Time-Series Analysis
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Time-Series Analysis
14.1 The AR(I) Model. Yt = PYt-l + Et with Ipi < 1 and Et-IIN(O,a~). Also,
Yo - N(0,a~l1-p2). a. By successive substitution
Yt =PYt-l + Et = P(PYt-2+Et-l) + Et = Plyt-2 + PEt-l + Et = p2(PYt_3+Et_2) + pEt_1 + Et = p3Yt_3+p2Et_2 + PEt-l + Et =.. = plyo + pt-l E1 + pt-2E2 +.. + Et Then, E(yJ = p~(y~ =0 for every t, since E(y~ =E(Et) =O. var(yJ = p2lvar(yo) + p2(t-1>Var(El) + p2(t-2>Var(E2) +.. +var(EJ
for every t. If P = 1, then var(yJ = a~/O-oo. Also, if Ipi > 1, then 1 - p2 < 0 and var(yJ
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