Subset ARMA selection via the adaptive Lasso

Model selection is a critical aspect of subset autoregressive moving-average (ARMA) modelling. This is commonly done by subset selection methods, which may be computationally intensive and even impractical when the true ARMA orders of the underlying model are high. On the other hand, automatic variable selection methods based on regularization do not directly apply to this problem because the innovation process is latent. To solve this problem, we propose to identify the optimal subset ARMA model by fitting an adaptive Lasso regression of the time series on its lags and the lags of the residuals from a long autoregression fitted to the time series data, where the residuals serve as proxies for the innovations. We show that, under some mild regularity conditions, the proposed method enjoys the oracle properties so that the method identifies the correct subset model with probability approaching 1 with increasing sample size, and that the estimators of the nonzero coefficients are asymptotically normal with the limiting distribution the same as the case when the true zero coefficients are known a priori. We illustrate the new method with simulations and a real application.