1. Analysis of Foreign Exchange Rates
with Functional Data Analysis
Statistics and Business project
Marina Doria G. Cortazar
Supervised by Pedro Galeano
June 2014
2.
3. Abstract
In this project volatility of 24 exchange rates against dollar is studied during the last 3 years.
The theory behind the statistical methods is explained for each section, graphs and tables
are provided for better understanding. First, the volatility is estimated through a model
used in financial time series called the GARCH model. Then, I make use of Functional Data
Analysis to convert the raw data, coming from the estimation, into functions of volatility.
After building 24 smooth curves, Principal Component Analysis is applied in order to
find main variables which describe volatility of exchange rates. The first two component
scores are plotted, and 4 different groups of currencies are defined. Finally, K-means and
Hierarchical clustering are used to show groups of currencies.
9. List of Tables
1 Classification of currencies by their exchange rate regime. . . . . . . . . . . 3
2 Exchange rates against dollar. . . . . . . . . . . . . . . . . . . . . . . . . . 4
10.
11. 1 Introduction
This project aims to discover different groups of currencies depending on their exchange
rate volatility. The exchange rate is used to convert one currency into another. Moreover,
it is also regarded as the value of one country’s currency in terms of another country’s
currency. There are a wide variety of factors which influence the exchange rate, such as
interest rates, inflation, and the state of politics and the economy in each country, among
others. When one currency has a lower value than others, it means that products in this
country are cheaper for foreigners with stronger currency. This will cause the increasing
in exports and the country will gain competitiveness against others with higher currency
value. Currencies are another kind of financial assets, people can buy and sell them at
market’s prices (exchange-rates). High volatility in any financial asset is related with high
risk and low volatility with stability and free risk. Usually, buyers are confident about a
currency because the politic and economic environment of the country using this currency
is reliable. By contrast, countries with weak politic institutions may have high volatile
currencies as the risk of buying this currency is high (debt ratios change fast and prices for
foreign investors are unstable). Rogoff and Reinhart in This time is different (2010) claim
that high volatility in the exchange rate can cause a financial crisis in a country. Also,
further research have been done and some economists think that changes in exchange rates
can measure politic risk, see, for instance, Bloomberg and Hess (1997).
Before starting, it is important to know that there are two main groups (to put it
simple)1
of exchange rate regimes deliberated by Central Banks: Fixed and Float. A fixed
exchange rate means that the Central Bank of one currency decide to follow the value of
another because it is suitable for the country’s economy and therefore the exchange rate is
controlled. A float exchange rate is the one who is free of intervention. However, evidence
1
I simplify the classification done by the International Monetary Fund. Mainly, I define Float ER to
”Free Floating” and ”Floating” regimes. In the group of Fixed ER I include the rest of regimes. (See
reference [9])
1
12. suggest that most of the flexible exchange rates are highly “managed float”. For instance,
although euro/dollar is a free exchange rate, the European Central Bank and the Federal
Reserve try to maintain the currency constant to avoid speculate attacks. Through this
project I want to analyze the volatility of 24 floating and fixed exchange rates to discover
how these two groups differ in terms of volatility. Is it true that Fixed and Float exchange
rates are completely different in terms of volatility? Which are the most similar currencies?
Are there any outlier with very low or very high volatility?
For answering these questions I will make use of the tools provided by conditional
heteroscedastic volatility models and the functional data analysis (FDA). In particular,
we use Generalized AutoRegressive Conditional Heteroskedasticity (GARCH) type models,
proposed by Bollerslev (1986), to obtain estimates of the volatility of the 24 exchange rates,
that are then fitted using a basis of b-splines. Once this is done, I obtain functional principal
components of the functional volatilities that are used to obtain clusters of exchange rate
regimes.
The rest of this project is as follows. In Section 2, I present the dataset of exchange rates.
In Section 3, I make use of GARCH models to estimate the volatilities of each exchange
rate. In Section 4, I use b-splines to fit the estimates volatilities obtained in Section 3. In
Section 5, I obtain the functional principal components of the fitted estimated volatilities
and I use them to obtain clusters of exchange rates. Finally, Section 6 concludes.
2
13. 2 Data
I chose 24 exchange-rates against dollar. The data were downloaded from the Bloomberg
Terminal at San Diego State University. The exchange rates are the corresponding to the
following currencies, see that are classified by their exchange rate regime (Table 1). As it
can be seen, the exchange rates include those of countries or regions with a high economic
level such as Great Britain, Switzerland, Canada, Japan and the Euro zone, those of
emergent countries such as Brazil or Turkey, and those of low economic level countries such
as Vietnam or Malaysia. There are also countries from all around the world, except Africa.
In particular, I collected three years of exchange rates, running from 27th January 2011
to 27th January 2014, a total number of 783 days. At the end, there is a matrix with 783
rows (days) and 24 columns (currencies). Table 2 shows a few of these exchange rates.
For instance, 1 euro was 1.0423 $ and 1 Mexican Peso was 12.03 $ on 27th Jan 2011. The
rest of currencies have the same data structure. However it can be seen that the exchange
rates can be quite different one from the other. Consequently, I am going to work with the
logarithm of the exchange rates because this transformation makes exchange rates more
stable and closer to each other. This transformation has been suggested in several empirical
studies, see, for instance, Wu and Chen (2003).
Regime Currency
Fixed Swiss Franc, Chinese Yen Malaysian Ringgit, Russian
Ruble, Indonesian Rupiah, Vietnamese Dong, Ukrainian
Hryvnia, Dominican Peso and Argentine Peso.
Float British Pound, Australian Dollar, Canadian Dollar,
Swedish Krona, Norwegian Krone, Mexican Peso,
Japanese Yen, Chilean Peso, Peruvian Sol, Brazilian
Real, Colombian Peso, Philippine Peso, Turkish Lira,
Hungarian Forint.
Table 1: Classification of currencies by their exchange rate regime.
3
14. Date Euro Mexican Peso Japanese Yen Chilean Peso
27 Jan 11 1.3734 12.0371 82.92 485.3
28 Jan 11 1.3611 12.2062 82.12 484.35
29 Jan 11 1.3694 12.1219 82.04 483.27
··· ··· ··· ··· ···
25 Jan 14 1.3696 13.4018 103.26 549.19
26 Jan 14 1.3678 13.46 102.31 550.6
27 Jan 14 1.3673 13.366 102.55 549.71
Table 2: Exchange rates against dollar.
Figure 1 shows a time plot of the of the Euro ER against dollar. In the plot, we can see
that the value of one euro in dollars grow during the first months of 2011. The peak was
reached in February when a euro was 1.483 $. Then, the value decreased constantly until
getting the minimum of 1.206 $ in June 2012.
Figure 2 shows the exchange rates logarithms, that allows to see the characteristics
of all currencies in the same graph, that was impossible with the original exchange rates.
Apparently, their fluctuations seem very similar, however, if we compare the Chilean ER
logarithm with the Brazilian ER they look completely different (Figure 3). This figures
motivates the study of volatility in ER that I conduct in Section 3. Look that now, the
difference between this two currencies in terms of fluctuations is more noticeable than in
the Figure 2.
4
15. Figure 1: Euro ER against dollar.
Figure 2: Logarithm of 24 exchange rates.
5
16. Figure 3: Chilean and Brazilian ER logarithms.
3 Volatility estimation
We want to know how exchange rates vary over time. For that, I first obtain the simple
returns of the exchange rates and then analyze the volatility of these simple returns. How-
ever, the volatilities of returns are not directly observable and should be estimated from the
observed returns. Although different methods to estimate volatility have been proposed,
we will use the very well known Generalized AutoRegressive Conditional Heteroskedasticity
(GARCH) models proposed by Bollerslev (1986). This is by far the most popular model to
estimate volatilities. Other possibilities are in order including models that consider leverage
effects such as the GJR-GARCH or the EGARCH models or stochastic volatility models,
but I focus on this model for simplicity. For a general review on the topic, see Tsay (2010),
for instance.
Let rt be the return of a given exchange rate logarithm. Then, the GARCH(p,q) model
6
17. assumes that rt is generated as follows:
rt = σtet
σ2
t = α0 +
q
i=1
αir2
t−i +
p
i=1
βiσ2
t−i
where σ2
t is the volatility of rt at time t and et are independent and identically distributed
random variables with mean 0 and variance 1. The et are usually called the shocks or
standardized return innovations.
As the idea is to fit 24 exchange rate logarithm returns, for simplicity I fitted the
usual GARCH(1,1) model assuming Gaussian innovations to all of them. For that, I have
used the garchFit function of the library fGarch. The usual Ljung-Box statistics on the
squared residuals showed that for most of the cases, the fits were appropriate and no model
discrepancy are found. The Ljung-Box statistics pointed out the presence of some model
discrepancy in only a few of the fits. However, the discrepancies were small enough and
probably due to the presence of some outliers. Therefore, I maintain the fitted GARCH(1,1)
models for the 24 exchange rates logarithm returns.
Figure 4 shows the estimated volatility ˆσ of Euro ER during 3 years. Logically, we see
that the maximum volatility was in the middle of 2011 when the euro was depreciating
quickly (see, Figure 1). This high volatility can reflect the uncertainty of the euro area
during this period. In January 2014, the level of volatility was the lowest since 2011.
Figure 5 shows the GARCH volatilities estimated in this section for all the ER during
the last 3 years. We observe that the lines are not smooth in some cases. Indeed, from
one day to another the line is straight and this produced rough peaks. However, volatility
should be smooth as the process is functional. This is one of the reasons why we make use
of FDA in the next sections.
7
18. Figure 4: Estimated volatility ˆσ of Euro ER.
Figure 5: Estimation of GARCH volatilities.
8
19. 4 Functional Data Analysis
4.1 What is Functional Data Analysis?
Functional Data Analysis is a collection of statistical techniques to estimate functions based
on a set of discrete observations taken over a period of time. The curve will give a smooth
approximation of how the data behave between observations.
One might ask why to estimate a function while the observed data is already smooth.
Well, for instance, it is important to have functions instead of raw data because it is easier
to compute derivatives. Information about the slopes and curvatures of curves may reveal
important aspects of the processes generating the data. This information is reflected in
their derivatives.
A clear example is series of daily temperatures in four different cities. The conversion of
these temperatures into curves invites an exploration of the ways in which the curves vary.
Furthermore, the rapidity temperature vary over time and the intensity these curves differ
from city to city. Another area where FDA can play an important role is in Economics. For
instance, lets think in a Non-durable good index, where you have monthly data recorded
over a long period of time, as Ramsay and Silverman (2002) propose in their book, “Applied
Functional Data Analysis. Methods and Case Studies”. One might be more interested in
the index rate of change between different time seasons and scales rather than its actual
size. Perhaps the evolution of seasonal variation can tell us something interesting about how
the economy evolves in normal times, and how it reacts to times of crisis and structural
change. If it is change that matters, it follows that we need to study whatever alters
velocity or the first derivative of the curve and the second derivative of the curve which
is the acceleration. Functional Data Analysis provides this information by transforming
raw data into functions. Additionally, many important techniques from the multivariate
analysis, such as principal components, classification and cluster analysis has been extended
9
20. to the functional framework, see, for instance, Ramsay and Silverman (2002) and Ferraty
and Vieu (2006).
In our example, we will be working with volatility of foreign exchange rates. The Foreign
Exchange Market (FX) is unique because its huge volume trade represents the largest asset
class in the world. All these financial transactions are public and recorded, therefore we
end with multiple data in a period of time. This study focuses in analyzing the volatility
of exchange rates. We assume that volatility is functional because from one observation
to the next the process is smooth. The goal is to transform discrete volatility records into
functions of volatility to have a deeply understanding of how this curves change along 3
years.
The goals of functional data analysis are essentially the same as those of any other
branch of statistics: represent the data, study patterns and variations among the data and
find groups with similar characteristics. In particular we are interested in:
• Represent exchange rates in ways that aid further analysis.
• Display data in order to find characteristics of each currency.
• Study sources of pattern and variation among exchange rates.
• Find groups of currencies with similar performance.
• Argument our results with economic reasoning.
4.2 Functional volatility
Now, for each one of the exchange rate logarithm returns, I have 783 estimated volatilities2
.
Therefore, for each return series, I have a series of estimated volatilities σ2
(1), σ2
(2), ..., σ2
(783).
2
A total of 784 exchange rate daily values are reduced to 783 volatility values because we need to
measure differences between days.
10
21. As mentioned before, the estimated volatilities are somehow rough. Therefore, my next
goal is to smooth the estimated volatilities, as basically, the volatilities can be seen as a
continuous smooth process in which jumps are rare. Therefore, for each return series, I am
looking for a function x(t) such that:
σ2
(tj) = x(tj) + (tj)
where tj = 1, . . . , 783 and (t) is a noise function evaluated at the observed points. In
particular, I assume that the ’s are independent and identically distributed with zero
mean and constant variance. Consequently, we want to obtain 24 sets of 783 pairs of the
form (tj, xi(tj)), where i = 1, . . . , 24 and j = 1, . . . , 783, where the xi are smooth functions
that approximate the estimated volatilities from exchange rate logarithm return i.
For that, I make use of the tools of the Functional Data Analysis (FDA). Note that I
consider this approach because posteriorly I want to perform cluster analysis on the smooth
functions to make groups of similar currencies in terms of volatilities. This goal can be
accomplished using the tools of the FDA as I will present in Sections 5.
The next section will explain the process of transforming the 24 discrete exchange rate
volatilities into smooth functions using the following two steps:
1. Representing functions by basic functions.
2. Fitting a curve to the data through a vector, matrix, or array of coefficients which
defines the function as a linear combination of these functions.
4.3 Basis functions
A basis function system is a set of known functions φk that are mathematically independent
of each other and that have the property that we can approximate any function by taking
11
22. a weighted sum or linear combination of a sufficiently large number K of these functions.
x(t) =
K
k=1
ckφk(t)
The parameters c1, c2, .., cK are the coefficients of the expansion and φ is the functional
vector whose elements are the basis functions φk. An exact representation is achieved
when K = n, (n = 783 days) in the sense that we can choose the coefficients ck to yield
x(tj) = yj for each j. For more details, see chapter 3 in Functional Data Analysis (2005)
by Ramsay and Silverman.
There are different ways to estimate these basic functions depending in the characteris-
tics of the data. For periodic data, Fourier series are used. However, as we see in the graph
our data does not follow a clear periodic pattern therefore we will be using B-spline basis.
They are defined by the range of validity, the knots, and the order. Splines are constructed
by dividing the interval of observation into subintervals, with boundaries at points called
break points or simply breaks. In our case because we want to make daily subintervals, we
need to set a range of validity between 0 and 783 days.
We use the term degree to refer the highest power in the polynomial. The order of
a polynomial is one higher than its degree. For example, a straight line is defined by a
polynomial of degree one since its highest power is one, it is of order two because it also
has a constant term. In our situation we can see that the data has troughs and peaks
constantly. We will need to set a higher order to describe our data, for example an order
of 6 will be sufficient.
The last parameter we need to fit is the number of basis k we want to use for building
the linear combination. The more number of basis one selects the more error you will
accumulate. On the other hand, the less basis you apply in the linear combination the less
accurate your smooth function will be. In this example, 100 k basic functions are used.
12
23. Through these 100 basic functions we will be able to plot smooth exchange rate volatilities.
Figure 6: 100 basic functions with an order of 6 during 783 days.
4.4 Fitting the data
Remember that our goal is to fit the discrete observations using the model σ2
= x(tj)+ (tj).
We saw that for the term x(tj) we apply a basis function expansion x(t) =
K
k=1
ckφk(t).
However we still need to obtain the coefficients of ck to complete our linear combination.
The simplest method Ramsay and Silverman give in their book (Functional Data Analysis,
2005) is by minimizing the least square criterion:
SMSSE(σ2
|c) =
n
j=1
[σ2
j −
K
k=1
ckφk(tj)]2
In matrix form,
SMSSE(σ2
|c) = (σ2
− Φc) (σ2
− Φc) = ||σ2
− Φc||2
13
24. Figure 7: Euro ER functional volatility
The criterion is minimized by the solution ˆc = Φ(Φ Φ)−1
Φ σ2
and the vector ˆy of fitted
values is:
ˆy = Φˆˆc = Φ(Φ Φ)−1
Φ σ2
If we use this method to find the coefficients we need to assume that the residuals j are
independent and equally distributed with mean 0 and constant variance.
This process has been done in Figure 7 for the Euro and in Figure 8 for all currencies.
A unique function ˆy is fitted for the Euro exchange-rate GARCH volatility, see the red line.
While in the next, 24 smooth curves are represented through the same process. Now the
lines are easy and points are linked smoothly, not as in Figures 4 and 5.
14
25. Figure 8: Exchange rate volatilities represented through 24 smooth curves.
5 Functional Principal Component and Cluster Anal-
yses
We have arrived to one of the most interesting parts of the study. In this section we apply
a powerful statistics tool to discover the relationship between our 24 currencies based on
their volatility.
An illuminating method of detecting connections between currencies is the analysis of
functional principle components (FPCA). Chapter 8 of Ramsay and Silverman (2005) pro-
vide a way of looking at covariance structure that can be much more informative and easier
to understand than the direct examination of covariance-variance matrices in multivariate
analysis or, in the case of functional analysis, covariance operators.
It will be helpful to approach Functional PCA from the multivariate example, which is
more common and easy to understand. The objective of both analysis is to describe the
maximum variability of the data with the minimum number of uncorrelated variables or
components.
15
26. Multivariate principal components are defined as
fi = β xi =
j=1
βjxij,
for i = 1, 2..., N where β is the vector (β1, ...., βp) and xi is the vector (xi1, ...., xip) con-
taining the values of the variables.
In our case we are working with functional data, where β(s) and x(s) are functions and
summations over j are replaced by integrations over s to define the inner product:
fi = βx = β(s)x(s)ds
In particular, we are interested in maximizing the variance in the fis. The objective
is finding the weight vector ξ1 = (ξ11, ..., ξp1) located in the linear combinations fi1 =
ξ1(s)x(s)ds. At the same time we maximize the square of N−1
i f2
i = N−1
i (ξ1x1)2
subject to the constraint (ξ1(s))2
= 1.
A helpful method to understand Functional Principal Components is to examine plots of
the overall mean function and the functions obtained by adding and subtracting a suitable
multiple of the principal component function. Figure 9 shows such a plot for the volatility
data. In each case, the solid curve is the overall mean volatility, and the dotted and dashed
curves show the effects of adding and subtracting a multiple of each principal component
curve. The first component explains almost half of the total variability. It seems that
this component gives importance to volatilities above and below the mean trend. The
second component reflects the mean during the first 400 days. While the third component
shows the average volatility. These first 4 components explain more than 80% of the total
variability of the exchange-rate GARCH volatilities.
The next step is plotting the first principal component scores against the second ones
for each currency (Figure 10). By looking at this graph we can see that there is a clear
16
28. Figure 10: Principal Component scores.
outlier corresponding to the Indonesian Rupiah. On the other hand, the other currencies
are spread all over the first principal component. In order to find groups of common
volatilities, we first consider the k-means clustering, which is probably the most usual
approach to perform cluster analysis in multivariate datasets. Using the sums of squares
within groups, we conclude that 4 groups are appropriate. Therefore, we run the k-means
with k = 4 and obtain the results shown in Figure 11.
Another very useful way to obtain groups of observations is hierarchical clustering.
Therefore, we perform hierarchical clustering with the average linkage criterion. The results
are shown in Figure 12. As in the previous case, following the sums of squares within
groups, we analyze the results obtained with 4 groups. We expect that the closer the
countries are, the more similar volatility they have during the last 3 years. It can be said
that proximity between countries and use of same language can cause the increasing of
trade between these countries and consequently similar volatility of exchange-rates. In the
clusters, it can be found pairs of currencies working in countries which are close or share the
same language: Chinese Yuan-Vietnamese Dong, Canadian Dollar-British Pound, Russian
18
29. Figure 11: K-means clustering.
Ruble-Turkish Lira and Swedish Krona-Norwegian Krone. However, we don’t know if
volatilities of different exchange rates are related with the proximity or business relation
between the countries using these currencies. We found similar volatilities in pairs of
currencies working in countries, conventionally, not very similar: Peruvian Sol–Ukrainian
Hryvnia, Philippine Peso-Dominican Peso, Malaysian Ringgit-Colombian Peso and Mexican
Peso-Swiss franc. Finally, the clusters shows clearly two outliers, the Hungarian Forint and
the Indonesian Rupiah.
The functional volatility of different groups of currencies found before is plotted in
Figure 13. It can be seen that this graph is the same as Figure 8, but each group of
currencies discovered on the cluster are plotted with different color; 1=red, 2=green, 3=blue
and 4=black. In average, the black group recorded higher values of volatility than the
green one. Notice that all the strongest currencies, with floating exchange rate, are in the
black group; British Pound, Euro, Canadian Dollar, Australian Dollar, Swiss Franc and
Swedish Krona. The green color is for the group compounded by; Argentine Peso (Fixed),
Chinese Yuan (Fixed), Peruvian Sol (Float), Ukrainian Krone (Fixed) and Vietnamese
19
30. Figure 12: Cluster of exchange-rate volatilities during 3 years.
Dong (Fixed). Notice that there is not such a big difference between Floating exchange
rate (supposedly more free to suffer changes in value) and Fixed exchange rate (apparently
less volatile). Actually, we can see that the Dominican Peso (Fixed) and the Philippine
Peso (Float) had very similar volatility during the last three years. Finally, the blue and
the red line represent Hungarian Forint and Indonesian Rupiah exchange-rate volatilities
respectively. Both of them have different pattern from the other groups. We can see that
the Indonesian Rupiah recorded the highest peaks of volatility during the last year, while
the Hungarian Forint started in 2011 with high volatility and it decreased constantly.
20
31. Figure 13: Four groups of functional volatilities.
6 Conclusions
After doing this analysis I conclude with the following points:
1. Foreign exchange rates are crucial to comprehend the economy of countries, they
have an impact on their trade balance and they are affected by the trust placed
in the country’s political institutions. I explained at the beginning, the difference
between an intervened currency (Fixed ER) and a free currency (Floating ER). The
Foreign Exchange Market is the biggest financial market, where currencies are traded
at the exchange rate price. We can find this data on Bloomberg terminals (daily) or
on Internet (monthly).
2. Economical and financial events can be treated as functional processes where data
(prices, indexes...) move smoothly up and down during some period of time. Volatility
is another functional index which measures instability. Usually, the volatility index
is used inside Finance to measure risk investment.
3. Functional Data Analysis helps to analyze functional processes by transforming raw
21
32. data into smooth functions. By working with functions, we are able to estimate
any value at any point in time. In particular, I showed how to built 24 volatility
functions taken from discrete exchange rate values during 3 years. First, computing
the GARCH volatility model (raw data), and then transforming 24 × 784 values into
24 unique functions.
4. The most important section of the paper was discovering different groups of currencies
depending on their volatility along 3 years. Knowing that currencies are intervened
by central banks, I decided to study volatility of exchange rates because in practice,
there is not such a difference between Floating exchange rate (supposedly more free to
suffer changes in value) and Fixed exchange rate (apparently less volatile). Actually,
after doing the Functional Principal Component Analysis, I saw in the cluster that
the Dominican Peso (Fixed ER with the dollar) and the Philippine Peso (Floating
ER) had very similar volatility during the last three years. Also, it is important
to notice that through this analysis we can find outliers like the Indonesian Rupee
(Floating ER) and the Hungarian Forint (Floating ER).
5. Functional Data Analysis can be helpful to discover patterns in Finance and Eco-
nomics. Each day the financial market produces huge amount of data, sometimes full
of noise. It would be useful to find trends by working with smooth functions. As well
as, finding second derivatives to see acceleration in some indexes or prices. This is
an ambitious topic, because data in Finance and Economics is produced by people
actions and expectations, which I personally think are difficult to estimate.
References
[1] Reinhart, C. M. and Rogoff, K. S. (2010) This Time Is Different: Eight Centuries of
Financial Folly. Princeton University Press.
22
33. [2] Bloomberg, S. B. and Hess, G. (1997) Politics and Exchange Rate Forecasts. Journal
of International Economics, 43, 89-205.
[3] Bollerslev, T. (1986) Generalized Autoregressive Conditional Heteroskedasticity. Jour-
nal of Econometrics, 31, 307-327.
[4] Ramsay, J. O. and Silverman, B. W. (2002) Applied Functional Data Analysis. Methods
and Case Studies. Springer.
[5] Ramsay, J. O. and Silverman, B. W. (2005) Functional Data Analysis. Springer.
[6] Ramsay, J. O., Hooker, G. and Graves, S. (2009) Functional Data Analysis with R and
MATLAB. Springer.
[7] Tsay, R. S. (2010) Analysis of Financial Time Series. John Wiley and Sons.
[8] Wu, J.-L. and Chen, S.-L. (2003) Real Exchange-Rate Prediction over Short Horizons.
Review of International Economics, 9, 401-413.
[9] International Monetary Fund (2013) Annual Report on Exchange Arrangements and
Exchange Restrictions.
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