Euro-Dollar Exchange Rate: An

Econometric ApproachEconometrics

Afonso Brites, 12564

Sebastião Fernandes, 12479

Miguel Leal, 12609

Pedro Tomé, 12723

Lisbon, 2015

Introduction

The issue of forecasting the euro-dollar exchange rate is obvious. At the moment

it’s the most important currency pair in the FX market and its fluctuations are important

for both major economic transactions between the two biggest game players (Eurozone

and US) and for the peripheral economies that rely on this exchange rate for some parity

maintenance of their currency. The euro was introduced on January 1, 1999, but its use

by consumers in retail transactions started on January 1, 2002. That means that only

sixteen years have passed since the first date and thirteen since the second.

Consequently, both ECB and the economic agents have been involved in a learning

process about the mechanisms of transmission of the monetary policy and its effects on

the economy of its own countries as in the economy of member countries and the role of

the euro as an international currency.

This paper's study centres on the following question: "Is it possible to predict

next month's exchange rate with some accuracy?" In order to do so, we focused on

macroeconomic fundamentals that represent the best arguments in our possession to

figure out what would be the best-fit set of explanatory variables. Its investigation relies

on the relation between each variable according to the one in the other zone. To solve

this problem we based our explanatory variables in differentials: money supply

differentials, interest rate differentials, inflation differentials, 3m bond yields

differentials, and GDP differentials. Only this set of variables would allow us to relate

the movements of the economies across seas in a parallel way.

The main limitation the literature has faced was in the measure of the GDP as its

data is quarterly based, but we could verify that a mixture of frequencies was already

used in past papers.

Literature Review

Exchange rates forecast has been a theme of enormous interest regarding the

international finance area, considering that both professional and academic resources

are focused on this task. In order to fulfill all of the areas that we wanted to cover, it was

of crucial importance to analyze several papers. We went from the ones that broadcast

our theme in a more general way to those that tried to explain the relevance of the

inclusion of our first set of variables. Such analysis is subscript below in a very

segmented manner:

First of all let's focus our attentions in what variables to include in the model.

Regarding Maeso-Fernandez, Osbat and Schnatz's study that relies on the determinants

of the euro effective exchange rate the results indicated that the differentials between

real interest rates and productivity have significant influence on the euro effective

exchange rate as well as the terms of trade shocks due to the oil dependence of the euro

area. In some instances, the relative fiscal stance and terms of trade also have a

noteworthy effect. Interest rate differentials can easily be included in our regression,

although it would make more sense to include nominal rates as opposed to real interest

rates. As for the productivity, this paper uses differentials of an index, which divides

real GDP by the number of employees of a particular important trading partner. This

data is easily obtained for the EU and US and can also be included in our study as an

independent variable. An alternative method of measuring productivity would be

through GDP growth differentials. Synthetic data was compiled for the euro area by

aggregating the data from different EU countries using trade weights.

Concerning the short run prediction we looked for fundamentals-based models

for the euro-dollar rate mixing economic variables quoted at different frequencies. This

mixture of frequencies allowed assessing the influence of macroeconomic variables

quoted at monthly frequency and not available at weekly frequency over weekly

movements in the FX rate. Such robust choice was made under the contributions of

Mariano and Murasawa (2003) and Camacho and Perez-Quiros (2010) who use

maximum likelihood factor analysis of time series with mixed frequencies, handling

quarterly series as monthly series with missing observations. We also believe that this is

an important factor backing up our quarterly data regarding the GDP. The model

acquires great improvement and advantage against random walk models by the use of

the trend of change metric, much more appropriate than any other loss measures.

Following the study of Jansen and De Haan on the Effect of European Central

Bank and National Central Banks statements on the euro-dollar exchange rate it is

possible to conclude that ECB statements have a large influence in the conditional

volatility and in some cases, the conditional mean as well. Such study might be of

interest to our model if ECB statements were to be classified as dummy variables upon

the degree of importance. However, given that the analysis of ECB's statements for this

paper concerned a time frame half as big as ours it would be too time-consuming to

include such a hard working variable. In order to verify one of the most important inferences, out-of-sample

forecasting accuracy, Meese and Rogoff concluded that structural models fail to

improve on the random walk model despite the fact of the forecasts being based upon

realized values of future explanatory variables. This derives from the failure of out-of-

sample estimates by the time series models. The vector auto regression implies that

major-country exchange rates are well predicted by a random walk model (without

drift). However, the failure of structural models to outforecast the random walk model is

less certain to be attributed to sampling error.

Ehrmann, Fratzscher and Rigobon estimate the financial transmission between

money, bond and equity markets and exchange rates within and between the USA and

the euro area. US financial markets explain on average around 30% of movements in

euro area financial markets, whereas euro area markets account only for about 6% of

US asset price changes. The methodology used, allowed them to identify indirect

spillovers through other asset prices, which are found to increase substantially the

international transmission of stocks within asset classes. In this case, international cross-

market spillovers are significant. However, regarding our model, we believe that these

kind of shocks are well represented by inflation differentials and bond differential that

sustain the major information of financial markets in the most broad and stable manner.

Conceptual framework

To explain the variation in the exchange rate between the US Dollar and the

Euro, we applied economic fundamentals that are based on macroeconomic reasoning.

There are many different theories of exchange rate determination, but most perform

quite poorly at forecasting. For this reason, we included variables from each of these

approaches and added other variables we considered to be important too. (When we use

the term spot rate or exchange rate, we will always be referring to the exchange rate

between the Euro and the US Dollar - €/$)

Prices of goods are a crucial determinant of the exchange rate. If US prices are

lower than EU’s, people from the EU will start buying products outside because it's

cheaper. This increases the demand for US products driving their prices up, and, at the

same time, as their products are dollar denominated, we have to exchange euros for

dollars to buy them, which appreciates the dollar. Both of these effects contribute to

more expensive foreign products from the EU's point of view, though they would still

be cheaper than ours. This process occurs several times, gradually reducing the price

gap until the exchange rate (€/$) stabilizes at a higher level. Assuming that

transportation costs are very low, if an economic area has a higher level of inflation than

the other, the exchange rate will move in order to offset that disequilibrium, meaning

that the currency should depreciate relative to currencies with a lower level of inflation.

Interest rates have two different effects, but both that take place in different time

periods. In the short-term, if, for example, the EU has lower interest rates than the US,

there will be a capital outflow to the latter, as people can easily get higher returns there.

That outflow would mean that investors exchange euros for dollars to buy the desired

securities, pushing the dollar's value up (the current spot rate increases). On the other

hand, as investors want to bring the money back home to buy goods, they will trade the

dollars back to euros. If arbitrage is still possible, investors will continue to buy dollars

now and selling them in the future, sending the forward rate (the price at which foreign

exchange is quoted for delivery at a specific future date) downwards. It's easier to

understand with an example. Assume that both the current spot rate and forward rate

one year from now are 1€/$ and that the EU and US interest rates are respectively, 2%

and 6%. If we have 1 euro we can exchange it for dollars, invest it in the US, sell 1,06

dollars forward at a rate of 1€/$, wait one year, collect the 1,06 dollars (principal plus

interest payments) and trade it back to euros at the arranged forward rate. This method

would yield €1,06 instead of €1,02 we would get investing in the EU. As other investors

see the opportunity for riskless profit, they would use the same mechanism, selling more

and more dollars forward and decreasing the forward rate. This would gradually reduce

the margin for profit and subsequently eliminate it altogether. Since the forward rate is,

under the efficient market hypothesis, an unbiased estimator of the future spot rate, we

expect a spot rate decline. "If home and foreign bonds are perfect substitutes, and

international capital is fully mobile, the two bonds can only pay different interest rates if

agents expect there will be a compensating movement in the exchange rate"- Second

Annual IMF Research Conference Kenneth Rogoff. In conclusion, imagine for some

reason that ECB announces quantitative easing. As prices are temporarily fixed, and so

is the GDP, the only way to reach an equilibrium is for euro area interest rates to go

down. Consequently, there will be an instantaneous capital outflow that would cause an

excessive depreciation of the euro (appreciation of the dollar). But as EU interest rates

are lower than in the US, investors expect the currency to appreciate to compensate for

lower returns. A lower EU interest rate compared to its initial value implies a higher

quantity of transactions and, thereby a higher demand that drives prices up. That will

cause the real supply of money to go down, pushing interest rates up again. Higher

interest rates mean shrinking the interest rate differentials and lower pressure for

currency appreciation until there are no more arbitrage opportunities, which is when the

exchange rate reaches equilibrium. For this reason the initial fall in the currency value

overshoots its eventual equilibrium level.

The difference between the expected returns on stocks in the EU and the US too,

play a role in the exchange rate determination, because investing is not confined to

buying bonds. One can wonder whether the effects would be identical to that of interest

rates, but this isn’t entirely true. As we cannot observe returns before they “happen”, we

expect that the capital inflow in the short-run won’t be so abrupt. Therefore the short-

term effect is expected to be similar but weaker than what was previously described and

the long-term effect is expected to be same.

The growth of the EU versus US money supply is also one of the determinants

of the exchange rate. But why is it important? Let's focus again on a EU expansionary

monetary policy. If money supply goes up, "domestic" prices will go up as foreign

prices remain fixed, which, as it was earlier explained, causes a euro depreciation. This

implies that if the money supply is growing faster in the EU than in the US our currency

will depreciate over time (there should be no immediate effect because prices are

sticky).

The last factor that we assume to be relevant is the relative economic growth

rate. If a nation exhibits a high growth rate, it will attract investment that seeks to

acquire domestic assets. So if that same nation has a higher growth rate relative to

others it will attract more investment than the latter, increasing the demand for its

currency by more than the demand for foreign currency, leading to an appreciation of

the first.1

Data

Only post-2008 euro-dollar exchange rates are part of the sample that we used to

estimate the model. The sample has monthly frequency, starts on January 1, 2009 and

ends on December 31, 2014, encompassing a total of 72 observations. One may be wary

about the choice of such a small number of observations, but it wasn’t really a choice

because, only on October 8, 2008 the ECB announced that main refinancing operations,

would be carried out through a fixed-rate procedure, which didn’t happen before.

Therefore analyzing data before that point in time would be much more complicated as

rates were variable. At the same time, we also wanted to be able to compare our

predictions with the true values, and so we stop collecting data as far as December 31,

2014.

1 Consult figure 1 in the appendix as it represents a summary of all the effects

Regarding the Real GDP, as it’s only released quarterly, our data had different

frequencies. We tried to fix this problem by assuming a constant growth intra-quarter,

even though that might not be totally realistic.

The model that follows is the first we are going to estimate in the subsequent

section.

Logeurdolt = β0 + β1Logeuusmst-1 + β2Intdifft + β3Intdifft-1 + β4Infdifft-1 +

β5Indexdifft + β6Indexdifft-1 + β7Rgdpdifft-1 + ut

The explained variable is the natural log of the exchange rate, which is

expressed as a ratio of Euro per unit of US Dollar and the first explanatory variable is

the log of the proportion of Euro area money supply (M2) to US money supply (M2) –

with one period lag. We chose to use natural logs for both of these variables because it

leads to coefficients with more appealing interpretations – how a percentage change in

one variable will impact the other, also measured in the form of a percentage change.

For example, if the money supply in Europe outgrows United States’ by 1 percent that

would increase/decrease the exchange rate by β1 percent). As variables are strictly

positive taking logs won’t generate any problem2. Also, regarding the dependent

variable, taking the log can, to a certain extent, mitigate the problem of

Heteroskedasticity if there’s any.

The remaining explanatory variables are the interest rate, inflation rate, stock

index return and real GDP growth differentials between the EU and the US, respectively

(each of them lagged one period). Additionally, the variables Intdifft and Indexdifft were

included in the model because, as it was previously explained, they’re the only variables

that are expected to have immediate effects in the exchange rate.

We should also mention that all the variables, except the exchange rate, interest

rate differentials and stock index return differentials, are seasonally adjusted. They

shouldn’t be adjusted anyway because they don’t exhibit seasonality. Regarding this

adjustment, there’s a chance that it causes a distortion in the parameter estimates

because it’s unclear whether all the variables were adjusted by the same method or not,

as R. A. Meese and K. Rogoff denoted in Exchange rate models of the seventies.

In the next section, you might be unsure about whether we forgot about the

interest rate differential in our model. That’s not the case. We simply found out, through

the results of our analysis, that the yield differential of German and US 3-month bonds

2 There could have been a problem with the use of the logarithmic transformation if variables were too close to zero, as it would alter the magnitude of the variables excessively, which is not the case.

(we named that variable Bonddiff), would be the best proxy of the interest rates in both

economies, since the ECB and FED’s fund rates or base rates resulted in parameter

estimates that weren’t neither as consistent with our macroeconomic rationale nor as

statistically significant. But if we are comparing the interest rates in EU and US how

can Germany be representative of the entire Euro area? As Germany is one of the

largest bond markets in the world, it’s the closest thing to a “Euro area bond” that is.

Most of the data was collected from the ECB, FRED of St. Louis, OECD and

Investing.com databases.

Model

Logeurdolt = β0 + β1Logeuusmst-1 + β2Bonddifft + β3Bonddifft-1 + β4Infdifft-1 +

β5Indexdifft + β6Indexdifft-1 + β7Rgdpdifft-1 + β8t + ut

Before we estimated the model we noticed that both logeurdol and logeuusms

series had noticeable trend and as a result we might have run into a spurious relationship

between them, which would create a spurious regression problem. Adding a trend can,

to a certain extent, solve this problem.

We also created a dummy variable for the period after the famous Mario

Draghi’s speech where he pledged to do whatever it takes to save the Euro. The main

purpose was to find out if it caused an upward pressure in the Euro. Although

statistically we found that it was not significant, if we look at the exchange rate over

time we can see clearly that it had a great impact in the period immediately after the

speech and some months that followed3. We believe that the reason behind the non-

significance is that the trend inverted somewhere April 2014, probably because markets

started to think that Quantitative Easing was a possibility.

We proceeded with the removal of the variable Indexdifft and its lag because

they were not significant when introduced separately in the model. As they could be

jointly significant we also did an F-Test, which resulted in non-rejection of the null

hypothesis4. Regarding the economic significance, Indexdifft-1 had an unexpected sign,

which further supported our decision to remove the variable.

Since the main goal of our project is to predict future values we are going to use

an AR model.

3 See figure 24 See figure 3

In order to find a suitable number of lags we plotted a partial correlogram with

95% confidence bands and concluded that there should be no more than one.

Thus, here’s our “final” model:

Logeurdolt = β0 + β1Logeuusmst-1 + β2Bonddifft + β3Bonddifft-1 + β4Infdifft-1 +

β5Rgdpdifft-1 + β6Logeurdolt-1 + β7t + ut

Violation of Assumptions

After arriving at our model, it had to be tested for any possible violation of

assumptions and corrected accordingly. Firstly, the Classical Assumptions were tested.

The assumptions “linearity in parameters” and “no perfect collinearity” weren’t tested

because we already took these into account when making the model.

The important assumption to test was “strict exogeneity” because without it, we

cannot guarantee unbiasedness of the estimators. We ran a regression of the residuals

against the independent variables for different time periods5 and noticed that the

residuals were in fact correlated with the independent variables (in t-1, for example).

Because we now know that this assumption was violated, we had to move onto the

Asymptotic Assumptions.

The first test for asymptotic properties was stationarity. We ran the Dickey-

Fuller test for each of the variables6 and noticed that logeurdol, l1logeurdol and

l1logeuusms were not stationary. In order to correct this, we used first difference for

each non-stationary variable and repeated the Dickey-Fuller to confirm7. Conveniently,

the same Dickey-Fuller test also allowed us to conclude that we have weak dependence

and therefore the first assumption holds.

After guaranteeing stationarity and weak dependence, we moved onto

contemporaneous exogeneity. We ran the residuals of the new regression on the new

model8 (with first differences of the previously non-stationary variables) in period t

(only) and concluded that the assumption holds. Because we already know we have no

perfect collinearity, we can conclude we have consistency of the estimators.

In order to achieve valid inferences, we still had to check for serial correlation

and heteroskedasticity. For serial correlation, we couldn’t use Cochrane-Orcutt test

5 See figure 46 See figure 57 See figure 68 See figure 7

because we didn’t have strict exogeneity. Instead, we ran a regression of the residuals

(with robust standard errors) against the independent variables and lagged residuals9.

We found no evidence for serial correlation, as the p-values for the variables were too

high. This was probably corrected when we introduced the first differences into the

model.

Finally, we ran the Breusch-Pagan test and the White test10, finding no evidence

of heteroskedasticity, as well as an ARCH test11 to make sure no ARCH effect was

verified. Having TS.1’ – TS.5’ verified (asymptotic normality is implied), we now had

asymptotically efficient estimators and the usual inference procedures were

asymptotically valid.

Results

After fixing the problems of our model we still decided to drop two variables –

the first difference of the natural logarithm of the money supply ratio (lagged one

period) and the GDP growth rate differentials (also lagged one period) because, even

though the signs of the coefficients were as expected according to macroeconomic

theory they were neither statistically nor jointly significant).

This is our actual final model12:

13D1Logeurdolt = β0 + β1Bonddifft + β2Bonddifft-1 + β3Infdifft-1 + B4D1Logeurdolt-1 + ut

The interpretation of the coefficients is the following:

β1 – An increase in the current Bond yields differential by 1 percentage point leads to a

9,97 percentage points decrease on the growth rate of the spot rate, on average ceteris

paribus.

β2 – The increase in Bond yields differential a month ago by 1 percentage point leads to

an 8,46 percentage points increase on the growth rate of the spot rate this month, on

average ceteris paribus.

9 See figure 810 See figure 911 See figure 1012 Consult figure 11 in the appendix 13 When we write D1 before a variable’s name that represents its first difference

β3 – The increase in inflation differential a month ago by 1 percentage point leads to an

increase of 6,92 percentage points on the growth rate of the spot rate this month, on

average ceteris paribus.

Β4 – An increase of 1 percentage point on the growth rate of the spot rate from the

month before will decrease the current value of the growth rate of the spot rate in 0,34

percentage points, on average ceteris paribus.

We should mention that the dependent variable is the first difference of the log,

which is basically a growth rate measured in a percentage change. But when interpreting

a coefficient of an independent variable we are looking at how much this variable

changes the growth rate, and so we must interpret it as percentage points.

All the effects are as expected if we apply the reasoning that was described in

the previous sections. The reason behind the negative coefficient of the spot rate growth

(lagged one period) is that people tend to overreact to economic shocks. This

phenomenon is called overshooting and it was also explained previously.

In our final model, all variables are statistically significant (the P-values are less

than 5%) and the regression is overall significant (the P-value of the F-test test is equal

to 0.01%).14 The R2 is about 0.2954, which means that roughly 30% of the variation of

the spot rate growth can be explained by our model. Although this is not a very high

value we need to pay attention to the fact that it is extremely difficult to find excellent

models that predict variables related to financial markets. Otherwise, one could easily

get rich.

Since there were some papers that suggested that the exchange rates follow a

random walk we thought that it could be interesting to check how it would perform

against our model.

A random walk is the sum of the lagged dependent variable and an error. We

estimated that error by getting the residuals of our final model and creating an AR (1)

with them. After that, we added the resulting errors to the lagged dependent variable in

order to get a random walk.

The R2 of this random walk is equal to 0.379715, which is greater than our

model’s (0.2954). That is, a random walk can explain the exchange rate’s growth rate

than it.

To perform a forecast we needed to compute ARMA models in order to predict

future values for the independent variables.

14 Consult figures 11 and 12 in the appendix15 Consult figure 13 in the appendix

To test the accuracy of our forecast for the spot rate we generated a confidence

interval around it for January, 2015 (one-step ahead) and check if the real value was

within this range (where the null hypothesis is for the forecast and real value to be

equal). However we only had the standard errors for the predictions of the exchange rate

growth, not for the exchange rate itself.16 Therefore we estimated a confidence interval

for the growth and then applied the upper and lower limit of that growth to the spot rate

of the previous month.

The obtained confidence interval, with a confidence level of 95% resulted in

growth rates between [0.52%, 3.91%]. By applying these growth rates on the spot rate

of the previous month we obtain a confidence interval (with the same significance level)

for the predicted spot rate of [0.831; 0.859]. Given that the observed value was 0.886

(outside the interval) it means that we reject the null hypothesis. We also perform the

test for a higher confidence level (99%) and the result was the same. This means that

even with 99% confidence, our forecast for the spot rate of the following month failed.

This has most likely something to do with the unusual change in the spot rate between

December and January (over 7%). This sudden change was due to the fact that people

anticipated the quantitative easing policy, which would be conducted afterwards by the

ECB.

Conclusions

The objective of this work has been to propose a model that could estimate the

euro-dollar exchange rate. This paper is based on a very comprehensive and consistent

data set for both Eurozone and the US, which has been compiled for the period from the

beginning of 2009 until the end of 2014. The selection of the explanatory variables of

the euro exchange rate follows the most relevant theoretical models as shown in the

literature review. These take into account the money supply differentials, interest rate

differentials, inflation differentials, 3m bond yields differentials, and GDP differentials.

Too much differentials? It might sound like, but after careful revision, the reader can

conclude by himself that regarding the study of an exchange rate, two zones are

interacting. For this fact, it is quite natural to include the differentials as an explanatory

variable since it measures the contemporaneous difference between the two zones.

After ending up with a pretty significant model, the hypothesis test made to the

predicted value, in which was rejected that the predicted value and the real one were

equal, serve us as an example of the problem of these kind of models who seek to

16 Consult figure 14 in the appendix

explain and predict the values of financial variables. Many analysts and investors, who

deeply trust in quantitative models that have 95% and 99% confidence intervals, end up

loosing too much money by disregarding the market awareness of unpredictable

fundamentals.

However, these results are promising and might hopefully give place to further

research. In particular, we highlight some possible extensions of this paper: The

inclusion of other variables such as announcements made by the BCE or by country

representatives (of every country in the euro area) that we believed to be impactful.

Symmetric to this it would also be important to include announcements made by the

Federal Reserve. In a particular way, the inclusion of statements made by the two zones,

would allow to check the impact difference of each Central Bank.

The data didn't support our premise as we rejected the hypothesis that the

prediction for the exchange rate is equal to the actual exchange rate. Despite all this, we

are aware that such rejection can be related to the quantitative easing policies practised

by both Central Banks. It would be of interest to include some dummy variables

characterizing the kind of QE measure in use.

Econometrics can be useful, but models aren't perfect.

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Appendix

Figure 1-Determinants of the exchange rate

Figure 2 – Saving the Euro

Figure 3 – Testing multiple exclusion restrictions

Figure 4 – Strict exogeneity

Figure 5 - Stationarity

Figure 6 – Stationarity (corrected)

Figure 7 – Contemporaneous exogeneity

Figure 8 – Serial Correlation

Figure 9 - Heteroskedasticity

Figure 10 – ARCH test

Figure 11 - Final Model

Figure 12 - Three Models

Figure 13 - Random Walk

Figure 14 - Prediction One -Step Ahead