In empirical research, panel (and multinomial) probit models are leading examples for the use of maximum simulated likelihood estimators. The Geweke–Hajivassiliou–Keane (GHK) simulator is the most widely used technique for this type of problem. This chapter suggests an algorithm that is based on GHK but uses an adaptive version of sparse-grids integration (SGI) instead of simulation. It is adaptive in the sense that it uses an automated change-of-variables to make the integration problem numerically better behaved along the lines of efficient importance sampling (EIS) and adaptive univariate quadrature. The resulting integral is approximated using SGI that generalizes Gaussian quadrature in a way such that the computational costs do not grow exponentially with the number of dimensions. Monte Carlo experiments show an impressive performance compared to the original GHK algorithm, especially in difficult cases such as models with high intertemporal correlations.
Heiss, F. (2010), "The panel probit model: Adaptive integration on sparse grids", Greene, W. and Carter Hill, R. (Ed.) Maximum Simulated Likelihood Methods and Applications (Advances in Econometrics, Vol. 26), Emerald Group Publishing Limited, Bingley, pp. 41-64. https://doi.org/10.1108/S0731-9053(2010)0000026006
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