Modeling and Forecasting the US Dollar / Euro Exchange Rate

In theory, a currency's value should gravitate over time in the direction of its real long-run equilibrium value. The intent of this paper is to investigate the sustainability of basic exchange rate theory and to construct econometric models capable to generate consistent and rational forecasts for the dollar/euro exchange rate. Considering past values of dollar/euro exchange rate, we build first an ARIMA model and we study the volatility of this exchange rate time series. However, since macroeconomics variables influence the exchange rate, we construct a model for dollar/euro exchange rate determination including macroeconomic variables whose choices have been theoretically driven. The most important outcomes of this research are the specifications of an economic model for dollar/euro exchange rate as well the estimation of the model in The Vector Error Correction Model form.


Introduction
Since the Euro takes the place of the ECU as an accounting unit beginning 1999, the large fluctuations in the dollar/euro exchange rate (in terms of dollar per euro) request more explanation of the underlying interactions.Equilibrium exchange rate modeling is not purely an arcane academic exercise.From policy perspectives, it is significant to know if the exchange rate tends to its long term equilibrium level or going away from this level.If we were able to estimate this exchange rate, investors would be able to identify the likely path that an exchange rate will take on a long-term basis and position their portfolios accordingly.Furthermore, forecasting exchange rates is very important for market participants.However, the complexity of this forecasting is proved by the fact that only three out of every ten spot foreign exchange dealers make a profit in any given year.This proves also that the foreign exchange market is efficient (Carney & Cunningham, 1996).
In theory, a currency's value should gravitate over time in the direction of its real long-run equilibrium value.
Regrettably, what exchange rate level represents a long-run equilibrium value of a currency does not constitute unanimity among economist.No accord also concerning the method that should be used to estimate this value.For instance, the approach which establishes the long-run equilibrium between the exchange rate and the price level, the purchasing power parity (PPP), is the widest subsequent among economists and strategists but it is as well recognized to have severe limitations because other fundamental forces have impact on the long-term path of exchange rates.Sartore, Trevisan, Trova and Volo (2001); Rosenberg (2003); Brent and Schnatz (2006), affirm the breakdown (on observed proof) of the PPP.While, Taylor and Taylor (2004) assume that the debate in academic world about the PPP competence as an exchange rate standard is still open and animated.
In recent years, after the breakdown of Bretton Wood Agreements, the interest rate parity (IRP) model has revealed invalidate since we observe that higher interest rates implies capital inflows and currency appreciation as consequences.Isard (1995) discredits the uncovered interest rate parity (UIP) model and postulates that unexpected information plays a bigger role in predicting exchange rates fluctuations than interest rate differentials.This finding support the idea that unexpected information about economic developments, policies, or other relevant factors has the main impact on fluctuations in exchange rates.Furthermore, it is become recognized that only a small part of exchange rates fluctuations are explained by interest differentials . Li Wenhao (2004) postulates that, the credibility of the basic theories, the purchasing power parity (PPP) and the uncovered interest parity (UIP) is undermined since these two theories are based on many unrealistic assumptions.
However, explaining future exchange rate value using the past value of exchange rate and building Auto-Regressive Integrated Moving Average (ARIMA) model suffer also from serious limitations since fundamental forces have important influence on the exchange rate.
The failure of the basic theories and the limited explanation power of the ARIMA series motivate the multivariate co-integration analysis.This latter issue has recently analyzed in economic literature on the exchange rate and aim to answer whether the exchange rate is driven by fundamentals and, to what degree it can be scheduled?Regrettably, no model yet fruitfully forecasts the changes in exchange rate.
The concern of this paper is to present a validate model able to explain and forecast the dollar/euro exchange rate.We try to explain first, the current value of the dollar/euro exchange rate by their past values in terms of linear relationships.This can be realized modeling the dollar/euro exchange rate series in the form of ARIMA model.We study also the volatility of the dollar/euro exchange rate.However, as proved by economic literature, macroeconomic variables impact the dollar/euro exchange rate, we develop therefore, an economic model to explain the current behavior of dollar/euro exchange rate and we put to test the impact of fundamental variables on this exchange rate.This paper is prepared as follows: Literature reviews are offered in section two.In section three we present the movement of dollar/euro exchange rate through the period beginning of 1999 to the end of 2012 and we assess the validity of the basic theories of exchange rate.In section four we model the dollar/euro exchange rate as ARIMA model and we evaluate the volatility of this exchange rate.In section five we identify the economic factors affecting the dollar/euro exchange rate.Section six explains the co-integration analysis and exposes the results of tests.Finally conclusions are present in section seven.

Literature Review
Although the economic literature includes wide studies on exchange rates and many economists proposed different theories to explain their movements, no theory can enthusiastically and winningly elucidate the changes of the dollar/euro exchange rate.All in all, the economic literature can be grouped in two classes of advances, class of expectation advances and class of fundamental advances.The latter class who explains the dollar/euro exchange rate fluctuations in terms of macro-economic variables is by far the mainstream.
Among the first group, where economists explore the capacity of ARIMA series to predict future exchange rate level, Bellgard and Goldschmidt (1999) investigated (aud/dollar) exchange rate using half hourly data during 1996.They studied the trading performance and forecasting precision of some conventional techniques, including random walk, exponential smoothing, and ARMA models with recurrent neural network (RNN) models.They brought to a close that statistical forecasting precision measures do not impact directly on profitability and foreign exchange time series show nonlinear patterns that are better explained by neural network models.Tyree and Long (1995) deviated from the latter path.They analyzed the case of the daily dollar/dem price variations from 1990 to 1994 and they discovered that the studied NNR models are less effective than the random walk model.They also found from a forecasting point of view, that what little structure is really present may well be too insignificant to be of any use although, price changes are not strictly random.They acknowledged, that it is expected that the optimal forecasting technique is the random walk.
However, Dunis and Huang (2001), estimated an ARMA (4, 4) model forecasting the dollar/euro exchange rate.Their estimation was not good enough as some elasticity were insignificant at the 95% confidence level.
For their part, to model the dollar/euro exchange rate series for the period starting 1994 to October 2007, Weisang and Awazu (2008) presented three ARIMA models with fundamental economic variables.They revealed that, the best model for the monthly series is a linear relationship between its past three values and the current and past three values of the difference of the log-levels of the share prices indices between the Euro zone and United States.
Among the second group, many economists considered various models to examine the fundamentals affecting the dollar/euro exchange rate.The mainly studied variables were the gross domestic product (GDP) and the interest rate differential.Maccauley (1997) predicted that the dollar/euro price will reveal inflation outcomes, growth performance, and long-term developments in net foreign-asset positions on both sides of the Atlantic.In short term however, the relation between business cycles and associated cycles in monetary policy will figure particularly in the dollar/euro fluctuations.Alberola et al. (1999) estimated the dollar/euro exchange rate considering the ratio of non-traded/traded goods prices and net foreign assets as explanatory variables.However, Chinn and Alquist (2000) recognized the narrow monetary aggregate M1, GDP, short-term interest rates, consumer price index (CPI) and the ratio of non-traded/traded goods prices as key explanatory variables.While, Alquist and Chinn (2002) used Johansen and Stock-Watson procedures to project co-integrating relationships between the real exchange rate, productivity, and the real price of oil.
However, Clostermann and Schnatz (2000) applied co-integration approaches and construct a synthetic euro/dollar price from 1975 to 1998.They identified the real euro/dollar exchange rate, the international real interest differentials, the real long term yield spread oil prices, government spending and the ratio of non-traded/traded goods prices as fundamentals factors affecting exchange rate.Lorenzen and Thygessen (2000) constructed an econometric model using net foreign assets, R&D spending, demographics and ratio of non-traded/traded goods prices.While, Duval (2001) identified consumption, multifactor productivity, real long term yield spread and the ratio of non-traded/traded goods prices.Also, similarly, Telletech (2000) identified the productivity, the government spending, real long-term yield spread, M1 and industrial production, as explanatory variables for his model.He found that the difference in real interest rates and productivity, and (in some conditions) the relative fiscal attitude and the real price of oil, have a significant influence on the effective dollar/euro price.
The impact of the flow of funds to stock market on exchange rate is identified by Bailley and Millard (2000), Meredith (2001) and Brooks et al. (2001).They advanced, explaining the shock on the demand that capital inflow increases share prices, as consequences, consumption and investment increase.On the supply side however, potential output is increased as consequences of the raise in labor productivity caused by the increase in capital stock cause in turn by higher investment.
Finally, Moosa (2002) presumed that market participants do not cause events actively; they rather forecast events passively since he found that short-term and medium-term expectations primarily impact the exchange rates.He also found that rational expectations have a small impact on the real world.Inherited from Keynes idea, Moosa postulated that since speculation is impulsive, the exchange rate movement will become random, if speculation dominates the short-term expectations.
In this paper, we aim to present insights on the configuration of the dollar/euro exchange rate time series in order to construct an adequate model for exchange rate forecasting.We use therefore in first steep, past values of the dollar/euro exchange rate time series to predict current value (ARIMA model) and we evaluate the volatility of this series.In second steep, we provide an economic model and we estimate the impact of theoretically chosen variables.

The Market Movement and the Failure of Basic Exchange Rate Theories
For long time the exchange rate theories were dominated by two main equilibrium theories of exchange rate: The PPP theory and the UIP theory.But can these theories explain on empirical ground the dollar/euro exchange rate fluctuations?

Purchasing Power Parity Theory (PPP)
In 1918, Gustav Cassel presented the PPP theory.This theory states the long-run equilibrium between the exchange rates and the price levels.The PPP exchange rate is the rate that equates the two currencies by eliminating the differences in the price levels between countries.The main theory element is the non-existence of price arbitrage.In other terms, the same article of trade must have exactly the same price in two different markets.If the same article of trade does not have the same price in two different markets, arbitrage transactions would occur.Therefore, the goods will be transported from lower price market to higher price market.As results, the supply of the article of trade on the lower price market rises.While the inverse would happen in the higher price market, prices would rapidly be equalized.
Suppose S t to be the nominal exchange rate dollar /euro (in terms of dollar per euro) (Note 1) and P t the price level in t-country.The PPP relationship is: Or more generally Where, Q t is the dollar/euro real exchange rate.Q t is the purchasing power of a currency relative to another.It is Therefore, the unrealistic assumptions of the purchasing power parity and the interest rate theories undermine their credibility.In order to estimate the current dollar/euro real exchange rate we use in the next section the past value of this exchange rate.

ARIMA Model and Volatility Estimation
The methodology of Autoregressive Integrated Moving Average (ARIMA) estimation and model selection, also called Box-Jenkins (1984) models, is a classical topic covered in most textbooks on time series analysis (e.g.Brockwell & Davis, 2003;Hamilton, 1994;Tsay, 2005;Wei, 2006).
We do not aim in this section to replicate the existing literature but rather to estimate the current value of dollar/euro exchange rate using its own past value as explanatory variables.The Autocorrelation Function (ACF) and Partial Autocorrelation Function (PCF) are used to estimate the ARIMA order by means of the regulations presented in Wei, 2005 p.109.

Model Building
The Correlogram of the real dollar/euro exchange rate (Q t ) series shows that only the ACF and the PACF of the first-differenced Q t series feature a decaying pattern with reasonable cut-off points.Conjointly, the guidelines of Wei regulations and the correlogram suggest an ARIMA (2,1,0) structure.The evolution equation is then as follows: ( This can be written: ) where Q t represents the real dollar/euro exchange rate, B is the backshift operator, and a t is random noise.

Stationary Results
The dollar per euro real exchange rate series (Q t ) presented in Fig 1 exhibits some non-stationarity.This non-stationarity is confirmed by the results of Augmented Dickey Fuller (ADF) test presented in table 2. These results prove that the Q t series is not stationary in level but stationary in first difference.Therefore the Q t series is integrated of order 1 (I(1)).

Model Estimation
The model estimation in least square method made by Eviews shows that the R-square is equivalent to (0.97) indicating an excellent fit for the model.The estimation results show also that Both Q t-1 and Q t-2 coefficients are significantly different from 0 with p values respectively (0.0000) and (0.0006).The constant and the Q t-3 coefficient are not significantly different from 0, since the p-value are (0.1838) and (0.1282) respectively (over 0.5%).The model estimation is as follows: Equation ( 5) establishes the evolution of the dollar per Euro exchange rate as a weighted sum of its past three values plus a random shock where Q * t is the estimation of Q t and the value in parentheses are the coefficient probabilities.We note the particularly small value of the constant in the latter evolution equation reflecting the short of significance of the estimate μ*.Furthermore, one interesting feature is, that because the time series of the exchange rate needs to be differenced once to be made stationary, the coefficients of this linear combination of past values sums to one.That is, the current value of the exchange rate can be interpreted as a weighted average of its three past values.Notice also that the coefficient of t-1 is greater in absolute value of t-2 and t-3, therefore giving proportionally more weight to the most recent values.

Volatility Evaluation
The analysis of the volatility of an exchange rate series is based on the returns of the data, which are the period by-period changes in the data.For example, returns on monthly exchange rate are the differences between exchange rates in two consecutive months.In this study the measure of the return is the difference in the exchange rate over two consecutive periods: R t = Q t -Q t-1 .
Generalized Autoregressive Conditional Heteroskedasticity (GARCH) (Note 2) formulation was used in this study to test whether the variance of returns is stationary and if price levels eventually revert back to a mean and, if they do, over what time period.The GARCH formulation tests an equation specification for the mean of the return series ( 6) and an equation for the conditional variance (7) of the returns: where ε t ~ N(0, σ 2 t ) and σ 2 t = Ε(ε 2 t ).This equation specification is often interpreted in a financial context, when an agent trader predicts this period's variance by forming a weighted average of a long term average (the constant), the forecasted variance from last period (the GARCH term: α), and information about volatility observed in the previous period (the ARCH term: β).If the asset return was unexpectedly large in either the upward or the downward direction, than the trader will increase the estimate of the variance for the next period.Since the Q t series is integrated of order 1, the Return series is stationary; we proceed then to measure the volatility.

Volatility Measure
The ARCH and GARCH test by Eviews gives the following estimation of equation 6: The values in parentheses are the coefficient probabilities.The sum of ARCH and GARCH (α + β) is very close to one, indicating that volatility shocks are quite persistent that is often observed in high frequency financial data.We proceed then to test if there is autocorrelation between the residuals.

Autocorrelation in the Residuals Test Results
The Lagrange multiplier (LM) tests for serial correlation by testing whether the serial correlation coefficients are significantly different from zero.The null hypothesis of the test is that there is no serial correlation in the residual up to the specified order.The LM test by Eviews indicates that we reject the hypothesis of no remaining significant autocorrelation in the residuals of the model up to order 2, since the Observed R-squared probability is 0.0041(below 0.5%).The presence of serial correlation suggests that the model do not seem to adequately capture the correlation information in the time series and the equation cannot be used for hypothesis test or forecasting.
ARIMA model fails to provide an explanation of the causal structure behind the evolution of the time series.Therefore, to get the structural relationships we specify the economic behavior of the dollar/euro exchange rate.
In other terms we specify the long-term relationships between the dollar/euro exchange rate movement and the changes of fundamental variables.

Fundamentals Affecting Exchange Rate
In section 3, we found that the IRP and the PPP explain a part of the dollar/euro exchange rate variation but these two theories fail to explain all this variation.To model the dollar/euro exchange rate we consider then in addition to the impact of the IRP and the PPP explained in section 3, the impact of the Money Aggregates, and the Business Cycles. www.ccsen

Money
Since the implement of exchang approach, countries.em of ew, a m the long-term trend will be corrected.The purpose of the co-integration test is to determine whether a group of non-stationary series is co-integrated or not.

Money
If the similarly integrated series in any given model are co integrated, then linear combinations of these variables will converge to stationary long-run equilibrium relationships.Thus, the non-stationary property of the series must be considered first.Testing for co integration is a second stage of pre-testing.The presence of a co-integrating relation forms the basis of the VEC specification.Only when this stage has been passed should we move on to the model building.To test for cointegration we use the method developed by Johansen (1991;1995a).This method allows knowing the number of cointegrating vectors; it allows also using the vector error correction model (VEC) to estimate the equation ( 9).The VEC (Note 3) has co integration relations built into the specification so that it restricts the long-run behavior or the endogenous variables to converge to their co integrating relationships while allowing for short-run adjustment dynamic.The co integration term is known as the error correction term since the deviation from long-run equation is corrected gradually through a series of partial short-run adjustment.

Stationary Test
The stationary test results according to Augmented Dickey Fuller (ADF) test are reported in table 3.These results indicate that all series are not stationary in their level but stationary in their first difference.All series are then integrated of order one I(1).

Co-Integration Test
Since All variables being I(1), we proceed to test for co integration.We estimate a multivariate co integration relationship to establish the existence of a long-run equilibrium relationship.The Johansen's Maximum Likelihood co integration test relations were estimated with intercept and linear deterministic trend in a Vector Auto Regression (VAR) model of order 1 with a lag length of 4, which was found to be the most parsimonious for the data series.The Johansen co integration tests are based on the Maximum Eigenvalue of the stochastic matrix as well as the Likelihood ratio test which is in turn based on the Trace of the stochastic matrix.Table 4 below shows the summary results of the Johansen's Maximum Likelihood co integration test.For the null hypothesis of r=0, the calculated trace statistics was larger than its critical value and calculated maximum Eigenvalue was also larger than its critical value at 5% level of significance.From the results, it is evident that both the trace test and the maximum Eigenvalue test indicate one co-integrating equation as the null hypothesis of r = 0 is rejected.Thus, it may be concluded that there is a unique long-run equilibrium relationship between the variables.

Vector Error Correction Model
These results imply that a long-run association exists among the exchange rate and the four explanatory variables.
Then, the long-run estimated coefficients are reported in Table 5.In the long run, all the model variables are statistically significant with the expected sign coefficient.We note however the high elasticity of inflation, in fact the model estimation shows that in the long run the variable INF has the more influence on the nominal exchange rate (elasticity -1.8), follows by MD (elasticity 0.9) and BC (elasticity -0.45) and INT(0.06).
It may be observed also from the VEC estimation by Eviews that the model fits the observed data fairly and significance of estimated relationships as indicated by the adjusted R 2 (0.268534) and F-statistic (3.546295) of the relevant error correction equation.The error correction coefficient (-0.068599), which measures the speed of adjustment towards long-run equilibrium carries the expected negative sign and it is highly significant at the 1% level.It shows however that about 6% of disequilibrium is "corrected" each month by changes in nominal exchange rate (NEX).The estimation of short term model is reported in Table 6.The short run equilibrium estimation shows that the variable NEX in time t-1 is statistically significant and affects the NEX in time t.It shows also that the variable BC in time t-1 and in time t-4 is statistically significant and affects the NEX in time t.The other variables at different time are not statically significant; we note therefore the low elasticity of all exogenous variables.

Conclusion
In one hand, the basic theories, the Power purchasing parity and the Interest rate parity explain partially the dollar/euro exchange rate.In other hand, we have seen above that a simple ARIMA model can provide an evolution equation with a simple interpretation.Although the exchange rate series presents a high volatility, the presence of serial correlation suggests that the model do not seem to adequately capture the correlation information in the time series and the model cannot be used for hypothesis test or forecasting.Furthermore, ARIMA model can be criticized because it fails to provide an explanation of the causal structure behind the evolution of the time series.
Therefore, the model developed in this study consider in addition to the two variables proposed by basic theories, two other variables, one variable representing the Money Aggregates, and one variable representing the Business Cycles.

Table 1 .
Annual variation of monthly real interest rate differential (d(RID)) and annual variation of monthly real dollar/euro exchange rate (d(Q t )) from 1999 to 2012

Table 2 .
Results of ADF test for Q t series by eviews Note: *, **, and *** show the statistical significance at the 10%, 5%, and 1% level of significance respectively.

Table 3 .
Results of ADF tests

Table 4 .
Results of cointegration testsTrace and max-eigenvalue tests indicate 1 cointegrating equation at the 0.01 level of significance.r indicates the number of cointegrating vector and λ trace and λ max are tests statistic of trace and maximum eigenvalue tests respectively.

Table 6 .
Estimated short-run model Dependent Variable D( NEX t )