Question

Problem 5. Consider the time series described in Problem 3 (a) Express each of Y. Yg, Y. Y. and YS in terms of {el, e2, ea, e4, e5} (b) Use (a) to compute cor(Yi, Ys) and cor(Y,Ys)

Answer 1

(a)

$Y_1 = -0.9 Y_{-1} + e_1 = 0.9 * 0 + e_1 = e_1$

$Y_2 = -0.9 Y_{0} + e_2 = 0.9 * 0 + e_2 = e_2$

$Y_3 = -0.9 Y_{1} + e_3 = -0.9 e_1 + e_3$

$Y_4 = -0.9 Y_{2} + e_4 = -0.9 e_2 + e_4$

$Y_5 = -0.9 Y_{3} + e_5 = -0.9 (-0.9e_1 + e_3) + e_5 = 0.81e_1 - 0.9e_3 + e_5$

(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) / $\sqrt{Var[Y_1]Var[Y_5] }$

= 4.2849 / $\sqrt{5.29 * 13.04567 }$

= 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) / $\sqrt{Var[Y_4]Var[Y_5] }$

= 0 / $\sqrt{Var[Y_4]Var[Y_5] }$

= 0