Exam Instructions

Assigned: Monday May 4, 2020 (5:00pm)

Due: Uploaded to Canvas on Friday, May 8, 2020, before 11:59pm (Central Time). Upload your exam as a

single pdf file.

This is a takehome exam. You may not work (give or receive help) with any one else on this exam; that

includes other students, as well as friends, family, colleagues, faculty, or anyone else. If you have ques￾tions you should contact the professor. You may reference all of your course materials (as well as outside

textbooks or the internet). (However: for full credit you will generally need to give an answer that refers

to what you learned specifically in class. If you use ideas or material from outside class, you will need to

explain what you are doing, and they may not yield full credit.)

Note: statistics students have been caught cheating by working on takehome exams together and sadly have

been forced to leave our program and university. Any students caught cheating in such a manner will fail

the course and have a mark put on their academic record, as a minimum punishment. Being caught cheating

can lead to expulsion from the university.

General formatting guidelines:

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.

• 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

On this takehome exam, you will analyze three data sets. Find final-data.rsav on the course web￾page. Load it into R by running load("path/final-data.rsav") where “path/final-data.rsav” is

replaced by the full path on your hard drive to the file final-data.rsav (the syntax for which is operating

system dependent). The file contains three data objects: dat1, dat2, and dat3. Each dataset corresponds

to a separate exam question. (Thus, this exam has three questions.) Your output should be in the following

format. (Points will be deducted if it is not.)

• The analysis for each question/dataset should begin on a new page and should have as label the name

of the dataset (“dat1”, “dat2”, or “dat3”).

• On the first page of output for each problem, you should first have a summary (labeled “Summary”)

that provides a succinct answer to the question.

E.g., for SARIMA modeling, provide the model chosen, whether any transformation was used, pa￾rameter estimates, standard errors, and p-values in that model. Specify explicitly if you exclude a

constant term. For example, “For the series Yt = X

1/2

t

, I chose an ARIMA(1, 2, 3) model, includ￾ing the intercept term. The parameter estimates were ...”. If you believe the data cannot distinguish

between two (or more) models you should describe both (all) of them in this manner here.

1

• After the summary, should be an explanation (labeled “Explanation”). Here you should provide a

clear explanation of the methodology or reasoning that lead to your answer in the Summary above.

Refer to the output of your analysis, which will be below. This part is still succinct (and does not

include code output) but explains in words why you got the answer(s) you got.

For SARIMA modeling, the model selection and diagnostic techniques we have discussed in class

can be discussed here. You do not need to (and should not) provide an exhaustive list of all possible

models, but should rather provide explanation for which models were reasonable contenders (and

why), and which model (or models) were the best out of those contenders (and why).

• After the explanation is the “Output” you refer to in your summary. (The output may/will be plots or

output from various commands.) All of it should be clearly labeled or described. You do not need to

provide exhaustive output from every single command you have run, but you should include enough

to justify all the arguments you make in your summary.

Finally, in Questions 1 and 2 please refer to the original/raw (untransformed) time series as Xt

in your

descriptions and as xx in your code. Refer to any transformed series as Zt

in your descriptions and zz in

your code. In Question 3, the two series are named xx and yy and you should not rename them.

Dat1

For dat1, your job is to fit the best SARIMA(p, d, q) × (P, D, Q)s model you can to the dataset.

Dat2

Consider dat2. There are two parts to this question.

(A). Pretend we have a “baseline” model, which is AR(1) with φ1 = .7. Your job is to assess whether

the true spectral density of the data match the spectrum from this AR(1) model or not, specifically at

the following frequencies: ω = 0.027, 0.049, 0.25, 0.35, 0.45. That is, using (two-sided) confidence

intervals, test the hypotheses that the true f(ω) equals fAR(1)(ω) (with φ1 = .7) at the 5 omega

values given. (Do not adjust for multiple comparisons.)

(B). Remove just the final observation in the time series (so the length will decrease by just 1). Now again

answer the previous question – but do this just for ω = .027.

To format your answers, follow the same general formatting used for our analysis of SARIMA models: a

Summary (describe just what your result is for each test), Explanation (the broad overview of the method￾ology that led you to your summary results), and Output. (Note: you may use any R functions you wish for

this question, you do not need to do anything ’by hand’.)

Dat3

dat3 has two columns, yy and xx, which are both time series. Your job is to regress yy on xx using the

(“transfer modelling”) methodology we discussed in class.