A Unified Theory of Time Varying Models with Applications to Economics and Finance

The paper examines the problem of representing the dynamics of stochastic time series models with time dependent coefficients. We provide the closed form of the fundamental solutions to time varying (TV) autoregressive moving average (ARMA) models, which is a long standing open question. This enable us to characterize the TV-ARMA(p,q) specification by deriving  i) its multistep ahead predictor; ii) the first two unconditional moments; and iii) its covariance structure. In addition, our theorem yields an explicit formula for product of companion matrices. To illustrate the practical significance of our results we consider AR models with multiple structural breaks and also apply our unified approach to a variety of processes such as periodic and smooth transition AR and GARCH models.

Models with time varying coefficients can better capture the dynamics of time series. But despite their importance for our understanding of especially series subject to structural breaks there is a lack of a general method that can be applied to examine these models. There exist some disperse contributions in both the mathematics literature on difference equations (from the perspective of linear algebra), and mathematical statistics results, but they lack either generality and/or applicability.

As we argue below, the Achilles' heel of this multidisciplinary and disperse literature is that it does not provide a general method, for finding the fundamental solutions for homogeneous linear models with time dependent coefficients. The main theoretical contribution of the present paper is the development of such a general method that provides the closed form of the p linearly independent solutions to time varying linear homogeneous difference equations of order p, which is a long standing open question.

In the mathematics literature the problem under study is the solution of a p order linear difference equation with variable coefficients. Kittappa (1993) gives such a general solution in terms of determinants but he does not provide the fundamental solution set, that is the p linearly independent homogeneous solutions. Thus his formulation cannot be utilized to derive the stability conditions for dynamic systems. Mallik (2000) provides a closed form solution for the homogeneous difference equation, but it appears not to be computationally tractable. Lim and Dai (2011) point out that although explicit solutions for general linear difference equations are given by Mallik (2000), they appear to be unmotivated and no methods of solution are discussed. Lim and Dai (2011) connect the solutions of linear difference equations with variable coefficients to enumerative combinatorics in an associated digraph, and they also show that the solution space of a linear homogeneous difference equation is given by the linear combinations of the integer translates of a single function.

Our response and contribution to the econometrics literature is to introduce a unified methodology that combines our innovative results with the valuable insights of the existing literature on mathematical statistics and stochastic processes. In doing so, it opens the door to the unified treatment of time varying models. In subsequent sections we discuss prototype models of such a unified treatment: processes with multiple structural breaks, periodic and smooth transition AR and GARCH models, and generalized random coefficients AR models.