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# hw2sol - 1 Xt = 1 Zt k k =0 m 1 1 = Z t Z t 1 m 1 m Zt m So...

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1 0 1 1 1 1 ( ) 1 m t t k k t t t m X Z m Z Z Z m = = + = + + + + " So, 1 1 ( ) ( ( ) ( ) ( )) 1 0 t t t t m E X E Z E Z E Z m = + + + + = " Then, 1 1 1 1 2 ( ) ( , ) ( , ) 1 1 [ ( ), ( )] 1 1 1 [( ),( )] ( 1) t t h t t h t t t m t h t h t h m t t t m t h t h t h m h Cov X X E X X E Z Z Z Z Z Z m m E Z Z Z Z Z Z m γ + + + + − + − + + − + − = = = + + + + + + + + = + + + + + + + " " " " i) If, h>m, this means t+k m>t, there is no overlap between 1 ( ) t t t m Z Z Z + + + " and 1 ( ) t h t h t h m Z Z Z + + − + − + + + " , since , , (0, ) i i d k z Z N σ ⎯⎯⎯→ , ( ) 0 h γ = ii) If, h<m, this means t+k m<t, the overlap between 1 ( ) t t t m Z Z Z + + + " and 1 ( ) t k t k t k m Z Z Z + + + + + + " is 1 ( ) t h m t h m t Z Z Z + − + − + + + + " 1 1 2 1 ( ) [( ),( )] ( 1) t k m t k m t t k m t k m t k E Z Z Z Z Z Z m γ + + + + + − + = + + + + + + + " " The correlation of rest items is equal to 0.

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