420Pr8 - Practice Problems 1 Consider the MA(2 process for...

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Unformatted text preview: Practice Problems 1. Consider the MA(2) process for which it is known that μ = 0, Y t = e t – θ 1 e t – 1 – θ 2 e t – 2 where { e t } is zero-mean white noise ( i.i.d. N ( 0, 2 e σ ) ). 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 θ 1 , θ 2 , and 2 e σ . b) Find the expression for ρ 1 , ρ 2 , and ρ 3 in terms of θ 1 and θ 2 . 2. Consider the AR(2) process for which it is known that μ = 0, Y t – φ 1 Y t – 1 – φ 2 Y t – 2 = e t where { e t } is zero-mean white noise ( i.i.d. N ( 0, 2 e σ ) ). Find the expression for ρ 1 and ρ 2 in terms of φ 1 and φ 2 . 3. Consider the MA(2) process for which it is known that μ = 0, Y t = e t – θ 1 e t – 1 – θ 2 e t – 2 where { e t } is zero-mean white noise ( i.i.d. N ( 0, 2 e σ ) ). 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 ( θ 1 , θ 2 ) = & = N t t e 1 2 for θ 1 = 0.3, θ 2 = 0.4. b) For θ 1 = 0.3, θ 2 = 0.4, forecast y 7 , y 8 , y 9 , and y 10 . c)* For θ 1 = 0.3, θ 2 = 0.4, given 2 ˆ e σ = 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 zero-mean white noise ( i.i.d. N ( 0, 2 e σ ) ), Y & t = Y t – μ . 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 Yule-Walker equations to find ρ 1 and ρ 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 = μ + φ 1 ( Y t – 1 – μ ) + φ 2 ( Y t – 2 – μ ) + 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 Yule-Walker equations to estimate φ 1 and φ 2 . b) If φ 1 and φ 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 04/29/2010 for the course STAT stat 420 taught by Professor Stepanov during the Spring '07 term at University of Illinois at Urbana–Champaign.

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420Pr8 - Practice Problems 1 Consider the MA(2 process for...

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