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# hwk2 - X t = 0 5 X t-1 Z t 4 Compute the autocorrelation...

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Time Series Analysis Problem Sheet 2 Hand in solutions to questions 3, 4 and 6 on Thursday 6th November at the 10.00 lecture. LATE solutions will NOT be accepted unless you have very good reasons supported by evidence. Your solutions to this sheet COUNT TOWARDS ASSESSMENT . 1. Find the mean, variance and autocovariance function of the following MA(2) process X ( t ) = Z t + 0 . 2 Z t - 1 - 0 . 4 Z t - 2 , where Z t is a purely random process with zero mean and σ 2 Z = 1. 2. Deﬁne the MA( m ) process by X t = m X i =0 a - i Z t - i , where a (0 , 1) is a constant and Z t is a purely random process with zero mean and σ 2 Z = 1. Compute the autocovariance function of X t . Check your answer is correct for k = m where γ ( m ) = a - m . 3. Compute the autocovariance function of the following autoregressive process:
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Unformatted text preview: X t = 0 . 5 X t-1 + Z t . 4. Compute the autocorrelation function of the following autoregressive process: X t = 1 3 X t-1 + 2 9 X t-2 + Z t . 5. Suppose the stationary process { X t } has autocovariance function γ X ( k ). Deﬁne a new sta-tionary process { Y t } by Y t = X t-X t-1 . Find the autocovariance function of { Y t } in terms of γ X ( k ) and obtain γ Y ( k ) when γ X ( k ) = ν | k | . 6. Show that the autocorrelation function of the mixed ARMA(1,1) model X t = αX t-1 + Z t + βZ t-1 is given by ρ (1) = (1 + αβ )( α + β ) 1 + β 2 + 2 αβ ρ ( k ) = αρ ( k-1) for k = 2 , 3 , 4 , . . . 1...
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