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

TL;DR

This paper introduces a penalized inference method for autoregressive moving average models with time-dependent predictors, simplifying complex models and ensuring consistent estimation in high-dimensional financial data analysis.

Contribution

It proposes an $l_1$ penalized likelihood approach with adaptive lasso penalties for ARMA models with time-dependent predictors, demonstrating oracle properties and consistency.

Findings

Successful application to stock index data

Estimator enjoys oracle property in high-dimensional settings

Model captures complex dynamic relationships effectively

Abstract

Linear models that contain a time-dependent response and explanatory variables have attracted much interest in recent years. The most general form of the existing approaches is of a linear regression model with autoregressive moving average residuals. The addition of the moving average component results in a complex model with a very challenging implementation. In this paper, we propose to account for the time dependency in the data by explicitly adding autoregressive terms of the response variable in the linear model. In addition, we consider an autoregressive process for the errors in order to capture complex dynamic relationships parsimoniously. To broaden the application of the model, we present an $l_1$ penalized likelihood approach for the estimation of the parameters and show how the adaptive lasso penalties lead to an estimator which enjoys the oracle property. Furthermore, we prove the consistency of the estimators with respect to the mean squared prediction error in high-dimensional settings, an aspect that has not been considered by the existing time-dependent regression models. A simulation study and real data analysis show the successful applications of the model on financial data on stock indexes.