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Linear models that contain a time-dependent response and explanatoryvariables have attracted much interest in recent years. The most general formof the existing approaches is of a linear regression model with autoregressivemoving average residuals. The addition of the moving average component resultsin a complex model with a very challenging implementation. In this paper, wepropose to account for the time dependency in the data by explicitly addingautoregressive terms of the response variable in the linear model. In addition,we consider an autoregressive process for the errors in order to capturecomplex dynamic relationships parsimoniously. To broaden the application of themodel, we present an $l_1$ penalized likelihood approach for the estimation ofthe parameters and show how the adaptive lasso penalties lead to an estimatorwhich enjoys the oracle property. Furthermore, we prove the consistency of theestimators with respect to the mean squared prediction error inhigh-dimensional settings, an aspect that has not been considered by theexisting time-dependent regression models. A simulation study and real dataanalysis show the successful applications of the model on financial data onstock indexes.

Penalised inference for autoregressive moving average models with time-dependent predictors