Time Series Analysis HW 04 Answer keys - 1 TIME SERIES...

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1 TIME SERIES ANALYSIS Assignment 4 answer keys 2.14 ° ± = ²³´µ¶·¸¹ + ºµ»¼¶·¸¹ , where ² and º are uncorrelated random variables with mean 0 and variance 1. Its ACVF ½¶ℎ¹ = ³´µ¶·ℎ¹ . (a) ¾ ¿ ° À = Á ¿¿ ° ¿ , where ½¶0¹Á ¿¿ = ½¶1¹ ⟹ Á ¿¿ = ³´µ¶·¹ . The corresponding MSE is Â¶° À − ¾ ¿ ° À ¹ À = ½¶0¹ − Á ¿¿ ½¶1¹ = 1 − ³´µ À ¶·¹ = µ»¼ À ¶·¹ . (b) ¾ À ° Ã = Á À¿ ° À + Á ÀÀ ° ¿ , where Ä ½¶0¹ ½¶1¹ ½¶1¹ ½¶0¹ Å Ä Á À¿ Á ÀÀ Å = Ä ½¶1¹ ½¶2¹ Å. Solving for Á À¿ and Á ÀÀ , we obtain Ä Á À¿ Á ÀÀ Å = Ä 1 ³´µ¶·¹ ³´µ¶·¹ 1 Å !¿ Ä ³´µ¶·¹ ³´µ¶2·¹ Å = " 2³´µ¶·¹ −1 #. The corresponding MSE is Â¶° Ã − ¾ À ° Ã ¹ À = ½¶0¹ − ¶2³´µ¶·¹ −1¹ Ä ³´µ¶·¹ ³´µ¶2·¹ Å = 1 − 2³´µ À ¶·¹ + ³´µ¶2·¹ = 0. (c) From part (b) and stationarity, ¾¶° \$%¿ \$ , ° \$!¿ ¹ = 2³´µ¶·¹° \$ − ° \$!¿ with MSE = 0. Since 2³´µ¶·¹° \$ − ° \$!¿ is a linear combination of ) , −∞ < µ ≤ ¼- , and in addition it is impossible to find a predictor of this form with a smaller MSE, we conclude that ¾ . \$ ° \$%¿ = 2³´µ¶·¹° \$ − ° \$!¿ with MSE = 0. 2.21 Â¶° ± ¹ = 0 and ACVF ½¶ℎ¹ = / 0 À ¶1 + 1 À ¹ if ℎ = 0 0 À 1 if ℎ = ±1 0 if |ℎ| > 1 . (a) Write the best linear estimate as 6 ¿ ° ¿ + 6 À ° À . Then, 6 ¿ and 6 À are determined by Ä ½¶0¹ ½¶1¹ ½¶1¹ ½¶0¹ Å " 6 ¿ 6 À # = Ä ½¶2¹ ½¶1¹ Å, or Ä 1 + 1 À 1 1 1 + 1 À Å " 6 ¿ 6 À # = " 0 1 #.
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