A random parameter approach to modeling and forecasting time series

Abstract

The dependence structure of a stationary time series can be described by its autocorrelation function ρ^k. Consider the simple autoregressive model of order 1: y_t = αy_t-1 + u_t where α ε (-1, 1) is a fixed constant and the u_t's are i.i.d. N(O,σ²). Here ρ^k = α^|k|, k = 0, ± 1, ± 2, . . . . It can be argued that as α ranges from 1 to -1, the behavior of the corresponding AR(1) model changes from that of a slowly changing, smooth time series to that of a rapidly changing time series. This motivates a generalized AR(1) model where the coefficient itself changes stochastically with time: y_t = α(t)y_t-1 + u_t where α(t) is a random function of time. This dissertation gives necessary and sufficient conditions for the existence of a mean zero stochastic process with finite second-order moments which is a solution to the generalized AR(1) model and gives sufficient conditions for the existence of a weakly stationary solution. The theory is illustrated with a specific model structure imposed on the random coefficient α(t); α(t) is modeled as a strictly stationary, two-state Markov chain with states taking on values between 0 and 1. The resulting generalized AR(1) process is shown to be weakly stationary. Techniques are provided for estimating the parameters of this specific model and for obtaining the optimal predictor from the estimated model.