STAT758
Final Project
Time series analysis of daily exchange rate between the British Pound and the
US dollar (GBP/USD)
Theophilus Djanie and Harry Dick Thompson
UNR • May 14, 2012
INTRODUCTION
Time Series Analysis accounts for the fact that data points taken over time may have an
internal structure (such as autocorrelation, trend or seasonal variation) that should be
accounted for. The behavior of time series variables such as exchange rates is not
consistent and to forecast it is irrational. Despite these assertions, many multinational
corporations, dealers in foreign exchange, exporters, importers and speculators continue
to make hedging decisions based on forecasted rates using ex-post data as their basis.
These hedging decisions are made under the premise that patterns exist in the ex-post
data and these patterns provide an indication of future movement of exchange rates, at
least in the short run. If such patterns exist, then it is possible in principle to apply
modern mathematical tools and techniques such as ARIMA and GARCH to forecast the
ex-ante exchange rates (Hamilton 1994, Klaassen 1998).
The ARMA model is made up of two processes: the Autoregressive AR and the Moving
Average MA. Given a series Xt, we can model the level of its current observations
depends on the level of its lagged observations. This way of thinking can be represented
by the AR model.
Also we can model that the observations of a random variable at time t are not only
affected by the shock at time t, but also the shocks that have taken place before time t.
This way of thinking can be represented by the MA model.
AutoRegressive Integrated Moving Average (ARIMA) models intend to describe the current
behavior of variables in terms of linear relationships with their past values. It has an
Integrated (I) component (d), which represents the amount of differencing to be performed
on the series to make it stationary. The second component of the ARIMA consists of an
ARMA model for the series rendered stationary through differentiation. The ARMA
component is further decomposed into AR and MA components which are explained
above. The Autocorrelation Function (ACF) and Partial Autocorrelation Function (PACF) are
used to estimate the values of the orders of the AR and MA processes respectively. The
statistical package R is used to analyze the data.
OBJECTIVE
The goal of this study is to perform statistical analysis on the foreign exchange data
between the GBP (Great Britain Pound) and the USD (United States dollar). The properties
of the data are described and basic time series techniques are applied to the data. Plots
of the series, autocorrelation function and the partial autocorrelation function are some
of the graphical tools used to analyze the series. We also aim to fit a model (ARMA,
ARCH, and GARCH) to the data in order to make credible forecasts from the model.
The data used for analysis is the close of business (COB) day value of the daily
exchange rate between the British Pound and the US dollar (GBP/USD). The data was
downloaded from the Oanda Corporation financial services website (http://
www.oanda.com/currency/historical-rates/), from 1st January, 2000 to 4th April, 2012.
Values on weekday holidays are assumed to be the value from the previous business
day. For any other unexpected closure of the market of a single weekday, or several
consecutive weekdays, the data points are filled in with the last COB value. A year of
data is considered to be 52 weeks with five business days per week which equals 366
data points per year.
EXPLORATIVE DATA ANALYSIS AND DESCRIPTIVE OF STATISTICS
The data set is made up of 4478 daily quotes of foreign exchange rate between the GBP
and the USD for the period year 2000 to 2012. The time series analysis for the
exchange rate data is plotted using the logarithm of the returns of the rates. That is, if
is the rate at time t, and is the rate at time t-1; then the logarithm of the returns is
given as:
Rt = Return
Pt = Current rate
Pt-1 = Previous day rate

This is because, the log returns of exchange rates have interesting statistical properties;
thus making analysis easier. Also, the log returns are assumed to be normally distributed,
which simplifies computations on the probabilistic aspect of the data. A histogram
illustrating this assumption is shown in fig a below. Some basic (descriptive) statistics of
this data set is given fig1. below.
Minimum 1st Quartile Median Mean 3rd Quartile Maximum
-1.733e-02 -8.595e-04 0 -1.067e-06 8.822e-04 1.359e-02
Table 1. descriptive statistics of exchange rate of GBP/USD data
A histogram illustrating a bell shape, an indication of the normality assumption.
Fig 1. Histogram of the log return of the exchange rate of GBP/USD
Fig 2. The plot of GBP/USD foreign exchange rates from 2000 to 2012
The time series plot for the returns is shown in fig. 2. It will be noted that this plot
exhibits no periodicity but we do have a trend effect due to the random walk nature.
After transformation using log returns, the trend effect is eliminated as seen in fig3. Thus,
we do not have to apply any differencing technique to make the series stationary;
therefore, the integration order is zero. Generally, the volatility of the series is fairly
uniform for the years 2000 to 2006. For the years of 2006 to 2012, volatility of the
series was highly non-uniform and more pronounced around 2009.
Fig 3. The plot of daily log return of GBP/USD foreign exchange rates from 2000 to 2012
One way to determine the order of the above series is to construct the ACF and PACF
plots. The ACF plot is shown below in fig 3. with a two spikes out of the confidence
range. ACF spikes for subsequent lags decay faster to zero with none being significant
after lag one. This implies that the series has an MA order.
Fig 4. ACF of log return
Fig 5. PACF of log return
A similar argument can be made for the PACF plot of the series exhibited in fig 5. Here,
the lags are well contained with a little spill at lag 20.
From the two plots above, it may be inferred that an ARMA or ARIMA model would be a
moving average model as the graphs displays its characteristics. Nevertheless, we apply
other techniques that may help us establish a model for the series. We do this by
constructing a matrix of models (with orders up to 4x4) and we choose the pair with the
lowest Akaike information criterion (AIC) score. The residuals are also tested using the
Box-Ljung test. This is to verify the residuals are white noise which is the desired
objective. The hypothesis for this test is formulated as follows:
H0: R ~ WN(0,σ2) vs Ha: R ≁ WN(0,σ2)
where R is the residuals
σ2 is the varainace

Under this test, H0 is the null hypothesis and Ha is the alternative hypothesis.
ARIMA ANALYSIS
Tables 1-3 in the appendix shows the output of ARIMA models with orders p and q
values ranging from zero to four. In table 1 we realize that the minimum AIC value is
-3.5779.39 and belongs to the model ARIMA (0, 0, 1). The standard error at the specified
p and q values also exhibits a minimum gradient to higher order. The box test for this
model passed the random test.
An estimation of the model can be seen below in table 2. The standard error and the
residual variance for our model is quite low (0.0146 and 1.978E−05) . This would be due
to the relatively small changes that occur with foreign exchange rates in general.

Estimation MA1 Mean
Coefficient 0.1659 0
Standard Error 0.0146 0.0001
Table 2 Estimated values for the ARIMA(0,0,1) model
ARIMA(0,0,1) Model:
Xt = Zt + θ Zt-1 , where Zt ~ WN(0,σ2)
Xt = Zt + 0.1659 Zt-1    , Zt ~ WN(0,1.978E−05)
Fig 6. Residuals plot over time
The residual of this model is shown below in fig.6. It can be observed that residuals have
a value of zero but displays a non-constant spread especially during 2006 to 2012.
Fig 7. 1-step prediction of ARIMA(0,0,1) model
With our current model ARIMA(0,0,1) we tried to do 1-step prediction for the period of
2001-to 2012 to validate our model. This is shown Fig 7. The predicted values can be
seen in the lighter color and it is an under prediction compared to the back drop true
data (true returns). It is obvious that these prediction are not good enough. Due to the
presence of a non-constant spread, we proceed to investigate the volatility of the time
series as it is the most important factor for portfolio risk management.
GARCH ANALYSIS
GARCH (Generalized Autoregressive Conditional Heteroskedasticity) processes is a process
where the conditional mean is constant but the conditional variance is non-constant and
hence an uncorrelated but dependent process. The dependence of the conditional
variance on the past causes the process to be dependent.
Conducting many GARCH models to fit out, we concluded that ARMA (1,0)-GARCH(1,2) was
the best as it had the lowest AIC value of -8.162937 ( see Tables 4 in the appendix).
ARMA (1,0)
An Autoregressive process Xt of order (lag) 1, AR (1), is defined as
Xt = c + φ Xt-1 + Zt, where Zt ~ WN(0,σ2)
GARCH (1,2)
A Generalized Autoregressive Conditional Heteroskedasticity process of Xt order (1,2),
GARCH(1,2) , is defined as
Xt = σt Zt ,where Zt ~ WN(0,σ2)
σ2 = ω + αX2t-1 + β1σ2t-1 + β2σ2t-1
ARMA (1,0)-GARCH(1,2)
A generalization of the GARCH(1,2) model, the ARMA(1,0)-GARCH(1,1) process, is defined
Xt = σt Zt ,where Zt ~ WN(0,σ2)
μt = φ Xt-1
σ2t = ω + α (Xt-1 - μt-1)2 + β1σ2t-1 + β2σ2t-1
Where
Xt is the log return
Zt is the shock or innovation
σ2t is the volatility
μt is the mean
and the coefficients and constant
α ≥ 0, β1 ≥ 0, β2 ≥ 0,
(α+ β1) < 1 and β2 < 1,
ω ≥ 0
Estimating the coefficients of our model yields the results in table 3. It is noticeable that
the mean μ is not significant. Hence, we can drop it and the modified model is
represented below.
ARMA(1,0)-GARCH(1,2) process, is defined
Xt = σt Zt, where Zt ~ WN(0,σ2)
μt = 0.1653 Xt-1
σ2t = 0.05088 Xt-12 + 0.154 σ2t-1 + 0.7888 σ2t-2
Parameters Estimates Standard Error t value p-value
μ 2.46E-05 5.68E-05 0.433 0.66513
φ 0.16530 1.59E-02 10.412 < 2e-16
ω 1.20E-07 4.13E-08 2.903 0.00369
α 0.05088 5.48E-03 9.286 < 2e-16
β1 0.15400 5.70E-02 2.703 0.00686
β2 0.78880 5.65E-02 13.962 < 2e-16
Table 3 Estimated values for the ARIMA(1,0,0)-GARCH(1,2) model
The residuals and the standard residuals were also tested for significance using Ljung-Box
Test. The squared of the residuals and the standard residuals were also accounted for
(see table 5 in appendix).
The results shows that the residuals and the standard residuals were non-significant but
their squared values are. This is evident in the residual plots in fig 8. Looking at the ACF
of the squared standard residuals (lower right of fig 8), we can say the spikes out of the
confidence region are moderate. These are due to small autocorrelations that should not
be of practical importance and therefore can be ignored. Also, the plot of the standard
residuals turn to have a constant spread compared to the actual residuals.
Fig 8. Residuals plots:
Plotting the volatility against time (fig 9), we find major turbulence around the period of
the economic recession (2008 -2009).
Fig 9. Volatility plot over time
A volatility prediction made between the period of 2010 to 2012 to validate our GARCH
model indicates a better fit to the actual volatility as seen below. The actual plot is
shown by the black line while the predicted values are represented by the lighter color.
Fig 10. Forecasting volatility from 2010 to 2012
INTERPRETATION
To apply the ARMA modeling technique to a time series data, the data should be
stationary. The logarithm of returns of the exchange rates are stationary, as seen in the
ACF plots, thus, the integration part of the ARIMA process is zero.
Volatility in the data was fairly homogeneous from 2000 to 2006 with 2008 to 2012
being inconsistent. This condition makes fitting ARMA to financial data quite difficult. This
was evident in the model orders - MA(1). The ARMA models obtained were not credible
and therefore producing poor prediction outcomes.
Due to the changing variance observed in the series from 2007 to 2010, it was
imperative that we analyze the variance of the series using ARMA(0,1)-GARCH(1,2). This
resulted in a better prediction of the volatility.

CONCLUSION
Time series analysis and modeling is a very popular technique in mathematics and
statistics used to explore the hidden details in time dependent data. ARIMA (ARMA)
modeling is one of the basic time series methods employed in practice. In this study, we
examined the foreign exchange rate between the GBP and the USD. Due the nature of
the data, the logarithms of the returns of the rates are used in the analysis instead of
the actual data. This is due to the favorable statistical properties of the logarithm of the
returns provides for analysis.
As noted previously, ARIMA (ARMA) modeling fails to effectively capture the process being
followed and subsequent forecasting chosen observations for validation. An application of
an alternative modeling technique (ARCH/GARCH) is used be to analyze the volatility in
the series and turn out to be of significant analysis due to the fact that it provides
bankers and traders information about the “value at risk” for a given portfolio.
APPENDIX
p,q 0 1 2 3 4
0 -35656.95 -35779.39 -35778.08 -35776.1 -35776.46
1 -35778.99 -35778.07 -35776.08 -35774.08 -35774.59
2 -35778.13 -35776.08 -35774.11 -35772.07 -35772.61
3 -35776.26 -35774.18 -35774.57 -35773.46 -35770.61
4 -35776.87 -35774.8 -35772.82 -35771.44 -35769.32
Table 1 AIC values for estimated ARIMA(p,0,q)
p,q 0 1 2 3 4
0 2.0333E-05 1.9776E-05 1.9773E-05 1.9773E-05 1.9762E-05
1 1.9778E-05 1.9773E-05 1.9773E-05 1.9773E-05 1.9762E-05
2 1.9773E-05 1.9773E-05 1.9773E-05 1.9773E-05 1.9762E-05
3 1.9772E-05 1.9772E-05 1.9762E-05 1.9758E-05 1.9762E-05
4 1.9761E-05 1.9761E-05 1.9761E-05 1.9758E-05 1.9759E-05
Table 2 Standard error values for estimated ARIMA(p,0,q)
p,q 0 1 2 3 4
0 FALSE TRUE TRUE TRUE TRUE
1 TRUE TRUE TRUE TRUE TRUE
2 TRUE TRUE TRUE TRUE TRUE
3 TRUE TRUE TRUE TRUE TRUE
4 TRUE TRUE TRUE TRUE TRUE
Table 3 Box-Test for estimated ARIMA(p,0,q) with p-value > 0.01 as True
Model AIC
GARCH(1,0) -8.029915
GARCH(2,0) -8.036744
GARCH(3,0) -8.038956
GARCH(1,1) -8.134582
GARCH(2,1) -8.134145
GARCH(1,2) -8.139603
GARCH(2,2) -8.139157
GARCH(3,2) -8.138769
ARMA(1,0)/GARCH(1,0) -8.04498
ARMA(1,0)/GARCH(2,0) -8.055509
ARMA(1,0)/GARCH(3,0) -8.053061
ARMA(1,0)/GARCH(1,1) -8.158708
ARMA(1,0)/GARCH(2,1) -8.158275
ARMA(1,0)/GARCH(1,2) -8.162937
ARMA(1,0)/GARCH(2,2) -8.16249
ARMA(1,0)/GARCH(3,2) -8.162107
Table 4 Estimation of GARCH model using AIC
Test Data Lags Statistic p-Value
Ljung-Box Test
R Q(10) 6.027875 0.8129159
R Q(15) 10.56042 0.7830879
R Q(20) 18.80854 0.5343004
R2 Q(10) 103.3104 0
R2 Q(15) 145.0879 0
R2 Q(20) 175.7589 0
Arch Test R TR2 91.06061 3.08E-14
Table 5 Standard residual tests for GARCH model
REFERENCES
• Tsay, Ruey S, (2001), “Analysis of Financial Time Series”, Third Edition
• Chatfield, Chris (2005), “The Analysis of Time Series, An Introduction”, Sixth Edition
• Brockwell, Peter., Davis, Richard.,”Introduction to Time Series and Forecasting”,
Second Edition
• Kedem, Benjamin., Fokianos, Konstantinos., (2002)“Regression Models for Time
Series Analysis”, Wiley and Sons Inc

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