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# H3 - H3 For MA(q processes 2 q-h j =0 j j h 0 h q(h = h > q...

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H3 For MA( q ) processes, 2 0 ,0 , () 0, . σθ θ γ + = = ⎪> qh jjh j hq h 0 22 1 , 1 0, . θθ ρ + = = + +⋅⋅⋅+ > j q h The ACF of a MA( q ) process “cuts off” after the point q . This is a benchmark property for MA processes. AR(1): 1 0 i ttt t i i X XZ Z φφ −− = =+ = . 2 4 2 2 0 ()0 ,V a r ( 1 ) 1 σ σφ φ . ϕ = == + + + = = L i tt i EX X / ( 1 ) hh X h 2 φσ =− = , provided ||1 < ( ) , 0,1, 2, . ... h ρφ ⇒= = . Example 1 1 0.9 t X X Z t and 1 0.9 X = −+ . A process { } t X is said to be invertible if the random disturbance at time t can be expressed as a convergent sum of present and past values of the process in the form 0 tj j t j Z X π = = , where || j < ∞ . This effectively means that the process can be rewritten in the form of an AR process whose coefficients form a convergent sum. By using the backward shift operator 1 ,, j j B BX X B X X , MA( q ) , 11 t q t q X ZZ Z = + , can be presented as X BZ = . where 1 () 1 q q B B θθθ + + B is a polynomial of order q in B. MA( q ) process is invertible
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