and variances are var(x)=∑var(m-)→ Var Var(t-)=t·σ The autocovariance of the two series are E(XtXt-n out E[(u+2u-1+…+u-r+…+9-1u1)(u1-+2u-r-1+…+d-r-hu) ∑φ E(YYt-r)=E Ut-T-i Ut+Ut u1( (t-r) We may expect that the autocorrelation functions are and The means of Xt and Yt are the same, but the variances(including autoco- variance) are different. The important thing to note is that the variances andand variances are V ar(Xt) = X t−1 i=0 φ 2iV ar(ut−i) −→ 1 1 − φ2 σ 2 u and V ar(Yt) = X t−1 i=0 V ar(vt−i) = t · σ 2 v . The autocovariance of the two series are γ X τ = E(XtXt−τ ) = E " X t−1 i=0 φ iut−i ! t−Xτ−1 i=0 φ iut−τ−i !# = E[(ut + φ 1ut−1 + ... + φ τut−τ + ... + φ t−1u1)(ut−τ + φ 1ut−τ−1 + ... + φ t−τ−1u1) = t−Xτ−1 i=0 φ iφ τ+i σ 2 u = σ 2 uφ τ ( t−Xτ−1 i=0 φ 2i ) −→ φ τ 1 − φ 2 σ 2 u = φ τ γ X 0 . and γ Y τ = E(YtYt−τ ) = E " X t−1 i=0 vt−i ! t−Xτ−1 i=0 vt−τ−i !# = E[(vt + vt−1 + ... + vt−τ + vt−τ−1 + ... + v1)(vt−τ + vt−τ−1 + ... + v1)] = (t − τ )σ 2 v . We may expect that the autocorrelation functions are r X τ = γ X τ γ X 0 = φ τ −→ 0 and r Y τ = γ Y τ γ Y 0 = (t − τ ) t −→ 1 ∀ τ. The means of Xt and Yt are the same, but the variances (including autoco￾variance) are different. The important thing to note is that the variances and 2