"Frontmatter". In: Analysis of Financial Time Series

28 LINEAR TIME SERIES ANALYSIS AND ITS APPLICATIONSwhere µ is the mean of r t , ψ 0 = 1and{a t } is a sequence of independent and identicallydistributed random variables with mean zero and a well-defined distribution(i.e., {a t } is a white noise series). In this book, we are mainly concerned with thecase where a t is a continuous random variable. Not all financial time series are linear,however. We study nonlinearity and nonlinear models in Chapter 4.For a linear time series in Eq. (2.4), the dynamic structure of r t is governed bythe coefficients ψ i , which are called the ψ-weights of r t in the time series literature.If r t is weakly stationary, we can obtain its mean and variance easily by using theindependence of {a t } asE(r t ) = µ,Var(r t ) = σ 2 a∞∑ψi 2 ,i=0where σ 2 a is the variance of a t. Furthermore, the lag-l autocovariance of r t isγ l = Cov(r t , r t−l ) = E[( ∞∑i=0)( )]∞∑ψ i a t−i ψ j a t−l− jj=0( )∞∑= E ψ i ψ j a t−i a t−l− ji, j=0=∞∑ψ j+l ψ j E(at−l− 2 j ) = σ a2j=0∞∑ψ j ψ j+l .j=0Consequently, the ψ-weights are related to the autocorrelations of r t as follows:ρ l = γ ∑ ∞i=0l ψ i ψ i+l=γ 0 1 + ∑ ∞i=1 ψi2 , l ≥ 0, (2.5)where ψ 0 = 1. Linear time series models are econometric and statistical models usedto describe the pattern of the ψ-weights of r t .2.4 SIMPLE AUTOREGRESSIVE MODELSThe fact that the monthly return r t of CRSP value-weighted index has a statisticallysignificant lag-1 autocorrelation indicates that the lagged return r t−1 might be usefulin predicting r t . A simple model that makes use of such predictive power isr t = φ 0 + φ 1 r t−1 + a t , (2.6)where {a t } is assumed to be a white noise series with mean zero and variance σ 2 a .This model is in the same form as the well-known simple linear regression model in