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Unformatted text preview: Practice Problems 1. 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 zeromean 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 . 2. Consider the AR(2) process for which it is known that = 0, Y t 1 Y t 1 2 Y t 2 = e t where { e t } is zeromean white noise ( i.i.d. N ( 0, 2 e ) ). Find the expression for 1 and 2 in terms of 1 and 2 . 3. 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 zeromean 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. b) For 1 = 0.3, 2 = 0.4, forecast y 7 , y 8 , y 9 , and y 10 . c)* For 1 = 0.3, 2 = 0.4, given 2 e = 16.3, calculate 95% probability limits for y 7 , y 8 , y 9 , and y 10 . 4. Consider the AR ( 2 ) processes Y & t 0.3 Y & t 1 0.1 Y & t 2 = e t where { e t } is zeromean white noise ( i.i.d. N ( 0, 2 e ) ), Y & t = Y t . a) Based on a series of length N = 100, we observe , y 98 = 152, y 99 = 156, y 100 = 147, y = 150. Forecast y 101 and y 102 . b) Use YuleWalker equations to find 1 and 2 . c) Is this process stationary? 5. 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 c) Y t 0.7 Y t 1 0.3 Y t 2 = e t + 0.5 e t 1 d) Y t 0.9 Y t 1 0.9 Y t 2 = e t 1.4 e t 1 6. Consider the AR(2) process Y t = + 1 ( Y t 1 ) + 2 ( Y t 2 ) + e t Based on a series of length N = 60, we observe , y 59 = 190, y 60 = 215, y = 200. a) Suppose r 1 = 0.40, r 2 = 0.26. Use YuleWalker equations to estimate 1 and 2 . b) If 1 and 2 are equal to your answers to part (a), is this process stationary? c) Use your answers to part (a) to forecast y 61 , y 62 , and y 63 . 7. The following sample ACF and PACF are from 3 simulated stationary time series....
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This note was uploaded on 02/10/2009 for the course STAT 420 taught by Professor Stepanov during the Spring '08 term at University of Illinois at Urbana–Champaign.
 Spring '08
 STEPANOV
 Statistics

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