formula_sheet - Concordia University Department of...

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Unformatted text preview: Concordia University Department of Mathematics & Statistics Formula Sheet 1. For an MA(1) process { X t = Z t + Z t- 1 ,t Z } , { Z t } WN(0 , 2 ), X (0) = 2 (1 + 2 ), X (1) = 2 , X ( h ) = 0 if h > 1 For an AR(1) process given by { X t = X t- 1 + Z t } , where { Z t } WN(0 , 2 ), | | < 1, X ( h ) = 2 | h | / (1- 2 ), h Z 2. Let { Y t } be a stationary time series with mean 0 and covariance function Y . If j =- | j | < , then the time series X t = j =- j Y t- j is stationary with mean 0 and ACVF X ( h ) = X j =- X k =- j k Y ( h + k- j ) . Further, if { Y t } has spectral density f Y ( ), then the spectral density of { X t } is given by f X ( ) = | ( e- i ) | 2 f Y ( ) where ( e- i ) = j =- j e- ij 3. The best linear predictor of Y in terms of 1 , W = ( W n ,...,W 1 ) is P ( Y | W ) = Y + a ( W- W ), where Y = E [ Y ] , W = (...
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