In binary QRM there is little basis to choose between the logit and probit models because of the similarity of the cumulative normal and the logistic distributions. In the multinomial situation this is not the case. The multinomial logit (MNL) model has a closed form representation and is computationally tractable but lacks flexibility due to the IIA property. The multinomial probit (MNP) model gives flexibility but is computationally burdensome because of the need to evaluate multidimensional integrals. Another problem which involves high dimensional integrals is the probit (or tobit) model with serially correlated errors.
Until a few years ago, only 3 or 4 dimensional integrals could be evaluated. However, developments during the last decade on simulation based estimation allow us to estimate otherwise intractable models by approximating highdimensional integrals. A simple exposition of the econometric literature on simulation methods can be found in Stern (1997).
The generic problem for simulation is to evaluate expressions like E[ g(x)] = / g(x)f (x)dx where x is a random variable with density f(x) and g(x) is a function of x. The simulation method consists of drawing samples x1, x2,…, xN from f(x) and computing g(x;). Then, E[g(x)] = N £N1 g(x.) is an unbiased estimator of E[g(x)], and its variance is var(g(x;))/N. As N ^ ™, the variance of the simulator goes to zero. f (x) can be a discrete distribution.
Different applications of the simulation method depend on refinements in the way the sampling of f(x) is done. These refinements in probability simulators fall into the following categories: importance sampling, GHK simulator, Gibbs sampling, and antithetic variables. Importance sampling involves oversampling some "important" parts of f(x) from a well-chosen distribution g(x) from which it is easy to sample. The GHK simulator algorithm decomposes the joint density
of the errors into a product of conditional densities. It has been found to perform very well for simulating multinomial probit probabilities. Gibbs sampling is an iterative sampling method from conditional densities. Finally, the method of antithetic variables can be used on the above probability simulators to reduce sampling costs and the variance of the simulator through the sampling of pairs of random variables which are negatively correlated.8
The estimation methods commonly used are the method of simulated moments (MSM), the method of simulated likelihood (MSL), and the method of simulated scores (MSS). A detailed review of the above estimation methods can be found in Hajivassiliou and Ruud (1994) and in Geweke, Houser, and Keane, Chapter 22, in this companion.
The MSM is based on the GMM (generalized method of moments) method of estimation. The least squares method is a simple example of the method of moments. The GMM method depends on orthogonality conditions. For example, Avery, Hansen, and Hotz (1983) suggest how to estimate the multiperiod probit model by a set of within period orthogonality conditions that allows consistent estimation of the parameters and their standard errors under serial correlation. In many problems, the orthogonality conditions cannot be evaluated analytically. This is where simulation methods help. The resulting estimator is the MSM estimator. Let us illustrate this method with the MNP model. The loglikelihood function for a sample of size n with m alternatives is given by (17.9). The MSM consists of simulating P(j/x, P) using the multivariate normal distribution and substituting it in the moment equation:
XX – P(j/x, P)j = 0, (17.20)
for some weighting function w (k is an index taking on the values 1,…, K where K is the dimension of P). The estimator SMSM is obtained by solving (17.20).
The MSL is based on the maximum likelihood method of estimation and is useful when the likelihood function is not easily computed by analytical methods. In the context of multinomial choice models, MSL consists in simulating the actual choice probabilities given by Pjx, P) in (17.9). SMSL is obtained by maximizing (17.9) once the simulated choice probabilities are substituted.
The MSS is based on the idea that the score statistic (the derivative of the likelihood function) should have an expected value of zero at the true value of p. The potential advantage of the MSS relative to MSM is that it uses the efficiency properties of ML, but it is computationally more difficult than the MSM since the weight function itself has to be simulated, while it is analytically computed on the MSM. To see this, note that the MSS implies simulating P( j/x, P) in the following expression:
which makes clear that the weight function itself has to be simulated.
There are several papers using simulation methods but there are very few that compare the different methods. Geweke, Keane, and Runkle (1994) compare different methods in the estimation of the multinomial probit model, based on two Monte Carlo experiments for a seven choice model. They compare the MSL estimator using the GHK recursive probability simulator, the MSM method using the GHK recursive probability simulator and kernel-smoothed frequency simulators, and Bayesian methods based on Gibbs sampling. Overall, the Gibbs sampling algorithm had a slight edge, while the relative performance of the MSM and MSL (based on the GHK simulator) was difficult to evaluate. The MSM with the kernel-smoothed frequency simulator was clearly inferior.
In another paper, Geweke, Keane, and Runkle (1997) again compare the Bayesian methods based on Gibbs sampling and the MSM and MSL methods based on the GHK simulator. They do Monte Carlo studies with AR(1) errors in the multinomial multiperiod probit model, finding that the Gibbs sampling algorithm performs better than the MSM and MSL, especially under strong serial correlation in the disturbances (e. g. an AR(1) parameter of 0.8). They also find that to have root mean squared errors (RMSEs) for MSL and MSM within 10 percent of the RMSEs by the Gibbs sampling method one needs samples of 160 and 80 draws respectively (much higher than the 20 draws normally used). Thus, with serially correlated errors, the performance ranking is Gibbs sampling first, MSM second, and MSL last.
Keane (1993) discusses in detail the MSM estimator of limited dependent variable models with general patterns of serial correlation. He uses the recursive GHK algorithm. He argues that the equicorrelation model (implicit in the random effects probit model), for which the Butler and Moffitt algorithm works, is not empirically valid.
* We would like to thank three anonymous referees, the editor, Stephen Cosslett, and Kajal Lahiri for helpful comments. Remaining errors are our own. A. Flores-Lagunes gratefully acknowledges financial support from the National Council for Science and Technology of Mexico (CONACYT).
1 In (17.11) we need some normalization, like setting the first element of P to 1.
2 In the case of the binary choice models (logit and probit), Davidson and MacKinnon (1989) show that the score test statistic based on the exact information matrix can be computed easily using a particular artificial regression. See also Davidson and MacKinnon Chapter 1 in this companion.
3 We draw here from Gourieroux (1989).
4 Note that the approximating density is positive for all u.
5 GLL suggest, based on simulations, that using K = 3 allows for considerable flexibility in the distribution.
6 The asymptotic bias of the estimator will become negligible as K increases. GLL suggest starting with K = 3 (see previous fn.) and use score tests to determine whether such a value of K is appropriate.
7 The corresponding theorems can be found in Gallant and Nychka (1987).
8 References on probability simulators are Geweke, Keane, and Runkle (1994, 1997), Stern (1997), and Tanner (1996). For the reader interested in applying these methods,
Vassilis Hajivassiliou offers GAUSS and FORTRAN routines for the GHK simulator and other simulators at the following electronic address: http://econ. lse. ac. uk/~vassilis/ pub/simulation/gauss/
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