Risk properties of a Stein-like estimator for multinomial choice models

Abstract

Stein-rule estimators, also known as shrinkage estimators, combine sample and non-sample information in a way that improves the precision of the estimation process or the quality of subsequent predictions. A Stein-rule estimator is a weighted average of a restricted and an unrestricted estimator, where the weights determine the degree of shrinkage, i.e. the importance that we place on the non-sample information. The existing literature shows that Stein-rule estimators may lead to squared error risk improvements in the linear regression, and in a number of non-linear models. The dissertation explores Stein-rule estimation in the context of multinomial choice models. It consists of three main parts. First, a Monte Carlo study is conducted to examine the properties of a Stein-rule estimator for the orthonormal conditional logit model. The shrinkage estimator is compared to the maximum likelihood estimator based on different measures of risk, namely squared error risk, weighted error risk, risk of marginal effects, and mean squared error of prediction in-sample and out-of-sample. Secondly, the analysis is extended to a more general data generation process by introducing various degrees of collinearity within alternatives, or between alternative-specific variables. Finally, there are three applications of Stein-rule estimation in multinomial choice models using marketing data. The main results of the study show that Stein-rule estimators offer significant risk improvement over the maximum likelihood estimator when certain conditions are met. The importance of this research is that shrinkage estimation is an easy to implement alternative to maximum likelihood estimation, which should be preferred in cases where we have good non-sample information, or when we are not sure of the performance of the MLE. The latter refers to data with small number of observations, or collinearity among the regressors, which is often a problem in practical applications.