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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

Contents

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

1 Introduction

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.

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Figure 1: Line Graph of the Exchange Rate e/$

0.5

0.6

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0.9

1.0

1.1

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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

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Figure 2: Line Graphs of the Additional Variables (Solid Line: EU, DashedLine: US)

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80 82 84 86 88 90 92 94 96 98 00 02

Money Supply

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80 82 84 86 88 90 92 94 96 98 00 02

Price Level

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80 82 84 86 88 90 92 94 96 98 00 02

3 Months Interest Rate

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80 82 84 86 88 90 92 94 96 98 00 02

Industrial Production

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

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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

or multiplicative:

Lt = yt

Sts+ (1 )(Lt1 + Tt1)

Tt = (Lt Lt1) + (1 )Tt1St =

ytLt

+ (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

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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

3.1 Unit-Roots

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

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