(a)
(b)
Cov(Y1, Y5) = E[Y1 * Y5] - E[Y1] * E[Y5]
= E[e1 * (0.81e1 - 0.9 e3 + e5)] - E[e1] * E[0.81e1 - 0.9 e3 + e5]
= E[0.81e12 - 0.9e1 e3 + e1 e5] + 0 * E[0.81e1 - 0.9 e3 + e5]
= 0.81 E[e12 ] - 0.9 E[e1] * E[e3] + E[e1] E[e5]
= 0.81 (Var[e1] + (E[e1])2) - 0 + 0
= 0.81 * 5.29 = 4.2849
Var[Y1] = Var[e1] = 5.29
Var[Y5] = Var[0.81e1 - 0.9 e3 + e5] = 0.812 Var[e1] + (-0.9)2 Var[e3] + Var[e5]
= 0.812 * 5.29+ (-0.9)2 * 5.29 + 5.29 = 13.04567
Cor(Y1, Y5) = Cov(Y1, Y5) /
= 4.2849 /
= 0.515798
Cov(Y4, Y5) = E[Y4 * Y5] - E[Y4] * E[Y5]
= E[(-0.9e2 + e4) * (0.81e1 - 0.9 e3 + e5)] - E[-9e2 + e4] * E[0.81e1 - 0.9 e3 + e5]
= E[-0.9 * 0.81 e1 e2 + 0.9 * 0.9 e2 e3 - 0.9 e2 e5 + 0.81 e1 e4 - 0.9 e2 e4 + e4 e5] + 0
= -0.9 * 0.81 E[e1] E[e2] + 0.9 * 0.9 E[e2] E[e3] - 0.9 E[e2] E[e5] + 0.81 E[e1] E[e4] - 0.9 E[e2] E[e4] + E[e4] E[e5]
= 0 (E[ei] = 0)
Cor(Y4, Y5) = Cov(Y4, Y5) /
= 0 /
= 0
Problem 5. Consider the time series described in Problem 3 (a) Express each of Y. Yg,...
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