  This paper introduces a new data-driven methodology for estimating sparse
covariance matrices of the random coefficients in logit mixture models.
Researchers typically specify covariance matrices in logit mixture models under
one of two extreme assumptions: either an unrestricted full covariance matrix
(allowing correlations between all random coefficients), or a restricted
diagonal matrix (allowing no correlations at all). Our objective is to find
optimal subsets of correlated coefficients for which we estimate covariances.
We propose a new estimator, called MISC, that uses a mixed-integer optimization
(MIO) program to find an optimal block diagonal structure specification for the
covariance matrix, corresponding to subsets of correlated coefficients, for any
desired sparsity level using Markov Chain Monte Carlo (MCMC) posterior draws
from the unrestricted full covariance matrix. The optimal sparsity level of the
covariance matrix is determined using out-of-sample validation. We demonstrate
the ability of MISC to correctly recover the true covariance structure from
synthetic data. In an empirical illustration using a stated preference survey
on modes of transportation, we use MISC to obtain a sparse covariance matrix
indicating how preferences for attributes are related to one another.
