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Ex3_NEW

# Ex3_NEW - Time Series(M30085 2002 Exercises 3 In these...

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Time Series (M30085) 2002 Exercises 3 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 = 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 the 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. Find the ACF of the first order AR process defined by X t - μ = 0 . 7( X t - 1 - μ ) = t . Plot ρ τ for k = - 6 , - 5 , ..., - 1 , 0 , +1 , ..., +6 5. If X t = μ + t + β t - 1 , where μ is a constant, show that the ACF does not depend on μ . 6. Find the values of λ 1 and λ 2 such that the second order AR process defined by X t = λ 1 X t - 1 + λ t X t - 2 + t is stationary. If λ 1 = 1 / 3 and λ 2 = 2 / 9, show that the ACF of X t is given by ρ τ = 16 / 21(2 / 3) | τ | + 5 / 12( - 1 / 3) | k | , k

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Ex3_NEW - Time Series(M30085 2002 Exercises 3 In these...

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