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Assume you have a model of the form $y = x_1\Phi(\beta_0 + \beta_2x_2 +\ldots+ \beta_nx_n)+u$ where $y_i\in(0,x_{1i}]$ and $\Phi$ is the probit function.

How can we estimate $\beta$s in e.g. Stata?

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    $\begingroup$ I am a little confused about this model since I don't know what error distribution would be sensible - you'd need to do something strange, or transform the response, to keep the $y \in (0,x_1]$ constraint. Anyway, I doubt this model is pre-implemented in any software package so you'll probably have to "make your own". If you know the distribution of $u$, you can calculate the (log) density of $y|x_1,...,x_n$ and maximize it as a function of $\beta_0, ...., \beta_n$ to get the maximum likelihood estimate. $\endgroup$ – Macro Feb 19 '13 at 16:58
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    $\begingroup$ maybe it's more helpful to think about it in a standard probit sense. let $y^*=y/x_1$. the model $y^*=\Phi(\beta_0+\beta_2x_2+\ldots+\beta_nx_n)+u$ is more natural in this case, as $y^*\in(0,1]$. we can estimate this using probit in stata. instead, im concerned with the case where instead we are concerned with separating the numerator and denominator of $y^*$...hope that helps $\endgroup$ – sam Feb 19 '13 at 17:02
  • $\begingroup$ in the model i posit, $E(u|x_1,\ldots,x_n)=0$. $\endgroup$ – sam Feb 19 '13 at 17:04
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    $\begingroup$ The usual probit model arises when you assume some underlying latent variable follows a usual Gaussian-error regression model and that latent variable is binned into categories. When there are only two categories and the threshold at $0$ separates them, you have the usual binary probit regression model. With more thresholds (and therefore more categories) you have the ordinal probit model. I'm failing to see the connect between that and this. What am I missing? $\endgroup$ – Macro Feb 19 '13 at 17:07
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    $\begingroup$ In that case, I'm not sure what this model buys you. You may want to look into beta regression and, if you have the number of "successes" and "failures" that produced the proportion, then you could use binomial regression (such as probit). $\endgroup$ – Macro Feb 19 '13 at 17:11
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For proportions as the outcome, take a look at Chris Baum's Stata Tip #63. Another related approach is user-written betafit from SSC. For many of these, the two extremes of 0 and 1 will be problematic. The glm method discussed by Baum is a notable exception.

Personally, I would avoid going the nonlinear least squares route with nl, but that may be feasible. In any case, you can do that in the following way (toy example):

use http://www.ats.ucla.edu/stat/data/hsbdemo, clear
nl (honors = math * normal({_cons} + {read}*read +{female}*female)), robust

One issue for comparisons is that ordinary probit estimates will not exactly match nl, even without using math as $x_1$:

. probit honors read female, nolog

Probit regression                                 Number of obs   =        200
                                                  LR chi2(2)      =      61.31
                                                  Prob > chi2     =     0.0000
Log likelihood = -84.990569                       Pseudo R2       =     0.2651

------------------------------------------------------------------------------
      honors |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
        read |   .0856048   .0130065     6.58   0.000     .0601126    .1110971
      female |   .6340312   .2300876     2.76   0.006     .1830678    1.084995
       _cons |  -5.672047   .7798024    -7.27   0.000    -7.200432   -4.143662
------------------------------------------------------------------------------

. nl (honors = normal({_cons} + {read}*read +{female}*female)), robust nolog
(obs = 200)

Nonlinear regression                                 Number of obs =       200
                                                     R-squared     =    0.4734
                                                     Adj R-squared =    0.4653
                                                     Root MSE      =   .376412
                                                     Res. dev.     =  173.7243

------------------------------------------------------------------------------
             |               Robust
      honors |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      /_cons |  -5.277098   .8583791    -6.15   0.000    -6.969889   -3.584307
       /read |   .0782903   .0138473     5.65   0.000     .0509824    .1055982
     /female |   .6814231   .2597361     2.62   0.009      .169203    1.193643
------------------------------------------------------------------------------
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  • $\begingroup$ I'd be interested in you elaborating as to why you don't like nl. That would be the first thing I would try. $\endgroup$ – StasK Feb 19 '13 at 21:32
  • $\begingroup$ Non-linear estimation outside toy examples always makes me nervous. With the probit, the log-likelihood is globally concave, so numerical methods should quickly converge to a unique optimum. I am worried that multiplying by $x_1$, a variable that we have no information on, may mess things up here. $\endgroup$ – Dimitriy V. Masterov Feb 20 '13 at 0:06
  • $\begingroup$ Dimitriy, I would say that these are different enough models to diverge in small samples ($n=200$ ain't a terrific sample size). nl is assumes that the errors are homoskedastic around the mean (given by the probit), while probit is smarter and knows that they aren't. I am actually more worried about the standard errors here -- I would argue that you need to use nl, robust standard errors here (and in pretty much any other nl analysis, I dare say). $\endgroup$ – StasK Feb 21 '13 at 16:44

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