Assignment 2 Can Yang Department of Mathematics Hong Kong Baptist University April 1, 2015 1. A covariance-stationary AR(2) process: Yt = 1.5Yt−1 − 0.75Yt−2 + ǫt , has a M A(∞) representation given by Yt = ψ(L)ǫt , where ψ(L) = ψ0 + ψ1 L + ψ2 L2 + . . . and ǫt is a white noise process. Find out a way to generate ψ0 , ψ1 , . . . . Verify your results with the R function ARMAtoMA. You can use “?ARMAtoMA” to get information about how to use function ARMAtoMA. 2. Consider a M A(1) process Yt = ǫt + θǫt−1 , where θ is a fixed parameter and ǫt is a white noise process with E(ǫ2t ) = σ 2 . ˆ (1) Find out the auto-covariance γj and auto-correlation ρj of this M A(1) process, j = 0, 1, 2, . . . . ˆ (2) Show that ρj is the same for θ = 5 and θ = 1/5. In addition, the parameter pair σ 2 = 1 and θ = 5 yields the same autocovariance function γj as the pair σ 2 = 25 and θ = 1/5. ˆ (3) Which parameter setting in (2) can be represented by a finite AR process? ˆ (4) Using the R function ARMAacf to verify (2). Again, you may use “?ARMAacf” for detailed information.  Pq ˆ (5) (optional) Can you design a simulation to verify that V ar(ˆ ρj ) = T1 1 + 2 i=1 ρ2i , j = 2, 3, . . . . Hint: you may use the R function acf.

3. Consider a covariance-stationary AR(2) process: Yt = 1.5Yt−1 − 0.75Yt−2 + ǫt , where ǫt is a white noise process with E(ǫ2t ) = 1. Suppose we have known that Y0 = 1 and Y1 = −1 at time t = 0 and t = 1. ˆ (1) Write down the Yule-walker equation. ˆ (2) Calculate its auto-correlation ρj for (j = 1, 2, 3, 4, 5). ˆ (3) Find out the optimal prediction Yˆt=2|t=1 , Yˆt=3|t=1 and Yˆt=4|t=1 . ˆ (4) Write MATLAB or R code to simulate this time series and compute the average value of Yt at time t = 2, t = 3 and t = 4, denoted as y¯2 , y¯3 and y¯4 (here we use lower case letter y to indicate that we are considering samples). Note that you need to generate data many times (e.g., 1000 times) because ǫt is random and then take average. Whether y¯2 , y¯3 and y¯4 equal to your predicted values, i.e., Yˆt=2|t=1 , Yˆt=3|t=1 and Yˆt=4|t=1 ?

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ˆ (5) (Optional) Can you design a simulation study to verify that V ar(ˆ αm ) ≈ 1/T for m = 3, 4, . . . , where αm is the mth partial autocorrelation. Hint: you may use the R function pacf. m

4. Assigned reading. Time series analysis by James D. Hamilton Chapters 1 and 2. 5. Suggested reading. ˆ Algorithm. We have discussed triangular factorization for solving the matrix inversion problem in prediction using finite observed Y ’s. In fact, some efficient algorithms exist for this purpose, e.g., the DurbinLevinson algorithm and the innovation algorithm. See Chapter 2 of the book Introduction Time Series and Forecasting (2nd) by Peter J. Brockwell and Richard A. Davis, 2002. ˆ More R examples: Chapters 1-3 of Time series analysis and Its applications with R examples by Robert H. Shumway and David S. Stoffer. 2011.

You may find these e-books from our library. Please let me know if you couldn’t. This assignment is due to April 18, 2015. You are encouraged to do the optional exercise but not required. You are NOT required to submit anything for the fourth and fifth exercise. The Rule for assignment: NO copy from others.

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