"Frontmatter". In: Analysis of Financial Time Series

30 LINEAR TIME SERIES ANALYSIS AND ITS APPLICATIONSThis result has two implications for r t . First, the mean of r t exists if φ 1 ̸= 1. Second,the mean of r t is zero if and only if φ 0 = 0. Thus, for a stationary AR(1) process, theconstant term φ 0 is related to the mean of r t and φ 0 = 0 implies that E(r t ) = 0.Next, using φ 0 = (1 − φ 1 )µ, the AR(1) model can be rewritten asr t − µ = φ 1 (r t−1 − µ) + a t . (2.8)By repeated substitutions, the prior equation implies thatr t − µ = a t + φ 1 a t−1 + φ1 2 a t−2 +···∞∑= φ1 i a t−i . (2.9)i=0Thus, r t − µ is a linear function of a t−i for i ≥ 0. Using this property and theindependence of the series {a t }, we obtain E[(r t − µ)a t+1 ]=0. By the stationarityassumption, we have Cov(r t−1 , a t ) = E[(r t−1 − µ)a t ]=0. This latter result canalso be seen from the fact that r t−1 occurred before time t and a t does not depend onany past information. Taking the square, then the expectation of Eq. (2.8), we obtainVar(r t ) = φ 2 1 Var(r t−1) + σ 2 a ,where σa2 is the variance of a t and we make use of the fact that the covariancebetween r t−1 and a t is zero. Under the stationarity assumption, Var(r t ) = Var(r t−1 ),so thatσ 2 aVar(r t ) =1 − φ12provided that φ1 2 < 1. The requirement of φ2 1< 1 results from the fact that thevariance of a random variable is bounded and non-negative. Consequently, the weakstationarity of an AR(1) model implies that −1