Likelihood Approximation for Discrete Choice Models with Sparse Grids

离散选择模型用于解释、分析和预测离散备选方案之间的选择。最广泛使用的参数估计方法是最大似然法，但似然函数只能对最基本的模型进行解析评估。对于更灵活的形式，如多项概率或混合多项Logit模型，它涉及到没有闭合形式解的多元积分。最常见的方法是使用蒙特卡罗模拟来近似似然函数，但足够精确的近似可能在计算上代价高得令人望而却步。这限制了在实证研究中的实际应用。这个项目的目的是开发基于模拟的常用估计方法的替代方法，这些方法通过有效地利用被积函数的给定光滑性来保证更快的收敛。为此，我们将采用已在许多不同领域成功使用的稀疏网格方法，并针对手头的具体问题进行调整。除了对似然函数的有效逼近外，我们还将研究所得到的估计量的性质。最后，我们将为以前受到计算成本显著限制的卫生和竞争经济学的具体实证应用实施方法。

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.