Maximum Likelihood Estimation and Unit Root Test for First Order Random Coefficient AutoRegressive Models

Abstract

Random Coefficient AutoRegressive (RCAR) models are obtained by introducing random coefficients to an AR or more generally AutoRegressive Moving Average (ARMA) model. For a weakly stationary first order RCAR model, it has been shown that the Maximum Likelihood Estimators (MLEs) are strongly consistent and satisfy a classical Central Limit Theorem. A broader class of first order RCAR models allowing the parameters to lie in the region of strict stationarity and ergodicity is developed. Asymptotic properties are established for this extended class of models which includes the unit root first order RCAR model as a special case. The existence of a unit root in a first order RCAR process has practical impact on data analysis especially in the context of model forecasting. A Wald type criterion based on the MLEs is also developed to test unit root hypothesis. The asymptotic normality of the Wald statistic under the null hypothesis is validated using a simulation study.

Key Words:

• Ergodicity
• Hypothesis testing
• Strict stationarity
• Time series