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MATH/STAT 4540/5540 Spr 2022 Time Series Midterm 2
发布时间:2022-04-02
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Midterm 2
MATH/STAT 4540/5540 Spr 2022 Time Series
Instructions Time: 60 minutes. Open note/book. Total points possible: 100.
The exam is open-note, open-book. This includes:
1. your notes, as well as my notes from when the class was zoom only
2. either of our two textbooks (Brockwell & Davis, or Shumway & Stoffer)
3. homework, including your solutions and my posted solutions
4. for the programming portion of the exam, you can use basic internet functionality like looking up R documentation
This excludes:
1. anything else from the internet (other than downloading/uploading the exam/solutions, and using R)
2. communication with any other person
You have approximately 24 hours to return the test (turn it in via upload to Gradescope). Within that window, take the pen-and-paper questions within a 60 minute contiguous block of time. Do not peek at the
pen-and-paper questions until you are ready to start the 60 minute block.
The computer question is untimed and has slightly different rules. When you are all finished with the entire test, please combine into a single PDF and upload to Gradescope (this step does not count toward the 60 minutes). You may take the pen-and-paper portion of the exam using a tablet if you wish; if you take it on paper, please scan your answers when all done.
As a reminder, the CU Boulder honor code is
“On my honor, as a University of Colorado Boulder student I have neither given nor received unauthorized assistance.”
The mandatory honor code syllabus statement is:
All students enrolled in a University of Colorado Boulder course are responsible for knowing and adhering to the Honor Code. Violations of the policy may include: plagiarism, cheating, fabrication, lying, bribery, threat, unauthorized access to academic materials, clicker fraud, submitting the same or similar work in more than one course without permission from all course instructors involved, and aiding academic dishonesty. All incidents of academic misconduct will be reported to the Honor Code ([email protected]; 303-492-5550). Students who are found responsible for violating the academic integrity policy will be subject to nonacademic sanctions from the Honor Code as well as academic sanctions from the faculty member. Additional information regarding the Honor Code academic integrity policy can be found on the Honor Code website.
The instructor must report all violations of the honor code. Please indicate and sign on your sub- mitted work that you acknowledge and agree to the honor code.
Problem 1: (25 points) (Computer Problem) From the website https://github.com/stephenbeckr/time- series-class/tree/main/Data use either Exam2data.RData or Exam2data.csv to load in the vec-
tor x which is a time series with 500 observations. Do you think this time series is plausibly white noise? Justify your answer. Grading is based on your reasoning.
Note: the rules for this problem are different. You may use a computer of course, and you may
use the internet for looking up documentation for programming. You may also look at anything
on the course github website, and you may use common R packages, including the stats and the book’s itsmr package. You may not look at any forums nor post to any forums, and you may not correspond with anyone else (in person, via email or chat, etc.).
Furthermore, this problem does not count toward the time limit. You may take as long as you want (within the 24 hour window) for this problem.
Test continues on the back side
Problem 2: (15 points) (Short Answer)
a) For the prediction/forecasting discussed in class and in the book, explain what were the criteria and constraints we used to derive the prediction operator?
b) (Graduate students only) If Γ is a n × n matrix, using standard algorithms, how many basic computer operations (“flops”) are needed to solve a linear system involving Γ? Now, if Γ is a Toeplitz n × n matrix, using the Durbin-Levinson algorithm, how many flops needed to solve a linear system involving Γ ?
Problem 3: (35 points) (Spectral density)
a) Let (X芒 ) be a zero-mean stationary time series with autocovariance function
,.γ〇
.
γ(h) =...γ(γ)ī2
. .
.0
h = 0
|h| = 1
|h| = 2
|h| > 2
for some non-zero values γ〇 , γī , γ2 . Derive an expression for the spectral density f(λ) of (X芒 ). Simplify the expression as much as possible to receive full credit.
b) Let (X芒 ) be a zero-mean stationary time series with autocovariance γx and spectral density fx . Let (Y芒 ) be a zero-mean stationary time series with autocovariance γY and spectral density fY . Assume X芒 and Yζ are uncorrelated for all t, s e N, and assume both γx and γY are absolutely summable. Let Z芒 = X芒 + Y芒 .
i. Find the autocovariance of (Z芒 )
ii. Find the spectral density of (Z芒 )
iii. Would your answer to the above questions change if we instead defined Z芒 = X芒 +Y芒 − ī ?
iv. (Graduate students only) How could you create a new time series (possibly using (X芒 )) that has spectral density fx (.) + 2?
Test continues on the back side
Problem 4: (25 points) You are given a time series x with n = 105 observations, and you plot the sample partial autocorrelation function which gives you the following plot:
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Furthermore, you use the arima function from the stats package in R to estimate the pa- rameters of the time series assuming that it is an ARMA model. In particular, you have two hypotheses: the first is that this is an AR(1) process, and hence you run the following fit (the order=c(1,0,0) tells it to fit to an ARMA(1,0) model):
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which gives the following output:
Coefficients:
ar1
0.4996
s.e. 0.0027
sigma^2 estimated as 1.002: log likelihood = -141990.9, aic = 283985.7
Your second hypothesis is that this is a MA(4) process, and hence you run the following fit (the order=c(0,0,4) tells it to fit to an ARMA(0,4) model):
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which gives the following output:
Coefficients:
ma1 ma2 ma3 ma4
0.4995 0.2524 0.1292 0.0663
s.e. 0.0032 0.0035 0.0035 0.0032
sigma^2 estimated as 1.001: log likelihood = -141935.3, aic = 283880.5
Using the provided information, which of the two hypothesis is more reasonable? Justify your
answer.