Time Series Analysis HW 05 Answer keys.pdf - 1 TIME SERIES...

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1TIME SERIES ANALYSISAssignment 5 answer keys3.2(for Problem 3.1(b, d), as 3.1(b) is not causal)Lagh12345678910!"ℎ$-0.9940.980-0.9590.932-0.9000.866-0.8300.792-0.7530.715%"ℎ$-0.994-0.810000000003.3(for Problem 3.1(b, d), as 3.1(b) is not causal)&= 1&*= "−1.8$&= −1.8&.= "−1.8$&*+ "−0.81$&= 2.43&4= "−1.8$&.+ "−0.81$&*= −2.916&7= "−1.8$&4+ "−0.81$&.= 3.2805&9= "−1.8$&7+ "−0.81$&4= −3.542943.4ACF: Multiply both sides of:;− 0.8:;<.= =;by:;<>, ℎ ≥ 0, and take expectations. We have!"0$ − 0.8!"2$ = !.!"1$ − 0.8!"1$ = 0!"2$ − 0.8!"0$ = 0!"ℎ$ − 0.8!"ℎ − 2$ = 0for ℎ ≥ 3
2TIME SERIES ANALYSISSolve the first 3 equations, we obtain!"0$ =.9'(),!"1$ = 0, and!"2$ =.’'().Moreover, with!"ℎ$ = 0.8!"ℎ − 2$forℎ ≥ 3, we obtain!"ℎ$ = *"0.8$>/.25!.9|ℎ| = 0,2, ⋯0|ℎ| = 1,3, ⋯It follows that ACF."ℎ$ = /"0.8$>/.|ℎ| = 0,2, ⋯0|ℎ| = 1,3, ⋯PACF: By definition,%"0$ = 1,%"1$ = 0!"0$1<*!"1$ = ."1$ = 0, and%"2$is the lastcomponent of/."0$."1$."1$."0$2<*/."1$."2$2 = 310014<*300.84 = 300.84 ;that is,%"2$ = 0.8 = 6..On the other hand,%"ℎ$ = 0forℎ > 2.3.8By equation (2.2.11) (textbook, page 54), the stationary solution of the difference equations,:;= 6:;<*+ =;with|6| > 1 & 9=;:~<="0, !.$, is given by:;= − > 6<?=;@?A?B*.It follows thatC":;$ = 0and!D"ℎ$ ='(E(<*6<|>|.Now,<;= :;− 6<*:;<*.Then,C"<;$ = C":;$ − 6<*C":;<*$ = 0andFGH"<;@>, <;$ = FGH":;@>− 6<*:;@><*, :;− 6<*:;<*$= "1 + 6<.$!D"ℎ$ − 6<*

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Term
Fall
Professor
Irene Hueter
Tags
ACF

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