Chamberlain Random Effect Probit (Correlated Random Effect Probit) in Multinomial Choice Model

[Minchul Park]

Hi, I am finding a command running the Chamberlain random effect probit model in multinomial choice model.
I know the fact that we can run the model by using "xtprobit" command and the option "re" with averaged regressors.
But, I have no idea how to implement the estimation in "multinomial choice model".
Is there anyone who know the solution?

p.s. the Chamberlain random effect probit is a model assuming the random effect error term follows a normal distribution and the mean is correlated with time-averaged regressors.

Thank you for your time to read this question.

[Jeff Wooldridge]

Minchul: In Chapter 16 of my MIT Press book I recommend simply adding the the time-averaged regressors, along with the original regressors and time dummies, to a standard MNL. So it is a pooled MLE rather than a joint MLE (xtprobit, re is a joint MLE). But the pooled MLE is preferred for robustness reasons and computational reasons; it is much easier and it allows any form of serial correlation. Be sure to cluster the standard errors using the cross-sectional identifier.

JW

[Minchul Park]

Jeff Wooldridge Thank you for a good comment! (from your huge fan)

[Minchul Park]

Jeff Wooldridge
Dear Professor:
I am writing to ask a question about the correlated multinomial logit model.

As professor mentioned, we can estimate the parameters in the correlated MNL model by applying pooled MNL with time-averaged regressors. (The MIT press book 653-654 page)
Here, I think, the underlying idea of this approach is that we just use a sufficient statistic of the unobserved heterogeneity, instead of integrating out the individual effects with the assumption that the individual effects fully depend on the time-averaged regressors.
But, I am not sure whether this is an exact understanding.
So, would you mind if I ask whether the understanding is correct?

Thank you for your time to read this question and a really good reference.
Regards,
Minchul Park.

[Mike Guan]

Minchul: In Chapter 16 of my MIT Press book I recommend simply adding the the time-averaged regressors, along with the original regressors and time dummies, to a standard MNL. So it is a pooled MLE rather than a joint MLE (xtprobit, re is a joint MLE). But the pooled MLE is preferred for robustness reasons and computational reasons; it is much easier and it allows any form of serial correlation. Be sure to cluster the standard errors using the cross-sectional identifier.

JW

Dear Professor Wooldridge,

I would be very greatful if you could share your thoughts on my problem. I am working on a Correlated Random Effect approach on a multinomial logit model with endogenous variables and its interactions. The main reference on the econometric side is your paper: Nonlinear Correlated Random Effects Models with Endogeneity and Unbalanced Panels https://economics.ucr.edu/repec/ucr/wpaper/202214.pdf

In your paper, you treat log(avgrexp_it) as endogenous because of the potential correlation with the unobserved heterogeneity as well as the idiosyncratic error term. I would like to do the same.

Here is the setup:

1) y2_it is the important variable which I believe to be endogenous by having correlation with the unobserved heterogeneity as well as the idiosyncratic error term. It is interacted with 2 other exogenous variables, x1_it x2_it. This makes these interactions endogenous too.

2) z1_it is the valid instrument that I found, and hence I will use z1_it as instrument for y2_it, the interaction of z1_it and x1_it as instrument for the interaction of y2_it and x1_it and so on.

3) z2_it is the control that I have in the mlogit.

So here is my plan:

1) I will try to model y2_it as well as the endogenous interactions like you did in the paper. I will regress y2_it on z1_it x1_it x2_it z2_it mean(x1_it) mean(x2_it) mean(z2_it). This is done using a simple OLS with robust standard error. Obtain the residual, v1_it

2) To model y2_it*x1_it, I will regress it on z1_it z1_it*x1_it x2_it z2_it mean(x1_it) mean(x2_it) mean(z2_it). This is again using a simple OLS with robust standard error. Obtain the residual, v2_it. Use similar way to get v3_it.

3) Finally, for the mlogit, I will have y2_it x1_it x2_it y2_it*x1_it y2_it*x2_it z2_it mean(y2_it) mean(x1_it) mean(x2_it) mean(y2_it*x1_it) mean(y2_it*x2_it) mean(z2_it) v1_it v2_it v3_it as well as time dummy.

Please let me know if you think this would work. And I do apologize if this looks messy.

Thank you very much for your time in advance.

Best

Mike