Pairwise likelihood inference for the random effects probit model

Abstract

This paper proposes a pairwise likelihood estimator based on an analytic approximation method for the random effects probit model. It is widely known that the standard approach for the random effects probit model relies on numerical integration and that its likelihood function does not have a closed form. When the number of time periods or the serial correlation across periods is large, the resulting estimator is likely to become biased. This study derives an analytic approximation for the likelihood function of one pair of time periods without relying on typical numerical-integral procedures. We then apply this formula in a pairwise likelihood estimation procedure to derive our estimator, which is obtained as the product of the analytic approximation of the likelihood function for all possible pairs of time periods. A simulation study is conducted for the comparison of our proposed estimator with the estimators for the pooled probit model and Gaussian quadrature procedure. The evidence shows that our proposed estimator enjoys desirable asymptotic properties. In addition, compared to the estimator based on the Gaussian quadrature procedure, our proposed estimator exhibits comparable performances in all the configurations considered in the simulation study and shows superiority for the cases of a large number of time periods and high serial correlation across periods. We apply our proposed estimator to British Household Panel Survey data so as to characterize the trend of working probabilities.

Appendix

$$\begin{aligned}&g_{1} = \frac{1-c_{2}\left( a_{1}^2 + a_{2}^2\right) }{2};&g_{2}&= \frac{c_{1}(a_{1} + a_{2}) - \sqrt{2}\,c_{2}(a_{1}b_{1}+a_{2}b_{2})}{2\sqrt{2}};\\&g_{3} = \frac{ \sqrt{2}\,c_{1}(b_{1}+b_{2})-c_{2}\left( b_{1}^2 + b_{2}^2\right) }{2};&g_{4}&= \frac{1-a_{2}^2 c_{2}}{2};\\&g_{5} = \frac{a_{2}(c_{1}-\sqrt{2}b_{2}c_{2})}{2\sqrt{2}};&g_{6}&= \frac{b_{2}(\sqrt{2}c_{1}-b_{2}c_{2})}{2};\\&g_{7} = \frac{-c_{1}(a_{1} - a_{2}) -\sqrt{2}\,c_{2}(a_{1}b_{1}+a_{2}b_{2})}{2\sqrt{2}};&g_{8}&= \frac{-2\,c_{1}(b_{1} - b_{2}) -\sqrt{2}\,c_{2}\left( b_{1}^2+b_{2}^2\right) }{2\sqrt{2}};\\&g_{9} = \frac{1-a_{1}^2 c_{2}}{2};&g_{10}&= \frac{-a_{1}(c_{1}+\sqrt{2}b_{1}c_{2})}{2\sqrt{2}};\\&g_{11} = \frac{-b_{1}(\sqrt{2}\,c_{1}+b_{1}c_{2})}{2};&g_{12}&= \frac{-a_{2}(c_{1}+\sqrt{2}b_{2}c_{2})}{2\sqrt{2}};\\&g_{13} = \frac{-b_{2}(\sqrt{2}\,c_{1}+b_{2}c_{2})}{2};&g_{14}&= \frac{-c_{1}(a_{1} + a_{2}) - \sqrt{2}\,c_{2}(a_{1}b_{1}+a_{2}b_{2})}{2\sqrt{2}};\\&g_{15} = \frac{-2\,c_{1}(b_{1} + b_{2}) - \sqrt{2}\,c_{2}\left( b_{1}^2+b_{2}^2\right) }{2\sqrt{2}};&g_{16}&= \frac{a_{1}(c_{1}-\sqrt{2}b_{1}c_{2})}{2\sqrt{2}};\\&g_{17} = \frac{b_{1}(\sqrt{2}c_{1}-b_{1}c_{2})}{2};&g_{18}&= \frac{c_{1}(a_{1} - a_{2}) - \sqrt{2}\,c_{2}(a_{1}b_{1}+a_{2}b_{2})}{2\sqrt{2}};\\&g_{19} = \frac{2\,c_{1}(b_{1} - b_{2}) - \sqrt{2}\,c_{2}(b_{1}^2+b_{2}^2)}{2\sqrt{2}}.&\end{aligned}$$