{smcl}
{* 15jan2003}{...}
{hline}
help for {hi:mvprobit}{right:Cappellari and Jenkins (15jan2003)}
{hline}
{title:Multivariate probit models, estimated by Simulated Maximum Likelihood}
{p 4 12}{cmd:mvprobit} {it:equation1} {it:equation2} {it: ...} {it:equationM}
[{it:weight}] [{cmd:if} {it:exp}] [{cmd:in} {it:range}] [{cmd:,}
{cmdab:dr:aws(}{it:#}{cmd:)} {cmdab:s:eed(}{it:#}{cmd:)}
{cmdab:b:eta0} {cmdab:a:trho0(}{it:matrix_name}{cmd:)}
{cmdab:r:obust} {cmdab:cl:uster(}{it:varname}{cmd:)}
{cmdab:const:raints(}{it:numlist}{cmd:)}
{cmdab:l:evel(}{it:#}{cmd:)} {it:maximize_options} ]
{p}where each equation is specified as
{p 12 12}{cmd:(} [{it:eqname}{cmd::}] {it:depvar} [{cmd:=}] [{it:varlist}]
[{cmd:,} {cmdab:nocon:stant}] {cmd:)}
{p}{cmd:by} {it:...} {cmd::} may be used with {cmd:mvprobit}; see help
{help by}.
{p}{cmd:pweight}s, {cmd:aweight}s, {cmd:fweight}s, and {cmd:iweight}s
are allowed; see help {help weights}.
{p}{cmd:mvprobit} shares the features of all estimation commands; see help
{help est}.
{p}{cmd:mvprobit} typed without arguments redisplays the last estimates.
The level option may be used.
{p}Predictions based on {cmd:mvprobit} estimates, including predicted joint
and marginal probabilities, can be derived using {help mvppred}.
{title:Description}
{p}{cmd:mvprobit} estimates {it:M}-equation probit models, by the method of
simulated maximum likelihood (SML). (Cf. {cmd:probit} and {cmd:biprobit}
which estimate 1-equation and 2-equation probit models by maximum likelihood.)
The variance-covariance matrix of the cross-equation error terms has values of
1 on the leading diagonal, and the off-diagonal elements are correlations
to be estimated (rho{it:ji} = rho{it:ij}, and rho{it:ii} = 1, for all
{it:i} = 1,...,{it:M}).
{p}{cmd:mvprobit} uses the Geweke-Hajivassiliou-Keane (GHK) simulator
to evaluate the {it:M}-dimensional Normal integrals in the likelihood
function. For each observation, a likelihood contribution is calculated
for each replication, and the simulated likelihood contribution is the
average of the values derived from all the replications. The simulated
likelihood function for the sample as a whole is then maximized using
standard methods ({cmd:ml} in this case). For a brief description of the
GHK smooth recursive simulator, see Greene (2000: 183-185), who also
provides references to the literature.
{p}Under standard conditions, the SML estimator is consistent as the number
of observations and the number of draws tend to infinity, and is
asymptotically equivalent to the true maximum likelihood estimator as the
ratio of the square root of the sample size to the number of draws tends
to zero. Thus, other things equal, the more draws, the better. In practice,
however, it has been observed that a relatively small number of draws may
work well for `smooth' likelihoods. An integer number corresponding to
the square root of the number of observations is often used for the number
of random draws. For small sample sizes, a larger number of draws may be
required.
{p}Estimation is numerically intensive, and may be very slow if the data
set is large, if the number of draws is large, or (especially) if the
number of equations is large. Users may also need to {cmd:set matsize}
and {cmd:set memory} to values above the default ones. (See help for
{help matsize} and {help memory}.) Use of the {cmd:atrho0} option may
speed up convergence.
{p}Models for which the matrix of rhos is close to not being positive
definite are likely to be difficult to maximize. (The Cholesky factorization
used by SML requires positive definiteness.) This is more likely if
|rho{it:ji}| is close to one. In these cases, {cmd:ml} may report
difficulties calculating numerical derivatives and a non-concave
log-likelihood. In difficult maximization problems, the message
"Warning: cannot do Cholesky factorization of rho matrix" may appear
between iterations. It may be safely ignored if the maximization proceeds
to a satisfactory conclusion. Results may differ depending on the sort order
of the data, because the sort order affects which values of the random variable(s)
get allocated to which observation. (Be assured, however, that {cmd:mvprobit}
does not change the sort order of the data.) This potential problem is reduced,
the larger the number of random draws that is used.
{title:Options}
{p 0 4}{cmd:draws(}{it:#}{cmd:)} specifies the number of random variates
drawn when calculating the simulated likelihood. The default is 5. (See the
discussion above concerning the number of draws.)
{p 0 4}{cmd:seed(}{it:#}{cmd:)} specifies the initial value of the
(pseudo-)random-number seed used by the {cmd:uniform()} function
in the simulation process. The value should be an integer (the
default value is 123456789). Warning: if the number of draws is 'small',
changes in the seed value may lead to surprisingly large changes in
estimates.
{p 0 4}{cmd:beta0} specifies that the estimates of the marginal probit
regressions (used to provide starting values) are reported.
{p 0 4}{cmd:atrho0(}{it:matrix_name}{cmd:)} allows users to specify
starting values for the off-diagonal elements of the rho matrix
that are different from the default values (which are all zero).
More precisely, the matrix {it:matrix_name} contains values of the
incidental parameter in each /atrho{it:ji} equation, i.e.
atanh(rho{it:ji}) = .5*ln((1+rho{it:ji})/(1-rho{it:ji})). Matrix
{it:matrix_name} must have properly named column names. E.g. if
a starting value in /atrho{it:21} is being set, one would first use the
command {cmd:matrix {it:matrix_name} = ({it:value})}, followed by
{cmd:matrix colnames {it:matrix_name} = atrho21:_cons}. Between 1 and
{it:M}({it:M}-1)/2 /atrho{it:ji} starting values may be specified,
where {it:j} = 2,...,{it:M}, and {it:i} < {it:j}. One likely source
for a non-default starting value for atrho{it:ji} is the /athrho
parameter estimate from the {cmd:biprobit} model corresponding to
equations {it:j} and {it:i} of the full {cmd:mvprobit} model.
{p 0 4}{cmd:robust} specifies that the Huber/White/sandwich estimator of
variance is to be used in place of the traditional calculation; see
{hi:[U] 23.11 Obtaining robust variance estimates}. {cmd:robust} combined
with {cmd:cluster()} allows observations which are not independent within
cluster (although they must be independent between clusters). If you
specify {help pweight}s, {cmd:robust} is implied.
{p 0 4}{cmd:cluster(}{it:varname}{cmd:)} specifies that the observations are
independent across groups (clusters) but not necessarily within groups.
{it:varname} specifies to which group each observation belongs; e.g.,
{cmd:cluster(personid)} in data with repeated observations on individuals.
See {hi:[U] 23.11 Obtaining robust variance estimates}. {cmd:cluster()} can be
used with {help pweight}s to produce estimates for unstratified
cluster-sampled data. Specifying {cmd:cluster()} implies {cmd:robust}.
{p 0 4}{cmd:noconstant} suppresses the constant term (intercept) in the
relevant regression.
{p 0 4}{cmd:constraints(}{it:numlist}{cmd:)} specifies the linear constraints
to be applied during estimation. Constraints are defined using the
{cmd:constraint} command and are numbered; see help {help constraint}. The
default is to perform unconstrained estimation.
{p 0 4}{cmd:level(}{it:#}{cmd:)} specifies the confidence level, in percent,
for the confidence intervals of the coefficients; see help {help level}.
{p 0 4}{it:maximize_options} control the maximization process; see help
{help maximize}. Use of them is likely to be rare.
{title:Saved results}
{p} In addition to the usual results saved after {cmd:ml}, {cmd:mvprobit} also
saves the following:
{p 0 4}{cmd:e(draws)} is the number of random draws used when simulating
probabilities.
{p 0 4}{cmd:e(seed)} is the initial seed value used by the random-number
generator.
{p 0 4}{cmd:e(neqs)} is the number of equations in the {it:M}-equation model.
{p 0 4}{cmd:e(ll0)} is the log-likehood for the comparison model (the sum of
the log-likelihoods from the marginal univariate probit models corresponding
to each equation).
{p 0 4}{cmd:e(chi2_c)} is chi-square test statistic for the likelihood ratio
test of the multivariate probit model against the comparison model.
{p 0 4}{cmd:e(nrho)} is the number of estimated rhos (the degrees of freedom
for the likelihood ratio test against the comparison model).
{p 0 4}{cmd:e(rho{it:ji})} is the estimate of correlation {it:ji} in the
variance-covariance matrix of cross-equation error terms.
{p 0 4}{cmd:e(serho{it:ji})} is the estimated standard error of
correlation {it:ji}.
{p 0 4}{cmd:e(rhs{it:i})} is the list of explanatory variables used in
equation {it:i}. This list does not include the constant term, regardless
of whether there is one is implied by equation {it:i}.
{p 0 4}{cmd:e(nrhs{it:i})} is number of explanatory variables in equation {it:i}.
This number includes the constant term if there is one implied by equation {it:i}.
{title:Examples}
{p 8 12}{inp:. use http://www.stata-press.com/data/r7/school.dta, clear}
{p 8 12}{inp:. biprobit (private = years logptax loginc) (vote=years logptax loginc) }
{p 8 12}{inp:. mvprobit (private = years logptax loginc) (vote = years logptax loginc), dr(15) }
{p 8 12}{inp:. mvprobit (private = years logptax loginc) (vote = years logptax, nocons), nolog }
{p 8 12}{inp:. mvprobit (private years logptax loginc) (vote years logptax, nocons), beta0 }
{p 8 12}{inp:. constraint define 1 [private]loginc = 0.4 }
{p 8 12}{inp:. mvprobit (private = years logptax loginc) (vote = years logptax loginc, nocons), constraint(1) }
{p 8 12}{inp:. mvprobit (private = years logptax loginc) (vote = years logptax loginc) (pub12 = years loginc) }
{p 8 12}{inp:. mvprobit (private = loginc logptax) (vote = loginc logptax)(school = logptax)(pub5 = ), dr(10) }
{title:Authors}
{p 4 4}Lorenzo Cappellari, Universita del Piemonte-Orientale, Italy{break}
{p 4 4}Stephen P. Jenkins, ISER, University of Essex, U.K.{break}
{title:Acknowledgements}
{p 4 4}Thanks to Nick Cox and Weihua Guan for comments and suggestions.
Much of our code for syntax handling and display of results was inspired
by code used in {cmd:biprobit}.
{title:References}
{p 4 8} Greene, W.H. (2000), {it:Econometric Analysis, Fourth edition},
Prentice-Hall International, Upper Saddle River NJ.
{title:Also see}
{p 1 14}Manual: {hi:[U] 23 Estimation and post-estimation commands},{p_end}
{p 10 14}{hi:[U] 29 Overview of model estimation in Stata},{p_end}
{hi:[R] biprobit}
{p 0 19}On-line: help for {help constraint}, {help est}, {help postest},
{help ml}, {help biprobit}, {help probit}, and (if installed)
{help triprobit}.{p_end}