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Homework 3

STAT 5511 (Spring 2020)

Due: Fri, Mar 20

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.)

– For example: for output from regression, the main things we would want to see are the estimates for each coe!cient (with appropriate labels

of course) together with the computed OLS/linear regression standard errors and p-values.

• 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.)

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.

2. Linear representation of ARMA: For those models of Question 1 that are causal, compute the first

five coe!cients 0,..., 4 in the causal linear process representation Xt = P1

j=0 jWt!j

3. Autocorrelation function of ARMA model: For those models of Question 1 that are causal,

(a) Compute theoretically the ACF.

(b) Simulate 100 observations from each model. Compute and plot the sample ACF together with

the theoretical ACF.

4. (Comparing time series with periodic behavior)

(a) Find an AR(2) process whose periodic ACF ⇢(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

iid⇠ N(0, #2) with #2 = 0.01 (or

# = .1), and where the signal µt is given by the ACF ⇢ you found in the first part. That is:

µt = ⇢(t). Plot the simulated data series. (If you want you may also plot the underlying true

signal µt = ⇢(t) on the same plot.)

(d) The point of what we’ve done so far is to compare di↵erent 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

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