# st: Assumptions of unit variance in multivariate probit (and panel selection models)

 From "Stephen P. Jenkins" To statalist@hsphsun2.harvard.edu Subject st: Assumptions of unit variance in multivariate probit (and panel selection models) Date Sat, 3 Feb 2007 12:51:42 -0000 (GMT)

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Date: Fri, 2 Feb 2007 09:00:53 -0800 (PST)
From: Kam Kup <kamkupan@yahoo.com>
Subject: Re: st: Assumptions of unit variance in multivariate probit: Stephen Jenkins?

Professor Jenkins,

Thanks for your response.  In fact, I did place the
same restrictions on the diagonal elements of the
matrices.  However, when the independent variables in
the selection and outcome equations overlap by more
than one variable (as the theory suggests), Stata
estimates the sum of the non-diagonal elements in
certain rows to be more than one, thus violating the
restriction.

Heckprob  estimates the selection and first outcome
equation just fine.

My questions are:

1. I am unclear on which elements of the cholesky
Matrix need to be restricted, since, if an observation
is selected, I have multiple observations on the
outcome variable over time.  (I'm using d0 as a
result.)

For example, in your final panel example in that
paper, you do not impose any restrictions on the
diagonal elements except for the first one.

I am unclear on why this is different from the earlier
multivariate probit cases.  Perhaps this would help me
understand which restrictions I should place.

2. Heckprob explicitly uses atanh(rho) in it's the
likelihood evaluator to ensure that constrain rho to
the appropriate interval.  Would you recommend doing
htis?  (I notice that you did not use this in your
programs, perhaps to avoid the issues with calculating
confidence intervals.)

Thanks again,
Kam

============================

Your questions all relate to the nature of the assumptions placed on the underlying
variance-covariance matrix of the model.  In the standard MV Probit model,
estimated with cross-sectional data, each of the error variances is normalised to
unity (cf. standard univariate probit) -- this is required for identification.  In
the model in our final example, we estimated a type of panel probit, and the
cross-time variation means that we can identify the error variances (aside from one
normalisation.)

Given the relationship between a variance covariance matrix and the corresponding
Cholesky matrix, restrictions on the former translate into restrictions on the
elements of the latter.

[The "standard" Heckman selection model is estimated differently, and I think using
atanh transformations is not the key point here.]

panel data (3 waves of observations) subject to another variable being observed.
This is more complicated than the specifications we considered in our article as
you are combining panel data and selection.

I understand that there is some literature on this but I have not followed it.
Look e.g. at the paper by Jeonghoon Ahn "Panel Data Sample Selection Model: an
Application to Employee Choice of Health Plan Type and Medical Cost Estimation"
at http://ideas.repec.org/p/ecm/feam04/560.html, and references therein. I note
that the abstract says "... Since longitudinal data structure is common in health
economics data, especially medical claims data, the correction of selection bias in
the longitudinal sense is especially valuable for health economics related
researches. The complicated modeling and extensive computer programming needs,
however, resulted to only a few health economics researches in this direction. ...
" On the more positive side, note the author states on p. 10 that all his
estimation was done in Stata 7.0. There are references to first using Gauss-Hermite
integration methods (not simulation), and then a non-parametric "Kyriazidou"
estimator, which seems to use Weighted OLS estimators. (Kyriazidou, E., 1997.
Estimation of a panel data sample selection model. Econometrica 65, 1335–1364.) See
also http://www.stata.com/statalist/archive/2005-06/msg00456.html and references
therein. And "Selection Correction in Panel Data Models: An Application to Labour
Supply and Wages" by Dustmann, Christian and Rochina-Barrachina, María Engracia

If you wish to use a simulation estimation approach, then, following first
principles, I guess that you will have to think about the variance covariance
matrix of your model and the restrictions placed on the elements of it in this
particular specification, and thence the restrictions implied by those on the
Cholesky matrix.

I have not done this myself yet in my own work, and I am afraid that I am now
devoid of further inspiration (and time).

It is an interesting application. I hope that you can post me and/or the list your
eventual solution!  Good luck

Stephen

- --- "Stephen P. Jenkins" <stephenj@essex.ac.uk> wrote:

>
=======================================================
>
> Date: Thu, 1 Feb 2007 08:12:31 -0800 (PST)
> From: Kam Kup <kamkupan@yahoo.com>
> Subject: st: Assumptions of unit variance in
> multivariate probit:
> Stephen Jenkins?
>
> I have a multivariate probit model with one
> selection
> equation and three other outcome variable across
> time.
>
> That is,
> selection equation: y1*=x*b1
> if y1*>0 for a given individual, we also observe the
> following over three points in time:
> z1*=w1*theta z2*=w2*theta z3*=w3*theta
>
> To compute the multivariate normal probabilities, I
> am
> using Stephen Jenkin's excellent mvnp progrma.  My
> question is about the restrictions that should be
> placed on the 4x4 Cholesky matrix.
>
> The problem: the Cholesky matrix in my case seems to
> converge for some specifications and not others.  In
> particular, convergence is not achieved if even one
> variable is included in both stages, even if there
> are
> several exclusion restrictions.  When the variables
> are totally different, the model converges and the
> results make sense.
>
> What is apparently happening is that the sum of the
> squares of the cholesky elements (other than the
> diagonal) sum to more than 1.
>
> Would you suggest using atanh to transform the
> elements of the Cholesky matrix, as heckprob.ado
> does
> to the covariance matrix?
>
> Thank you,
> Kam
>
==========================================================
>
> Kam refers to the code distributed with Stata
> Journal 6-2 article by
> Lorenzo Cappellari and myself: package st0101
>   accompanying
> "Calculation of multivariate normal probabilities by
> simulation, with
> applications to maximum
> simulated likelihood
> estimation".  [Preprint version of article available
> as ISER WP at
>
http://www.iser.essex.ac.uk/pubs/workpaps/pdf/2006-16.pdf]
> The main
> programs are -_gmvnp()-, an egen function with
> associated plug-in for
> calculating multivariate normal probabilities using
> the GHK simulator,
> and -mdraws- for creating pseudo-random and Halton
> draw variables.
>
> Your penultimate paragraph suggests that you have
> not imposed the
> appropriate constraints on the elements of the
> Cholesky matrix.  Look
> at the code for the trivariate probit with one
> selection (p. 178 of SJ
> article, and note the lines placing restrictions on
> scalars `cf22',
> `cf33'
>
>
>
> Stephen
>

--
Professor Stephen P. Jenkins <stephenj@essex.ac.uk>
Institute for Social and Economic Research
University of Essex, Colchester CO4 3SQ, U.K.
Tel: +44 1206 873374.  Fax: +44 1206 873151.
http://www.iser.essex.ac.uk
Survival Analysis using Stata:
http://www.iser.essex.ac.uk/teaching/degree/stephenj/ec968/