Many economic time series, however, are not stationary, but may often be transformed to stationarity by the simple operation of differencing. Chapter 3 examines some informal methods of dealing with non-stationary data and consequently introduces the key concept of an integrated process, of which the random walk is a special case, so leading to the class of autoregressive-integrated-moving average (ARIMA) models. As is demonstrated by way of examples, although the informal methods proposed in Chapter 3 often work well in practice, it is important that formal means of testing whether a series is integrated or not and, if it is, of testing what order of integration it might be, are available. Chapter 4 thus develops the theory and practice of testing for one or more unit roots, the presence of which is the manifestation of ‘integratedness’ of a time series. An alternative to differencing as a means of inducing stationarity is to detrend the series using a polynomial, typically a linear, function of time. How to distinguish between these two methods of inducing stationarity by way of generalised unit root tests and the differing implications of the two methods for the way the series reacts to shocks are also discussed in this chapter.

Up to this point we have assumed that the errors, or innovations, in the various models have constant variance. For many economic time series, particularly financial ones observed at relatively high frequencies, this assumption is untenable, for it is well known that financial markets go through periods of excessive turbulence followed by periods of calm, a phenomenon that goes under the general term ‘volatility’. The manifestation of market volatility is that error variances change over time, being dependent upon past behaviour. Chapter 5 therefore introduces the class of autoregressive conditionally heteroskedastic (ARCH) processes. These are designed to incorporate volatility into models and, indeed, to provide estimates of such volatility.

An important aspect of time series modelling is to forecast future observations of the series being analysed. Chapter 6 develops a theory of forecasting for all the models introduced so far, emphasising how the properties of the forecasts depend in important ways on the model used to fit the data.

Only individual time series have been analysed so far, and hence the models have all been univariate in nature. Chapter 7 extends the analysis to consider a set of stationary time series, brought together as a vector and modelled as an autoregressive process, thus introducing the vector autoregression (VAR). With a vector of time series, the multivariate linkages between the individual series need to be investigated, so leading to the concept of Granger-causality, impulse response analysis and innovation accounting, all of which are discussed in this chapter.

Of course, assuming that the vector of time series is stationary is far too restrictive, but allowing the individual series to be integrated raises some interesting modelling issues for, as we demonstrate, it is possible for a linear combination of two or more integrated time series to be stationary, a concept known as cointegration. Cointegration is related to the idea of a dynamic equilibrium existing between two or more variables and, if it exists, it enables multivariate models to be expressed not only in terms of the usual differences of the series but also by the extent to which the series lie away from equilibrium: incorporating this ‘equilibrium error’ leads to the class of vector error correction models (VECMs).

Chapter 8 thus focuses on the consequences for conventional regression analysis when the variables in a regression are non-stationary, thus introducing the idea of spurious regression, before considering the implications of the variables in the regression being cointegrated. Tests for cointegration and estimation under cointegration are then discussed. Chapter 9 explicitly considers VECMs and how to test for and model cointegration within a VAR framework.

The final chapter, Chapter 10, explicitly recognises that this is only an introductory text on time series econometrics and so briefly discusses several extensions that more advanced researchers in the modelling of economic and financial time series would need to become familiar with. To keep within the remit of a ‘concise introduction’, however, no mention is made of the increasingly important subject of panel data econometrics, which combines time series with cross-sectional data, for which several textbooks are available.²

Several examples using actual data, typically from the UK, are developed throughout the book. The content is thus suitable for final year undergraduate and postgraduate students of economics and finance wishing to undertake an initial foray into handling time series data.