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

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Time Series (MATH5/30085) 2004 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 = epsilon1 t - epsilon1 t - 1 (c) For an MA(1) process, X t = epsilon1 t - θ 1 epsilon1 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 + epsilon1 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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