A Computationally Efficient Analytic Procedure for the Random Effects Probit Model

It is found in Lee (2000) and Rabe-Hesketh et al. (2005) that the typical numerical-integral procedure suggested by Butler and Moffitt (1982) for the random effects probit model becomes biased when the correlation coefficient within each unit is relatively large. This could possibly explain why Guilkey and Murphy (1993, p. 316) recommend that if only two points (T=2) are available, then one may as well use the probit estimator. This paper tackles this issue by deriving an analytic formula for the likelihood function of the random effects probit model with T=2. Thus, the numerical-integral procedure is not required for the closed-form approach, and the possible bias generated from numerical integral is avoided. The simulation outcomes show that the root of mean-squared-error (RMSE) of the random effects probit estimator (MLE) using our method could be over 40% less than that from the probit estimator when the cross correlation is 0.9.