This chapter considers the estimation of sample selection (SEL) models with polychotomous choices or multiple selection criteria.The focus is on the estimation of models which require estimation via simulation techniques. An example of such models is a model with multivariate normal disturbances. SEL models with polychotomous choices have been considered in Dubin and McFadden (1984), Lee (1983), and others, under extreme-value distributions. Schmertmann (1994) discussed limitations of these model specifications. In some empirical studies, in order to achieve simplicity, researchers imposed restrictive and unrealistic assumptions such as independence across disturbances of equations. Such practices resulted in controversies and debates (e.g., Duan et al. 1984, Hay and Olsen 1984). In many circumstances with complex selection criteria, multivariate normal disturbances may be the preferred specification (Amemiya 1974, Hausman and Wise 1978). With the development of simulation estimation methods, complex models can, in principle, be estimated (McFadden 1989, Pakes and Pollard 1989). Methods of simulated moments (MSM), simulated pseudolikelihood (SPL), and simulated maximum likelihood (SML) have been proposed in the literature (see McFadden 1989, Pakes and Pollard 1989, Lee 1992, Gourieroux and Monfort 1993, and Borsch–Supan and Hajivassiliou 1993). Parameter estimates may, however, be sensitive to simulators used (McFadden and Ruud 1994). The GHK simulator introduced by Geweke (1991), Borsch-Supan and Hajivassiliou (1993), and Keane (1994) provides an adequate probability simulator for the multinomial probit model. Monte Carlo results on comparing various simulation methods and simulators have mostly focused on discrete choice (DC) models (Hajivassiliou, McFadden, and Ruud 1996, Borsch-Supan and Hajivassiliou 1993, Geweke, Keane, andRunkle 1994, 1997).