This book provides a self-contained account of periodic models for seasonally observed economic time series with stochastic trends. Two key concepts are periodic integration and periodic cointegration. Periodic integration implies that a seasonally varying differencing filter is required to remove a stochastic trend. Periodic cointegration amounts to allowing cointegration paort-term adjustment parameters to vary with the season. The emphasis is on useful econrameters and shometric models that explicitly describe seasonal variation and can reasonably be interpreted in terms of economic behaviour. The analysis considers econometric theory, Monte Carlo simulation, and forecasting, and it is illustrated with numerous empirical time series. A key feature of the proposed models is that changing seasonal fluctuations depend on the trend and business cycle fluctuations. In the case of such dependence, it is shown that seasonal adjustment leads to inappropriate results.

This text provides a self-contained account of periodic models for seasonally observed economic time series with stochastic trends. The analysis considers econometric theory, Monte Carlo simulation and forecasting, and it is illuminated with empirical time series.

This book considers periodic time series models for seasonal data, characterized by parameters that differ across the seasons, and focuses on their usefulness for out-of-sample forecasting. Providing an up-to-date survey of the recent developments in periodic time series, the book presents a large number of empirical results. The first part of the book deals with model selection, diagnostic checking and forecasting of univariate periodic autoregressive models. Tests for periodic integration, are discussed, and an extensive discussion of the role of deterministic regressors in testing for periodic integration and in forecasting is provided. The second part discusses multivariate periodic autoregressive models. It provides an overview of periodic cointegration models, as these are the most relevant. This overview contains single-equation type tests and a full-system approach based on generalized method of moments. All methods are illustrated with extensive examples, and the book will be of interest to advanced graduate students and researchers in econometrics, as well as practitioners looking for an understanding of how to approach seasonal data.

This book provides a formal analysis of the models, procedures, and measures of economic forecasting with a view to improving forecasting practice. David Hendry and Michael Clements base the analyses on assumptions pertinent to the economies to be forecast, viz. a non-constant, evolving economic system, and econometric models whose form and structure are unknown a priori. The authors find that conclusions which can be established formally for constant-parameter stationary processes and correctly-specified models often do not hold when unrealistic assumptions are relaxed. Despite the difficulty of proceeding formally when models are mis-specified in unknown ways for non-stationary processes that are subject to structural breaks, Hendry and Clements show that significant insights can be gleaned. For example, a formal taxonomy of forecasting errors can be developed, the role of causal information clarified, intercept corrections re-established as a method for achieving robustness against forms of structural change, and measures of forecast accuracy re-interpreted.

In seminars and graduate level courses I have had several opportunities to discuss modeling and analysis of time series with economists and economic graduate students during the past several years. These experiences made me aware of a gap between what economic graduate students are taught about vector-valued time series and what is available in recent system literature. Wishing to fill or narrow the gap that I suspect is more widely spread than my personal experiences indicate, I have written these notes to augment and reor ganize materials I have given in these courses and seminars. I have endeavored to present, in as much a self-contained way as practicable, a body of results and techniques in system theory that I judge to be relevant and useful to economists interested in using time series in their research. I have essentially acted as an intermediary and interpreter of system theoretic results and perspectives in time series by filtering out non-essential details, and presenting coherent accounts of what I deem to be important but not readily available, or accessible to economists. For this reason I have excluded from the notes many results on various estimation methods or their statistical properties because they are amply discussed in many standard texts on time series or on statistics.

Modelling trends and cycles in economic time series has a long history, with the use of linear trends and moving averages forming the basic tool kit of economists until the 1970s. Several developments in econometrics then led to an overhaul of the techniques used to extract trends and cycles from time series. In this second edition, Terence Mills expands on the research in the area of trends and cycles over the last (almost) two decades, to highlight to students and researchers the variety of techniques and the considerations that underpin their choice for modelling trends and cycles.