David, Mushfiq: While Mushfiq advice on calculating the Inverse Mills
Ratios for an mlogit model is correct, your approach of calculating the IMR
and using them as regressors in a second-stage logit model will not give you
consistent estimates of this model. The two-step approach to estimating
selection models using IMR only produces consistent estimates when the
second-stage is a linear regression model. If as in your case the second
stage model is a non-linear model, the system of equations needs to be
estimated using ML-methods. This is why the stata command heckprob has no
two-step option.
Steve
> -----Original Message-----
> From: A. Mushfiq Mobarak [SMTP:[email protected]]
> Sent: Wednesday, April 09, 2003 5:20 AM
> To: [email protected]; [email protected]
> Subject: RE: st: Inverse Mills Ratio after MLOGIT
>
> David,
>
> Let me answer the second question first. To my knowledge, the IMR are a
> function of the predicted probabilities of the various outcomes in your
> first stage mlogit regression.
>
> For your four outcomes, you first have to create variables for your
> predicted probabilities of each outcome. Right after running the mlogit,
> you can type:
>
> predict p0 if e(sample), outcome(0);
> predict p1 if e(sample), outcome(1);
> predict p2 if e(sample), outcome(2);
> predict p3 if e(sample), outcome(3);
>
> You then have to use p0,p1,p2,p3 to create the 3 mills ratio terms.
> According to formulas given by Dubin and McFadden (Econometrica circa
> 1984), the following code would create the mills terms: (you should
> check whether the formulas below are appropriate for your particular
> problem)
>
> gen trnsp0=(p0*ln(p0))/(1-p0);
> gen trnsp1=(p1*ln(p1))/(1-p1);
> gen trnsp2=(p2*ln(p2))/(1-p2);
> gen trnsp3=(p3*ln(p3))/(1-p3);
>
> gen millsp1=3*ln(p1)+ trnsp0 +trnsp2 +trnsp3;
> gen millsp2=3*ln(p2)+ trnsp0 +trnsp1 +trnsp3;
> gen millsp3=3*ln(p3)+ trnsp0 +trnsp1 +trnsp2;
>
> You can plug in millsp1-millsp3 in your second stage logit. If you are
> interested in the standard errors for the mills ratio terms in the
> second stage logit, then more work has to be done - you should probably
> bootstrap errors.
>
> As to your first question, in my opinion, this is a fine thing to do, as
> long as you have a variable that helps identify the covariance in the
> first and second stage error terms. If you're using the exact same set
> of variables in your first and second stages, then only the
> non-linearity of the Mills ratio terms is used for identification, and
> according to the literature on selection correction, this is not a good
> way to proceed. There's a paper in the Journal of Economic Surveys on
> the Heckman selection correction that discusses these issues.
>
> -Mushfiq
>
> A. Mushfiq Mobarak
> Assistant Professor of Economics
> University of Colorado at Boulder
> 303-492-8872
>
> Date: Mon, 07 Apr 2003 08:52:28 -0600
> From: David Leblang <[email protected]>
> Subject: st: Inverse Mills Ratio after MLOGIT
>
> Listers,
>
> I am trying to estimate a selection type model in the tradition of the
> heckprob command however where the first stage has multiple outcomes
> (four) and the second stage is a standard logit/probit. My approach to
> this is to estimate the first stage as a multinomial logit, get the
> predicted probabilities, and plug them into the second stage logit.
> However, because I assume that the errors from the first and second
> stage models are correlated, I want to generate the inverse mills ratio
> (IMR) from the first stage multinomial logit and add those in the second
>
> stage equation (this is discussed in Millimet's faq on endogeniety).
> Here are my questions:
>
> 1. from a statistical point of view, does this make sense?
>
> 2. how can I obtain the IMR after the mlogit? I have searched the
> faqs, etc but cannot find an answer.
>
> Thanks,
>
> David Leblang
> University of Colorado
>
>
>
>
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