The GHK algorithm 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. Train has well documented steps for implementing this algorithm for a multinomial probit model. What follows here will applies to the binary multivariate probit model. Consider the case where one is attempting to evaluate the choice probability of where and where we can take as choices and as individuals or observations, is the mean and is the covariance matrix of the model. The probability of observing choice is Where and, Unless is small there is no closed form solution for the integrals defined above. 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. Evaluating is simplified by recognizing that the latent data model can be rewritten using a Cholesky factorization,. This gives where the terms are distributed. Using this factorization and the fact that the 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 then the task becomes Note: , 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 will be a number between 0 and 1 because the above is a CDF. This suggests to generate random draws from the truncated distribution one has to solve for giving, where and and is the standard normal CDF. With such draws one can reconstruct the 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, Where is the multivariate normal distribution. Because conditional on is restricted to the set 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, has distribution, where is the standard normal pdf for choice. Because the above standardization makes each term mean 0 variance 1. Let the denominator and the numerator where 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.
