Obtaining reliable likelihood ratio tests from simulated likelihood functions

Models allowing for continuous heterogeneity by assuming that value of one or more parameters follow a specified distribution have become increasingly popular. This is known as 'mixing' parameters, and it is standard practice by researchers--and the default option in many statistical programs--to base test statistics for mixed models on simulations using asymmetric draws (e.g. Halton draws). PROBLEM 1: INCONSISTENT LR TESTS DUE TO ASYMMETRIC DRAWS: This paper shows that when the estimated likelihood functions depend on standard deviations of mixed parameters this practice is very likely to cause misleading test results for the number of draws usually used today. The paper illustrates that increasing the number of draws is a very inefficient solution strategy requiring very large numbers of draws to ensure against misleading test statistics. The main conclusion of this paper is that the problem can be solved completely by using fully antithetic draws, and that using one dimensionally antithetic draws is not enough to solve the problem. PROBLEM 2: MAINTAINING THE CORRECT DIMENSIONS WHEN REDUCING THE MIXING DISTRIBUTION: A second point of the paper is that even when fully antithetic draws are used, models reducing the dimension of the mixing distribution must replicate the relevant dimensions of the quasi-random draws in the simulation of the restricted likelihood. Again this is not standard in research or statistical programs. The paper therefore recommends using fully antithetic draws replicating the relevant dimensions of the quasi-random draws in the simulation of the restricted likelihood and that this should become the default option in statistical programs. JEL classification: C15; C25.