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.

翻译：我们考虑线性随机系数回归模型，其中允许回归量具有有限支撑。首先，我们研究可识别性，并证明若协变量（不含截距项）的支撑集包含一个笛卡尔积，且每个坐标至少包含三个点，则随机系数的均值、方差与协方差可由给定协变量条件下响应的前两个条件矩唯一确定。我们还讨论了高阶混合矩的识别问题，以及存在二元回归量时的部分识别情况。其次，我们证明了自适应LASSO在有限维及中等高维条件下对随机系数方差与协方差的变量选择一致性。这意味着所得的协方差矩阵估计量实际上为半正定矩阵，因此是有效的协方差矩阵——这与简单最小二乘拟合得到的估计量形成对比。我们通过模拟研究对提出的方法进行了说明。