Generalized autoregressive errors in the multinomial probit model

In discrete choice analysis, the multinomial probit provides the most flexible framework to allow for general interdependencies among the alternatives. These interdependencies are usually modeled through the correlation structure of the error term. This framework suffers from two serious impediments, however. The first and major one is computational and is related to the evaluation of the response probabilities, which are multidimensional normal integrals. In the past, this has restricted its utilization to studies involving less than five alternatives where using numerical integration remains practical. A recent solution to the dimensionality problem consists in replacing the choice probabilities with easy to compute efficient simulators. The second impediment arises in models with large choice sets when a fully unconstrained error correlation structure is postulated. In that case, the large number of nuisance parameters to estimate in the error covariance matrix becomes a problematic issue that can exacerbate the estimation process. To tackle that problem, a first-order generalized autoregressive [GAR(1)] error approach is suggested. The approach enables one to approximate general correlation structures with parsimonious parametric specifications. The key feature of the approach is that the number of nuisance parameters grows linearly with the number of alternatives considered. The methodology is most useful in models with large choice sets where the estimation also requires to use probability simulators. The paper focuses on the GAR(1) solution to the error covariance matrix estimation problem. The issue of identification of the nuisance parameters is examined in detail and a rank condition is suggested. Some theoretical and numerical examples based on synthetic data are presented.