Characteristics of Time Series

Abstract

Statistical inference is about learning something that is unknown from the known. Time series analysis is no exception in this aspect. In order to achieve this, it is necessary to assume that at least some features of the un-derlying probability law are sustained over a time period of interest. This leads to the assumptions of different types of stationarity, depending on the nature of the problem at hand. The dependence in the data marks the fundamental difference between time series analysis and classical statisti-cal analysis. Different measures are employed to describe the dependence at different levels to suit various practical needs. In this chapter, we intro-duce the most commonly used definitions for stationarity and dependence measures. We also make comments on when those definitions and measures are most relevant in practice. 2.1 Stationarity 2.1.1 Definition We introduce two types of stationarity, namely (weak) stationarity and strict stationarity, in this section. Both of them require that time series exhibit certain time-invariant behavior. Definition 2.1 A time series {X t , t = 0, ±1, ±2, · · · } is stationary if E(X 2 t) < ∞ for each t, and (i) E(X t) is a constant, independent of t, and 30 2. Characteristics of Time Series (ii) Cov(X t , X t+k) is independent of t for each k. Definition 2.2 A time series {X t , t = 0, ±1, ±2, · · · } is strictly stationary if (X 1 , · · · , X n) and (X 1+k , · · · , X n+k) have the same joint distributions for any integer n ≥ 1 and any integer k. The stationarity, which is often referred to as the weak stationarity in textbooks, assumes that only the first two moments of time series are time-invariant and is generally weaker than the strict stationarity, provided that the process has finite second moments. Weak stationarity is primarily used for linear time series, such as ARMA processes, where we are mainly con-cerned with the linear relationships among variables at different times. In fact, the assumption of stationarity suffices for most linear time series anal-ysis, such as in spectral analysis. In contrast, we have to look beyond the first two moments if our focus is on nonlinear relationships. This explains why strict stationarity is often required in the context of nonlinear time series analysis.