quiz_1 - X ( ) and autocorrelation function (acf) X ( )....

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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)
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Unformatted text preview: X ( ) and autocorrelation function (acf) X ( ). Show by the denition of acvf and acf that, X ( ) = X (- ) for = 1 , 2 ,... and X (0) = 1. 5. Write down the denition 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 2 Z . (a) Compute its autoco-variance function Z ( ) and autocorrelation function Z ( ) for = 0 , 1 , 2 ,... . (b) Is this process weakly statioanry? 7. Consider the MA(1) process: X t = Z t + Z t-1 where | | < 1 and Z t are independent and identically distributed random variables with mean 0 and variance 2 Z . (a) Compute its autocovariance function Z ( ) and autocorrelation function Z ( ) for = 0 , 1 , 2 ,... . (b) Is this process weakly statioanry?...
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This note was uploaded on 02/28/2012 for the course AMS 316 taught by Professor Xing during the Fall '09 term at SUNY Stony Brook.

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