Estimating a constrained hackman model

[Ben Thompson]

Hello,

I'm interested in estimating a Heckman selection model via maximum likelihood (if possible via the "hackman" command), however, I'd also like to impose a constraint on the variables being estimated (specifically on the beta's and the rho being estimated). Is it possible to impose such constraints using the constraints option? If not, is there a work-around?

Thanks.

[Alfonso Sánchez-Peñalver]

Hi Ben,

you don't provide much detail about what you want to do exactly, so since the question is general, the answer is also general: yes the heckman maximum likelihood estimator in Stata allows for linear constraints of the parameters being estimated. See help heckman and scroll down to heckman_ml_options.

[Ben Thompson]

Hi, my apologies. What I meant was that I want to impose a constraint in the form of a relationship between the variables being estimated -- i.e. I would like to impose the condition that the rho is actually a function of the betas. Is this possible within the heckman command?

[Alfonso Sánchez-Peñalver]

Ben, yes. Consider the following:

clear all
set more off
webuse womenwk

const 1 [athrho]_cons = 0.84 * ([wage]education + [select]education)
heckman wage educ age, select(married children educ age) const(1)

di .84*(_b[wage:education] + _b[select:education])

This produces the following results:

Heckman selection model                         Number of obs      =      2000
(regression model with sample selection)        Censored obs       =       657
                                                Uncensored obs     =      1343

                                                Wald chi2(1)       =    122.12
Log likelihood = -5178.305                      Prob > chi2        =    0.0000

 ( 1)  - .84*[wage]education - .84*[select]education + [athrho]_cons = 0
------------------------------------------------------------------------------
        wage |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
wage         |
   education |    .989368   .0511711    19.33   0.000     .8890746    1.089661
         age |   .2134143   .0193119    11.05   0.000     .1755637    .2512649
       _cons |    .476678   1.051715     0.45   0.650    -1.584646    2.538002
-------------+----------------------------------------------------------------
select       |
     married |    .445116   .0673295     6.61   0.000     .3131526    .5770795
    children |    .438425   .0268442    16.33   0.000     .3858114    .4910386
   education |   .0556047   .0102393     5.43   0.000      .035536    .0756734
         age |   .0365247   .0041351     8.83   0.000     .0284201    .0446294
       _cons |  -2.489352   .1845662   -13.49   0.000    -2.851095   -2.127608
-------------+----------------------------------------------------------------
     /athrho |    .877777   .0463081    18.96   0.000     .7870147    .9685393
    /lnsigma |   1.793242   .0215357    83.27   0.000     1.751033    1.835451
-------------+----------------------------------------------------------------
         rho |   .7053039    .023272                      .6567146    .7480617
       sigma |   6.008901   .1294057                      5.760549    6.267961
      lambda |   4.238102   .1978183                      3.850385    4.625818
------------------------------------------------------------------------------
Wald test of indep. eqns. (rho = 0): chi2(1) =   359.30   Prob > chi2 = 0.0000

. di .84*(_b[wage:education] + _b[select:education])
.87777704

I hope this is what you're asking for.

[Ben Thompson]

Yes, thanks very much. That is exactly what I'm looking for. One more question -- I'm interested in estimating rho and beta's while setting rho equal to a function of the covariances of the estimated beta's and their respective variable vectors, i.e. rho = f(cov(X'b)). Would this be at all possible?

[Alfonso Sánchez-Peñalver]

I don't think so. The covariances of the estimated betas are due to sampling they are not population parameters. The reason you want to impose a constraint is because the constraint holds in the population. In the population there are no standard errors of the beta estimates: they are zero. So the constraint you're talking about doesn't make much sense.

[Ben Thompson]

Thanks. I think I was mistaken when I said I wanted the covariances of the estimated beta's. The constraint I'm attempting to include in my estimation can be expressed as:

rho = cov(x'b, x'g)/var(x'g)

where b is the (parameter) vector of coefficients from the main equation and g is the (parameter) vector of coefficients on the selection equation. Just wondering how I might go about including this constraint.

I realize that the equation is also saying that rho is the projection of x'b onto x'g, so I was wondering if that would be useful knowledge at all.

[Alfonso Sánchez-Peñalver]

No, I don't believe that can be done.