Instructions:

. This assignment contains 3 questions and worth a total of 50 points which will count towards 20% of the final mark for the subject.

. You have two option to submit this assignment:

1.  Convert the R Markdown file (with the answers to all of the questions) into a HTML File and submit the HTML File.

OR

2.  Convert the R Markdown file (with the answers to Question 1 (Except Part (B)) and Question 2 (Except Parts (A) and (B))) into a HTML File and sub- mit the HTML File along with the pdf file with answer to Question 1(B), Question 2(A,B) and Question 3.

No other forms of submission will be accepted.   Late  assignments  will not be marked.

. Make sure your HTML file contains the answer to questions. You will not get the mark if you just include the code with no output.

. Cover sheet not required.  Submission implies you have adhered to UNSW plagia- rism and student misconduct regulations.

. A correct submission will require correct answers and sufficient detail in order that the key steps in mathematics and reasoning can be followed.

Question 1:[21 Points Total]

The monthly price (per pound) of a chicken in the US from mid-2001 to mid-2016 (180 months), called xt , is provided in the data set chicken in astsa package.

(A)  (3 Points) Plot the data along with their ACF and PACF, and describe the patterns in the graphs.

(B)  (3 Points) We apply the transformation yt = more stable and then we decide to fit the following model to the series:

yt = yt − 1  + ϕ1 (yt − 1  − yt −2 ) + ϕ2 (yt −2  − yt −3 ) + ϕ3 (yt −3  − yt −4 ) + wt + Θ 1wt − 12

where wt is a white noise series. What sort of SARIMA model is this (i.e., what are p, q, d, P, Q, D and s)?

(C)  (3 Points) Fit the model in Part (B) to the data and estimate the parameters.

(D)  (3 Points) Do the residual diagnostics for your final model. Does your model fit the data well?

(E)  (3 Points) Do the forecast for the next 12 month.

(F)  (6 Points) Take difference of order 1 from the original data set. Plot the periodogram for the ‘diff(chicken)‘ .  Aside from the obvious annual cycle, is there any cycles with spectrum greater than 0.5? If yes, form the confidence interval for the spectral density of the related frequencies.

Question 2:[18 Points Total]

Consider two processes

xt = wt and yt = ϕxt −D + vt

where wt and vt are independent white noise processes with common variance σ2 , ϕ is a constant, and D is a fixed integer delay.

(A)  (3 Points) Compute the coherency between xt and yt .

(B)  (3 Points) Compute the phase between xt and yt .

For the following simulation study, use the 4 last digits of your student ID as set.seed. For example, if your student ID is 1234567 then use set.seed(4567) and if it is 1230567, use set.seed(567).

(c)  (6 Points) Simulate n = 1024 normal observations from xt andyt for ϕ = .9, σ2  = 1, and D = 0. Then estimate and plot the coherency between the simulated series for the following values of L and comment:

(i)  L = 1,