st: ml program in which likelihood function depends on value of independent var

[Malgosia Madajewicz <mm1174@columbia.edu>]

Hi,

I'm trying to do maximum likelihood estimation in Stata and am having a 
problem. I believe that I can use the lf method. However, my model is 
different from anything discussed in the Gould, Pitblado, Sribney book, 
because it is actually a two-stage model and what the second-stage 
likelihood function looks like depends on the value of the dependent 
variable from the first stage. I think there are two different ways to go 
about implementing this, but there is a step I don't know how to do in each 
one.

I write the model below. I don't describe the full underlying structure, 
the correlations between the error terms, etc, because that would take too 
long. Please assume that the way I'm solving the endogeneity problems is 
correct, since I am pretty sure that it is. Take this just as a question 
about how to implement what I want to do.

I write the model assuming that I am going to implement it by first doing 
the first stage probit regression using the probit command. Then, I save 
the predicted values from this stage as a variable in the data. I write an 
ml routine only for the second stage. Then the model is the following:

1st stage: probit regression  delta = z * gamma + error
delta is binary, z is the vector of independent vars, gamma is the 
coefficient vector

Let deltahat denote the predicted values from the first stage which are the 
linear combination (xb option) not predicted probabilities.

2nd stage likelihood function:

lnf = delta * ln[1 � norm(xb/sigma)] + (1- delta) * ln[1 � norm(x'b/sigma)] 
if $ML_y1 == 0
lnf = delta * {ln[normden(($ML_y1, xb, sigma)] � ln sigma} + (1- delta) * 
{ln[normden(($ML_y1, x'b, sigma)] � ln sigma} if $ML_y1 > 0

In this likelihood x is a vector of independent variables, one of which is 
normden(deltahat)/norm(deltahat). x' is a vector of independent variables, 
one of which is normden(deltahat)/(1 - norm(deltahat)).

Two elements in this model are different from what I have seen. First, what 
the likelihood function looks like for each observation depends on the 
value of delta for that observation. Delta is not a dependent variable, it 
is just one of the vars in the data set. Also, the vector of independent 
variables differs in one position depending on whether delta is 1 or 0. 
Could someone tell me how to handle these issues in the ml program?

Alternatively, I could imagine writing an ml program which estimates both 
equations. Then delta is just the dependent variable for the first 
equation. However that is a ml program which estimates two different max 
likelihoods and uses the result of the first for the second. I have no idea 
how to do this.

My eternal gratitude for any help.

Malgosia


Assistant Professor of Economics and International Affairs
Columbia University
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