formula_sheet

# 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 + θZ t- 1 ,t ∈ Z } , { Z t } ∼ WN(0 ,σ 2 ), γ X (0) = σ 2 (1 + θ 2 ), γ X (1) = σ 2 θ , γ X ( h ) = 0 if h > 1 For an AR(1) process given by { X t = φX t- 1 + Z t } , where { Z t } ∼ WN(0 ,σ 2 ), | φ | < 1, γ X ( h ) = σ 2 φ | h | / (1- φ 2 ), h ∈ Z 2. Let { Y t } be a stationary time series with mean 0 and covariance function γ Y . If ∑ ∞ j =-∞ | ψ j | < ∞ , then the time series X t = ∑ ∞ j =-∞ ψ j Y t- j is stationary with mean 0 and ACVF γ X ( h ) = ∞ X j =-∞ ∞ X k =-∞ ψ j ψ k γ Y ( h + k- j ) . Further, if { Y t } has spectral density f Y ( λ ), then the spectral density of { X t } is given by f X ( λ ) = | Ψ( e- iλ ) | 2 f Y ( λ ) where Ψ( e- iλ ) = ∑ ∞ j =-∞ ψ j e- ijλ 3. The best linear predictor of Y in terms of 1 , W = ( W n ,...,W 1 ) is P ( Y | W ) = μ Y + a ( W- μ W ), where μ Y = E [ Y ] ,μ W = (...
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