SAUSS: Stochastic Approximation with Unbiased Simulated Scores for Limited Dependent Variable Models

Simulated maximum likelihood is widely used in latent-variable models where likelihood evaluation requires intractable integrals or probabilities. With a fixed number of simulation draws, its estimating equation is generally biased. We develop Stochastic Approximation with Unbiased Simulated Scores (SAUSS), combining score simulation with stochastic gradient descent. The Hajivassiliou--McFadden identity, an instance of Fisher's identity, expresses the likelihood score as the conditional expectation of the complete-data score given the observed data. Exact conditional draws yield an unbiased simulated score for any fixed number of draws. The estimator averages stochastic gradient iterates computed using mini-batches of fixed size regardless of sample size, avoiding repeated full-sample likelihood evaluations. Under local regularity and initialization conditions, we establish asymptotic normality and a functional central limit theorem for the partial-sum process of the iterates under multi-pass sampling from an observed sample. These results support plug-in and random-scaling confidence intervals that remain asymptotically valid when algorithmic uncertainty is non-negligible relative to sampling uncertainty. We develop multinomial probit as a leading application. In simulations and two applications, SAUSS returns estimates in every reported run, including cases where the simulated maximum likelihood implementation returns none. When both return estimates, SAUSS gives comparable estimates in less time.