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Assignment #3

ARIMA and Regression Models ## Instructions:

### 1. This assignment can be done by a single student.

2. Make sure your submitted assignment is neat, readable, and well-organized. Assignment marks will be adjusted for sloppiness, poor grammar and spelling as well as for technical errors.

3. Front page of the document has to include the title of the assignment, course code and section, student name and student number. The second page is the statement of integrity that must be signed.

4. Plagiarism on assignments will not be accepted.

5. Questions related to the assignment should be sent to the Teaching Assistant, Zahra Abtahi ([email protected]).

6. The assignment is to be submitted electronically as a single Word/PDF document file via Brightspace by Friday November 29th prior to 23:59. ## Total 100 points

1. Consider the Australian production of electricity (from aus_production). (32 marks)

a) Do the data need transforming? If so, find a suitable transformation.

b) Are the data stationary? If not, find an appropriate differencing which yields stationary data.

c) Identify a couple of ARIMA models that might be useful in describing the time series. In addition, apply the automatic model using ARIMA() function. Which of your models is the best according to their AIC values?

d) Estimate the parameters of your best model and do diagnostic testing on the residuals. Do the residuals resemble white noise? If not, try to find another ARIMA model which fits better.

e) Forecast the next 24 months of data using your preferred model.

f) Compare the forecasts obtained using ETS().

2. For the United States GDP series (from global_economy): (30 marks)

a) If necessary, find a suitable Box-Cox transformation for the data

b) Fit a suitable ARIMA model to the (transformed) data using ARIMA()

c) Try some other plausible models (at least 5 neighbour models) by experimenting with the orders chosen

d) Choose what you think is the best model and check the residual diagnostics

e) Produce forecasts of your fitted model. Do the forecasts look reasonable?

f) Compare the results with what you would obtain using ETS() (with no transformation).

3. The population of Switzerland from 1960 to 2017 is in data set global_economy. (24 marks)

a) Produce a time plot of the data.

b) You decide to fit the following model to the series:

������ = ��� + ������−1 + ∅1(������−1 − ������−2) + ∅2(������−2 − ������−3) + ∅3(������−3 − ������−4) + ������

where ������ is the Population in year ��� and ������ is a white noise series. What sort of ARIMA model is this (i.e., what are ���, ��� and ���)?

c) Explain why this model was chosen using the ACF and PACF of the differenced series.

d) The last five values of the series are given below. Year 2013  2014 2015 2016 2017 Population (millions) 8.09 8.19 8.28 8.37 8.47 The estimated parameters are ��� = 0.0053, ∅1 = 1.64, ∅2 = −1.17, and ∅3 = 0.45. Without using the forecast() function, calculate forecasts for the next three years (2018–2020).

e) Now fit the model in R and obtain the forecasts from the same model (show your forecasts). Are they different than the ones you calculated in part (d)? If yes, why?

4. The annual population of Afghanistan is available in the global_economy data set. (14 marks)

a) Plot the data and comment on its features. Can you observe the effect of the Soviet-Afghan war?

b) Fit a linear trend model and compare this to a piecewise linear trend model with knots at 1980 and 1989. Plot the fitted values and the actual data. Which model seems to be better based on the plot.

c) Calculate and report the prediction accuracy of the models and compare the models based on the

���2 values.

d) Generate forecasts from these two models for the five years after the end of the data, and comment on the results.