Burberry PLC Time Series Analysis and Cointegration (ACFI3308 Project)

This capstone project analyzes the financial time series data of Burberry PLC, encompassing various statistical and econometric techniques. The analysis begins with an exploration of the Augmented Dickey-Fuller (ADF) test to assess the stationarity of the price series and its first difference, followed by a discussion of the results in the context of financial theory, including the Efficient Market Hypothesis and the Random Walk Theory. The project then proceeds to estimate the best-fitting ARIMA model using the Box-Jenkins approach, assess its adequacy, and formulate diagnostic procedures. Forecasts are generated, and their accuracy is evaluated using relevant measures. Furthermore, the report estimates and discusses the preferability of ARCH and GARCH models for volatility modeling. Part two extends the analysis to cointegration, pairing the FTSE-100 index with other country's stock indices and applying the Engle-Granger (EG) approach to test for cointegration. The report concludes with an executive summary of the key findings and references supporting literature.

P2573201
Assignment II: Capstone Project
ACFI3308
Word Count: 1281

Introduction
This report will be split into two parts – one and two.
Part one will be exploring the performance of the ADF test of stationarity on the original
price series (adjusted close/close price) and its first difference for my firm – Burberry PLC. I
will state your ADF test hypotheses and test results. I will also be making comments on my
results in line with relevant financial theory. I will also be estimating the best fitting ARIMA
model of the price series, following the Box-Jenkins. I will also assess the adequacy of my
results and formulate appropriate diagnostic procedures. Forecasts will have been produced
for the price series and discussions around the accuracy of my forecasting using the relevant
forecast accuracy measures will be made. I will also be estimating both models (ARCH and
GARCH) and attempt to discuss the preferability of the model.
Part two will involve the pairing of the FTSE-100 index with two other country’s stock
indices and test for cointegration among the pairs of variables by applying the Engle-Granger
(EG) approach. I will critically comment on these results.

Part 1
ADF
In R studio, I used the adfTest from the fUnitRoots package. This conducted an ADF test
with no constant term – for Burberry PLC. I ran the command by typing the name of the
object that holds the series, along with a $ sign which served as an indicator. I then looked to
the object and picked the variable it was named.
My first initial P value came to 0.9717, which indicates to us that we cannot reject the null
hypothesis, as it indicates the price trending upwards. Another test was also performed to find
the constant term. This highlights that given the P value is more than 0.05 we do not reject
the null hypothesis, and instead conclude that our series is not stationary. In my result
including a constant time in the regression, but even in that case the P value resulted in
0.9365, so we will not reject the null hypothesis in this instance either. For my DHP series in
R (up to 12 log), my P value resulted in 0.01, so as it is less than 0.05, in this case we reject
the null hypothesis, and go for the alternate hypothesis (which indicates the series is
stationary). In my second case, my P value resulted in 0.02092 again, it is less than 0.05 so
we reject the null hypothesis. For constant and trend (CT), is only significant at the 10%
significance level.
As for relevant finance theories, both ‘Efficient Market Hypothesis’ and ‘Random Walk
Theory of Asset Prices’ are applicable. This is since if a series is shown to be a random walk
(stationary), then it can be extrapolated that the exact process is a random walk – and
ultimately less probable to be predictable – but still theoretically predictable to some extent.
Secondly, if, from an image point of view of the market hypothesis, it implies that we cannot
supply any information from the past. In efficient market hypothesis, there are three levels to
determine the quality of efficiency – being weak, semi strong, and strong form. If there is no
information from the past, then the market must either be semi strong, or even potentially
weak strong form.

The price series depicts a non-stationary pattern, as it is trending upwards and not ideal for
time series analysis.

ARMA / ARIMA Modelling & Box Jenkins Approach
ARMA modelling only provides the variable that you are interested in. it is strictly not a
relationship model in which you are attempting to relate a Y variable to an X variable -
instead it essentially uses the past values or the past focus areas and try and predict what the
future values would be.
One of the key aspects within the ‘Jenkins Approach’ (1970) is that it uses the least auto
regressive or moving average terms do not over-parameterise the data. This essentially
translates to not needing up to 12 different terms to be able to obtain all the dynamics within
the given data. This model also suggests that smaller (parsimonious) models produce better
forecasts, as additional options does not necessarily render into added value.
This procedure helps to reduce the number of auto regressive and / or moving average time
for when it comes to forecasting the data.
My time series plot exhibits mostly an upward trend, with a slight random walk in the latter
half. This would suggest that it is non-stationary. This indicates that the series needs to be
differenced to achieve stationarity.
My autocorrelation function ( ACF) plot demonstrates that the autocorrelation is slow
decaying – meaning it has a high persistence. From lag one to six, the ACF of the price series
is easily above the 0.75 threshold. Even at the likes of lag of eighteen, the autocorrelation is
above 0.5, which confirms the already announced slow decaying (persistence) of the non-
stationary series.

My partial autocorrelation function (PACF) of Burberry PLC displays that only the first lag is
significant (as it has exceeded the blue dashed lines). PACF calculates the correlation
between the current value of a series and lag K – which therefore ignores all lags
consequently found in between.

For my combined plot, it shows an upward trend, with minor decaying – which advises us
that there is an AR structure. PACF indicates two AR terms. This would suggest that up to
two AR terms would be recommended. There could potentially be MA structure, however it
is inconclusive. No real significant spike, but still maintains a steady climb (upward trend).

ARCH & GARCH
The Autoregressive Conditional Heteroskedasticity method (ARCH) provides a way to model
a change in variance in a time series that is time dependent, such as increasing or decreasing
volatility. An extension of this approach named GARCH or Generalized Autoregressive
Conditional Heteroskedasticity allows the method to support changes in the time dependent
volatility, such as increasing and decreasing volatility in the same series (Jason Brownlee,
2018).
One negative is that a change in variance or volatility over time can cause issues when
modelling time series with traditional methods such as ARIMA, which has been applied
within this report already.
Part 2
Engle-Granger (EG) Approach (Cointegration)
The Engle-Granger methodology follows a two-step estimation. The first step generates the
residuals and the second step employ generated residuals to estimate a regression of first
differenced residuals on lagged residuals. Hence, any possible error from the first step will be
carried into second step (Bilgili, F., 1998).
In my findings, there was no correlation between the cointegration between FTSE, CAC and
DAX. The critical values, which were obtained through running the script through R studio,
indicated to not reject the null hypothesis.
Executive Summary
To conclude, I have explored the performance of the ADF test of stationarity on the original
price series (adjusted close/close price) and its first difference for my firm – Burberry PLC. I
have published the ADF test hypotheses and test results, with comments on my results in line
with relevant financial theory. I estimated the best fitting ARIMA model of the price series,
following the Box-Jenkins. I also assessed the adequacy of my results and formulate
appropriate diagnostic procedures. Forecasts have been produced for the price series and
discussions around the accuracy of my forecasting using the relevant forecast accuracy
measures have been discussed. I also estimated both models (ARCH and GARCH) and tried
to discuss the preferability of the model.
In part two I involved the pairing of the FTSE-100 index with two other country’s stock
indices and tested for cointegration among the pairs of variables by applying the Engle-
Granger (EG) approach, with comments on these results.