A Unified Theory for Time Varying Models: Foundations and Applications in the Presence of Breaks and Heteroscedasticity (and some results on Companion and Hessenberg Matrices)

The paper develops an integrated approach to examine the dynamics of stochastic time series models with time dependent coefficients. We provide the closed form of the general solution for time varying autoregressive moving average models which is a long standing research topic. This enable us to characterise these models by deriving i) its multistep ahead predictor; ii) the first two unconditional moments; and iii) its covariance structure. In addition, capitalising on the connection between linear difference equations and the product of companion matrices, we employ our general methodology to obtain an explicit formula for the latter. We also apply our method to obtain results on generalised continuant matrices. To illustrate the practical significance of our results we consider autoregressive models with multiple breaks and also apply our unified approach to a variety of processes such as i) periodic, cyclical and smooth transition autoregressive models, ii) time varying generalised autoregressive conditional heteroscedasticity specifications, and iii) generalised random coefficient autoregressive models.

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