Analysis of Financial Time Series, Third Edition

3.11 Random Coefficient Autoregressive Models

In the literature, the random coefficient autoregressive (RCA) model is introduced to account for variability among different subjects under study, similar to the panel data analysis in econometrics and the hierarchical model in statistics. We classify the RCA model as a conditional heteroscedastic model, but historically it is used to obtain a better description of the conditional mean equation of the process by allowing for the parameters to evolve over time. A time series rt is said to follow an RCA(p) model if it satisfies

(3.39)

where p is a positive integer, {δt} = {(δ1t, … , δpt)′} is a sequence of independent random vectors with mean zero and covariance matrix Ωδ, and {δt} is independent of {at}; see Nicholls and Quinn (1982) for further discussions of the model. The conditional mean and variance of the RCA model in Eq. (3.39) are

which is in the same form as that of a CHARMA model. However, there is a subtle difference between RCA and CHARMA models. For the RCA model, the volatility is a quadratic function of the observed lagged values rt−i. Yet the volatility is a quadratic function of the lagged innovations at−i in a CHARMA model.