STAT 5511 (Fall 2022) Homework 3
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Homework 3
STAT 5511 (Fall 2022)
The usual formatting rules:
• Your homework (HW) should be formatted to be easily readable by the grader .
• You may use knitr or Sweave in general to produce the code portions of the HW . However, the output from knitr/Sweave that you include should be only what is necessary to answer the question, rather than just any automatic output that R produces . (You may thus need to avoid using default R functions if they output too much unnecessary material, and/or should make use of invisible() or capture .output() .)
– For example: for output from regression, the main things we would want to see are the estimates for each coefficient (with appropriate labels of course) together with the computed OLS/linear regression standard errors and p-values . If other output is not needed to answer the question, it should be suppressed!
• Code snippets that directly answer the questions can be included in your main homework document; ideally these should be preceded by comments or text at least explaining what question they are answering . Extra code can be placed in an appendix .
• All plots produced in R should have appropriate labels on the axes as well as titles . Any plot should have explanation of what is being plotted given clearly in the accompanying text .
• Plots and figures should be appropriately sized, meaning they should not be too large, so that the page length is not too long . (The arguments fig .height and fig .width to knitr chunks can achieve this .)
• Directions for “by-hand” problems: In general, credit is given for (correct) shown work, not for final answers; so show all work for each problem and explain your answer fully.
Questions:
1. ARMA models: Several ARMA models are written below. You can assume in all cases that Wt N(0, 1). For each of the ARMA models, find the roots of the AR and MA polynomials. Identify any parameter redundancy: find the values of p and q for which each model is ARMA(p,q) and write the model in its correct (non-redundant) form. Determine whether each model is causal, and determine whether it is invertible.
(a) Xt + 0.81Xt−2 = Wt + Wt−1
(b) Xt − Xt−1 = Wt − Wt−1 − Wt−2
(c) Xt − 3Xt−1 = Wt + 2Wt−1 − 8Wt−2
(d) Xt − 2Xt−1 + 2Xt−2 = Wt − Wt−1 (e) Xt − 4Xt−2 = Wt − Wt−1 + Wt−2 (f) Xt − Xt−1 − Xt−2 = Wt
2. Linear representation of ARMA: For those models of Question 1 that are causal, compute the first five coefficients ψ0 , . . . ,ψ4 in the causal linear process representation Xt = 对 ψjWt−j
3. Autocorrelation function of ARMA model: For the models in parts (a) and (b) of Question 1: Simulate 100 observations from each model. Compute and plot the sample ACF together with the theoretical ACF. You can have R compute the theoretical ACF using the ARMAacf function.
4. (Comparing time series with periodic behavior)
(a) Find an AR(2) process whose periodic ACF ρ(h) has period 9.
(b) Simulate a time series following the distribution you found in the previous part. Plot both the
true ACF and the simulated data series (not the sample ACF!).
(c) Simulate a signal-in-noise time series Yt = µt + Wt where Wt N(0,σ2 ) with σ 2 = 0.01 (or σ = .1), and where the signal µt is given by the ACF ρ you found in the first part. That is: µt = ρ(t). Plot the simulated data series. (If you want you may also plot the underlying true signal µt = ρ(t) on the same plot.)
(d) The point of what we’ve done so far is to compare different models that lead to periodic behavior. So: discuss/compare both the locations and the frequency/number of the peaks (maxima) and valleys (minima) of the simulated AR(2) data series to the locations of the maxima and minima of the true ACF and to the locations of the maxima and minima of the signal in noise data series.
2022-10-24