structures of a set of outcomes, while modeling the effect of covariates. We assume a linear mixed model formulation on the outcomes as response to covariates, a model corresponding to spiked covariance matrices. Since the subject-specific covariance matrices and the effects of covariates are believed to be sparse, we penalize coefficients using an oracle penalty function. Under some assumptions on the parameters and the likelihood, we show that the maximum likelihood estimator of the parameters is asymptotically consistent and is uniformly sparse ("sparsistent"), even when the number of parameters is small. We propose using the Bayesian Information Criterion (BIC) for tuning parameter selection and show that it is consistent for model selection. Using a simple iterated least squares procedure we are able to recover the model parameters with high accuracy. The method is implemented to study the effect of smoking on the covariances of gene methylations in the asthma pathway in smokers and non-smokers US veterans from the Normative Aging Study (NAS).