30.51163 |
4.244791 |
-7.19 |
0.000 |
-38.84481 |
-22.17846 |
|
educ |
28.76112 |
13.03905 |
2.21 |
0.028 |
3.163468 |
54.35878 |
|
exper |
65.67251 |
10.79419 |
6.08 |
0.000 |
44.48186 |
86.86316 |
|
expersq |
-.7004939 |
.3720129 |
-1.88 |
0.060 |
-1.430812 |
.0298245 |
|
_cons |
1330.482 |
274.8776 |
4.84 |
0.000 |
790.8556 |
1870.109 |
|
Tobit estimation with left censoring at zero is represented by the option ll(0) |
. tobit hours nwifeinc kidslt6 kidsge6 ‘control’ ‘E’, ll(0)
|
Tobit regression |
Number of obs |
= 753 |
|
LR chi2 (7) |
= 271.59 |
|
|
Prob > chi2 |
= 0.0000 |
|
|
Log likelihood = -3819.0946 |
Pseudo R2 |
= 0.0343 |
|
hours |
Coef. |
Std. Err. |
t |
P>|t| |
[95% Conf. Interval] |
|
|
nwifeinc |
-8.814243 |
4.459096 |
-1.98 |
0.048 |
-17.56811 |
-.0603724 |
|
kidslt6 |
-894.0217 |
111.8779 |
-7.99 |
0.000 |
-1113.655 |
-674.3887 |
|
kidsge6 |
-16.218 |
38.64136 |
-0.42 |
0.675 |
-92.07675 |
59.64075 |
|
age |
-54.40501 |
7.418496 |
-7.33 |
0.000 |
-68.96862 |
-39.8414 |
|
educ |
80.64561 |
21.58322 |
3.74 |
0.000 |
38.27453 |
123.0167 |
|
exper |
131.5643 |
17.27938 |
7.61 |
0.000 |
97.64231 |
165.4863 |
|
expersq |
-1.864158 |
.5376615 |
-3.47 |
0.001 |
-2.919667 |
-.8086479 |
|
_cons |
965.3053 |
446.4358 |
2.16 |
0.031 |
88.88528 |
1841.725 |
|
/sigma |
1122.022 |
41.57903 |
1040.396 |
1203.647 |
Obs. summary: 325 left-censored observations at hours<=0
428 uncensored observations 0 right-censored observations
c. This replicates Table 17.1 of Wooldridge (2009, p. 585) using Stata
. reg inlf nwifeinc kidslt6 kidsge6 ‘control’ ‘E’, r
Linear regression Number of obs = 753
F(7,745) = 62.48
Prob > F = 0.0000
R-squared = 0.2642
Root MSE = .42713
|
inlf |
Coef. |
Robust Std. Err. |
t |
P>|t| |
[95% Conf. Interval] |
|
|
nwifeinc |
-.0034052 |
.0015249 |
-2.23 |
0.026 |
-.0063988 |
-.0004115 |
|
kidslt6 |
-.2618105 |
.0317832 |
-8.24 |
0.000 |
-.3242058 |
-.1994152 |
|
kidsge6 |
.0130122 |
.0135329 |
0.96 |
0.337 |
-.013555 |
.0395795 |
|
age |
-.0160908 |
.002399 |
-6.71 |
0.000 |
-.0208004 |
-.0113812 |
|
educ |
.0379953 |
.007266 |
5.23 |
0.000 |
.023731 |
.0522596 |
|
exper |
.0394924 |
.00581 |
6.80 |
0.000 |
.0280864 |
.0508983 |
|
expersq |
-.0005963 |
.00019 |
-3.14 |
0.002 |
-.0009693 |
-.0002233 |
|
_cons |
.5855192 |
.1522599 |
3.85 |
0.000 |
.2866098 |
.8844287 |
The Logit estimates yield:
. logit inlf nwifeinc kidslt6 kidsge6 ‘control’ ‘E’, r Iteration 0: log pseudolikelihood = -514.8732 Iteration 1: log pseudolikelihood = -402.38502 Iteration 2: log pseudolikelihood = -401.76569 Iteration 3: log pseudolikelihood = -401.76515 Iteration 4: log pseudolikelihood = -401.76515
|
Logistic regression |
Number of obs |
= 753 |
|
Wald chi2 (7) |
= 158.48 |
|
|
Prob > chi2 |
= 0.0000 |
|
|
Log pseudolikelihood = -401.76515 |
Pseudo R2 |
= 0.2197 |
|
inlf |
Coef. |
Robust Std. Err. |
z |
P>|z| |
[95% Conf. Interval] |
|
|
nwifeinc |
-.0213452 |
.0090782 |
-2.35 |
0.019 |
-.039138 |
-.0035523 |
|
kidslt6 |
-1.443354 |
.2031615 |
-7.10 |
0.000 |
-1.841543 |
-1.045165 |
|
kidsge6 |
.0601122 |
.0798825 |
0.75 |
0.452 |
-.0964546 |
.2166791 |
|
age |
-.0880244 |
.0144393 |
-6.10 |
0.000 |
-.1163248 |
-.0597239 |
|
educ |
.2211704 |
.0444509 |
4.98 |
0.000 |
.1340482 |
.3082925 |
|
exper |
.2058695 |
.0322914 |
6.38 |
0.000 |
.1425796 |
.2691594 |
|
expersq |
-.0031541 |
.0010124 |
-3.12 |
0.002 |
-.0051384 |
-.0011698 |
|
_cons |
.4254524 |
.8597308 |
0.49 |
0.621 |
-1.259589 |
2.110494 |
. estat classification
Logistic model for inlf
|
Classified |
D |
~D |
Total |
|
+ |
347 |
118 |
465 |
|
- |
81 |
207 |
288 |
|
True- |
|
Total |
|
428 |
|
325 I 753 |
Classified + if predicted Pr(D) >= .5 True D defined as inlf!= 0
|
Sensitivity Specificity Positive predictive value Negative predictive value |
Pr(+I D) Pr(-| ~D) Pr(D| +) Pr(~D| -) |
81.07% 63.69% 74.62% 71.88% |
|
False + rate for true ~D |
Pr(+I ~D) |
36.31% |
|
False - rate for true D |
Pr(- D) |
18.93% |
|
False + rate for classified + |
Pr(~D| +) |
25.38% |
|
False - rate for classified - |
Pr( Dj -) |
28.13% |
|
Correctly classified |
73.57% |
|
. mfx |
|
Marginal effects after logit y = Pr(inlf) (predict) = .58277201
|
|
Average partial effects after logit y = Pr(inlf) |
||||||
|
variable |
Coef. |
Std. Err. |
z |
P>jzj |
[95% Conf. Interval] |
|
|
nwifeinc |
-.0038118 |
.0015923 |
-2.39 |
0.017 |
-.0069327 |
-.0006909 |
|
kidslt6 |
-.240805 |
.0262576 |
-9.17 |
0.000 |
-.292269 |
-.189341 |
|
kidsge6 |
.0107335 |
.0142337 |
0.75 |
0.451 |
-.017164 |
.038631 |
|
age |
-.0157153 |
.0023842 |
-6.59 |
0.000 |
-.0203883 |
-.0110423 |
|
educ |
.0394323 |
.0074566 |
5.29 |
0.000 |
.0248176 |
.0540471 |
|
exper |
.0367123 |
.0051935 |
7.07 |
0.000 |
.0265332 |
.0468914 |
|
expersq |
-.0005633 |
.0001767 |
-3.19 |
0.001 |
-.0009096 |
-.0002169 |
. probit inlf nwifeinc kidslt6 kidsge6 ‘control’ ‘E’, r Iteration 0: log pseudolikelihood = -514.8732 Iteration 1: log pseudolikelihood = -402.06651 Iteration 2: log pseudolikelihood = -401.30273 Iteration 3: log pseudolikelihood = -401.30219 Iteration 4: log pseudolikelihood = -401.30219
|
Probit regression |
Number of obs |
= 753 |
|
Wald chi2 (7) |
= 185.10 |
|
|
Prob > chi2 |
= 0.0000 |
|
|
Log pseudolikelihood = -401.30219 |
Pseudo R2 |
= 0.2206 |
|
inlf |
Coef. |
Robust Std. Err. |
z |
P>|z| |
[95% Conf. Interval] |
|
|
nwifeinc |
-.0120237 |
.0053106 |
-2.26 |
0.024 |
-.0224323 -.0016152 |
|
|
kidslt6 |
-.8683285 |
.1162037 |
-7.47 |
0.000 |
-1.096084 -.6405735 |
|
|
kidsge6 |
.036005 |
.0452958 |
0.79 |
0.427 |
-.0527731 .124783 |
|
|
age |
-.0528527 |
.0083532 |
-6.33 |
0.000 |
-.0692246 -.0364807 |
|
|
educ |
.1309047 |
.0258192 |
5.07 |
0.000 |
.0803 .1815095 |
|
|
exper |
.1233476 |
.0188537 |
6.54 |
0.000 |
.086395 .1603002 |
|
|
expersq |
-.0018871 |
.0006007 |
-3.14 |
0.002 |
-.0030645 -.0007097 |
|
|
_cons |
.2700768 |
.505175 |
0.53 |
0.593 |
-.7200481 1.260202 |
|
|
. mfx |
||||||
|
Marginal effects after probit |
||||||
|
y = Pr(inlf) (predict) |
||||||
|
= .58154201 |
||||||
|
variable |
dy/dx |
Std. Err. |
z |
P>|z| |
[95% C. I.] |
X |
|
nwifeinc |
-.0046962 |
.00208 |
-2.26 |
0.024 |
-.008766 -.000626 |
20.129 |
|
kidslt6 |
-.3391514 |
.04565 |
-7.43 |
0.000 |
-.428628 -.249675 |
.237716 |
|
kidsge6 |
.0140628 |
.01769 |
0.80 |
0.427 |
-.020603 .048729 |
1.35325 |
|
age |
-.0206432 |
.00327 |
-6.31 |
0.000 |
-.027056 -.014231 |
42.5378 |
|
educ |
.0511287 |
.01011 |
5.06 |
0.000 |
.031308 .07095 |
12.2869 |
|
exper |
.0481771 |
.00739 |
6.52 |
0.000 |
.033694 .06266 |
10.6308 |
|
expersq |
-.0007371 |
.00024 |
-3.14 |
0.002 |
-.001198 -.000276 |
178.039 |
|
margeff |
Average partial effects after probit y = Pr(inlf)
|
Variable |
Coef. |
Std. Err. |
z P>|z| |
[95% Conf. Interval] |
|
|
nwifeinc |
-.0036162 |
.0015759 |
-2.29 0.022 |
-.0067049 -.0005275 |
|
|
kidslt6 |
-.2441788 |
.0257356 |
-9.49 0.000 |
-.2946198 -.1937379 |
|
|
kidsge6 |
.0108274 |
.0135967 |
0.80 0.426 |
-.0158217 .0374765 |
|
|
age |
-.0158917 |
.0023447 |
-6.78 0.000 |
-.0204873 -.011296 |
|
|
educ |
.0393088 |
.0073669 |
5.34 0.000 |
.02487 .0537476 |
|
|
exper |
.037046 |
.0051959 |
7.13 0.000 |
.0268621 .0472299 |
|
|
expersq |
-.0005675 |
.0001775 |
-3.20 0.001 |
-.0009154 -.0002197 |
|
|
. dprobit inlf nwifeinc kidslt6 kidsge6 ‘control' ‘E', r |
|||||
|
Iteration 0: log pseudolikelihood = -514.8732 |
|||||
|
Iteration 1: log pseudolikelihood = -405.78215 |
|||||
|
Iteration 2: log pseudolikelihood = -401.32924 |
|||||
|
Iteration 3: log pseudolikelihood = -401.30219 |
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|
Iteration 4: log pseudolikelihood = -401.30219 |
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|
Probit regression, reporting marginal effects |
Number of obs = 753 |
||||
|
Wald chi2 (7) = 185.10 |
|||||
|
Prob > chi2 = 0.0000 |
|||||
|
Log pseudolikelihood = —401.30219 |
Pseudo R2 = 0.2206 |
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|
Robust |
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|
inlf |
dF/dx |
Std. Err. |
z P>|z| |
x-bar [ 95% C. I. ] |
|
|
nwifeinc |
-.0046962 |
.0020767 |
-2.26 0.024 |
20.129 -.008766 -.000626 |
|
|
kidslt6 |
-.3391514 |
.045652 |
-7.47 0.000 |
.237716 -.428628 -.249675 |
|
|
kidsge6 |
.0140628 |
.0176869 |
0.79 0.427 |
1.35325 -.020603 .048729 |
|
|
age |
-.0206432 |
.0032717 |
-6.33 0.000 |
42.5378 -.027056 -.014231 |
|
|
educ |
.0511287 |
.010113 |
5.07 0.000 |
12.2869 .031308 .07095 |
|
|
exper |
.0481771 |
.0073896 |
6.54 0.000 |
10.6308 .033694 .06266 |
|
|
expersq |
-.0007371 |
.000235 |
-3.14 0.002 |
178.039 -.001198 -.000276 |
|
|
obs. P |
.5683931 |
||||
|
pred. P |
.581542 |
(at x-bar) |
|
z and P> |z| correspond to the test of the underlying coefficient being 0 |
. estat classification
Probit model for inlf
|
True-
|
Classified + if predicted Pr(D) >= .5 True D defined as inlf!= 0
|
Sensitivity Specificity Positive predictive value Negative predictive value |
Pr(+| D) Pr(-| ~D) Pr(D| +) Pr(-D| -) |
81.31% 63.08% 74.36% 71.93% |
|
False + rate for true —D |
Pr(+| ~D) |
36.92% |
|
False - rate for true D |
Pr(-| D) |
18.69% |
|
False + rate for classified + |
Pr(-D| +) |
25.64% |
|
False - rate for classified - |
Pr(D| -) |
28.07% |
|
Correctly classified |
73.44% |
d. Wooldridge (2009, Chapter 17) recommends one obtain the estimates of (fi/a2) from a probit using an indicator of labor force participation. Then comparing those with the Tobit estimates generated by dividing fi by a2. If these estimates are different or have different signs, then the Tobit estimation may not be appropriate. Part (c) gave such probit estimates. For (kidslt6) this was estimated at —0.868. From part (b) the tobit estimation gave a fi estimate for (kidslt6) of —894 and an estimate of a2 of 1122. The resulting estimate of (fi/a2) is —0.797. These have the same sign but with different magnitudes.
13.18 Heckit Estimation of Married Women’s Earnings
a. OLS on this model yields
|
. reg Iwage educ exper expersq
|
b. The inverse mills ratio coefficient lambda is estimated to be.032 with a standard error of 0.134 which is not significant. This does not reject the
null hypothesis of no sample selection.
c. The MLE of this Heckman (1976) sample selection model.
. heckman Iwage educ exper expersq, select (educ exper expersq age kidslt6 kidsge6 nwifeinc)
Iteration 0: log likelihood = -832.89776 Iteration 1: log likelihood = -832.88509 Iteration 2: log likelihood = -832.88508
|
Heckman selection model |
Number of obs = |
753 |
|
(regression model with sample selection) |
Censored obs = |
325 |
|
Uncensored obs = |
428 |
|
|
Wald chi2(3) = |
59.67 |
|
|
Log likelihood = -832.8851 |
Prob > chi2 = |
0.0000 |
|
lwage |
Coef. |
Std. Err. |
z |
P>|z| |
[95% Conf. Interval] |
|
|
lwage |
||||||
|
educ |
.1083502 |
.0148607 |
7.29 |
0.000 |
.0792238 |
.1374767 |
|
exper |
.0428369 |
.0148785 |
2.88 |
0.004 |
.0136755 |
.0719983 |
|
expersq |
-.0008374 |
.0004175 |
-2.01 |
0.045 |
-.0016556 |
-.0000192 |
|
_cons |
-.5526973 |
.2603784 |
-2.12 |
0.034 |
-1.06303 |
-.0423651 |
|
select |
||||||
|
educ |
.1313415 |
.0253823 |
5.17 |
0.000 |
.0815931 |
.1810899 |
|
exper |
.1232818 |
.0187242 |
6.58 |
0.000 |
.0865831 |
.1599806 |
|
expersq |
-.0018863 |
.0006004 |
-3.14 |
0.002 |
-.003063 |
-.0007095 |
|
age |
-.0528287 |
.0084792 |
-6.23 |
0.000 |
-.0694476 |
-.0362098 |
|
kidslt6 |
-.8673988 |
.1186509 |
-7.31 |
0.000 |
-1.09995 |
-.6348472 |
|
kidsge6 |
.0358723 |
.0434753 |
0.83 |
0.409 |
-.0493377 |
.1210824 |
|
nwifeinc |
-.0121321 |
.0048767 |
-2.49 |
0.013 |
-.0216903 |
-.002574 |
|
_cons |
.2664491 |
.5089578 |
0.52 |
0.601 |
-.7310898 |
1.263988 |
|
/athrho |
.026614 |
.147182 |
0.18 |
0.857 |
-.2618573 |
.3150854 |
|
/lnsigma |
-.4103809 |
.0342291 |
-11.99 |
0.000 |
-.4774687 |
-.3432931 |
|
rho sigma lambda |
.0266078 .6633975 .0176515 |
.1470778 .0227075 .0976057 |
-.2560319 .6203517 -.1736521 |
.3050564 .7094303 .2089552 |
||
|
LR test of indep. eqns. (rho |
= 0): |
chi2(1) = |
0.03 Prob > chi2 = |
0.8577 |
This yields the same results as the two-step Heckman procedure and the LR test for (rho = 0) is not significant.
References
Wooldridge, J. M. (2009), Introductory Econometrics: A Modern Approach (SouthWestern: Ohio).
CHAPTER 14
