A numerically stable quadrature procedure for the one-factor random-component discrete choice model

The Gaussian quadrature formula had been popularized by Butler and Moffitt (1982 Econometrika 50, 761–764) for the estimation of the error component probit panel model. Borjas and Sueyoshi (1994, Journal of Econometrics 64, 164–182) pointed out some numerical and statistical difficulties of applying it to models with group effects. With a moderate or large number of individuals in a group, the likelihood function of the model evaluated by the Gaussian quadrature formula can be numerically unstable, and at worst, impossible to evaluate. Statistical inference may also be inaccurate. We point out that some of these difficulties can be overcome with a carefully designed algorithm and the proper selection of the number of quadrature points. However, with a very large number of individuals in a group, the Gaussian quadrature formulation of integral may have large numerical approximation errors.