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# quiz_1 - γ X τ and autocorrelation function(acf ρ X τ...

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AMS316 Sample Midterm This is the AMS316 miderm in fall 2010. 1. Suppose we have a seasonal seris of monthly observations { X t } , for which the seasonal factor at time t is denoted by { S t } . We further assume that the seasonal pattern is constant through time so that S t = S t - 12 for all t . Denote a stationary seris of random variables by { t } . Consider the model X t = bt + S t + t having a global linear trend and additive seasonality. Show that the seasonal difference operator 12 acts on X t to produce a stationary series. (Hints: Please use only heuristic arguments about stationarity as you did in your HW.) 2. Consider the series X t = a + bt + ct 2 , where a , b and c are constants. Compute its first- and second-order differences X t and 2 X t . 3. Given N observations x 1 , . . . x N , on a time series. (a) Write down the estimation formula for autocorrelation coefficient at lag 1. (b) In general, what is your estimate for autocorrelation coefficient at lag k . 4. Suppose that a stationary stochastic process X ( t ) have autocovariance function (acvf) γ
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Unformatted text preview: γ X ( τ ) and autocorrelation function (acf) ρ X ( τ ). Show by the deﬁnition of acvf and acf that, γ X ( τ ) = γ X (-τ ) for τ = 1 , 2 ,... and ρ X (0) = 1. 5. Write down the deﬁnition of the second-order (or weakly) stationary process. 6. Consider the purely random processes { Z t } , where Z t are independent and identically distributed random variables with mean 0 and variance σ 2 Z . (a) Compute its autoco-variance function γ Z ( τ ) and autocorrelation function ρ Z ( τ ) for τ = 0 , 1 , 2 ,... . (b) Is this process weakly statioanry? 7. Consider the MA(1) process: X t = Z t + θZ t-1 where | θ | < 1 and Z t are independent and identically distributed random variables with mean 0 and variance σ 2 Z . (a) Compute its autocovariance function γ Z ( τ ) and autocorrelation function ρ Z ( τ ) for τ = 0 , 1 , 2 ,... . (b) Is this process weakly statioanry?...
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