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

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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 μ ) = φ ( Y t – 1 μ ) + e t where e t is a mean zero white noise process. The model has been fitted to a time series giving φ ˆ = 0.8 , μ ˆ = 10.2 , and 2 σ ˆ 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 σ ) ) , Y & t = Y t μ . a) Is this process stationary? Justify your answer. b) Use Yule-Walker equations to find ρ 1 and ρ 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
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5.* 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 . 6.** 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.
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