Generalizations of Linear Time Series Models

Autoregressive moving-average models have become the predominant approach in the analysis of economic, especially macroeconomic time series. The success of these parametric models is due to a mature and by now well-understood statistical theory which has been the subject of this book. The main assumption behind this theory is its linear structure. Although convenient, the assumption of a constant linear structure turned out to be unrealistic in many empirical applications. The evolution of economies and the economic dynamics are often not fully captured by constant coefficient linear models. Many time series are subject to structural breaks which manifest themselves as a sudden change in the model coefficients by going from one period to another. The detection and dating of such structural breaks is the subject of Sect. 18.1. Alternatively, one may think of the model coefficients as varying over time. Such models have proven to be very flexible and able to generate a variety of non-linear features. We present in Sects. 18.2 and 18.3 two variants of such models. In the first one, the model parameters vary in a systematic way with time. They are, for example, following an autoregressive process. In the second one, the parameters switch between a finite number of states according to a hidden Markov chain. These states are often identified as regimes which have a particular economic meaning, for example as booms and recessions. Further parametric and nonparametric methods for modeling and analyzing nonlinear time series can be found in Fan and Yao (2003).