Time-Varying Parameters in Continuous and Discrete Time

We consider models for both deterministic one-time and continuous stochastic parameter change in a continuous time autoregressive model around a deterministic trend function. For the latter we focus on the case where the autoregressive parameter itself follows a first-order autoregression. Exact discrete time analogue models are detailed in each case and compared to corresponding parameter change models adopted in the discrete time literature. The relationships between the parameters in the continuous time models and their discrete time analogues are also explored. For the one- time parameter change model the discrete time models used in the literature can be justified by the corresponding continuous time model, with a only a minor modification needed for the (most likely) case where the changepoint does not coincide with one of the discrete time observation points. For the stochastic parameter change model considered we show that the resulting discrete time model is characterised by an autoregressive parameter the logarithm of which follows an ARMA(1,1) process. We discuss how this relates to models which have been proposed in the discrete time stochastic unit root literature. The implications of our results for a number of extant discrete time models and testing procedures are discussed.