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

• Homework Help
• 6
• 100% (1) 1 out of 1 people found this document helpful

This preview shows page 1 - 6 out of 6 pages.

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<*

Course Hero member to access this document

End of preview. Want to read all 6 pages?