[Limdep Nlogit List] Testing heterosc. sample selection model

[William Greene]

JK. Interesting question.  The reason for the discrepancy is as follows:
The maximum likelihood estimator for the homoscedasticity model as you
have specified it is reestimating the probit equation. You will notice
that the output for this model includes results for the probit equation.
The MLE for the heteroscedasticity model is a two step estimator that does
not reestimate the probit equation. It uses the previously computed
probit model as is.  Thus, the output for it does not
include a set of results for the probit equation. Unfortunately, that means
that your strategy for testing for heteroscedasticity won't produce
the right results.  What you can do, as you suggested below, is constrain
the parameters in the variance equation to equal zero in the 
heteroscedasticity model, using Rst=kxw_b,0,b19,b20.  I think, however, 
that if you have more than one variable in the XH namelist, you need
to change the list in this restriction. In any event, this will get you
an appropriate test statistic.
Regards,
Bill Greene



----- Original Message -----
From: "JK" <dwakotki at wp.pl>
To: "limdep" <limdep at limdep.itls.usyd.edu.au>
Sent: Wednesday, October 8, 2008 3:58:10 AM GMT -05:00 US/Canada Eastern
Subject: [Limdep Nlogit List] Testing heterosc. sample selection model

Dear LimDep Users,
I'm testing a heteroscedasticity in the sample selection model, so my 
procedure is:

? model homosk. (restr.)
 PROBIT; Lhs=SELEKCJA; Rhs=W; Hold$
 SELECT; Lhs=ZAL; Rhs=XW; MLE$
  CALC; LOGhom=LOGL$
? model heterosk. (unrestr.)
 PROBIT; Lhs=SELEKCJA; Rhs=W; Hold$
 SELECT; Lhs=ZAL; Rhs=XW; Hfn=XH; MLE$
  CALC; khe=Col(XH)$
  CALC; LOGhet=LOGL; StatLR=2*(LOGhet-LOGhom)
   ;pvLR=1-Chi(StatLR,khe)$
(...)

The problem is that StatLR<0. I was looking for a mistake in my 
procedure and I find that there are two possible restricted logL:
1)
--> PROBIT; Lhs=SELEKCJA; Rhs=W; Hold$
--> SELECT; Lhs=ZAL; Rhs=XW; MLE$
+---------------------------------------------+
| ML Estimates of Selection Model
| Maximum Likelihood Estimates
| Dependent variable ZAL
| Weighting variable None
| Number of observations 6988
| Iterations completed 30
| Log likelihood function -24312.41    <=====
| Number of parameters 28
| Info. Criterion: AIC = 6.96635
| Finite Sample: AIC = 6.96638
| Info. Criterion: BIC = 6.99380
| Info. Criterion:HQIC = 6.97581
| Model estimated: Oct 01, 2008, 01:49:07PM
| FIRST 9 estimates are probit equation.
+-------------------------------------------+

2)
--> PROBIT; Lhs=SELEKCJA; Rhs=W; Hold$
--> SELECT; Lhs=ZAL; Rhs=XW; Hfn=XH; MLE; Matrix; Rst=kxw_b,0,b19,b20$
+---------------------------------------------+
| Selection with heteroscedasticity
| Maximum Likelihood Estimates
| Dependent variable ZAL
| Weighting variable None
| Number of observations 6988
| Iterations completed 2
| Log likelihood function -24997.32   <====
| Number of parameters 19
| Info. Criterion: AIC = 7.15979
| Finite Sample: AIC = 7.15981
| Info. Criterion: BIC = 7.17842
| Info. Criterion:HQIC = 7.16621
| Restricted log likelihood -24997.32
| McFadden Pseudo R-squared .0000000
| Chi squared .1236913E-09
| Degrees of freedom 1
| Prob[ChiSqd > value] = .9999911
| Model estimated: Oct 01, 2008, 01:25:51PM
+---------------------------------------------+

In the sample selection model with hetroskedasticity, the restricted 
logL is the same as in the model with Rst=kxw_b,0,b19,b20 above:
--> PROBIT; Lhs=SELEKCJA; Rhs=W; Hold$
--> SELECT; Lhs=ZAL; Rhs=XW; Hfn=XH; MLE; Matrix$
+---------------------------------------------+
| Selection with heteroscedasticity
| Maximum Likelihood Estimates
| Dependent variable ZAL
| Weighting variable None
| Number of observations 6988
| Iterations completed 27
| Log likelihood function -24991.37    <====
| Number of parameters 20
| Info. Criterion: AIC = 7.15838
| Finite Sample: AIC = 7.15839
| Info. Criterion: BIC = 7.17799
| Info. Criterion:HQIC = 7.16514
| Restricted log likelihood -24997.32  <====
| McFadden Pseudo R-squared .0002379
| Chi squared 11.89333
| Degrees of freedom 1
| Prob[ChiSqd > value] = .5633534E-03
| Model estimated: Oct 01, 2008, 01:30:10PM
+---------------------------------------------+

I would like to know which version is correct and why the restricted 
logL, which is computed in the homoscedastic model, is different from 
the restricted logL which is computed in model with restrictions 
Rst=kxw_b,0,b19,b20. I thought it should be the same because the 
likelihood function of the homoscedastic sample selection model is a 
special case of the likelihood function of the heteroscedastic sample 
selection model (E30-12; E30-19).
Have you got any suggestions for me?
Thanks,

Jadwiga Kostrzewska
Zakład Teorii Prognoz
Katedra Statystyki
Uniwersytet Ekonomiczny w Krakowie
Polska

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