# Pairwise likelihood inference for the random effects probit model

- 64 Downloads

## 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.

## Keywords

Discrete choice Panel probit model Error function## Supplementary material

## References

- Abramowitz M, Stegun IA (1972) Handbook of mathematical functions with formulas, graphs, and mathematical tables, 10th edn. Dover, New YorkMATHGoogle Scholar
- Aigner D, Lovell CAK, Schmidt P (1977) Formulation and estimation of stochastic frontier production function models. J Econom 6:21–37MathSciNetCrossRefMATHGoogle Scholar
- Arnold BC, Strauss D (1991) Pseudolikelihood estimation: some examples. Sankhy\(\overline{a}\) Indian J Stat Ser B 53:233–243Google Scholar
- Besag JE (1975) Statistical analysis of non-lattice data. J R Stat Soc Ser D 24:179–195Google Scholar
- Booth AL, Jenkins SP, Serrano CG (1999) New men and new women? A comparison of paid work propensities from a panel data perspective. Oxf Bull Econ Stat 61:167–197CrossRefGoogle Scholar
- Borjas GJ, Sueyoshi GT (1994) A two-stage estimator for probit models with structural group effects. J Econom 64:165–182MathSciNetCrossRefGoogle Scholar
- Butler JS, Moffitt R (1982) A computationally efficient quadrature procedure for the one-factor multinomial probit model. Econometrica 50:761–764CrossRefMATHGoogle Scholar
- Cox DR, Reid N (2004) A note on pseudolikelihood constructed from marginal densities. Biometrika 91:729–737MathSciNetCrossRefMATHGoogle Scholar
- Guilkey DK, Murphy JL (1993) Estimation and testing in the random effects probit model. J Econom 59:301–317CrossRefGoogle Scholar
- Heagerty PJ, Lele SR (1998) A composite likelihood approach to binary spatial data. J Am Stat Assoc 93:1099–1111MathSciNetCrossRefMATHGoogle Scholar
- Heckman JJ, Willis RJ (1976) Estimation of a stochastic model of reproduction: an econometric approach. In: Terleckyj NE (ed) Household production and consumption. National Bureau of Economic Research, New YorkGoogle Scholar
- Kuk AYC, Nott DJ (2000) A pairwise likelihood approach to analyzing correlated binary data. Stat Probab Lett 47:329–335CrossRefMATHGoogle Scholar
- Lee LF (2000) A numerically stable quadrature procedure for the one-factor random-component discrete choice model. J Econom 95:117–129MathSciNetCrossRefMATHGoogle Scholar
- Mardia KV, Kent JT, Hughes G, Taylor CC (2009) Maximum likelihood estimation using composite likelihoods for closed exponential families. Biometrika 96:975–982MathSciNetCrossRefMATHGoogle Scholar
- Press WH, Teukolsky SA, Vetterling WT, Flannery BP (1992) Numerical recipes, 2nd edn. Cambridge University Press, New YorkMATHGoogle Scholar
- Renard D, Molenberghs G, Geys H (2004) A pairwise likelihood approach to estimation in multilevel probit models. Comput Stat Data Anal 44:649–667MathSciNetCrossRefMATHGoogle Scholar
- Stroud A, Secrest D (1966) Gaussian quadrature formulas. Prentice-Hall, Englewood CliffsMATHGoogle Scholar
- Tsay WJ, Huang CJ, Fu TT, Ho IL (2013) A simple closed-form approximation for the cumulative distribution function of the composite error of stochastic frontier models. J Product Anal 39:259–269CrossRefGoogle Scholar
- Varin C, Vidoni P (2009) Pairwise likelihood inference for general state space models. Econom Rev 28:170–185MathSciNetCrossRefMATHGoogle Scholar