Using penalized likelihood to select parameters in a random coefficients multinomial logit model

The multinomial logit model with random coefficients is widely used in applied research. This paper is concerned with estimating a random coefficients logit model in which the distribution of each coefficient is characterized by finitely many parameters. Some of these parameters may be zero or close to zero in a sense that is defined. We call these parameters small. The paper gives conditions under which with probability approaching 1 as the sample size approaches infinity, penalized maximum likelihood estimation (PMLE) with the adaptive LASSO (AL) penalty function distinguishes correctly between large and small parameters in a random-coefficients logit model. If one or more parameters are small, then PMLE with the AL penalty function reduces the asymptotic mean-square estimation error of any continuously differentiable function of the model's parameters, such as a market share, the value of travel time, or an elasticity. The paper describes a method for computing the PMLE of a random-coefficients logit model. It also presents the results of Monte Carlo experiments that illustrate the numerical performance of the PMLE. Finally, it presents the results of PMLE estimation of a random-coefficients logit model of choice among brands of butter and margarine in the British groceries market.