CHAPTER 12 TIME SERIES ANALYSIS Remark 1 If IXt, tez is statio nary, then =(r, s)=%x(r x(r-s,0) Hence, we may define the autoco variance function of a statio nary process as a function of just one variable, which is the difference of two time inder. That is, instead of ya(T, s), we may write 7 ane way Pn(h)=7x2(h)/x2(0) Example 3 Xt=et+8et-l, et N iid(0, a2) Xt is stationary e nple 4 Xt=Xt-1+ et, et niid(0, 02) T ei+ xo Xt is not stationary, since Var(Xt)=to(assume Xo=0) 3 xample5X≡N(0,a2) X is not stationa The time series IXt, tezi is said to be strict ly stationary if the joint distribution of tk+h)are the same for all posit ive integers k and for all th,h∈Z. 12.3 Autoregressive processes 3t=C13-1+.+ ap3h-p+et: AR(p)process t=s EC=0 and Eees{=0,t≠CHAPTER 12 TIME SERIES ANALYSIS 2 Remark 1 If {Xt ,t ∈ Z} is stationary, then γx (r, s) = γx (r − s, s − s) = γx (r − s, 0). Hence, we may define the autocovariance function of a stationary process as a function of just one variable, which is the difference of two time index. That is, instead of γx (r, s), we may write γx (r − s) = γx (h). To be more precise, γx (h) = Cov (Xt+h, Xt). In the same way, ρx (h) = γx (h) /γx (0). Example 3 Xt = et + θet−1, et ∼ iid (0, σ2 ). Xt is stationary. Example 4 Xt = Xt−1 + et , et ∼ iid (0, σ2 ). Then Xt =  t i=1 ei + X0. Xt is not stationary, since V ar (Xt) = tσ2 (assume X0 = 0). Example 5 Xt ≡ N (0, σ2 t ). Xt is not stationary. The time series {Xt , t ∈ Z} is said to be strictly stationary if the joint distribution of (Xt1 , · · · , Xtk ) ′ and (Xt1+h, · · · , Xtk+h) ′ are the same for all positive integers k and for all t1, · · · , tk, h ∈ Z. 12.3 Autoregressive processes yt = α1yt−1 + · · · + αpyt−p + et : AR (p) process where Eet = 0 and Eetes  = σ 2 , t = s = 0, t = s