A RANDOM PARAMETER PROCESS FOR MODELING AND FORECASTING TIME SERIES

Abstract. A generalized autoregressive (GAR) process {Z ( t ) ; t = 0 , ±1, …} is defined to satisfy the recurrence relation Z(t) = A_θ (t)Z (t -l)+ u( t ), where {A_θ(t); t = 0,±1, …} is itself a stochastic process depending on a vector parameter θ and where {u( t ); t = 0, ±1, …} is white noise with Eu ² ( t ) = a ². This paper develops theory and methodology and implementing the class of GAR processes for time series modeling and forecasting. Conditions on the 'parameter process' { A _θ ( t ); t = 0, ±1, …} are obtained for the existence of a GAR process; necessary and sufficient conditions on { A_θ ( t ) ; t = 0, ±1, …} for existence of a stationary GAR process are also obtained. Procedures are developed for computing maximum likelihood estimates of the parameters 0 and u² and for computing the minimum mean squared error forecasts for GAR processes.