# Maximum Approximated Likelihood Estimation

## TL;DR

This paper extends the theoretical foundation of maximum approximated likelihood (MAL) estimators, demonstrating their consistency and asymptotic normality, and compares their performance with other approximation methods through examples and simulations.

## Contribution

It generalizes existing results for maximum simulated likelihood to a broader class of MAL estimators, providing conditions for their statistical properties.

## Key findings

- MAL estimators are consistent under general conditions.
- MAL estimators are asymptotically normal.
- Simulation results show MAL performs well compared to other methods.

## Abstract

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.