Ex1_2005

# Ex1_2005 - tionary giving full justiﬁcation X t = 13 4 X...

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Time Series (MATH5/30085) 2005 Exercises 1 1. Properties of covariance. Using the deﬁnition Cov ( X, Y ) = E [( X - μ X )( Y - μ Y )] Prove the following: (a) Cov ( X, Y ) = Cov ( Y, X ) (b) Cov ( a + bX, c + dY ) = bdCov ( X, Y ) (c) Cov ( X, Y ) = E ( XY ) - μ X μ Y 2. Find the auto-correlation sequences for the following processes (a) A white noise process with E ( X t ) = μ , V ar ( X t ) = σ 2 t (b) X t = ± t - ± t - 1 (c) For an MA(1) process, X t = ± t - θ 1 ± t - 1 , show that you cannot identify an MA(1) process uniquely from the auto-correlation by comparing the results using θ 1 with those if you replaced θ 1 by θ - 1 1 3. Considering a ﬁrst order AR process X t = φ 1 X t - 1 + ± t AR(1) – Markov process (a) Find the mean and an expression for the variance (b) Show that for the variance to be ﬁnite, | θ 1 | must be less than one (c) Find the auto-correlation sequence (you may assume stationarity) 4. (a) What is meant by saying that a stochastic process is second-order sta- tionary? (b) Determine whether the following stochastic process is second-order sta-

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Unformatted text preview: tionary, giving full justiﬁcation X t = 13 4 X t-1-3 4 X t-2 + ± t . 1 5. HARDER QUESTION Let { X t } be the zero mean autoregressive process of order 2 deﬁned by X t-( g 1 + g 2 ) X t-1 + g 1 g 2 X t-2 = ± t , where | g 1 | , | g 2 | < 1 , and { ± t } is white noise with mean zero and variance σ 2 ± . (a) Explain why { X t } is stationary. (b) Show that { X t } can be written in the general linear model form X t = ± 1 g 2-g 1 ! ∞ X k =0 ² g k +1 2-g k +1 1 ³ ± t-k . (c) Hence show that the autocovariance sequence takes the form s τ = ± σ 2 ± g 2-g 1 ! g | τ | +1 2 (1-g 2 1 )-g | τ | +1 1 (1-g 2 2 ) (1-g 2 1 )(1-g 2 2 )(1-g 1 g 2 ) τ = 0 , ± 1 , ± 2 , ... 2...
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Ex1_2005 - tionary giving full justiﬁcation X t = 13 4 X...

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