Ex2_2005 - Y t = p X j =1 j,p Y t-j + t . Show that the...

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Time Series (MA5/30085) 2005 Exercises 2 In these questions, { ± t } is a discrete, purely random process, such that E ( ± t ) = 0, V AR ( ± t ) = σ 2 ± , COV ( ± t t + τ ) = 0 for τ 6 = 0. 1. Find the ACF of the second order MA process given by X t = ± t + 0 . 7 ± t - 1 - 0 . 2 ± t - 2 2. Show that the ACF of he m th order MA process is given by X t = m k =0 ± t - k / ( m + 1) is ρ τ = ( m + 1 - k ) / ( m + 1) k = 0 , 1 ,...,m 0 k > m 3. Show that the infinite MA process { ± } defined by X t = ± t + C ( ± t - 1 + ± t - 2 + ... ) where C is a constant is non-stationary. Also show that the series of first differences { Y t } defined by Y t = X t - X t - 1 is a first order MA process and is stationary. Find the ACF of { Y t } 4. The stationary process { X t } has ACVF s τ . A new stationary process { Y t } is defined as Y t = X t - X t - 1 . Obtain the ACVF of { Y t } in terms of s τ and find form when s τ = λ | k | ( λ is a constant). 5. (a) What is meant by saying that a linear process is invertible? (b) How may we check whether such a process is invertible? (c) Determine whether the following process is invertible: X t = 4 25 ± t + 4 25 ± t - 1 + 1 25 ± t - 2 . 6. Suppose that { Y t } is a zero mean stationary AR( p ) process:
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Unformatted text preview: Y t = p X j =1 j,p Y t-j + t . Show that the process { X t } given by X t = q X k =0 k Y t-k , where = 1, can be written as an ARMA( p , q ) process. 1 7. HARDER QUESTION (a) Let { X t } be a real zero mean stationary process with variance 2 X and let X = 1 N N t =1 X t . Show that we can write var { X } = 2 X N 1 + 2 N N-1 X i =1 N X j>i j-i . where { } is the autocorrelation sequence of { X t } . (b) Now consider the AR(1) process X t = 1 , 1 X t-1 + t with variance 2 X = 2 / (1- 2 1 , 1 ) and autocorrelation sequence = | | 1 , 1 . Show that for this process N X j>i j-i = 1 , 1 (1- N-i 1 , 1 ) 1- 1 , 1 . (c) Hence show that for the AR(1) process var { X } = 2 N (1- 2 1 , 1 ) 1 + 2 1 , 1 N (1- 1 , 1 ) " N-(1- N 1 , 1 ) (1- 1 , 1 ) #! . 2...
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Ex2_2005 - Y t = p X j =1 j,p Y t-j + t . Show that the...

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