  We consider linear random coefficient regression models, where the regressors
are allowed to have a finite support. First, we investigate identifiability,
and show that the means and the variances and covariances of the random
coefficients are identified from the first two conditional moments of the
response given the covariates if the support of the covariates, excluding the
intercept, contains a Cartesian product with at least three points in each
coordinate. We also discuss ientification of higher-order mixed moments, as
well as partial identification in the presence of a binary regressor. Next we
show the variable selection consistency of the adaptive LASSO for the variances
and covariances of the random coefficients in finite and moderately high
dimensions. This implies that the estimated covariance matrix will actually be
positive semidefinite and hence a valid covariance matrix, in contrast to the
estimate arising from a simple least squares fit. We illustrate the proposed
method in a simulation study.
