A Computationally Efficient Quadrature Procedure for the One-Factor Multinomial Probit Model

Abstract

A PROBLEM OF ESTIMATION that has long confronted many economists is the difficulty of estimating the parameters of equations with limited dependent variables on cross-section time-series (i.e., panel) data. While there are widely available packaged computer programs for estimating either (a) cross-section probit and Tobit models or (b) simple permanent-transitory, random-effects panel models with continuous dependent variables, there are no available computationally feasible methods of combining these two models. This is because the likelihood function that arises in such a combined model contains multivariate normal integrals whose evaluation is quite difficult, if not impossible, with conventional approximation methods. There is a widespread feeling among those working in the area that one possible method of evaluation, the use of quadrature techniques, is in principle possible but is in practice computationally too burdensome to consider (e.g., Albright et al. [2, p. 13]; Hausman and Wise [6, p. 12]). In this note we point out that this is true only of standard quadrature techniques such as trapezoidal integration or its improved variants; Gaussian quadrature, on the other hand, is extremely efficient and is well within the bounds of computational feasibility on modern computers. In what follows, we state the nature of the integrals that need to be evaluated, provide a brief exposition of Gaussian quadrature, and provide a numerical illustration of its use in