Limited Dependent Variables
- PDF / 329,857 Bytes
- 49 Pages / 439.36 x 666.15 pts Page_size
- 86 Downloads / 185 Views
Limited Dependent Variables
13.1 The Linear Probability Model yi ui Prob. 0 1 1 xi “ i 0 x0i “ 1 i a. Let i D PrŒyi D 1, then yi D 1 when ui D 1 x0i “ with probability i as shown in the table above. Similarly, yi D 0 when ui D x0i “ with probability 1 i . Hence, E.ui / D i 1 x0i “ C .1 i / x0i “ . For this to equal zero, we get, i i x0i “ C i x0i “ x0i “ D 0 which gives i D x0i “ as required. 0 2 2 0 2 b. var.ui / D E.ui / D 1 xi “ i C xi “ .1 i / h 2 i 2 D 1 2x0i “ C x0i “ i C x0i “ .1 i / D i 2x0i “ i C .x0i “/2 D i 2i D i .1 i / D x0i “ 1 x0i “ using the fact that i D x0i “. 13.2 a. Since there are no slopes and only a constant, x0i “ D ’ and (13.16) becomes n P log ` D fyi log F.’/C.1yi/ logŒ1F.’/g differentiating with respect iD1
to ’ we get X yi X .1 yi / @ log ` D f.’/ C .f.’//. @’ F.’/ 1 F.’/ n
n
iD1
iD1
Setting this equal to zero yields O Therefore, F.’/ D yi D 1.
n P
n P
.yi F.’//f.’/ D 0.
iD1
yi =n D yN . This is the proportion of the sample with
iD1
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
333
334
Badi Baltagi
O b. Using F.’/ D yN , the value of the maximized likelihood, from (13.16), is log `r D
n X
fyi log yN C .1yi / log.1Ny/g D nNy log yN C .nnNy/ log.1Ny/
iD1
D nŒNy log yN C .1 yN / log.1 yN /
as required.
c. For the empirical example in Sect. 13.9, we know that yN D 218=595 D 0.366. Substituting in (13.33) we get, log `r D nŒ0.366 log 0.366 C .1 0.366/ log.1 0.366/ D 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 EX WKS OCC IND SOUTH SMSA MS FEM ED BLK
1.195872 -0.001974 -0.017809 0.318118 0.030048 -0.170130 0.084522 0.098953 -0.108706 -0.016187 0.050197
0.227010 0.001726 0.003419 0.046425 0.038072 0.039801 0.038464 0.063781 0.079266 0.008592 0.071130
5.267922 -1.143270 -5.209226 6.852287 0.789229 -4.274471 2.197419 1.551453 -1.371398 -1.883924 0.705708
0.0000 0.2534 0.0000 0.0000 0.4303 0.0000 0.0284 0.1213 0.1708 0.0601 0.4807
R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat
0.233548 0.220424 0.425771 105.8682 -330.6745 1.900963
Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion F-statistic Prob(F-statistic)
0.366387 0.482222 -1.689391 -1.608258 17.79528 0.000000
Chapter 13: Limited Dependent Variables
335
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 EX WKS OCC IND SOUTH SMSA MS FEM ED BLK
4.380828 -0.011143 -0.108126 1.658222 0.181818 -1.044332 0.448389 0.604999 -0.772222 -0.090799 0.355706
1.338629 0.009691 0.02
Data Loading...
