Time Series Analysis HW 06 Answer keys - TIME SERIES...

This preview shows page 1 - 3 out of 4 pages.

1 TIME SERIES ANALYSIS Assignment 6 answer keys 4.1 If ° = ℎ ± ² ³´µ¶·¸¹ º» ¼ ¶¼ = ± 1º» ¼ ¶¼ = ½ − ´−½¸ = 2½. If ° ≠ ℎ ± ² ³´µ¶·¸¹ º» ¼ ¶¼ = ² ³´µ¶·¸¹ ¾´° − ℎ¸ ¿ ¶¼ ¼ = ÀÁÂô° − ℎ¸½Ä − ÀÁ´−´° − ℎ¸½¸ ¾´° − ℎ¸ = 0. 4.4 Since Å´∙¸ is even and is nonzero only at lags 0 , ±2 , and ±3 , it follows from Corollary 4.1.1 that Å is an ACVF if and only if the function "´»¸ = 1 # ² ¶³·¹ Å´ℎ¸ $ ·%¶$ = 1 &1 − 0.5ò ¶(³¹ + ² (³¹ Ä − 0.25ò ¶*³¹ + ² *³¹ Ä+ = 1 ,1 − ÀÁ´2»¸ − 0.5ÀÁ´3»¸- is nonnegative for all » ∈ ´−½, ½- . But "´»¸ < 0 when » = 0 . So, Å cannot be an ACVF. 4.5 By Problem 1.7, 12 3 = 4 3 + 5 3 6 is stationary with ACVF Å 7 ´ℎ¸ = Å 8 ´ℎ¸ + Å 9 ´ℎ¸ . It follows that ± ² ³·¹ º: 7 ´»¸ ¼ ¶¼ = ± ² ³·¹ ºÃ: 8 ´»¸ + : 9 ´»¸Ä ¼ ¶¼ . To ensure the above equality holds for all integers , it is necessary to have : 7 = : 8 + : 9 . 4.6 Let ; 3 = <ÀÁ´½=/3¸ + ?¾@´½=/3¸ . By Problem 1.7, Å 8 ´ℎ¸ = Å A ´ℎ¸ + Å 9 ´ℎ¸ , where Å A ´ℎ¸ = B ( ÀÁ´½ℎ/3¸ = B ( 2 ² ¶³¼·/* + B ( 2 ² ³¼·/* , Å 9 ´ℎ¸ = C 7.25E ( ℎ = 0 2.5E ( ℎ = ±1 0 Á/F .
2 TIME SERIES ANALYSIS Moreover, by Problem 4.5, : 8 ´»¸ = : A ´»¸ + : 9 ´»¸ , where : A ´»¸ = G 0 » < −½/3 B ( /2 −½/3 ≤ » < ½/3 B ( » ≥ ½/3 , : 9 ´»¸ = ± E ( Ã7.25 + 5ÀÁ´»¸Äº» ¹ ¶¼ = E ( ,7.25´» + ½¸ + 5¾@´»¸-.

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture