Tuning parameter selection for the adaptive LASSO in the autoregressive model

We study the adaptive least absolute shrinkage and selection operator (LASSO) for the sparse autoregressive model (AR). Here, the sparsity of the AR model implies some of the autoregression coefficients are exactly zero, that must be excluded from the AR model. We propose the modified Bayesian information criterion (MBIC) as a way of selecting an optimal tuning parameter for the adaptive LASSO, which must be the most critical point in using the adaptive LASSO for the AR model. We prove that the adaptive LASSO obtained by minimizing the MBIC correctly distinguishes the true autoregression coefficients from zero asymptotically. The results hold even when the numbers of zero and nonzero true autoregression coefficients are diverging to infinity and the minimum of the absolute values of nonzero true autoregression coefficients decreases toward zero as the sample size increases. A small number of numerical studies are conducted to confirm the theoretical results.