Modelling Trends and Cycles in Economic Time Series

This chapter begins with a historical perspective on the analysis of trends and cycles in economic time series, beginning with the early studies of business cycles in the late nineteenth century, before moving on to the introduction of the concept of trends in the early years of the twentieth century. Formal models of the business cycle began to emerge in the 1930s, and a range of approaches, some using mathematical developments from other scientific fields, were employed to take the models in various directions. After the ‘measurement without theory’ debate of the late 1940s and early 1950s, technical developments in time series analysis from the 1970s enabled great progress to be made in the statistical and econometric modelling of stochastic and common trends and cycles existing in observed economic data. The chapter then provides an overview of the remaining chapters of the book.

This chapter begins by introducing the ‘classical’ trend-cycle decomposition, in which a time series is split additively into trend and cyclical components that are assumed to be statistically independent of each other. Deterministic models for the trend are then introduced, which may be nonlinear and may have breaks in them. Trends may also be modelled using moving averages of various varieties. The cyclical component is then modelled using autoregressive processes before several issues involved in detrending using these approaches are discussed. Further reading and background material is provided.

This chapter begins by introducing the concept of stochastic trends through the use of autoregressive-integrated-moving average (ARIMA) models. It is shown that the order of integration, or degree of differencing, that a time series exhibits is of fundamental importance for determining the properties of stochastic trends and methods for selecting the order of integration are developed through detailed examples of ARIMA modelling. The crucial distinction between trend and difference stationarity is then introduced, and methods to distinguish between the two using unit root tests and to estimate trends robustly are discussed. Breaking and segmented trends in the possible presence of unit roots are then analysed. Unobserved component models are investigated, including the Beveridge-Nelson decomposition, and the signal extraction approach via the Kalman filter is used to estimate such ‘structural’ models. Further reading and background material is provided.

This chapter begins by examining the use of linear filters for detrending, with particular attention being paid to the frequency domain properties of symmetric linear filters. Low-pass, high-pass and band-pass filters are introduced, and their design and properties analysed. A popular technique for extracting the cyclical component, known as the Hodrick-Prescott filter, is then discussed, this focusing on both its infinite and finite sample variants and the various critiques that have been levelled against it. The relationship between filters and structural time series models is then considered, with model-based filter design being examined. Trends and cycles that are contained within unobserved structural models are finally considered. Further reading and references are provided.

This chapter begins with the development of regime shift models, starting with Markov processes and then considering threshold and smooth transition regime models. Model selection issues are a focus here, as are the different implications of the alternative models for trend and cycle behaviour. Nonparametric modelling of trends is then considered. Here attention is focused on smoothing estimators, along with kernel and local polynomial regression. Finally, nonlinear stochastic trends are introduced. Further reading and references are provided.

This chapter begins by introducing the concept of common features in time series, focusing on their specification and testing. The modelling of common cycles and codependence and of common deterministic trends is then discussed. The concept of common stochastic trends is then introduced, along with its manifestation as the property of cointegration. This leads on to the specification of the vector error correction model (VECM), with its consequent issues of estimation and tests for cointegrating rank. Stochastic common cycles are then introduced within the VECM framework. The chapter ends with a discussion of multivariate filtering and of co-breaking. Further reading and references are provided.

This chapter provides brief conclusions and some remarks about the understanding of trend behaviour and the occurrence of structural breaks at the present time.