Likelihood Approximation for Discrete Choice Models with Sparse Grids

离散选择模型用于解释、分析和预测离散方案之间的选择。最广泛使用的方法估计的参数是最大似然，但似然函数只能分析评估最基本的模型。对于更灵活的版本，如多项probit或混合多项logit模型，它涉及没有封闭形式解的多变量积分。最常见的方法是使用蒙特卡罗模拟来近似似然函数，但是足够精确的近似可能在计算上代价高昂。这限制了实证研究的实际应用。本项目旨在开发替代常见的基于模拟的估计方法，通过有效地使用给定的被积函数的平滑性，保证更快的收敛。为此，我们将采用稀疏网格的方法，它已成功地用于许多不同的领域，并将其调整到手头的具体问题。除了似然函数的有效逼近外，我们还将研究所得估计量的性质。最后，我们将实现健康和竞争经济学的具体实证应用的方法，这些方法以前受到计算成本的严重限制。

Discrete choice models are used to explain, analyze, and predict choices between discrete alternatives. The most widely used method for estimating the parameters is maximum likelihood, but the likelihood function can be analytically evaluated only for the most basic models. For more flexible versions like the multinomial probit or the mixed multinomial logit model, it involves multivariate integrals that do not have a closed-form solution. The most common approach is to approximate the likelihood function using Monte Carlo simulation, but a sufficiently accurate approximation can be computationally prohibitively costly. This restricts the practical use in empirical research.This project aims to develop alternatives to the common simulation-based estimation methods that promise a much faster convergence by efficiently using the given smoothness of the integrands. To this end we will adopt the method of sparse grids which has been used successfully in many different areas and tailor it to the specific problems at hand. Besides the efficient approximation of the likelihood function, we will also study the properties of the resulting estimators. Finally, we will implement the methods for specific empirical applications from health and competition economics that were significantly restricted by computational costs before.