  Empirical economic research frequently applies maximum likelihood estimation
in cases where the likelihood function is analytically intractable. Most of the
theoretical literature focuses on maximum simulated likelihood (MSL)
estimators, while empirical and simulation analyzes often find that alternative
approximation methods such as quasi-Monte Carlo simulation, Gaussian
quadrature, and integration on sparse grids behave considerably better
numerically. This paper generalizes the theoretical results widely known for
MSL estimators to a general set of maximum approximated likelihood (MAL)
estimators. We provide general conditions for both the model and the
approximation approach to ensure consistency and asymptotic normality. We also
show specific examples and finite-sample simulation results.
