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420Pr7 - Practice Problems 1 Consider the AR(1 model Yt =...

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Practice Problems 1. 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. 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 a) 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. ) b) Obtain the least-squares estimates for the AR(1) model parameters, ° ˆ and φ ˆ . 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 , ° ˆ = 10.2 , and 2 ± 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. ( Homework 13, Problem 4 ) Consider the ARMA ( 1, 1 ) model ( Y t – 60 ) + 0.3 ( Y t – 1 – 60 ) = e t – 0.4 e t – 1
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