420Pr5 - Practice Problems 1. Given the time series of 5...

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Unformatted text preview: Practice Problems 1. Given the time series of 5 observations: y 1 = 10.1, y 2 = 9.3, y 3 = 9.4, y 4 = 9.8, y 5 = 10.6 Calculate the first two sample autocorrelation coefficients, r 1 and r 2 . ( Note: In practice reliable autocorrelation estimates are only obtained from series consisting of approximately 50 observations or more. ) 2. Consider the AR(1) model: ( Y t ) = ( Y t 1 ) + e t where e t is a mean zero white noise process. The model has been fitted to a time series giving = 0.8, = 10.2, and 2 e = 0.25. The last five values of the series are y 96 = 10.1, y 97 = 9.3, y 98 = 9.4, y 99 = 9.8, y 100 = 10.6. Using the t = N = 100 as the forecast origin, forecasts the next three observations. Calculate the 95% probability limits for the next three observations. 3. Consider the AR ( 2 ) processes Y & t 0.98 Y & t 1 + 0.96 Y & t 2 = e t where { e t } is zero-mean white noise ( i.i.d. N ( 0, 2 e ) ), Y & t = Y t . a) Is this process stationary? Justify your answer. b) Use Yule-Walker equations to find 1 and 2 . c) Based on a series of length N = 100, we observe , y 99 = 22, y 100 = 14, y = 20. Forecast y 101 and y 102 . 4. Determine whether the following processes are stationary: a) Y t Y t 1 = e t 0.8 e t 1 b) Y t 0.39 Y t 2 0.16 Y t 4 = e t 0.8 e t 1 d) Y t 0.9 Y t 1 0.9 Y t 2 = e t 1.4 e t 1 5.* Consider the MA(2) process for which it is known that = 0, Y t = e t 1 e t 1 2 e t 2 where { e t } is zero-mean white noise ( i.i.d. N ( 0, 2 e ) ). a) Find the expression for Var ( Y t ) = Cov ( Y t , Y t ), Cov ( Y t , Y t + 1 ), and Cov ( Y t , Y t + 2 ), and Cov ( Y t , Y t + 3 ) in terms of 1 , 2 , and 2 e . b) Find the expression for 1 , 2 , and 3 in terms of 1 and 2 . 6.** Consider the MA(2) process for which it is known that = 0, Y t = e t 1 e t 1 2 e t 2 where { e t } is zero-mean white noise ( i.i.d. N ( 0, 2 e ) ). Based on a series of length N = 6, we observe y 1 y 2 y 3 y 4 y 5 y 6 4.4 2.0 6.3 4.1 5.6 6.1 a) Using e = 0, e 1 = 0, calculate S ( 1 , 2 ) = = N t t e 1 2 for 1 = 0.3, 2 = 0.4....
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This note was uploaded on 12/17/2010 for the course STAT 420 taught by Professor Stepanov during the Spring '08 term at University of Illinois, Urbana Champaign.

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420Pr5 - Practice Problems 1. Given the time series of 5...

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