ECOM90024
Forecasting in Economics and Business
Assignment 3


Question 1 (3 Marks)

Using the Case Shiller housing price data from your previous assignment, test for the
presence of a unit root using an appropriate form of the Augmented Dickey Fuller test. Once
you have determined the order of integration, repeat parts (b) and (c) of Question 2 from
Assignment 2. Describe how your forecasts have changed after this second attempt and
compare the forecasts using the RMSE and MAE metrics presented in lecture 3.

Question 2 (2 Marks)

Suppose that the innovations ! of a time series are governed by an ARCH(2) process,
! = !!
!~".".$.(0,1)
!% = & + '!('% + %!(%%

Where & > 0, ' ≥ 0 and % ≥ 0

a.) Derive the unconditional variance of the process and the associated restrictions it
imposes on the ARCH coefficients.

b.) Given the information set Ω!, derive expressions for the 1-step and 2-step forecasts
of the conditional variance in terms of the variables in the conditioning set.

Question 3 (5 Marks)

The file “aapl.csv” contains observations of the daily closing price of Apple from 15/05/2015
to 15/05/2019. Using R you are required to do the following:

a.) Generate the daily returns of Apple’s stock price as the log difference of the daily
closing price. Plot the data and provide a brief description of the observed time
series.

b.) Using the methodology outlined in previous lectures, specify and estimate an
appropriate and parsimonious model for the conditional mean of the daily
returns. Do not include any deterministic trend or seasonal dummies in your
specification.

c.) Using your estimates from part b, compute and plot ℎ-step ahead point and 95%
interval forecasts for the daily returns for ℎ = 1,2, … ,10.

d.) Using the methods described in lecture 9, test for the presence of ARCH effects
in the residuals obtained from the conditional mean estimation in part b.

e.) Estimate an appropriate ARCH/GARCH model for the innovations. Verify that
your specification is adequate.

f.) Using conditional variance forecasts from your ARCH/GARCH model, compute
and plot ℎ-step ahead point and 95% interval forecasts for the daily returns for ℎ = 1,2, … ,10.

g.) Compare the forecast intervals from part c with those from part f. Do you
observe any differences?




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