A COMPANION TO Theoretical Econometrics
Simulation estimation
The multinomial probit model has long been recognized as a useful discrete choice model. But, because its choice probability does not have a closed-form expression and its computation involves multiple integrals, it has not been a popular model for empirical studies until the 1990s. The advancement of computing technology and the recent development on simulation estimation techniques provide
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effective ways to implement this model. For a general description on various simulation methods, see Chapter 22 by Geweke, Houser, and Keane in this volume. Simulation estimation methods can be developed for the estimation of the multinomial probit sample selection model (Lee, 1996). The model can be estimated by two-stage methods and/or the method of maximum likelihood via Monte Carlo simulation. A simulated two-stage method is similar to the method of simulated moments (McFadden, 1989). For a two-stage method, the choice equations are first estimated. The likelihood function of the choice model can be simulated with the GHK (Geweke-Hajivassiliou-Keane) simulator. The probabilities of Uj - Ul > 0, l = 1,..., m but l Ф j are determined by the normal distribution of Ej = (v1 - Vj, ... , V-1 - Vj, Vj+1 - Vj, ... , vm - Vj). Denote Wj = (Zj - z1,..., Zj - Zj-1, Zj - z/+1,..., Zj - zm). The Ej can be represented as Ej = Яд-, where Hj is a lower triangular matrix of the Cholesky decomposition of the variance of Ej and n = (Пд,..., njm-1) is a standard normal vector. Define j = Wj1j/hj11 and Lji = [WjiY - Щhjlknk]/hjU for l = 2,..., m - 1. It follows that
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where ф(Яд) is a truncated standard normal density with support on (a, b). The GHK sampler is to generate sequentially truncated standard normal variables from Щ-^)(Пд). With S simulation runs, the GHK likelihood simulator of the choice model is tc§ (I) = nm= 1{SfXS^nm/^Lj?)}1'. The parameter у can be estimated by maximizing this simulated likelihood function. With the first-stage estimate y, the outcome equation can be estimated by the method of simulated moments. Simulated moment equations can be derived from observed outcome equations. Consider the bias-corrected outcome equation y = xв + E(u | x, I1 = 1) + П. As e1 has the Cholesky decomposition e1 = H1q1,
The GHK sampler П г=-1ф(-», Ll, )(n1l) can be used simultaneously to simulate both the numerator and denominator in (18.10). With S simulation runs from the GHK sampler, (18.10) can be simulated as ES(u |I1 = 1) = XSs=1 E(u | e1 = H1n1s))ra(s), where ra(s) = Пm=-1Ф(L(l! l))/IS=1П1=-1Ф(L^). ES(u|I1 = 1) is a consistent estimator of E(u 111 = 1) when S goes to infinity. The simulated method of moments can then be applied to estimate the equation y = xв + ES(u |I1 = 1) + -. The method of simulated moments is a generalized method of (simulated) moments. It is essentially an instrumental variable (IV) method. It is not desirable to apply a least squares procedure to the bias-corrected equation because ES(u|I1 = 1) with a finite S creates an error-in-variable (on regressors) problem.
The method of simulated maximum likelihood can be asymptotically efficient if the number of random draws S increases at a rate faster than the sjn - rate
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(Lee, 1992b). The likelihood function for an observation of the model with m - alternatives and with outcome for alternative 1 is
where u = y - xp. Each of the likelihood components can be simulated without bias with a generalization of the GHK simulator. The likelihood function can be simulated as
S m—1 і m і S m—1
/(U|£i = ЯіП(і8)) ^(A‘M ^
l=1
where the random variables are drawn from Щ—11ф( —„, Ll )(njl) for each j = 1,..., m. Lee (1996) compares these simulation methods and finds that the simulated maximum likelihood method can indeed be more efficient with a moderate amount of simulation draws. One may expect that simulation methods may play an important role in the future development of sample selection models with dynamic structures.
