Regression coefficient and autoregressive order shrinkage and selection via the lasso

The "least absolute shrinkage and selection operator" ('lasso') has been widely used in regression shrinkage and selection. We extend its application to the regression model with autoregressive errors. Two types of lasso estimators are carefully studied. The first is similar to the traditional lasso estimator with only two tuning parameters (one for regression coefficients and the other for autoregression coefficients). These tuning parameters can be easily calculated via a data-driven method, but the resulting lasso estimator may not be fully efficient. To overcome this limitation, we propose a second lasso estimator which uses different tuning parameters for each coefficient. We show that this modified lasso can produce the estimator as efficiently as the "oracle". Moreover, we propose an algorithm for tuning parameter estimates to obtain the modified lasso estimator. Simulation studies demonstrate that the modified estimator is superior to the traditional estimator. One empirical example is also presented to illustrate the usefulness of lasso estimators. The extension of the lasso to the autoregression with exogenous variables model is briefly discussed. Copyright 2007 Royal Statistical Society.