The GHK algorithm (Geweke, Hajivassiliou and Keane)[1] is an importance sampling method for simulating choice probabilities in the multivariate probit model.
These simulated probabilities can be used to recover parameter estimates from the maximized likelihood equation using any one of the usual well known maximization methods (Newton's method, BFGS, etc.).
Train[2] has well documented steps for implementing this algorithm for a multinomial probit model.
What follows here will apply to the binary multivariate probit model.
Consider the case where one is attempting to evaluate the choice probability of
is the covariance matrix of the model.
The probability of observing choice
is small (less than or equal to 2) there is no closed form solution for the integrals defined above (some work has been done with
The alternative to evaluating these integrals closed form or by quadrature methods is to use simulation.
GHK is a simulation method to simulate the probability above using importance sampling methods.
is simplified by recognizing that the latent data model
can be rewritten using a Cholesky factorization,
terms are distributed
are distributed independently one can simulate draws from a truncated multivariate normal distribution using draws from a univariate random normal.
For example, if the region of truncation
has lower and upper limits equal to
, substituting: Rearranging above, Now all one needs to do is iteratively draw from the truncated univariate normal distribution with the given bounds above.
This can be done by the inverse CDF method and noting the truncated normal distribution is given by, Where
This suggests to generate random draws from the truncated distribution one has to solve for
is the standard normal CDF.
by its simplified equation using the Cholesky factorization.
These draws will be conditional on the draws coming before and using properties of normals the product of the conditional PDFs will be the joint distribution of the
is the multivariate normal distribution.
by the setup using the Cholesky factorization then we know that
is a truncated multivariate normal.
The distribution function of a truncated normal is, Therefore,
is the standard normal pdf for choice
the above standardization makes each term mean 0 variance 1.
is the multivariate normal PDF.
Going back to the original goal, to evaluate the Using importance sampling we can evaluate this integral, This is well approximated by