forecasting usd_euro ex rates using arma model

7/27/2019 Forecasting Usd_euro Ex Rates Using ARMA Model

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An assignment

on

Forecasting of USD/Euro Exchange rate

using ARMA model

By,

Navin Poddar - 12BSPHH010595



Introduction

The exchange rate forecast is an important topic in international finance especially after the breakdown

of the Bretton Woods system in 1973. Firms that involve in international business need to know the

future exchange rate for various and accurate decision making in firms such as financing, investing and

hedging. An accurate exchange rate forecast is not only important to firms involved in internationalbusiness but also to households, governments and international organizations engage in international

transaction.

For decades, both academics and market participants diligently explore methods to forecast exchange

rates. Previous research extensively analyzes the predictive contact of two types of information: news

on macroeconomic fundamentals as used in structural exchange rate models, and information from

historical prices as used in technical trading rules.

Abstract

In this assignment we are forecasting the exchange rate of the USD/Euro using the technical rules which

specifies on the study being made with the use of the trend followed in the past; that is using the

historical exchange rates.

In theory, a currency's value should gravitate over time in the direction of its real long-run equilibrium

value. The intent of this assignment is to investigate the sustainability of basic exchange rate theory and

to construct econometric models capable to generate consistent and rational forecasts for the USD/Euro

exchange rate. Considering past values of USD/Euro exchange rate, we build first an ARMA model and

we study the volatility of this exchange rate time series.

The key motivation of this study is to examine the application of ARMA model for forecasting and

trading the USD/Euro exchange rates from Jan 01, 2010 to Dec 31, 2013 as in-sample and Jan 01, 2014

to Jan 21, 2014 as out of sample data set. Based on the observations acquired we would decide on

whether the models used for the prediction give a result that specifies the existence of a relation of the

future predictions to the past exchange rate data. And whether we could use the model for the

prediction of future exchange rates which can help us in benefitting from currency trading or could help

us in hedging the risks existing due to exchange rate fluctuations.



Backdrop

1. U.S. Dollar (USD)

Central Bank: Federal Reserve (Fed)

Current Interest Rate Information: http://www.federalreserve.gov/releases/h15/data.htm

The Almighty Dollar

Created in 1913 by the Federal Reserve Act, the Federal Reserve System (also called the Fed) is the

central banking body of the U.S. The system is itself headed by a chairman and board of governors, with

most of the focus being placed on the branch known as the Federal Open Market Committee (FOMC).

The FOMC supervises open market operations as well as monetary policy or interest rates.

The current committee is comprised of five of the 12 current Federal Reserve Bank presidents and seven

members of the Federal Reserve Board, with the Federal Reserve Bank of New York always serving on

the committee. Even though there are 12 voting members, non-members (including additional Fed Bank

presidents) are invited to share their views on the current economic situation when the committeemeets every six weeks.

Sometimes referred to as the greenback, the U.S. dollar (USD) is the home denomination of the world's

largest economy, the United States. As with any currency, the dollar is supported by economic

fundamentals, including gross domestic product, and manufacturing and employment reports. However,

the U.S. dollar is also widely influenced by the central bank and any announcements about interest rate

policy. The U.S. dollar is a benchmark that trades against other major currencies, especially the euro,

Japanese yen and British pound.

2. European Euro (EUR)

Central Bank: European Central Bank (ECB)

Current Interest Rate: http://www.ecb.int/stats/monetary/rates/html/index.en.html

The Dollar's Nemesis

Headquartered in Frankfurt, Germany, the European Central Bank is the central bank of the 17 member

countries of the Eurozone. In similar fashion to the United States' FOMC, the ECB has a main body

responsible for making monetary policy decisions, the Executive Council, which is composed of five

members and headed by a president. The remaining policy heads are chosen with consideration that

four of the remaining seats are reserved for the four largest economies in the system, which include

Germany, France, Italy and Spain. This is to ensure that the largest economies are always represented in

the case of a change in administration. The council meets approximately 10 times a year.

In addition to having jurisdiction over monetary policy, the ECB also holds the right to issue banknotes as

it sees fit. Similar to the Federal Reserve, policy makers can interject at times of bank or system failures.

The ECB differs from the Fed in an important area: instead of maximizing employment and maintaining

stability of long-term interest rates, the ECB works towards a prime principle of price stability, with



secondary commitments to general economic policies. As a result, policymakers will turn their focus to

consumer inflation in making key interest rate decisions.

Although the monetary body is somewhat complex, the currency is not. Against the U.S. dollar, the euro

(EUR) tends to be a slower currency compared to its colleagues (i.e., the British pound or Australian

dollar). On an average day, the base currency can trade between 30-40 pips, with more volatile swings

averaging slightly more, at 60 pips wide per day. Another trading consideration is time. Because the FX

market is open 24/7, forex traders must strategically set FX trading schedules. Trading in the euro-based

pairs can be seen during the London and U.S. sessions (which occur from 2am through 11am EST).

Trading - EUR-USD Pair

Due to the fact that the euro and U.S. dollar are the world's two largest currencies, representing the

world’s two largest economic and trading blocs, many multinational corporations conduct business in

both the United States and Europe. These corporations have an almost constant need to hedge their

exchange rate risk. Some firms, such as international financial institutions, have offices in both the

United States and Europe. Firms that fit this description are also constantly involved in trading the euro

and the U.S. dollar.

Because the euro/U.S. dollar is such a popular currency pair, arbitrage opportunities are next to

impossible to find. However, forex traders still love the pair. As the world's most liquid currency pair, the

euro/U.S. dollar offers very low bid-ask spreads and constant liquidity for traders wanting to buy or sell.

These two features are very important to speculators and have helped contribute to the pair's

popularity. In addition, the large number of market participants and the non-stop availability of

economic and financial data allow traders to constantly formulate and re-examine their positions and

opinions on the pair. This constant activity provides for relatively high levels of volatility, which can lead

to opportunities for profit.

The combination of liquidity and volatility makes the euro/U.S. dollar pair an excellent place to begin

trading for forex newcomers. However, keep in mind that it is always necessary to understand the role

of risk management when trading currencies or any other kind of instruments.

EUR/USD Facts

The United States and the European Union are the two largest economic powers in the world. The U.S.

dollar is both the world's most heavily traded and most widely held currency. The currency of the

European Union, the euro, is the world's second most popular currency. And since it contains the two

most popular currencies in the world, the EUR/USD pair is forex's most actively traded currency pair.



The Unique Role of the U.S. Dollar

The U.S. dollar plays a unique and important role in the world of international finance. As the world's

generally accepted reserve currency, the U.S. dollar is used to settle most international transactions.

When global central banks hold foreign currency reserves, a large fraction of those reserves are often

held in U.S. dollars. Also, many smaller countries choose either to peg their currency's value to that of

the U.S. dollar or just completely forgo having their own currency, choosing to use the U.S. dollar

instead. Additionally, the price of gold (and many other commodities) are generally set in U.S. dollars.

Not only this, but the Organization of Petroleum Exporting Countries (OPEC) transacts in, you guessed it,

U.S. dollars. This means that when a country buys or sells oil, it buys or sells the U.S. dollar at the same

time. All of these factors contribute to the dollar's status as the world's most important currency.

Since the dollar is the most heavily traded currency in the world, most foreign currencies trade against

the U.S. dollar more often than in a pair with any other currency. For this reason, it is important for

traders starting out in the currency markets to have a firm grasp of the fundamentals that drive United

States economy to gain a solid understanding of the direction in which the U.S. dollar is going.

The European Union Economy

Overall, the European Union represents the world's largest economic region with a GDP of more than

$13 trillion. Much like the United States, the economy of Europe is heavily focused on services,

manufacturing, however, represents a greater percentage of GDP in Europe than it does in the United

States. When economic activity in the European Union is strong, the euro generally strengthens; when

economic activity slows, as expected, the euro should weaken.

Why the Euro Is Unique

While the U.S. dollar is the currency of a single country, the euro is the single currency of 16 European

countries within the European Union, collectively known as the "Eurozone" or the European and

Economic Monetary Union (EMU). Disagreements arise from time to time among Europeangovernments about the future course of the European Union or monetary policy and when these

political or economic disagreements arise, the euro can be expected to weaken.

Factors Influencing the Direction of the EUR/USD

The primary issue that influences the direction of the euro/U.S. dollar pair is the relative strength of the

two economies. Holding all else equal, a faster-growing U.S. economy strengthens the dollar against the

euro, and a faster-growing European Union economy strengthens the euro against the dollar. As

previously discussed, one key sign of the relative strength of the two economies is the level of interest

rates. When U.S. interest rates are higher than those of key European economies, the dollar generally

strengthens. When Eurozone interest rates are higher, the dollar usually weakens. However, as we'vealready learned, interest rates alone cannot predict movements in currencies.

Another major factor that has a strong influence on the euro/U.S. dollar relationship is any political

instability among the members of the European Union. The euro, introduced in 1999, is also relatively

new compared to the world's other major currencies. Many economists view the Eurozone as a test

subject in economic and monetary policy. As the countries within the Eurozone learn to work with one



another, differences sometimes arise. If these differences appear serious or potentially threatening to

the future stability of the Eurozone, the dollar will almost certainly strengthen against the euro.

The list below shows the current members of the Eurozone as of January 1, 2009. When trading the

euro/U.S. dollar pair, investors should carefully watch for troublesome economic and political news

originating in these countries. If several Eurozone countries have weakening economies, or if newspaper

headlines are discussing political difficulties among the countries in the region, the euro is likely to

weaken against the dollar.

Members of the Eurozone

Austria

Belgium

Cyprus

Finland

France

Germany

Greece

Ireland

Italy

Luxembourg

Malta

Netherlands

Portugal

Slovakia

Slovenia

Spain

Objectives of the Study

Forecasting and trading of the USD/ Euro exchange rate with an autoregressive model is the key

objective of this study. In order to attain the core objective of this study, the specific objectives are as

follows:

To estimate the autoregressive model and the autoregressive moving average model.

To forecast the USD/ Euro exchange rate with an autoregressive model, an autoregressive

moving average model, a naïve strategy and a moving average model.

To evaluate the forecasting accuracy and trading simulation of the selected models.

Feasibility of the Objectives

The objectives in this study are feasible, since the specified objectives are relevant to fulfil the key

motivation of this study. Moreover, the required data and the relevant software are available as well.



Collect Data

Arrange (old to new)

Plot Series

Apply transformation

Is

Variance

stable?

YES

Obtain ACF & PACF

Apply regular &

seasonal differencin

Model Selection

Appropriate Model

Forecast

Is Mean

Stationary?

YES

Methodology



.64

.68

.72

.76

.80

.84

10 11 12 13

CLOSE

Volatile Trend of the USD/Euro Exchange Rate

The graph below shows the total dataset for the USD/ Euro and its volatile trend 1 st Jan 2010 to 21st Jan

2014:

From the above figure we can see that the data over the period is not stationary and as such it is very

difficult for us to proceed with the analysis of the proper model fit for the model.

Thus, we now will try stationarizing the data using the log of the exchange rates available to us.

Thus by using the software that we are using for the analysis for the forecast (i.e. Eviews) we will using

the following command:

genr lclose= log(close)

Note: lclose will represent the log values of the exchange rates while close represents the actual

exchange rates that we have acquired from the source.

The results are:



From the above figure we still can see that the data is still not stationary and as such we will have to go astep further with the analysis of stationarizing the data using the differential log.

For which we are using the command

genr dlclose=lclose- lclose(-1)

Note: dlclose represents the differential of the actual exchange rate while lclose represents the log of

the actual data. Also we are using -1 to signify that we are differencing the exchange rates with the one

day prior values.

The results for the same is as under

-.40

-.36

-.32

-.28

-.24

-.20

-.16

10 11 12 13

LCLOSE

-.03

-.02

-.01

.00

.01

.02

.03

10 11 12 13

DLCLOSE



So now we can see that the data is stationary and thus we can go ahead with the further analysis.

As per the histogram we can see that the probability is also below 0.05 and also that the stationarity has

a high Jarque Bera value thus we can say that the data is stationary.

Now we will go ahead with the test for Correlogram to obtain the ACF (Auto correlation) and PACF

(Passive Autocorrelation) factors. For which the results at level difference were not clear and such we

had to go ahead with the 1st difference Correlogram to obtain a clearer understanding. The results are as

under:

Date: 02/05/14 Time: 02:19

Sample: 1/01/2010 1/21/2014

Included observations: 1226

Autocorrelation Partial Correlation AC PAC Q-Stat Prob

****| | ****| | 1 -0.505 -0.505 313.01 0.000

| | **| | 2 0.023 -0.311 313.64 0.000

| | **| | 3 -0.030 -0.256 314.76 0.000

| | *| | 4 0.018 -0.195 315.16 0.000

| | **| | 5 -0.046 -0.229 317.82 0.000| | *| | 6 0.047 -0.179 320.56 0.000

| | *| | 7 0.009 -0.123 320.67 0.000

| | *| | 8 0.002 -0.081 320.67 0.000

| | *| | 9 -0.025 -0.089 321.47 0.000

| | *| | 10 0.012 -0.073 321.66 0.000

| | *| | 11 -0.037 -0.121 323.35 0.000

| | *| | 12 0.053 -0.069 326.81 0.000

| | | | 13 -0.021 -0.064 327.37 0.000

0

50

100

150

200

250

300

-0.02 -0.01 0.00 0.01 0.02 0.03

Series: DLCLOSE

Sample 1/01/2010 1/21/2014

Observations 1227

Mean 4.84e-05

Median 0.000000

Maximum 0.028723

Minimum -0.026282

Std. Dev. 0.005958

Skewness 0.140325

Kurtosis 4.814679

Jarque-Bera 172.3846

Probability 0.000000



| | *| | 14 -0.004 -0.076 327.38 0.000

| | *| | 15 -0.009 -0.095 327.49 0.000

| | | | 16 0.034 -0.051 328.90 0.000

| | | | 17 -0.021 -0.042 329.47 0.000

| | | | 18 0.011 -0.021 329.63 0.000

| | *| | 19 -0.045 -0.090 332.19 0.000

| | | | 20 0.065 -0.030 337.43 0.000

| | | | 21 -0.039 -0.037 339.34 0.000

| | | | 22 0.024 -0.013 340.08 0.000

| | | | 23 -0.023 -0.026 340.75 0.000

| | *| | 24 -0.019 -0.087 341.18 0.000

| | *| | 25 0.025 -0.077 341.95 0.000

| | | | 26 0.042 0.006 344.11 0.000

| | | | 27 -0.053 -0.020 347.65 0.000

| | | | 28 0.024 -0.011 348.39 0.000

| | | | 29 -0.014 -0.015 348.65 0.000

| | | | 30 0.011 0.005 348.80 0.000

| | | | 31 -0.042 -0.046 351.04 0.000| | | | 32 0.070 0.010 357.28 0.000

*| | | | 33 -0.067 -0.059 363.02 0.000

| | | | 34 0.052 -0.026 366.41 0.000

| | | | 35 -0.031 -0.036 367.62 0.000

| | | | 36 0.011 -0.039 367.78 0.000

As per the results we can take ACF as 1 and PACF can be taken upto 3 and even more. But seeing the

maximum stars I have considered only 3.

Thus from the above results we can assume that the best fit model for the data would lie among one of

the following models:

AR(1)

AR(1) MA(1)

AR(1) MA(1) MA(2)

AR(1) MA(1) MA(2) MA(3)

Therefore we will go ahead with the ARMA test to see which model fits the data the best.



From the above results we can see that ARMA(1,1), ARMA(1,2), ARMA(1,3) have the highlights of red

which means that the probability is not significant and is exceeding 5% level of significance and as such

we do not consider these models.

And the remaining models are significant as per the result of probability but if we see the rsquared value

it is the maximum under the AR(1) model and thus we will pic the model AR(1) as the best fit.

Now we will go ahead with the forecasting of the exchange rates for the period 1st Jan 2014 to 21st Jan

2014.

This is the graphical representation of the forecast which shows that there were 18 observations that

were to be forecasted which has a standard error of -2 to +2. Thus from this we can conclude that the

predicted values have not deviated much.

AR(1) MA(1) MA(2) MA(3) ARMA(1,1) ARMA(1,2) ARMA(1,3)

C 0.7548 0.7515 0.7515 0.7516 0.7548 0.7548 0.7551

AR(1) 0.9895 - - - 0.9896 0.9895 0.9902

MA(1) - 0.8915 1.3624 1.5301 -0.0061 -0.0059 -0.0072

MA(2) - - 0.7750 1.3291 - 0.0018 -0.0004

MA(3) - - - 0.5885 - - -0.0329

C 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

AR(1) 0.0000 - - - 0.0000 0.0000 0.0000

MA(1) - 0.0000 0.0000 0.0000 0.8329 0.8371 0.8029

MA(2) - - 0.0000 0.0000 - 0.9498 0.9892

MA(3) - - - 0.0000 - - 0.2551

0.9813 0.6931 0.8677 0.9187 0.9813 0.9813 0.9813

- 7. 9845 - 5. 1854 - 6. 0247 - 6. 5106 - 7. 9829 - 7. 9812 - 7. 9808

- 7. 9761 - 5. 1770 - 6. 0122 - 6. 4940 - 7. 9704 - 7. 9646 - 7. 9600

-15.9606 -10.3624 -12.0368 -13.0046 -15.9532 -15.9458 -15.9408

Akaike info criterion

Schwarz criterion

Sum (AIC+SBC)

Coeff

Prob

Rsquared

.69

.70

.71

.72

.73

.74

.75

.76

.77

1 2 3 5 6 7 8 9 10 12 13 14 15 16 17 19 20 21

2014m1

CLOSEF ± 2 S.E.

Forecast: CLOSEF

Actual: CLOSE

Forecast sample: 1/01/2014 1/21/2014

Included observations: 18

Root Mean Squared Error 0.004971

Mean Absolute Error 0.004376

Mean Abs. Percent Error 0.594668

Theil Inequality Coefficient 0.003394

Bias Proportion 0.721109

Variance Proportion 0.130665

Covariance Proportion 0.148226



Conclusion

The comparison of the actual market values to that of the predicted or forecasted values are shown

below:

Date Close Forecast Difference % diff

1/1/2014 0.7265 0.7279 -0.0014 0%

1/2/2014 0.7318 0.7282 0.0036 0%

1/3/2014 0.7359 0.7285 0.0074 1%

1/5/2014 0.7355 0.7287 0.0068 1%

1/6/2014 0.7336 0.7290 0.0046 1%

1/7/2014 0.7344 0.7293 0.0051 1%

1/8/2014 0.7369 0.7295 0.0074 1%

1/9/2014 0.7350 0.7298 0.0052 1%

1/10/2014 0.7315 0.7301 0.0014 0%

1/12/2014 0.7315 0.7303 0.0012 0%

1/13/2014 0.7315 0.7306 0.0009 0%

1/14/2014 0.7316 0.7308 0.0008 0%

1/15/2014 0.7354 0.7311 0.0043 1%

1/16/2014 0.7344 0.7314 0.0031 0%

1/17/2014 0.7386 0.7316 0.0070 1%

1/19/2014 0.7392 0.7318 0.0074 1%

1/20/2014 0.7382 0.7321 0.0061 1%

1/21/2014 0.7375 0.7323 0.0052 1%

Thus we can see that the values forecasted using the ARMA model is very close to that of the actuals inthe market with a variation of 0% to 1%. But if we see closely, the analysis on a weekly basis would be

much evident giving us a % difference of 0%.

forecasting usd_euro ex rates using arma model

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