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Ex3_2005

Ex3_2005 - φ θ(1 θ 2 φθ ρ k = φρ k-1,k = 2 3 6 For...

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Time Series (M5/30085) 2005 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 + k ) = 0 for k = 0. 1. Find the ACF of the first order AR process defined by X t - μ = 0 . 7( X t - 1 - μ ) + t . Plot ρ k for k = - 6 , - 5 , ..., - 1 , 0 , +1 , +2 , ..., +6. 2. If X t = μ + t + β t - 1 , where μ is a constant, show that the ACF does not depend on μ . 3. 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 k = 16 21 2 3 | k | + 5 21 - 1 3 | k | , k = 0 , ± 1 , ± 2 , ... 4. For each of the following models: (a) X t = 0 . 3 X t - 1 + t (b) X t = t - 1 . 3 t - 1 + 0 . 4 t - 2 (c) X t = 0 . 5 X t - 1 + t - 1 . 3 t - 1 + 0 . 4 t - 2 express the model in B notation and determine whether the model is stationary and/or invertible. For the first model, find the equivalent MA representation. 5. Show that the ACF of the ARMA(1,1) model X t = φX t - 1 + t + θ t - 1 is given by ρ 1 = (1 +

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Unformatted text preview: φ + θ ) / (1 + θ 2 + φθ ) ρ k = φρ ( k-1) ,k = 2 , 3 ,... 6. For the model (1-B )(1-. 2 B ) X t = (1-. 5 B ) ± t : (a) Classify the model as an ARIMA(p,d,q) process (i.e. ﬁnd p,d,q) (b) Determine whether the process is stationary 7. Show that the AR(2) process X t = X t-1 + cX t-2 + ± t is stationary provided-1 < c < 0. 1 (a) Find the autocorrelation function when c =-3 / 16. (b) Show that the AR(3) process X t = X t-1 + cX t-2-cX t-3 + ± t is non-stationary for all values of c . 8. Consider X t to follow a SARIMA (seasonal ARIMA) model of order (1 , , 0) × (0 , 1 , 1) 12 . (a) Write out the model in B notation (b) Expand this equation, and write the model in terms of X t and ± t . 2...
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Ex3_2005 - φ θ(1 θ 2 φθ ρ k = φρ k-1,k = 2 3 6 For...

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