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ECON920
Applied Macroeconometrics
TIME ALLOWED: 24 Hours
INSTRUCTIONS TO CANDIDATES
You must answer all four questions
Show necessary working for numerical questions
All statistical tests should be performed at the 5% significance level unless otherwise
stated in the questions
The 97.5th percentile of the standard normal distribution is 1.96
FINAL EXAMINATIONS 2022
Paper Code: ECON920 Page 2 of 7
Applied Macroeconometrics
QUESTION 1
a) Describe three key features that you may observe in time series data. For each feature,
provide one practical example in economics or finance, and write a data generating process
that can potentially capture the specific feature. (6 Marks)
b) Let ! denote an (0, ") and ! = #!"" + "!$%. Is ! weakly stationary? Is ! ergodic? Is !
an IID? Show all working and explain why or why not. (6 Marks)
c) For each of the following, derive the mean and autocovariance functions, and state if it is a
weakly stationary process. Here !~(0, ").
(i) ! = 2 + ! + 2.5!$% − 1.5!$"
(ii) ! = 0.5!$% + !
(iii) ! = 0.8!$" + ! (9 Marks)
d) Consider a Gaussian white noise process !~(0,1). If the process ! = !", derive the
mean and autocovariance function of ! . Is ! weakly stationary? (4 marks)
TOTAL [25 Marks]
- End of Question 1 -
Paper Code: ECON920 Page 3 of 7
QUESTION 2
Consider the following GARCH(1,1) process ! = !, !|!$%~(0, !") !" = + %!$%" + %!$%"
where !$% is the information available up to and including time − 1, is constant. Answer
part a) to d) for the above model.
a) Use the conditional variance process !", explain briefly how does the GARCH(1,1) model
capture the volatility clustering feature in time series !. (2 Marks)
b) Obtain the ARMA representation of the GARCH(1,1) process. Show all working and state the
stationarity condition of the conditional variance process. (4 Marks)
c) Obtain the ARCH(∞) representation of the GARCH(1,1) process. Show all working and state
the condition when the ARCH(∞) representation exist. (4 Marks)
d) Obtain the unconditional expectation of !", i.e. [!"]. (2 Marks)
Consider modelling a series of daily exchange rates between US dollar and British Pound using an
ARMA-GARCH-type model. Answer part e) to h) for this case study.
e) A plot of the USD/GBP daily exchange rates is shown below. What feature you may observe
from this time series plot? How to systematically verify the presence of this feature? What
kind of data transformation may be needed in order to estimate the model? (3 Marks)
Paper Code: ECON920 Page 4 of 7
f) The transformed time series has the following correlogram with ACF, PACF and Q-Stat. What
ARMA(p,q) process, or more specifically, what values for p and q you would suggest for the
mean equation of the model? Explain your answer. (2 Marks)
g) After estimating the mean equation by an ARMA(p,q) process, the presence of ARCH/GARCH
effect can be detected by some diagnostic checks. Suggest at least two diagnostic checks to
identify the presence of ARCH/GARCH effect. (4 Marks)
h) Suppose a GARCH(1,1) process is chosen according to some ARCH/GARCH effect diagnostic
checks, the following is the estimated variance equation of the model. Is the conditional
variance process stationary? What kind of post-estimation diagnostic checks you would
suggest in order to check the model adequacy? (4 Marks)
TOTAL [25 Marks]
- End of Question 2 -
Paper Code: ECON920 Page 5 of 7