An Improved Online Penalty Parameter Selection Procedure for $\ell_1$-Penalized Autoregressive with Exogenous Variables

Selecting relevant features in a time series model is a longstanding open problem in both statistics and econometrics. Traditional feature selection approaches, such as those proposed by Box and Jenkins (1994) rely on heuristic methods, such as visual inspection of diagnostic plots. Alternatively, popular data-driven approaches select features based upon the minimization of information criterion, such as Akaike's Information Criterion (AIC, Akaike 1974) or Bayesian Information Criterion (BIC, Schwarz et al. 1978) over a subset of potential models. Typically, because the space of all possible models is extremely large, such methods require imposing substantial restrictions on the feature space. More recent approaches have extended the lasso (Tibshirani, 1996) to a time dependent setting. The lasso has advantages over conventional methods in that it shrinks least squares estimates toward zero in addition to performing feature selection. It also allows for estimation under scenarios in which there are more potential features than observations.