Explain the concepts of non-stationarity and cointegration, and how are they connected. Illustrate how one can test for cointegration using the two-step Engle and Granger approach.

Financial Econometrics
QUESTION 1.

Conduct all your statistical tests at the 5% level for this question.

You are given the quarterly data of U.K. Consumer Price Index (CPI) over the
period 1960Q1 to 2019Q2. The data file name is “CPI.xls”. Calculate the
logarithmic change of the price series, i.e., ∆cpit= cpit – cpit-1, where cpit is the
natural logarithm of the Consumer Price Index at time t and ∆ is the first difference
operator, then:

a) Follow the Box-Jenkins approach in building an ARMA(p,q) model for ∆cpit;
specifically,

i. Obtain the autocorrelation function (ACF) and partial autocorrelation
function (PACF) for ∆cpit (specify the number of lags to be 8) using
data from 1960Q1 to 2017Q4 (Note that this is not the full sample).
Discuss the significance of the ACF and PACF coefficients and identify
the suitable models that may be appropriate for this time series.
[5%]

ii. Estimate all ARMA models from order (0, 0) to (4, 4) for ∆cpit over the
sample period 1960Q1 to 2017Q4. From your estimations, which is the
suitable model order? Explain why? (You would also need to report all
relevant information for the models you estimate, including the value of
the AIC and SBIC and other relevant required criteria in a Table).
[10%]

iii. Re-estimate the suitable model(s) from Question a(ii). Again, use only
the sample 1960Q1 to 2017Q4. Report and comment on the results.
Perform diagnostic checks on the residuals from these estimated
model(s). Do the model(s) fit the data well?
[10%]

b) Use the model(s) estimated in Question a(iii) to generate one step ahead
(static) forecasts for the period 2018Q1 – 2019Q2. Create a graph of the
actual ∆cpit series and the forecasts that you have generated over the
specified out-of-sample period. Comment on the results.
[10%]
1

Financial Econometrics

QUESTION 2.

Conduct all your statistical tests at the 10% level for this question. Support
your discussion for this question using appropriate mathematical equations
and references in the relevant area(s) of research.

You are given the monthly time series of the spot Japanese yen exchange rate
against the US dollar (denoted as JPYtoUSD) and the Consumer Price Indices,
which proxy the general price levels, for Japan and the US (denoted respectively
as JPCPI and USCPI) for the period of January 1991-August 2020. The data file
name is “PPP.xls”:

a) Explain the concepts of non-stationarity and cointegration, and how are they
connected. Illustrate how one can test for cointegration using the two-step
Engle and Granger approach.
[10%]

b) Test for long-run Purchasing Power Parity (PPP) using the two-step Engle
and Granger cointegration approach applied to the following regression:
s¥/$,t = α+β₁ptJP + β₂ptUS, (1)

where s¥/$,t is the natural logarithm of the spot exchange rate (the amount of
Japanese yen per 1 US dollar) and pt JP and pt US are the natural logarithms
of Japan and US price levels respectively. Under the long-run PPP, β₁=1 and
β₂ = -1.
[10%]

c) After determining whether Equation (1) is a cointegrating relationship or not,
estimate the respective Error Correction Model (ECM). Comment on your
results.
[10%]
2
Financial Econometrics

QUESTION 3.

Conduct all your statistical tests at the 5% level for this question. Support
your discussion for this question using appropriate mathematical equations
and references in the relevant area(s) of research.

You are given the daily prices of WTI Crude Oil Spot (Dollars per Barrel), namely
WTI, covering the period 01 January 1991-23 October 2020. The data file name is
“WTI.xls”:

a) Discuss the statistical properties of the series by (i) calculating relevant
summary statistics of the WTI returns (also known as log price changes),
and (ii) plotting the returns, as well as their histograms and quantile-quantile
(QQ) diagrams.
[5%]
b) Plot the ACF for returns, returns squared, and absolute returns, then
discuss whether any of these plots provide an indication about the
predictability of the series.
[5%]

c) Describe the ARCH-GARCH family of models and explain why it may be
useful in explaining the volatility of WTI returns.
[10%]

d) Use two univariate GARCH type models which nest ARCH (e.g. GARCH,
PGARCH, etc.) to estimate the volatility of returns, explaining the motivation
for their use.

Test for the differences between the models (e.g. parameter
significance and Likelihood Ratio ( LR) tests), and discuss how their volatility
estimates and residuals differ.

Last Completed Projects