[Limdep Nlogit List] Censored normal distribution in Mixed Logit and its correlation

[William Greene]

Dear Kar Lim.  First, I'll assume you mean that the coefficients on "A" and
"B" are random with normal and censored normal distributions.
(1).  beta(i) = Min(0,beta+sigma*w(i)) where w(i) ~ N[0,1].  In your
example,
beta = .5 and sigma = 2.  The mean E[beta(i)] is somewhere left of zero.
You can use the laws of probability, E[.] = 0*Prob(beta+sigma*w(i)>=0)
+ Prob(<0) * E[<=0], which is a little involved.  The actual value for your
example is about -0.6.  What is reported is the (beta,sigma).
(2) The correlation matrix applies to the underlying primitives.  For your
example, if the coefficient on A is gamma = .3 + 3*v(i), the correlation
reported is that between gamma and (beta+sigma*w(i)).  The correlation
of the truncated variable will be somewhat less, since after the truncation,
there is variation in gamma that is not accompanied by variation in
beta_sigma*w.
I'm not sure a positive correlation could always mean
"a person who likes A tends to like B."
For your specific examples, I would say that a positive correlation of the
primitives
would indeed accompany a positive correlation of gamma with beta.  I'm not
sure
about your qualitative interpretation, however.
Regards,
Bill Greene


On Fri, Oct 29, 2021 at 2:57 PM KH Lim <khlim2 at gmail.com> wrote:

> Hi all,
>
> Suppose I estimate a mixed logit model with two variables. Variable A is
> distributed normal (n), and B is distributed censored right normal (m).  My
> questions are:
>
> 1. Is the reported coefficient of B the mean or the mean of B's underlying
> normal distribution? For example, if the reported coefficient is 0.5, and
> standard deviation is 2, is the mean of B indeed 0.5? Or is it somewhere
> more to the right?
>
> 2. In the NLOGIT 5 manual, page N-530:
> "If you will be mixing distributions, the specification of correlated
> parameters, while allowable, produces ambiguous results. The nature of the
> correlation is difficult to define. However, the program will have no
> unusual difficulty estimating a model in which correlated parameters have
> different distributions. "
>
> Is the correlation matrix produced by the estimator subject to such
> complications in the case of censored normal distribution? If so, can a
> positive correlation still mean "a person who likes A tends to like B"?
>
> Thank you,
> Kar Lim
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-- 
William Greene
Department of Economics, emeritus
Stern School of Business, New York University
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