An extensive literature in econometrics and in numerical analysis has considered the problem of evaluating the multiple integral P(B; µ, Ω) = Integralab n(v - µ, Ω)dv =EV1(V c B), where V is a m-dimensional normal vector with mean µ, covariance matrix Ω, and density n(v - µ, Ω) and 1(V c B) is an indicator for the event B = {V|a < V < b}. A leading case of such an integral is the negative orthant probability, where B = {v|v < 0}. The problem is computationally difficult except in very special cases. The multinomial probit (MNP) model used in econometrics and biometrics has cell probabilities that are negative orthant probabilities, with µ and depending on unknown parameters (and, in general, on covariates). Estimation of this model requires, for each trial parameter vector and each observation in a sample, evaluation of P(µ;B) and of its derivatives with respect to µ and Ω. This paper surveys Monte Carlo techniques that have been developed for approximations of P(µ;Ω) and its linear and logarithmic derivatives that limit computation while possessing properties that facilitate their use in iterative calculations for statistical inference: the Crude Frequency Simulator (CFS), Normal Importance Sampling (NIS), a Kernel-Smoothed Frequency Simulator (KFS), Stern’s Decomposition Simulator (SDS), the Geweke–Hajivassiliou–Keane Simulator (GHK), a Parabolic Cylinder Function Simulator (PCF), Deák’s Chi-squared Simulator (DCS), an Acceptance/Rejection Simulator (ARS), the Gibbs Sampler Simulator (GSS), a Sequentially Unbiased Simulator (SUS), and an Approximately Unbiased Simulator (AUS). We also discuss Gauss and FORTRAN implementations of these algorithms and present our computational experience with them. We find that GHK is overall the most reliable method.
The method of simulated scores (MSS) is presented for estimating LDV models with flexible correlation structure in the unobservables. We propose simulators that are continuous in the unknown parameter vectors, and hence standard optimization methods can be used to compute the MSS estimators that employ these simulators. We establish consistency and asymptotic normality of the MSS estimators and derive suitable rates at which the number of simulations must use if biased simulators are used. The estimation method is applied to analyze a model in which the incidence and the extent of debt repayments problems of LDC’s are viewed as optimized choices of the central authorities of the countries in a framework of credit rationing. The econometric implementation of the resulting multi-period probit and Tobit models avoids the need for high dimensional integration. Our findings show that the restrictive error structures imposed by past studies may have led to unreliable econometric results.
Keywords: Simulation model, asymptotic theory, censored model