A5 - t = 1 12 Z t-1 1 12 Z t-2 a t where σ 2 a = 4 Suppose...

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STATISTICS 4005B ASSIGNMENT 5 Due date: April 18, 2008 1. Recall that the dataset “data3.xls”. Fit an ARI(1,1) model (1 - φB )(1 - B ) Z t = β + a t with a t NID (0 2 a ) to { Z t } . Find the estimates of β , φ and σ 2 a based on: (a) the method of moments; (b) conditional least squares. 2. From a series of length 144, we have computed r 1 = 0 . 8 , r 2 = 0 . 5 , r 3 = 0 . 4 ¯ Z = 2, and a sample variance of 9. If we assume that an AR(2) model Z t = μ + φ 1 ( Z t - 1 - μ ) + φ 2 ( Z t - 2 - μ ) + a t is appropriate, find the method of moments estimates of μ , φ 1 , φ 2 , and σ 2 a . 3. Find the conditional least-squares estimate of θ of the MA(1) process Z t = a t - θa t - 1 based on a series of length 4 with Z 1 = 0 , Z 2 = 0 , Z 3 = 3 and Z 4 = 2. 4. Consider the AR(2) model: Z
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Unformatted text preview: t = 1 12 Z t-1 + 1 12 Z t-2 + a t , where σ 2 a = 4. Suppose Z 2005 = 120, Z 2006 = 110 and Z 2007 = 100. (a) Find the roots of the characteristic equation. (b) Find Cov ( Z t-1 ,Z t-2 ). (c) Find the forecasts ˆ Z 2007 (1) and ˆ Z 2007 (3). (d) Calculate the 95% prediction interval for Z 2008 and the 95% prediction interval for Z 2010 . Note that if W is a standard normal random variable, P ( W ≤ 1 . 96) = 0 . 975. (e) Update your forecast for Z 2010 given Z 2008 = 120. NOTE: The data set ’data3.xls’ can be downloaded from http://www.cuhk.edu.hk/wbt...
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