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Re: st: Multinomial logit-type probability in log likelihood forML model


From   Bob Hammond <robert.g.hammond@vanderbilt.edu>
To   statalist@hsphsun2.harvard.edu
Subject   Re: st: Multinomial logit-type probability in log likelihood forML model
Date   Tue, 04 Sep 2007 07:24:19 -0500

Hey Maarten,

Thanks for your reply. The objective in the example you sent is to estimate a multinomial logit (MNL) model, but my objective is different. I want to estimate a binary choice model where the probability of Y=1 is governed by a log likelihood function that contains a MNL-type probability. I can't simply use a MNL model because I do not observe the outcome of the MN process. If the MN process is described by Binomial(n, j, q), then I observe n, but not j, preventing me from using Stata's built-in MNL. My simplification of the log likelihood function made this less clear than it should have been; sorry for the confusion. My approach is to model q as a function of observed covariates, allowing me to use the distribution of the unknown j.

After working more on this, I wonder if I can use the following (maintaining my simplification from the initial email that ignores the fact that j is unobserved):

gen a = .
...
egen double `tempvar1' = sum(exp(`theta'))
replace a = 1 + `tempvar1'
replace `lnf' = (Binomial(n,j,exp(`theta')/a) ///
- Binomial(n,j+1,exp(`theta')/a) * f(x) if $ML_y1 == 1

That is, can I simply sum up the `theta' for the entire data set into a new variable, then use this variable in the "replace `lnf'" statement? Thanks,

Bob

I posted a -ml- program that does a multinomial logit on statalist some
time ago:
http://www.stata.com/statalist/archive/2007-05/msg00449.html

Hope this helps,
Maarten


--- Bob Hammond <robert.g.hammond@vanderbilt.edu> wrote:

> All,
>
> I am having trouble coding an ML program for my log likelihood
> function.  I'll simplify several aspects of the log likelihood that
> seem
> unrelated to my question.  Define `lnf' for an individual observation
> as
> follows:
>
> replace `lnf' = (Binomial(n,j,q) - Binomial(n,j+1,q)) * f(x)  if
> $ML_y1 == 1
>
> In words, the probability that Y is 1 is the probability of observing
>
> exactly j successes out of n (where q is the probability of a success
> on
> an individual trial) times some function f(x).  The probability q
> takes
> a multinomial logit form:
>
> q_i = exp(X_i * `theta') / (1 + sum(exp(X_h * `theta')))
>
> where the sum goes from h=1, ..., J, so it sums the product of the
> covariate vector X times the parameter vector `theta' for all
> observations in the data set.
>
> It seems that I need some way to construct the summation in the
> denominator of q first, but my confusion is that this denominator
> contains the `theta' parameter vector.  Basically, for every
> observation
> in the data set, I need to construct a scalar that multiplies the 1 x
> k
> covariate vector X by the k x 1 parameter vector `theta' that needs
> to
> be estimated.  Then I need to sum these J scalars up (call this a,
> which
> is a function of `theta').  If I could do that, then, after
> constructing
> a, I would write:
>
> replace `lnf' = (Binomial(n,j, exp(`theta') / (1 + a) - Binomial(n,j,
>
> exp(`theta') / (1 + a)) * f(x)
>             if $ML_y1 == 1
>
> I don't know how to construct "a" because it contains the parameter
> vector `theta'.  I've tried taking a look at the ado file for mlogit,
>
> but didn't find the answer.  Thanks in advance,
>
> Bob
--
------------------------------------------------------------------------
Bob Hammond
Department of Economics
Vanderbilt University
http://people.vanderbilt.edu/~robert.g.hammond/
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