[Limdep Nlogit List] probit model -- mathematical definition
Erik Ferguson
eferguson at aus.edu
Fri Feb 1 20:34:44 EST 2008
The probit model is based on a 'cumulative distribution function' which
is rarely represented in discrete mathematical form. I have seen at
least one equation on the Internet that appears to be a partial
expansion of a Taylor series. Can anyone point me to a reputable source
on the derivation of this formula and its true and correct expression in
mathematical form? Thanks.
Erik Ferguson
Urban Planning Graduate Program
School of Architecture and Design
American University of Sharjah
PO Box 26666
Sharjah, UNITED ARAB EMIRATES
eferguson at aus.edu
971 6 515 2878 office
971 6 515 2800 fax
971 50 369 8355 mobile
Peter Glick wrote:
> Hello,
>
> I am using the 'cluster' option in Nlogit4 with an ordered probit
> model of years of schooling. The clusters are schools. The adjusted
> standard errors make sense as long as I don't also allow for
> censoring arising from some in the sample still being in school.
> Allowing for censoring without the adjustment is fine, but when
> 'Cluster' is used the program seems unable to compute the variances
> (see model output below: many coefficient standard errors are missing
> and labeled 'fixed parameters').
>
> Is there a logical reason for this (and any solution)? My thinking
> was that the censoring would somehow be affecting the number of
> within-cluster observations used to correct the standard errors, but I
> can't see why this would be the case. (there are 14 individuals on
> average per cluster, and censoring is high, over 60% of the sample).
>
> Thanks for any insight.
>
> Peter Glick
>
> Below, the model output without using the cluster option follows the
> model with the option.
>
>
> Normal exit from iterations. Exit status=0.
> +---------------------------------------------+
> | Ordered Probability Model |
> | Maximum Likelihood Estimates |
> | Model estimated: Jan 31, 2008 at 03:37:01PM.|
> | Dependent variable GRADATT1 |
> | Weighting variable None |
> | Number of observations 834 |
> | Iterations completed 49 |
> | Log likelihood function -879.4733 |
> | Number of parameters 38 |
> | Info. Criterion: AIC = 2.20018 |
> | Finite Sample: AIC = 2.20465 |
> | Info. Criterion: BIC = 2.41552 |
> | Info. Criterion:HQIC = 2.28274 |
> | Restricted log likelihood -1547.237 |
> | McFadden Pseudo R-squared .4315847 |
> | Chi squared 1335.528 |
> | Degrees of freedom 30 |
> | Prob[ChiSqd > value] = .0000000 |
> | Underlying probabilities based on Normal |
> +---------------------------------------------+
>
> +---------------------------------------------+
> | Ordered Probability Model |
> | Cell frequencies for outcomes |
> | Y Count Freq Y Count Freq Y Count Freq |
> | 0 1 .002 1 18 .021 2 46 .055 |
> | 3 74 .088 4 134 .160 5 263 .315 |
> | 6 108 .129 7 144 .172 8 45 .053 |
> | Censoring indicator is DROPOUT |
> | Total observations = 834.0 |
> | Uncensored = 306.0, censored = 528.0 |
> +---------------------------------------------+
> +---------------------------------------------------------------------+
> | Covariance matrix for the model is adjusted for data clustering. |
> | Sample of 834 observations contained 60 clusters defined by |
> | variable NUMECOLE which identifies by a value a cluster ID. |
> | Sample of 834 observations contained 1 strata defined by |
> | 834 observations (fixed number) in each stratum. |
> +---------------------------------------------------------------------+
> +--------+--------------+----------------+--------+--------+----------+
> |Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]| Mean of X|
> +--------+--------------+----------------+--------+--------+----------+
> ---------+Index function for probability
> Constant| 8.13441449 .97235688 8.366 .0000
> AGE1 | -.31250626 .13458868 -2.322 .0202 15.6175060
> GIRL1 | -.26554081 ......(Fixed Parameter).......
> RICH | .33973482 .01153289 29.458 .0000 -.02167237
> MISEDM | -.21837324 .10648913 -2.051 .0403 .10071942
> MOMED | -.00288493 .00580012 -.497 .6189 1.52673674
> DADED | .02934360 ......(Fixed Parameter).......
> MISEDFHH| -.26316948 .10422245 -2.525 .0116 .16546763
> REGION1 | -.63593108 ......(Fixed Parameter).......
> REGION2 | -.18266061 .12762061 -1.431 .1524 .05755396
> REGION3 | .05705597 .44192474 .129 .8973 .05875300
> REGION4 | -.15384265 .35685256 -.431 .6664 .11510791
> REGION5 | -.10319191 .32027725 -.322 .7473 .11630695
> REGION6 | -.15925307 .34317630 -.464 .6426 .04916067
> REGION8 | .02918756 ......(Fixed Parameter).......
> ETHN2 | .00497250 ......(Fixed Parameter).......
> ETHN3 | .39856760 .14538589 2.741 .0061 .21223022
> ETHN4 | .72539671 ......(Fixed Parameter).......
> ETHN5 | .29420208 ......(Fixed Parameter).......
> ETHN6 | -.05300107 .18236991 -.291 .7713 .04436451
> RURALF | -.27918343 ......(Fixed Parameter).......
> YRSDR95 | .00880528 ......(Fixed Parameter).......
> EDDRBAC1| -.03796261 ......(Fixed Parameter).......
> MISDIR95| 1.14741544 .62372666 1.840 .0658 .02038369
> YRSTCHR | -.00114542 ......(Fixed Parameter).......
> TCHBAC | .30397634 ......(Fixed Parameter).......
> FTSHARE | -.31230345 .10813592 -2.888 .0039 .24280575
> GIRL_FT | .34810059 .35422595 .983 .3258 .10971223
> F1_SUPPL| .04794771 .07133228 .672 .5015 .00838170
> F1_FAC1 | -.01430482 .07753319 -.184 .8536 -.06643672
> DISTC | .00036505 .00857017 .043 .9660 6.91998082
> ---------+Threshold parameters for index
> Mu(1) | .93564915 .14997002 6.239 .0000
> Mu(2) | 1.57033181 .10105539 15.539 .0000
> Mu(3) | 1.97796974 .06884951 28.729 .0000
> Mu(4) | 2.40797842 .08186716 29.413 .0000
> Mu(5) | 3.01425959 .12735666 23.668 .0000
> Mu(6) | 3.02337926 .12906982 23.424 .0000
> Mu(7) | 3.13413989 .14681578 21.347 .0000
>
>
>
>
> +---------------------------------------------+
> | Ordered Probability Model |
> | Maximum Likelihood Estimates |
> | Model estimated: Jan 31, 2008 at 03:36:50PM.|
> | Dependent variable GRADATT1 |
> | Weighting variable None |
> | Number of observations 834 |
> | Iterations completed 49 |
> | Log likelihood function -879.4733 |
> | Number of parameters 38 |
> | Info. Criterion: AIC = 2.20018 |
> | Finite Sample: AIC = 2.20465 |
> | Info. Criterion: BIC = 2.41552 |
> | Info. Criterion:HQIC = 2.28274 |
> | Restricted log likelihood -1547.237 |
> | McFadden Pseudo R-squared .4315847 |
> | Chi squared 1335.528 |
> | Degrees of freedom 30 |
> | Prob[ChiSqd > value] = .0000000 |
> | Underlying probabilities based on Normal |
> +---------------------------------------------+
>
> | Total observations = 834.0 |
> | Uncensored = 306.0, censored = 528.0 |
> +---------------------------------------------+
> +--------+--------------+----------------+--------+--------+----------+
> |Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]| Mean of X|
> +--------+--------------+----------------+--------+--------+----------+
> ---------+Index function for probability
> Constant| 8.13441449 .87643948 9.281 .0000
> AGE1 | -.31250626 .05308666 -5.887 .0000 15.6175060
> GIRL1 | -.26554081 .12228330 -2.172 .0299 .41247002
> RICH | .33973482 .08221790 4.132 .0000 -.02167237
> MISEDM | -.21837324 .19147133 -1.141 .2541 .10071942
> MOMED | -.00288493 .02098014 -.138 .8906 1.52673674
> DADED | .02934360 .01423680 2.061 .0393 2.70019039
> MISEDFHH| -.26316948 .15716786 -1.674 .0940 .16546763
> REGION1 | -.63593108 .18051299 -3.523 .0004 .18705036
> REGION2 | -.18266061 .22508431 -.812 .4171 .05755396
> REGION3 | .05705597 .24942158 .229 .8191 .05875300
> REGION4 | -.15384265 .21546830 -.714 .4752 .11510791
> REGION5 | -.10319191 .17454227 -.591 .5544 .11630695
> REGION6 | -.15925307 .23560296 -.676 .4991 .04916067
> REGION8 | .02918756 .21983493 .133 .8944 .06714628
> ETHN2 | .00497250 .13954754 .036 .9716 .19664269
> ETHN3 | .39856760 .15829931 2.518 .0118 .21223022
> ETHN4 | .72539671 .25392708 2.857 .0043 .06115108
> ETHN5 | .29420208 .17842763 1.649 .0992 .12110312
> ETHN6 | -.05300107 .22580284 -.235 .8144 .04436451
> RURALF | -.27918343 .18820756 -1.483 .1380 .56115108
> YRSDR95 | .00880528 .00672356 1.310 .1903 12.4088729
> EDDRBAC1| -.03796261 .12763570 -.297 .7661 .53965411
> MISDIR95| 1.14741544 .45896995 2.500 .0124 .02038369
> YRSTCHR | -.00114542 .00847133 -.135 .8924 11.5313749
> TCHBAC | .30397634 .14449243 2.104 .0354 .42865708
> FTSHARE | -.31230345 .18342327 -1.703 .0886 .24280575
> GIRL_FT | .34810059 .24338114 1.430 .1526 .10971223
> F1_SUPPL| .04794771 .03781105 1.268 .2048 .00838170
> F1_FAC1 | -.01430482 .05127098 -.279 .7802 -.06643672
> DISTC | .00036505 .00532954 .068 .9454 6.91998082
> ---------+Threshold parameters for index
> Mu(1) | .93564915 .11284646 8.291 .0000
> Mu(2) | 1.57033181 .07560000 20.772 .0000
> Mu(3) | 1.97796974 .06318754 31.303 .0000
> Mu(4) | 2.40797842 .05591468 43.065 .0000
> Mu(5) | 3.01425959 .05834226 51.665 .0000
> Mu(6) | 3.02337926 .05865874 51.542 .0000
> Mu(7) | 3.13413989 .07964772 39.350 .0000
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