Shrinkage estimation for linear regression with ARMA errors

In this paper, we extend the modified lasso of Wang et al. (2007) to the linear regression model with autoregressive moving average (ARMA) errors. Such an extension is far from trivial because new devices need to be called for to establish the asymptotics due to the existence of the moving average component. A shrinkage procedure is proposed to simultaneously estimate the parameters and select the informative variables in the regression, autoregressive, and moving average components. We show that the resulting estimator is consistent in both parameter estimation and variable selection, and enjoys the oracle properties. To overcome the complexity in numerical computation caused by the existence of the moving average component, we propose a procedure based on a least squares approximation to implement estimation. The ordinary least squares formulation with the use of the modified lasso makes the computation very efficient. Simulation studies are conducted to evaluate the finite sample performance of the procedure. An empirical example of ground-level ozone is also provided.