The previous chapter has demonstrated that the order of integration, d, is a crucial determinant of the properties that a time series exhibits. This chapter begins with an exposition of the techniques available for determining the order of integration of a time series, emphasising the importance of the chosen alternative hypothesis to the null of a unit root: in particular, whether the alternative is that of a constant mean, a linear trend, or a segmented trend. The importance of these models to finance is demonstrated through a sequence of examples.

We then move, in section 2, to examining methods of decomposing an observed time series into two or more unobserved components, emphasising the signal extraction approach to estimating these components. This approach is particularly suited to estimating, under assumptions of market efficiency, expected, or ex ante, values using only observed, or ex post, observations, and is illustrated by showing how expected real interest rates can be extracted from observed rates.

The final sections of the chapter focus attention on long-term properties of financial time series. A number of models of stock market behaviour yield the prediction that stock returns, far from being unpredictable, should exhibit negative autocorrelation over long time horizons, i.e., that they should be mean reverting. Section 3 thus develops techniques for measuring and testing for such mean reversion, or persistence as it is often referred to. Section 4 introduces an alternative method of modelling long-term memory in a time series, through the use of a fractional value of d. Fractionally integrated extensions of ARIMA models are developed and methods of testing for fractional integration and estimating such models are introduced.