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sample_final

# sample_final - a is a parameter(a For what values of real...

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AMS316 Final in 2010 (2-hour exam) Assume that z t are independent and identically distributed normal random variables with mean 0 and variance σ 2 . 1. Consider the process x t = a + bt + ct 2 + s t + y t , in which s t is a deterministic process satisfying s t = s t - 12 and s t + · · · + s t +11 = 0, y t is stationary process with mean zero. (a) What are the trend and seasonal components in the process x t ? (b) Let s t 0. Is the lag-2 difference (1 - B ) 2 X t stationary? 2. Consider a stationary process x t = z t + θz t - 1 + θ 2 z t - 2 . (a) What is the ACF of x t at lag k 0? (b) Suppose a new process y t is expressed as y t - αy t - 1 = x t , where | α | < 1. Find the ACF of y t at lag k . (It is OK to write ρ y ( k ) as functions of ρ x ( j )) 3. Consider the AR(2) process: x t = 1 a ( a + 1) x t - 1 + 1 a ( a + 1) x t - 2 + z t , in which the real number
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Unformatted text preview: a is a parameter. (a) For what values of real number a , the process is stationary? (b) What is the autocorrelation of the process at lag k , i.e., ρ ( k )? ( k ≥ 0). (c) What are the 1-step and 2-step ahead forecasts of the process at forecast origin x n ? 4. Consider the stationary process x t + φx t-1 = z t-θz t-1 . (a) What are the stationarity and invertbility conditions for x t ? (b) Compute the lag-1 and lag-2 autocorrelations ρ (1) and ρ (2). (c) Given the observations y 1 ,...,y n , what are the 1- and 2-step ahead forecast of y at the forecast origin y n ?...
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