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Forecasting the Exchange Rate
A forecasting application with the exchange rate betweenthe Euro and the US Dollar (1980-2003)
Term paper for the course Econometric Forecasting
Vienna, June 14, 2005
Gurkan BirerMatr.nr.: 0254010
Johannes HollerMatr.nr.: 9611729
Michael WeichselbaumerMatr.nr.: 9640369
2UK 406347 Okonometrische Prognose
Lehrveranstaltungsleiter: O. Univ.-Prof. Dr. Robert Kunst
1 Introduction 21.1 Descriptive Statistics . . . . . . . . . . . . . . . . . . 2
2 Model Free Methods 42.1 Exponential Smoothing . . . . . . . . . . . . . . . . . 4
3 Univariate Modelling 63.1 Unit-Roots . . . . . . . . . . . . . . . . . . . . . . . . 63.2 Autoregressive Models . . . . . . . . . . . . . . . . . 6
4 Multivariate Modelling 84.1 Multivariate Single Equation Model . . . . . . . . . . 94.2 VAR . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.2.1 Granger Causality . . . . . . . . . . . . . . . 12
5 Forecast Comparison 165.1 Plotted Forecasts . . . . . . . . . . . . . . . . . . . . 165.2 Mean Squared Error of Predictions . . . . . . . . . . 17
6 Conclusion 19
This paper is an empirical investigation in forecasting using theexchange rate between the U.S.-Dollar and the Euro. The mainpurpose is to compare different methods of forecasting.
In our analysis, the exchange rate U.S. Dollars in terms of Eurois our dependent variable. The sample encompasses the time periodfrom January 1980 to December 2003 and is available at monthlyfrequency. For the time span where the Euro was not in existence,the data represents the synthetic Euro, which is a weighted ex-change rate from the single countries in the Euro area.
Since a central question of this work is to compare the ability ofthe constructed models to forecast actual variations, we divide thesample in two parts: the range from the beginning until December2002 is used for model selection and estimation, whereas the twelvemonths of 2003 remained untouched as long as we did not reach thelast section of this work, where the comparison is done.
This introduction is extended by a subsection containing somedescriptive statistics. The next section captures model free forecast-ing methods. It is important to notice that the model free methodsare the only ones where we talk about the original series. For everymodel estimated later we use logarithms of every series. After exam-ining the stationarity properties of the logarithmic series, section 3presents univariate forecasting methods for the first differences ofthe logarithmic exchange rate. These results are used to calculatethe forecasts for the original exchange rate series. In section 4, ad-ditional variables are used in multivariate models. All the modelschosen then will be used to forecast the exchange rate and the resultsare compared in section 5.
1.1 Descriptive Statistics
With the intention to keep the structure of this work simple, weput graphs of all our variables, which encompasses the exchangerate our target for forecasting as well as the variables used inthe multivariate modelling section, together with some descriptivestatistics at the very beginning, though only the exchange rate datawill be used in section 2 and section 3.
Figure 1 shows the line graph of the exchange rate.The additional variables that we will use in section 4 are both
available for the US and the EU, which is indicated by the subscript:
Mcountry Money supply M2.
Figure 1: Line Graph of the Exchange Rate e/$
80 82 84 86 88 90 92 94 96 98 00 02
Pcountry Price level, Consumer Price Index (CPI)
Scountry Short term interest rate: three months.
Ycountry Industrial production.
Line graphs for the four pairs are given in figure 2. The solid linerepresents the data for the EU, the dashed line its US counterpart.
Finally, some descriptive key measures are given in table 1.
Table 1: Descriptive Statistics for all Variables
Variable Mean Median Max Min Std.Dev. Obs.Rate e/$ 0.903 0.866 1.389 0.591 0.163 276MEU 115.80 106.86 239.70 45.43 53.16 276MUS 107.49 108.13 152.99 48.13 33.50 276PEU 84.95 87.10 112.00 47.54 17.79 276PUS 77.63 79.04 105.46 45.30 16.88 276SEU 7.972 7.778 17.455 2.579 3.391 276SUS 7.449 6.391 19.629 1.407 3.758 276YEU 98.94 98.87 122.98 81.44 11.82 276YUS 108.72 101.06 151.78 76.81 22.63 276
Figure 2: Line Graphs of the Additional Variables (Solid Line: EU, DashedLine: US)
80 82 84 86 88 90 92 94 96 98 00 02
80 82 84 86 88 90 92 94 96 98 00 02
80 82 84 86 88 90 92 94 96 98 00 02
3 Months Interest Rate
80 82 84 86 88 90 92 94 96 98 00 02
2 Model Free Methods
Our first modelling of the the exchange rate consists of univariatestrategies. Thus, for this section, we neglect economic knowledgesuggesting the influence and thus inclusion of the other variables.
2.1 Exponential Smoothing
During the following analysis estimated values for , and areused. Estimation of parameters is done through calculation of valuesthat minimize the sum of squared errors. Starting value for thecalculation is always the mean of the observed series. For singleexponential smoothing, the estimated equation follows the recursiveformulation:
yt = yt + (1 )yt1Since the parameter estimate we get for is with 0.9990 virtually
one, the series can be considered a random walk when used for one-step forecasting. Since we want to see how well this basic smoothing
procedure works for a distinguishing alpha, we also estimate theparameter for the differentiated series, which is then 0.0980.
Continuing with double exponential smoothing, we go back tothe original values of the exchange rate. Now the method containsa local level estimate Lt and a local trend estimate Tt.
Lt = yt + (1 )Lt1Tt = Lt + (1 )Tt1
The forecast for period k after the estimation sample of N ob-servations ends is then given by a linear trend extrapolated k timesfrom level 2LN TN :
yN(k) = 2LN TN + (LN TN)1 k
Our conjecture that this method will better apply to the seriesproves correct: now we get an alpha of 0.5780.
Next, we report the estimates for the three Holt-Winters meth-ods, which require two parameters for the version without a season.An additional parameter has to be estimated now:
Lt = yt + (1 )(Lt1 + Tt1)Tt = (Lt Lt1) + (1 )Tt1
In this first case of the method, the forecast is produced by thesummand of the last calculation from the sample for the level, LN ,and k times the corresponding value from the trend, TN . For alphawe get a value of one, and beta is equal to 0.11.
When a seasonal component is added, a third parameter is intro-duced which can be additive:
Lt = (yt Sts) + (1 )(Lt1 + Tt1)Tt = (Lt Lt1) + (1 )Tt1St = (yt Lt) + (1 )Sts
Lt = yt
Sts+ (1 )(Lt1 + Tt1)
Tt = (Lt Lt1) + (1 )Tt1St =
+ (1 )StsIn both cases we have a third equation for the seasonal compo-
nent and s = 12 due to the monthly frequency. Accordant to theirname and formulation, the forecasts are obtained by either addingthe seasonal component to or by multiplying it with the level- andtrend-summand described for the first version of the method. Our
estimates for alpha and beta do not vary, i.e. remain one and 0.11,and for gamma we get a zero estimate. It has to be mentioned thatdespite being zero, both the additive and the multiplicative versiondo involve a seasonal factor. It only restricts the seasonal factorsfrom changing over time. Therefore the forecasts are different fromthe Holt Winters no seasonal method.
Visual inspection of Graph1 shows, that DES seems to producesthe best forecast. Since Holt Winters multiplicative and additiveforecasts are less acurate than HW no seasonal, one could follow,that including seasonality does not make the forecast better.
3 Univariate Modelling
To assess the transformation of the series we want to work with forevery one of them, we perform an Augmented Dickey-Fuller Test forunit roots and choose the step of differentiation of every series wherethe Null of unit root can be rejected at a level of significance of fivepercent. In every test we included an intercept. Though we onlyneed the results for the additional series in section 4, we put all ofthem together in this subsection. Table 2 gives the results, but somefurther explanations are necessary: in accordance with similar workon forecasting the exchange rate, we took the logarithm of everyseries; this is indicated by the use of lower case letters; additionally,the variables without subscripts are defined as m = mUS mEU symmetrically for p, s and y.
The test results suggest stationarity for most of the series aftertaking first differences. Thus, we do this for all series, in spite of theindication of using the second differences of the money supply (inthe case of the EU minus US variable m) and for the European pricelevel, since we believe that this is rather a peculiarity of the data, i.e.of the realisations of the errors, than a systematic relationship. Inthe case of the US price level, we also use the first differences for tworeasons: first, the surprising result of stationarity led us to havinga look at the line graph, which indeed does not seem stationary;second, fitting an AR(1)-model to pUS gives a parameter value ofslightly above one (t-statistic is 28289).
3.2 Autoregressive Models
Inspection of the correlogram shows us, that the autocorrelationfunction of the growth rate of the exchange rate converges to 0 at
Table 2: Unit Root Tests Results
Variable t-Statistic Probability Stationarye -2.49 0.119
e -11.44 0.000 *mEU 0.12 0.967
mEU -3.65 0.005 *mUS -3.32 0.015
mUS -5.94 0.000 *pEU -1.00 0.753
pEU -2.55 0.1052pEU -11.96 0.000 *
pUS -4.55 0.000 *sEU 0.023 0.959
sEU -9.58 0.000 *sUS -0.62 0.862
sUS -11.48 0.000 *yEU 0.37 0.982
yEU -8.82 0.000 *yUS -0.128 0.944
yUS -5.97 0.000 *m -0.486 0.891
m -2.13 0.2352m -10.95 0.000 *
p -1.22 0.667p -13.32 0.000 *
s -1.38 0.592s -13.71 0.000 *
y -0.77 0.825y -22.36 0.000 *
Note: The last column indicates stationarity by an asterisk.
a geometrical rate, which is a sign for stationarity. It further showsthat the partial correlation function becomes 0 for lag orders largerthan one, which suggests the usage of an AR(1) model. Analysis ofthe AIC confirm this fact.
For the specification of our ARMA-model it is difficult to use thecorrelogram since it does not reproduce stylised patterns. Thereforewe consult AIC, which yields the choice of an ARMA(1,2) model.Table 3 shows the estimation results for the two models. It should bementioned that we did not perform an ARCH model forecast, sincea test on autoregressive conditional heteroskedasticity was rejected.
Table 3: Regression Output for ARI(1,1) and ARIMA(1,1,2)
Explanat. variable Coefficient (Std. Err.) z P > |z|Const. 0.001 0.001 0.77 0.442L(e) 0.349 0.057 6.14 0.000
R-squared 0.122 Mean dependent var 0.002Adjusted R-squared 0.118 S.D. dependent var 0.026S.E. of regression 0.024 Akaike info criterion -4.570Sum squared resid 0.164 Schwarz criterion -4.544Log likelihood 628.080 Durbin-Watson stat 1.929
Explanat. variable Coefficient (Std. Err.) z P > |z|Const. -0.000 0.002 -0.18 0.856L(e) 0.963 0.044 21.78 0.000 -0.586 0.072 -8.11 0.000L() -0.366 0.059 -6.15 0.000
R-squared 0.149 Mean dependent var 0.002Adjusted R-squared 0.140 S.D. dependent var 0.026S.E. of regression 0.024 Akaike info criterion -4.587Sum squared resid 0.159 Schwarz criterion -4.534Log likelihood 632.436 Durbin-Watson stat 1.996Note: L() is the first lag of a variable.
4 Multivariate Modelling
After applying some univariate models to the series, we now want tomake use of multivariate specifications. As described in section 1,our data set includes four different additional variables for the EUand the US: mEU , mUS (the money supply), pEU , pUS (the consumerprice indexes), sEU , sUS (three months interest rate) and yEU , yUS(industrial production) all of them in logarithms. As suggestedby economic theory, we assume that these variables are stronglyconnected to the development of the exchange rate. Throughoutthis section, we are going to use two different types of models, clas-sified by the set of variables: the first one includes the series ofthe EU and the US separately, that yields a set of nine variables ifone includes the exchange rate (this set gets the number 1 assignedthroughout the remainder); for the second one we only use the differ-ence of every variable from its foreign counterpart, i.e. VariableEU -VariableUS (constitutes set number 2). This means, we impose therestriction that the influences of changes in the one variable areequal but opposite in sign to influences of changes of the other vari-
ables. Economically spoken, this means that the difference betweenthe two variables at time t counts, not their level.
4.1 Multivariate Single Equation Model
Assuming that all variables besides the exchange rate are strictlyexogenous, we try to model a multivariate single equation model.We estimate equations using the two sets of variables as describedabove and in both sets we only take the exchange rate as endogenous in the end we want two single equation models, one for each setof variables. All series are in first differences in order to achievestationarity.
We use the following steps as the strategy to choose the modelfor variable set 1 (2):
1. Write an equation consisting of all eight (four) exogenous vari-ables.
2. Include the lags number 1, 2, 3, 4 and 12 for the possibilityof a seasonality of 12 months for every exogenous variable.
3. Add lag 1, 2, 3, 4 and 12 of the endogenous variable.
4. Estimate the resulting equation; choose the lag length by con-secutively deleting the variables with the highest lag order andchoose the lag order with the smallest SIC we demand all thevariables to have the same lag length.
5. Consecutively drop the variables with the highest p-value untilall variables have a p-value smaller than 0.05.
Table 4 presents the results for the model with eight and withfour exogenous variables.
Discussion As we know from the lecture, this model is not extraor-dinary useful except the exogenous variables are especially easy toforecast, since we need some exogenous variables at time t to fore-cast exchange rate at time t in our specification. But there is noindication that we have a case of simplification in forecasting of anyvariable here. But the point we want to make is that this model isspecified incorrectly if there is feedback running from the exchangerate to any explanatory variable in the case of the second modelfrom table 4 from e to s. This can be tested to some extent ina VAR-framework using the concept of Granger causality.
Table 4: SIC-Best Single Equation Model for Set 1 and Set 2
Explanat. variable Coefficient (Std. Err.) z P > |z|L(e) 0.325 0.057 5.72 0.000pEU 0.936 0.370 2.53 0.012L(sEU ) -0.063 0.031 -2.03 0.043sUS 0.092 0.022 4.14 0.000L(yUS) -0.492...