Some Common Themes

Abstract

The first four sections of this chapter review some basic concepts that, in part, serve to establish a common notation and define some underlying concepts for later chapters. In particular Section 1.1 considers the nature of a stochastic process, which is the conceptualisation underlying the generation of time series data. Section 1.2 considers stationarity in its strict and weak forms. As in Volume 1, the parametric models fitted to univariate time series are often variants of ARIMA models, so these are briefly reviewed in Section 1.3 and the concept of the long-run variance, which is often a key component of unit root tests where there is weak dependency in the errors, is outlined in Section 1.4. Section 1.5 is more substantive and considers, by means of some simple illustrations, a problem and its solution that arises in Chapters 5, 6 and 8. The problem is that of designing a test statistic when there is a parameter that is not identified under the null hypothesis. The two cases considered here are variants of the Chow test (Chow, 1960), which tests for stability in a regression model. This section also considers how to devise a routine to obtain the bootstrap distribution of a test statistic and the bootstrap p-value of a sample test statistic.