Frequentist and Bayesian Analysis of Random Coefficient Autoregressive models

Abstract

Random Coefficient Autoregressive (RCA) models are obtained by introducing random coefficients to an AR or more generally ARMA model. These models have second order properties similar to that of ARCH and GARCH models. Historically an RCA model has been used to model the conditional mean of a time series, but it can also be viewed as a volatility model. In this thesis, we consider both Frequentist and Bayesian approaches to analyze the first order RCA models.

For a weakly stationary RCA(1), it has been shown that the Maximum Likelihood Estimates (MLEs) are strongly consistent and satisfy a classical Central Limit Theorem. We consider a broader class of RCA(1) models whose parameters lie in the region of strict stationarity and ergodicity. We show that similar asymptotic properties can be extended to this class of models which includes the unit-root RCA(1) as a special case. The existence of a unit root in an RCA(1) has significant impact on the inference of data especially in the aspect of model forecasting. We develop the Wald criterion based on MLEs for testing unit root and evaluate its power via simulation studies.

In addition to the Frequentist approach to RCA(1) models, Bayesian methods can also be used. We propose non-informative priors for the model parameters and apply them in Bayesian estimation procedure. Two model selection criteria are investigated to see their performance in choosing between RCA(1) and AR(1) models. We use two Bayesian methods to test for the unit-root hypothesis: one is based on the Posterior Interval (PI), and the other one is by means of Bayes Factor (BF). We apply both flat and mixed priors for the stationarity parameter in RCA(1) and compare the performance of different Bayesian unit-root testing criteria using these two types of prior densities through simulation. At the end of the thesis, two real life examples involving the daily stock volume transaction data are presented to show the applicability of the proposed methods.