Simulation estimation of mixed discrete choice models using randomized and scrambled Halton sequences

Chandra R. Bhat
2003 Transportation Research Part B: Methodological  
The use of simulation techniques has been increasing in recent years in the transportation and related fields to accommodate flexible and behaviorally realistic structures for analysis of decision processes. This paper proposes a randomized and scrambled version of the Halton sequence for use in simulation estimation of discrete choice models. The scrambling of the Halton sequence is motivated by the rapid deterioration of the standard Halton sequence's coverage of the integration domain in
more » ... dimensions of integration. The randomization of the sequence is motivated from a need to statistically compute the simulation variance of model parameters. The resulting hybrid sequence combines the good coverage property of quasi-Monte Carlo sequences with the ease of estimating simulation error using traditional Monte Carlo methods. The paper develops an evaluation framework for assessing the performance of the traditional pseudo-random sequence, the standard Halton sequence, and the scrambled Halton sequence. The results of computational experiments indicate that the scrambled Halton sequence performs better than the standard Halton sequence and the traditional pseudo-random sequence for simulation estimation of models with high dimensionality of integration. Keywords: Maximum simulated likelihood estimation, pseudo-random sequences, quasi-random sequences, hybrid sequences, multinomial probit model, mixed logit model, mixed probit model. dimensional integral that can be evaluated accurately using well-established quadrature techniques. However, quadrature formulas are unable to compute integrals with sufficient precision and speed for estimation of models with higher than 1-2 dimensions of integration (see Hajivassiliou and Ruud, 1994) . In fact, because of the curse of dimensionality, the quadrature method is literally useless in high dimensions. Two broad simulation methods are available in high dimensions: (a) Monte Carlo methods and (b) Quasi-Monte Carlo methods. Each of these is discussed in the next two sections. Section 1.3 discusses a hybrid of the Monte-Carlo and Quasi-Monte Carlo methods. The Monte-Carlo method The Monte-Carlo simulation method (or "the method of statistical trials") to evaluating multidimensional integrals entails computing the integrand at a sequence of "random" points and computing the average of the integrand values. The basic principle is to replace a continuous average by a discrete average over randomly chosen points. Of course, in actual implementation, truly random sequences are not available; instead, deterministic pseudo-random sequences which appear random when subjected to simple statistical tests are used (see Niederreiter, 1995 for a discussion of pseudo-random sequence generation). This pseudo-Monte Carlo (or PMC) method has a slow asymptotic convergence rate with the expected integration error of the order of N -0.5 in probability (N being the number of pseudo-random points drawn from the s-dimensional integration space). Thus, to obtain an added decimal digit of accuracy, the number of draws needs to be increased hundred fold. However, the PMC method's convergence rate is remarkable in that it is applicable for a wide class of integrands (the only requirement is that the integrand have a finite variance; see Spanier and Maize, 1994) . Further, the integration error can be easily estimated using the sample values and invoking the central limit theorem, or by replicating the evaluation of the integral several times using independent sets of PMC draws and computing the variance in the different estimates of the integrand. The quasi-Monte Carlo method The quasi-Monte Carlo method is similar to the Monte Carlo method in that it evaluates a multidimensional integral by replacing it with an average of values of the integrand computed at discrete points. However, rather than using pseudo-random sequences for the discrete points, the quasi-Monte Carlo approach uses "cleverly" crafted non-random and more uniformly distributed
doi:10.1016/s0191-2615(02)00090-5 fatcat:dy5lzxwqzzg3pf3wp4dbkeujou