Chapter 2

Linear Time Series Analysis and Its Applications

In this chapter, we discuss basic theories of linear time series analysis, introduce some simple econometric models useful for analyzing financial data, and apply the models to financial time series such as asset returns. Discussions of the concepts are brief with emphasis on those relevant to financial applications. Understanding the simple time series models introduced here will go a long way to better appreciate the more sophisticated financial econometric models of the later chapters. There are many time series textbooks available. For basic concepts of linear time series analysis, see Box, Jenkins, and Reinsel (1994, Chapters 2 and 3) and Brockwell and Davis (1996, Chapters 1–3).

Treating an asset return (e.g., log return rt of a stock) as a collection of random variables over time, we have a time series {rt}. Linear time series analysis provides a natural framework to study the dynamic structure of such a series. The theories of linear time series discussed include stationarity, dynamic dependence, autocorrelation function, modeling, and forecasting. The econometric models introduced include (a) simple autoregressive (AR) models, (b) simple moving-average (MA) models, (b) mixed autoregressive moving-average (ARMA) models, (c) seasonal models, (d) unit-root nonstationarity, (e) regression models with time series errors, and (f) fractionally differenced models for long-range dependence. For an asset return rt, simple models attempt to capture the linear relationship between rt and information available prior to time t. The information may contain the historical values of rt and the random vector in Eq. (1.14), which describes the economic environment under which the asset price is determined. As such, correlation plays an important role in understanding these models. In particular, correlations between the variable of interest and its past values become the focus of linear time series analysis. These correlations are referred to as serial correlations or autocorrelations. They are the basic tool for studying a stationary time series.