# 420Pr7 - Practice Problems 1 Consider the MA(2 process for...

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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 = 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 .

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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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