Penalized Quasi-likelihood Estimation and Model Selection in Time Series Models with Parameters on the Boundary

TL;DR

This paper develops a penalized likelihood approach for time series models with boundary parameters, ensuring standard Gaussian asymptotics and effective model selection, demonstrated through ARCH models and financial data.

Contribution

It extends penalized likelihood theory to models with boundary constraints and dependence, providing new asymptotic results and practical model selection methods for time series.

Findings

Penalized estimation performs well with many parameters.

Model selection accurately identifies suitable ARCH models.

Method confirms long-memory in financial time series.

Abstract

We extend the theory from Fan and Li (2001) on penalized likelihood-based estimation and model-selection to statistical and econometric models which allow for non-negativity constraints on some or all of the parameters, as well as time-series dependence. It differs from classic non-penalized likelihood estimation, where limiting distributions of likelihood-based estimators and test-statistics are non-standard, and depend on the unknown number of parameters on the boundary of the parameter space. Specifically, we establish that the joint model selection and estimation, results in standard asymptotic Gaussian distributed estimators. The results are applied to the rich class of autoregressive conditional heteroskedastic (ARCH) models for the modelling of time-varying volatility. We find from simulations that the penalized estimation and model-selection works surprisingly well even for a large number of parameters. A simple empirical illustration for stock-market returns data confirms the ability of the penalized estimation to select ARCH models which fit nicely the autocorrelation function, as well as confirms the stylized fact of long-memory in financial time series data.