Measurement of Simulation Variance in Parameter Estimation by Daniel McFadden and Kenneth Train November 15, 1995 I. Introduction Estimation by simulation involves drawing random terms for each observation and calculating an element (such as the probability or score) that enters an objective function (e.g. the simulated log-likelihood function) and/or a first-order condition (e.g., the sum of simulated scores or simulated moment conditions). The resulting parameter estimates depend on the particular values of the draws for each observation and necessarily vary over draws. We investigate the variance in parameter estimates that is induced by the drawing of random terms and propose a practical method for measuring this simulation variance and correcting the standard errors of the estimates to account for it. Our analysis is conducted in the context of a random-parameters logit model estimated on data regarding households’ choices among classes of new cars to buy. We find that the simulation variance in the estimated parameters is fairly large relative to the estimates themselves. In particular, the simulation standard deviation ranges from 3% to 41% of the estimated parameters when using 20 repetitions and, even with 1000 repetitions, ranges from 0.4% to 8% of the parameters. These large standard deviations in parameter estimates occur even though the simulation standard deviations in the average probability and the log-likelihood function are very small -- about one-twentieth of one percent of their values with as few as 20 repetitions. Apparently, small simulation variance in the probabilities does not guarantee small simulation variance in the estimated parameters. Even though the simulation variance in parameter estimates is sizeable relative to the parameters, it is smaller than the sampling variance in our application. With 20 repetitions, the simulation standard deviation ranges from 6% to 70% of the sampling 1