STAT/MATH 4/5540 – Project 2 – Spring 2020
by: Manuel Lladser
INSTRUCTIONS. Failure to follow these instructions may result in points discounted.
This lab is due at 10 AM on Friday, March 20, 2020. Discussing this project with anyone besides your group partner (if any), the Instructor, or the TA is not permitted. By submitting a report, all its participants agree to comply with the CU Honor Code Policy.
Students registered for APPM 4540 may work in groups of up to 2 members, and submit one project report with all participant names on it. Submit a single report in CANVAS. Due to the COVID- 19 (coronavirus) outbreak, avoid meeting in person and instead collaborate remotely using some video conferencing such as Facetime, Skype, WhatsApp, or Zoom.
Students registered for APPM 5540 must work on the project on their own.
Your report is limited to 5 pages with a minimum font size of 11 points and 1-inch margins; in particular, please provide complete but brief answers. Appendices do not count toward the page limit!
To receive full credit, you must submit a professional report addressing all the instructions and questions in the same order as requested. Be sure to include all figures or tables and to label them (e.g., Figure 1, Table 2, etc). Also, be sure to cite any sources (textbooks, papers, websites, etc) you consult. Your write-up should include brief but complete answers to all the questions listed below with appropriate references to labeled figures or tables. At the end of your report, include an appendix with the R code you used to address the project. The code must include annotations. Good luck!
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In this project you will simulate your own ARMA time series to test some of the concepts and methods discussed in lecture. In addressing the following questions, you may cite any result covered
in lecture as of Friday, March 13, 2020.
Let σ > 0 be finite and Zt, with t ∈ Z, be i.i.d. random variables with Zt ∼ Normal(0,σ2). Consider the time series defined as:
Xt :=∞ 2j+1Zt−j, foreacht∈Z. (⋆) j=0 2j
1. Determine the mean function of {Xt}.
2. Showthat4Xt−4Xt−1+Xt−2 =4Zt+2Zt−1,forallt∈Z.
3. What type of ARMA process is {Xt}? Justify!
4. Is {Xt} causal? Explain!
5. Is {Xt} invertible? Explain! In the affirmative case, represent each Zt as infinite linear combination of Xj , with j ≤ t.
6. Determine a homogeneous linear recursion with constant coefficients satisfied by the auto- covariance function γ(h) of {Xt}.
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7. Compute γ(0), . . . , γ(3).
8. ShowthatifZ∼Normal(0,σ2)thenE|Z|=σ2·2.
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9. Markov’s inequality states that for all random variable S and ε > 0: P(|S| ≥ ε) ≤ E|S|/ε. Using this, determine k such that the probability that | 2j+1Z | ≥ 0.01 is at most
j