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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 Pr {\displaystyle \Pr} where y i = , {\displaystyle \mathbf {y_{i}} =,\ } and where we can take j {\displaystyle j} as choices and i {\displaystyle i} as individuals or observations, X i β {\displaystyle \mathbf {X_{i}\beta } } is the mean and Σ {\displaystyle \Sigma } is the covariance matrix of the model. The probability of observing choice y i {\displaystyle \mathbf {y_{i}} } is
Where A = A 1 × ⋯ × A J {\displaystyle A=A_{1}\times \cdots \times A_{J}} and,
Unless J {\displaystyle J} 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.