Ex1_2004 - Time Series (MATH5/30085) 2004 Exercises 1 1....

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Unformatted text preview: Time Series (MATH5/30085) 2004 Exercises 1 1. Properties of covariance. Using the definition Cov (X, Y ) = E [(X − µX )(Y − µY )] 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 ) − µX µY 2. Find the auto-correlation sequences for the following processes (a) A white noise process with E (Xt ) = µ, V ar(Xt ) = σ 2 ∀t (b) Xt = t − t−1 t (c) For an MA(1) process, Xt = − θ1 t−1 , show that you cannot identify an MA(1) process uniquely from the auto-correlation by comparing the − results using θ1 with those if you replaced θ1 by θ1 1 3. Considering a first order AR process Xt = φ1 Xt−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, |θ1 | 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 stationary? (b) Determine whether the following stochastic process is second-order stationary, giving full justification Xt = 13 Xt−1 4 − 3 Xt−2 + t . 4 1 5. HARDER QUESTION Let {Xt } be the zero mean autoregressive process of order 2 defined by Xt − (g1 + g2 )Xt−1 + g1 g2 Xt−2 = t , where |g1 |, |g2 | < 1, and { t } is white noise with mean zero and variance σ 2 . (a) Explain why {Xt } is stationary. (b) Show that {Xt } can be written in the general linear model form Xt = 1 g2 − g 1 ∞ k k g2 +1 − g1 +1 k=0 t−k . (c) Hence show that the autocovariance sequence takes the form sτ = g |τ |+1 (1 − g 2 ) − g |τ |+1 (1 − g 2 ) σ2 2 1 1 2 2 2 g2 − g1 (1 − g1 )(1 − g2 )(1 − g1 g2 ) τ = 0, ±1, ±2, ... 2 ...
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This note was uploaded on 10/19/2009 for the course MATH 611 taught by Professor Jsdkasj during the Spring '09 term at Kansas.

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Ex1_2004 - Time Series (MATH5/30085) 2004 Exercises 1 1....

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