STA457H1 Lecture Notes - Lipschitz Continuity, Lag Operator, Polynomial
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X(h) = cov[xt, xt+h] correlation function: x(h) = corr[xt, xt+h] = cov[xt, xt+h]/ var(xt)var(xt+h) = (h)/ (0) (0) = | (h)| (0) for any h: (h) = (-h) for any h. Proof: cov[xt, xt+h] = (h) = (-h) = cov[xt, xt-h] White noise sequence is the building block for many more complicated time series models. Moving average process of order 1: ma(1) model. Suppose z0, z1, z2, , zn is a wn(0, 2) Cov[xi, xi+h] = cov[zi+azi-1, zi+h+azi+h-1] = cov[zi, zi+h] + acov[zi, zi+h-1] + acov[zi-1, zi+h] + a2cov[zi- If h=0, a= 2, b=0, c=0, d= 2 => cov[xi, xi] = 2 + a2 2 = 2(a2+1) If h=1, a=0, b= 2, c=0, d=0 => cov[xi, xi+1] = a 2. If h>1, a=0, b=0, c=0, d=0 => cov[xi, xi+h] = 0. To summarize, xi is a stationary process with covariance function. (h) = { 2(a2+1) if h=0 a 2 if h=1. Let {zi} be a wn(0, 2), i z .
