theoretical midsols11

theoretical midsols11 - Midterm Theoretical Part STAT 443...

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Unformatted text preview: Midterm Theoretical Part STAT 443 Spring 2011 Last Name First Name 1. Since X t and Y t are weakly stationary, the rst and second moments do not depend on t . Let E ( X t ) = X , Cov ( X t ,X t + h ) = X h , E ( X t ) = Y , Cov ( Y t ,Y t + h ) = Y h . Then, E ( X t + Y t ) = X + Y which does not depend on t . Also, Cov ( X t + Y t ,X t + h + Y t + h ) = Cov ( X t ,X t + h ) + Cov ( X t ,Y t + h ) + Cov ( X t + h ,Y t ) + Cov ( X t + h ,Y t + h ) = Cov ( X t ,X t + h ) + 0 + 0 + Cov ( X t + h ,Y t + h ) since X t and Y t are uncorrelated = X h + Y h which is not dependent on t since X h and Y h are not dependent on t . Additionally, we see above that the ACF of X t + Y t (i.e. Cov ( X t + Y t ,X t + h + Y t + h ) ) is equal to the sum of the ACF's of X t and Y t (i.e. X h + Y h ). 2. Given that X t = Z t Z t- 1 , where Z t is i.i.d. N (0 , 1) , we have E ( X t ) = E ( Z t Z t- 1 ) = E ( Z t ) E ( Z t- 1 ) by independence assumption = 0 since E ( Z t ) = 0 and Cov ( X t ,X t + k ) = E ( Z t Z...
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theoretical midsols11 - Midterm Theoretical Part STAT 443...

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