时序分析代写: MATH5845 Time Series Analysis

程序代写 CS代考 / R语言代写代考

MATH5845: Time Series Analysis
Session 1, 2018
Assignment 2
Due: By 4:00 pm, Monday April 23 via Moodle submission

Instructions:

  • This assignment consists of two parts: Part 1: Two (2) theory exercises – below, Part 2: One (1) Computer exercise – using R markdown.

    Submission of all files required must be via Moodle by the due time and date. Cover sheet not required. Submission implies you have adhered to UNSW plagiarism and student misconduct regulations. Late assignments will not be accepted of marked.

  • Theory exercises (2)– see following pages:
    • –  Solutions to all parts of both theory exercises must be limited to one page for each in a single pdf file named A2Theory.pdf. Material beyond a single page per exercise will not be marked. Typeset or neatly handwritten please – no small handwriting, or fonts, all page margins at least 2.5cm. Name and Student number at top of each page.
    • –  In order to meet the page limitations you will need to include only the most essential mathematical steps and reasoning. You can refer to results in notes (page number please) without repeating them in your submission.
    • –  A correct submission will require correct answers and sufficient detail in order that the key steps in mathematics and reasoning can be followed. Correct answers without mathematical steps or reasons will not be awarded any marks.
    • –  If you use your smart phone or tablet camera to scan your solutions to pdf please ensure that the material on the pages is properly aligned. Keep the file size manageable by using an image to pdf converter app.
  • Computer Exercise:
    Download the R markdown file A2Rcode.Rmd and complete the Computer

    exercise by answering all parts required in the file.

  • IT IS ESSENTIAL THAT YOU ENSURE YOUR STUDENT ID replaces the dummy ID in the lines of code 
         SID=3141593 ## replace by your own SID
     set.seed(SID)

    otherwise you will not receive marks for this exercise. You must submit two files to moodle: A2Rcode.Rmd containing the R code and comments required for you answers and the pdf knitted file produced A2Rcode.pdf.

    1

Exercise 1 Let {Xt} be a stationary, causal autoregression of degree p in weakly stationary white noise innovations Zt and let {Ut} be weakly stationary white noise uncorrelated with the process {Zt}. Let Wt = Xt + Ut

  1. Show that {Wt} has the autocovariance function of an ARMA(p,q) proces and specify q in terms of p.
  2. Can {Wt} be written as stationary causal and invertible ARMA process de- fined in terms of a weakly stationary white noise process {et}? Give brief reasons based on reference to lectures and notes.
  3. How many constraints are there (do not explicitly derive these) on the para- meters in ARMA(p, q) process representaion? Please explain.
  4. If {Ut} and {Zt} are each sequences of independent and identically distribu- ted normal random variables and are independent of each other, is the process {et} (used for the ARMA version of {Wt} in part b.) an I.I.D N(0,1) se- quence? Please explain.

SOME HINTS:

• Please answer part b) first and then DEDUCE part a) from that. • For part b) you will find the following useful:

  • –  Look at Exercise 3.13 solutions top of page 2 and consider a similar approach here. That is apply the autoregression to both sides of Wt = Xt + Ut. Determine the ACF of the new RHS.
  • –  Page 64 of notes has a fact about q-dependent processes. Use that and conclude the RHS can be written as a MA process (specify degree) in some new white noise.

    Exercise 2 For the ARMA(1, 1) model:

a. Using equation (4.14) of Chapter 4 Lecture Notes, show that the truncated

recursive predictors, initiated with Z ̃0n = 0, X0 = 0, are obtained as follows: X ̃n = φX + θZ ̃n, m = 1, and X ̃n = φX ̃n , m > 1

n+1 n n n+m n+m−1
b. Show that the truncated forecast errors, initiated with Z ̃0n = 0, X0 = 0, are

Z ̃n=X−φX −θZ ̃n . t=1,…,n. t t t−1 t−1

c. Using equation (4.10) of Chapter 4 Lecture Notes, show that the mean square error of forecasts is given by

̃ 2 2 􏰎 (φ+θ)2(1−φ2(m−1))􏰏 E(Xn+m−Xn+m) =σ 1+ 1−φ2 .

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d. Simplify the forecasts in part (a.) and their forecast mean square error of part(c.) forthespecialcasesp=1,q=0andp=0,q=1.

• Sketch the forecasts and prediction intervals as lead time m increases in these cases.

• Discuss whether the forecasts and forecast mean squared error seem in- tuitively reasonable.

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