Module Code MAT00045H BA, BSc and MMath Examinations 2019/20 Department: Mathematics Title of Exam: Time Series Time Allowed: You have 24 hours from the release of this exam to upload your solutions. However, this exam should take approximately 2 hours to complete. Allocation of Marks: The marking scheme shown on each question is indicative only. Question: 1 2 3 Total Marks: 32 30 38 100 Instructions for Candidates: Answer ALL questions. Please write your answers in ink; It is important to show your working and reasoning in order to demonstrate your knowledge and understanding. Do not use red ink. Queries: If you believe that there is an error on this exam paper, then please use the “Queries” link below the exam on Moodle. This will be available for the first four hours after the release of this exam. No corrections will be announced after four hours. After that, if a question is unclear, then answer it as best you can and note the assumptions you’ve made to allow you to proceed. Submission: Please write clearly and submit a single copy of your solution to each question. Any handwritten work in your electronic submission must be legible. Black ink is recommended for written answers. View your submission before uploading. Number each page of your solutions consecutively. Write the exam title, your candidate number, and the page number at the top of each page. Upload your solutions to the “Exam submission” link below the exam on Moodle (preferably as a single PDF file). If you are unable to do this, then email them to maths-submit@york.ac.uk. Page 1 (of 5) MAT00045H A Note on Academic Integrity We are treating this online examination as a time-limited open assessment, and you are therefore permitted to refer to written and online materials to aid you in your answers. However, you must ensure that the work you submit is entirely your own, and for the whole time the assessment is live (up to 48 hours) you must not: • communicate with departmental staff on the topic of the assessment (except by means of the query procedure detailed overleaf), • communicate with other students on the topic of this assessment, • seek assistance on this assessment from the academic and/or disability support services, such as the Writing and Language Skills Centre, Maths Skills Centre and/or Disability Services (unless you have been recommended an exam support worker in a Student Support Plan), • seek advice or contribution from any third party, including proofreaders, friends, or family members. We expect, and trust, that all our students will seek to maintain the integrity of the assessment, and of their awards, through ensuring that these instructions are strictly followed. Failure to adhere to these requirements will be considered a breach of the Academic Misconduct regulations, where the offences of plagiarism, breach/cheating, collusion and commissioning are relevant — see Section AM.1.2.1 of the Guide to Assessment (note that this supersedes Section 7.3). Page 2 (of 5) continued from previous page MAT00045H In this exam paper, Z is the set of all integers, B is the backward shift operator, and Zt, t ∈ Z, is white noise with variance σ2. 1 (of 3). (a) Let Xt = t+t 2+Z (1) t , Yt = 2t 2+Z (2) t . {Z(1)t , t ∈ Z} and {Z(2)t , t ∈ Z} are independent with each other, and both of them are white noise processes. Are {Xt, t ∈ Z} and {Yt, t ∈ Z} cointegrated? Why? [5] (b) What is the key feature of the autocorrelation function of an MA(q) model in order specification? What is the key feature of the partial au- tocorrelation function of an AR(p) model in order specification? Figure 1 is the plot of the autocorrelation function of a time series {Xt, t ∈ Z}. Based on this plot, which model would you use to fit this time series? [9] 0 5 10 15 20 − 0 . 2 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 Lag A C F The Autocorrelation Function Figure 1: The autocorrelation function of Xt. (c) Suppose Xt, t ∈ Z, is an ARMA(1, 1) process, and (1− 0.1B)Xt = 0.1Xt−1 + Zt − 0.1Zt−1 Find the autocorrelation function of Xt. [18] Page 3 (of 5) Turn over MAT00045H 2 (of 3). (a) Show that the following set of equations represents a VAR (vector au- toregressive) process{ Xt = 0.1Xt−1 + 0.2Yt−1 + Z (1) t Yt = 0.2Yt−1 − 0.1Xt−1 + Z(2)t where Z (1) t and Z (2) t are independent white noise processes. What is the order of this VAR process? [4] (b) Suppose Xt = 0.1Xt−1 + 0.2Xt−2 + Zt, t ∈ Z. Is the time series Xt, t ∈ Z, stationary? If it is stationary, find its autocorrelation function and partial autocorrelation function. [14] (c) Suppose Xt = (α + β)Xt−1 − αβXt−2 +Xt−3 − (α + β)Xt−4 + αβXt−5 + Zt Find the range of values of α and β, such that for a suitable choice of s, the seasonal difference series Yt = Xt −Xt−s is stationary. [12] Page 4 (of 5) MAT00045H 3 (of 3). (a) After fitting a model to a time series of 250 observations, the residual values were calculated. The number of turning points in the residual series is 150. Carry out a statistical test at the 0.05 significance level to test the hypothesis that the residual series is a white noise process. [10] (b) Suppose Xt, t ∈ Z, is an ARIMA process, and Xt = 2.3Xt−1 − 1.6Xt−2 + 0.3Xt−3 + Zt − 0.2Zt−1. What is the order of this ARIMA process? Why? Express the prediction of Xt+2 in terms of Xt, Xt−1, · · · , X1. [13] (c) Consider the process Xt = Zt + 0.2 ∞∑ k=1 0.1k−1Zt−k. Is this process an ARMA(p, q) process with p < ∞ and q < ∞? If it is, what is the order of this ARMA process? What is the AR(∞)- representation of this process? [15] Page 5 (of 5) End of examination.













































































































































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