This paper discusses estimation methods for limited dependent variable (LDV) models that employ Monte Carlo simulation techniques to overcome computational problems in such models. These difficulties take the form of high dimensional integrals that need to be calculated repeatedly but cannot be easily approximated by series expansions. In the past, investigators were forced to restrict attention to special classes of LDV models that are computationally manageable. The simulation estimation methods we discuss here make it possible to estimate LDV models that are computationally intractable using classical estimation methods.
We first review the ways in which LDV models arise, describing the differences and similarities in censored and truncated data generating processes. Censoring and truncation give rise to the troublesome multivariate integrals. Following the LDV models, we described various simulation methods for evaluating such integrals. Naturally, censoring and truncation play roles in simulation as well. Finally, estimation methods that rely on simulation are described. We review three general approaches that combine estimation of LDV models and simulation: simulation of the log-likelihood function (MLS), simulation of moment functions (MSM), and simulation of the score (MSS). The MSS is a combination of ideas from MSL and MSM, treading the efficient score of the log-likelihood function as a moment function.
We use the rank ordered probit model as an illustrative example to investigate the comparative properties of these simulation estimation approaches.
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.