Forecasting the Exchange Rate - Zentraler...

21
Forecasting the Exchange Rate A forecasting application with the exchange rate between the Euro and the US Dollar (1980-2003) Term paper for the course ”Econometric Forecasting” Vienna, June 14, 2005 urkan Birer Matr.nr.: 0254010 Johannes Holler Matr.nr.: 9611729 Michael Weichselbaumer Matr.nr.: 9640369 2UK 406347 ¨ Okonometrische Prognose Lehrveranstaltungsleiter: O. Univ.-Prof. Dr. Robert Kunst

Transcript of Forecasting the Exchange Rate - Zentraler...

Page 1: Forecasting the Exchange Rate - Zentraler …homepage.univie.ac.at/robert.kunst/prog05bihawei.pdf · Forecasting the Exchange Rate A forecasting application with the exchange rate

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

Page 2: Forecasting the Exchange Rate - Zentraler …homepage.univie.ac.at/robert.kunst/prog05bihawei.pdf · Forecasting the Exchange Rate A forecasting application with the exchange rate

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

Page 3: Forecasting the Exchange Rate - Zentraler …homepage.univie.ac.at/robert.kunst/prog05bihawei.pdf · Forecasting the Exchange Rate A forecasting application with the exchange rate

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.

2

Page 4: Forecasting the Exchange Rate - Zentraler …homepage.univie.ac.at/robert.kunst/prog05bihawei.pdf · Forecasting the Exchange Rate A forecasting application with the exchange rate

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

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

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

3

Page 5: Forecasting the Exchange Rate - Zentraler …homepage.univie.ac.at/robert.kunst/prog05bihawei.pdf · Forecasting the Exchange Rate A forecasting application with the exchange rate

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

40

80

120

160

200

240

280

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

Money Supply

40

50

60

70

80

90

100

110

120

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

Price Level

0

4

8

12

16

20

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

3 Months Interest Rate

70

80

90

100

110

120

130

140

150

160

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 α, β and γ 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 = αyt + (1− α)yt−1

Since the parameter estimate we get for α is with 0.9990 virtuallyone, the series can be considered a random walk when used for one-step forecasting. Since we want to see how well this basic smoothing

4

Page 6: Forecasting the Exchange Rate - Zentraler …homepage.univie.ac.at/robert.kunst/prog05bihawei.pdf · Forecasting the Exchange Rate A forecasting application with the exchange rate

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 = αyt + (1− α)Lt−1

Tt = αLt + (1− α)Tt−1

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 +α(LN − TN)

1− αk

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 = αyt + (1− α)(Lt−1 + Tt−1)Tt = β(Lt − Lt−1) + (1− β)Tt−1

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 = α(yt − St−s) + (1− α)(Lt−1 + Tt−1)Tt = β(Lt − Lt−1) + (1− β)Tt−1

St = γ(yt − Lt) + (1− γ)St−s

or multiplicative:

Lt = α yt

St−s+ (1− α)(Lt−1 + Tt−1)

Tt = β(Lt − Lt−1) + (1− β)Tt−1

St = γ yt

Lt+ (1− γ)St−s

In 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

5

Page 7: Forecasting the Exchange Rate - Zentraler …homepage.univie.ac.at/robert.kunst/prog05bihawei.pdf · Forecasting the Exchange Rate A forecasting application with the exchange rate

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 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

6

Page 8: Forecasting the Exchange Rate - Zentraler …homepage.univie.ac.at/robert.kunst/prog05bihawei.pdf · Forecasting the Exchange Rate A forecasting application with the exchange rate

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 0.105∆2pEU -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 0.235∆2m -10.95 0.000 *

p -1.22 0.667∆p -13.32 0.000 *

s -1.38 0.592∆s -13.71 0.000 *

y -0.77 0.825∆y -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.

7

Page 9: Forecasting the Exchange Rate - Zentraler …homepage.univie.ac.at/robert.kunst/prog05bihawei.pdf · Forecasting the Exchange Rate A forecasting application with the exchange rate

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 0.000ε -0.586 0.072 -8.11 0.000L(ε) -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-

8

Page 10: Forecasting the Exchange Rate - Zentraler …homepage.univie.ac.at/robert.kunst/prog05bihawei.pdf · Forecasting the Exchange Rate A forecasting application with the exchange rate

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.

9

Page 11: Forecasting the Exchange Rate - Zentraler …homepage.univie.ac.at/robert.kunst/prog05bihawei.pdf · Forecasting the Exchange Rate A forecasting application with the exchange rate

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 0.000∆pEU 0.936 0.370 2.53 0.012L(∆sEU ) -0.063 0.031 -2.03 0.043∆sUS 0.092 0.022 4.14 0.000L(∆yUS) -0.492 0.229 -2.15 0.033

R-squared 0.192 Mean dependent var 0.002Adjusted R-squared 0.180 S.D. dependent var 0.026S.E. of regression 0.024 Akaike info criterion -4.632Sum squared resid 0.151 Schwarz criterion -4.566Log likelihood 639.571 Durbin-Watson stat 1.958

Explanat. variable Coefficient (Std. Err.) z P > |z|L(∆e) 0.363 0.056 6.54 0.000∆s -0.072 0.020 -3.66 0.000

R-squared 0.161 Mean dependent var 0.002Adjusted R-squared 0.158 S.D. dependent var 0.026S.E. of regression 0.024 Akaike info criterion -4.616Sum squared resid 0.156 Schwarz criterion -4.590Log likelihood 634.382 Durbin-Watson stat 1.929Note: L(·) is the first lag of a variable.

4.2 VAR

As a starting point, we formulate the system of a VAR model thatrepresents the possibilities given by the number of variables and plags. The most general formulation of the models we use is:

Xt = A0 + A1Xt−1 + εt (1)

where Xt is – for set 1 – the vector:

X ′t = (et,mt,EU ,mt,US, pt,EU , pt,US, st,EU , st,US, yt,EU , yt,US)′

and – for set 2 – the vector:

X ′t = (et,mt, pt, st, yt)

A0 is a vector of constants:

A′0 = (A10, A20, . . . , An0)

10

Page 12: Forecasting the Exchange Rate - Zentraler …homepage.univie.ac.at/robert.kunst/prog05bihawei.pdf · Forecasting the Exchange Rate A forecasting application with the exchange rate

with dimension n either nine or five. A1 is a matrix consisting ofn× n entries of lag polynomals:

A1 =

A11(L) . . . A1n(L)A21(L) . . . A2n(L)

.... . .

...An1(L) . . . Ann(L)

The lag polynomal for the i-th row is of order p for all i in {1, · · · , n}and is defined as:

Aij(L) = (aij0 + aij1L + aij2L2 + . . . + aijpL

p) (2)

The last summand of equation (2) is a vector of error terms. Xt−1

is the first lag of the vector Xt. Thus, the right hand side of equa-tion (2) gives for every line of the left hand side, which representsone variable, an equation consisting of a constant, plus the sum ofp lags for each of the variables in the system and each of the result-ing p times n coefficients multiplied with a parameter, plus an errorterm.

In order to find out which lag to choose we apply SIC criterionfor selection. This will follow below, but first a few words aboutstationarity again. In a VAR-system it is possible that we havestationarity though the individual series are non-stationary. In ourmodel this means that it is possible that we have integration of orderone in every series, I(1), but the vector consisting of five variablesfor set 1 is I(0). That would mean the components of the vectorX ′

t = (et,mt, pt, st, yt)′ are CI(1,1) – individually integrated of order

one and together integrated of order one minus one, thus stationary.Depending on the result of the cointegration test, one should

proceed as follows: if the series are not cointegrated, we differen-tiate them to get them individually stationary and then estimatethe VAR. If they are cointegrated, one should use the Vector ErrorCorrection model. We use the test since we hope that our variablesare not cointegrated because then our model is correctly specified.Nevertheless, if we find cointegration we decide to estimate the VARafter differentiating individually.

For the cointegration test it is necessary that all series are inte-grated of the same order. As described in section 3.1 most of ourseries are indeed integrated of order one, whereas for the rest wefind it reasonable to make the according assumption. Next, we haveto choose lag length. We do this by minimising the SIC for everymodel that we construct in the remainder. The results of this min-imisation process are given in table 5. We compared the values of

11

Page 13: Forecasting the Exchange Rate - Zentraler …homepage.univie.ac.at/robert.kunst/prog05bihawei.pdf · Forecasting the Exchange Rate A forecasting application with the exchange rate

the SIC up to a lag order of eight. For all models the optimal lengthis given at order one.

Table 5: SIC per model of Lag Order

No. of Variable set 1 Variable set 2Lags full reduced full reduced0 -56.80 -23.20 -28.62 -10.631 -57.16* -23.69* -28.74* -10.80*2 -56.10 -23.57 -28.35 -10.733 -49.99 -23.41 -28.13 -10.714 -55.00 -23.14 -27.70 -10.645 -53.77 -22.85 -27.26 -10.566 -52.48 -22.59 -26.82 -10.487 -51.21 -22.32 -26.40 -10.428 -48.81 -22.08 -26.04 -10.38Note: The model chosen is indicated by an asterisk.

Table 6 gives the test results for the cointegration test for thetwo models with all variables from each of the two sets. Thus, theseries are undifferentiated but, as usual, in logarithms.

Table 6: Cointegration test results. H0: No cointegrating equation exists

Model Eigenvalue Trace Statistic Critical Value Prob.Set 1 0.397 371.227 197.371 0.000Set 2 0.185 108.862 69.819 0.000

The result indicates that we have at least one cointegration re-lation in each model. We set the results of those VAR-estimationsaside and proceed with the results of the estimations from the VARwith all variables and all series individually differentiated (table 7and table 8). Now a further cointegration test would not make anysense, since the individual series are stationary.

Considering the t-statistics, it is not evident that there are clear– in the sense of significant – relationships among all the variables.This is also true for the exchange rate, where we have some non-significant relationships in both models. With the wish to refine themodel, we want to find an instrument that makes it possible to findindication for enodgeneity.

4.2.1 Granger Causality

Since the VAR model relies on the assumption that all variables areendogenous, we want to apply an instrument for testing if this is

12

Page 14: Forecasting the Exchange Rate - Zentraler …homepage.univie.ac.at/robert.kunst/prog05bihawei.pdf · Forecasting the Exchange Rate A forecasting application with the exchange rate

Table 7: VAR Model set 1

∆e ∆mEU ∆yEU ∆pEU ∆sEU ∆mUS ∆yUS ∆pUS ∆sUS

L(∆e) 0.332 -0.034 -0.041 0.019 0.214 -0.023 0.007 0.000 -0.057t-stat 5.55 -1.51 -1.85 3.86 1.94 -1.46 0.47 0.02 -0.38L(∆mEU ) 0.265 -0.304 0.059 -0.005 0.106 -0.139 -0.066 -0.011 0.170t-stat 1.73 -5.19 1.03 -0.39 0.37 -3.48 -1.81 -0.89 0.44L(∆yEU ) -0.206 0.071 -0.429 0.018 -0.091 0.029 -0.006 0.003 -0.041t-stat -1.37 1.23 -7.61 1.44 -0.33 0.73 -0.16 0.23 -0.11L(∆pEU ) 0.888 -0.065 -0.274 0.448 0.986 0.271 -0.258 0.228 -0.039t-stat 1.23 -0.24 -1.02 7.64 0.74 1.45 -1.52 3.95 -0.02L(∆sEU ) -0.052 -0.044 0.018 0.004 0.040 0.010 -0.003 0.004 0.057t-stat -1.58 -3.47 1.43 1.51 0.66 1.22 -0.41 1.66 0.68L(∆mUS) 0.145 0.124 -0.123 0.039 -0.414 0.330 0.108 -0.025 1.040t-stat 0.67 1.50 -1.51 2.22 -1.04 5.84 2.11 -1.44 1.88L(∆yUS) -0.122 -0.068 0.013 -0.028 1.312 0.019 0.178 -0.011 3.005t-stat -0.48 -0.70 0.14 -1.34 2.81 0.29 2.98 -0.55 4.67L(∆pUS) 0.110 0.233 -0.430 0.166 2.715 -0.393 -0.337 0.417 0.485t-stat 0.16 0.86 -1.63 2.88 2.08 -2.14 -2.02 7.36 0.27L(∆sUS) 0.010 0.011 0.021 -0.002 -0.003 -0.024 0.031 0.006 0.242t-stat 0.42 1.18 2.30 -1.14 -0.07 -3.66 5.31 3.07 3.84Const. -0.004 0.007 0.004 0.001 -0.018 0.004 0.004 0.001 -0.019t-stat -1.32 6.20 3.86 4.42 -3.38 5.06 5.25 5.61 -2.55

R2 0.160 0.154 0.218 0.436 0.099 0.220 0.233 0.432 0.201R2 0.132 0.125 0.192 0.417 0.068 0.194 0.207 0.413 0.174SSR 0.157 0.023 0.022 0.001 0.533 0.011 0.009 0.001 1.013AIC -4.557 -6.477 -6.521 -9.573 -3.331 -7.250 -7.443 -9.603 -2.689SIC -4.425 -6.345 -6.389 -9.441 -3.199 -7.118 -7.311 -9.471 -2.557Note: Explanatory variables vertically, dependent horizontally.

at least possible. The Granger causality test does not allow to testfor endogeneity, but if a variable is endogenous it should Granger-cause the other and vice versa. If we can not find Granger causalitybetween a variable and the exchange rate, we will try a VAR withoutthe respective variable. The definite usefulness of this method willonly be visible when we compare the performance of this model’sforecast versus the full model in the last part, section 5.

Granger-Causality means that past values of one variable im-prove the forecast of another variable. The null hypothesis is thatthe coefficient in the VAR of the right hand side variable is zero.More exact, in terms of equation (2), if all the coefficients in Aij arezero, variable j does not Granger-cause variable i, since significancesuggests that the lags of j do not explain the variations in i. InTable 9 gives the results for both models.

Since the results of the Granger-causality tests are devastating

13

Page 15: Forecasting the Exchange Rate - Zentraler …homepage.univie.ac.at/robert.kunst/prog05bihawei.pdf · Forecasting the Exchange Rate A forecasting application with the exchange rate

Table 8: VAR Model set 2

∆e ∆s ∆m ∆p ∆yL(∆e) 0.331 0.256 -0.008 0.016 -0.058t-stat 5.64 1.48 -0.32 2.92 -2.35L(∆s) -0.013 0.216 -0.040 0.006 0.028t-stat -0.61 3.55 -4.30 3.16 3.31L(∆m) 0.106 0.254 -0.062 -0.015 0.142t-stat 0.78 0.64 -1.03 -1.21 2.52L(∆p) 0.568 0.110 -0.473 0.265 0.113t-stat 0.90 0.06 -1.68 4.53 0.43L(∆y) -0.150 0.407 0.066 0.013 -0.279t-stat -1.11 1.02 1.09 1.01 -4.95Const. 0.001 0.002 0.002 0.000 -0.001t-stat 0.58 0.50 3.33 0.17 -2.11

R2 0.130 0.055 0.081 0.146 0.155R2 0.114 0.037 0.064 0.130 0.139SSR 0.162 1.415 0.032 0.001 0.028AIC -4.551 -2.384 -6.163 -9.302 -6.300SIC -4.472 -2.305 -6.084 -9.222 -6.221Note: Explanat. variables vertically, dependent horizontally.

Table 9: Granger causality test results for ∆e

Variable Chi-sq df Prob.∆mEU 3.005 1.000 0.083∆yEU 1.878 1.000 0.171∆pEU 1.523 1.000 0.217∆sEU 2.487 1.000 0.115∆mUS 0.450 1.000 0.502∆yUS 0.232 1.000 0.630∆pUS 0.024 1.000 0.876∆sUS 0.179 1.000 0.672All 12.193 8.000 0.143

Variable Chi-sq df Prob.∆s 0.370 1.000 0.543∆m 0.610 1.000 0.435∆p 0.811 1.000 0.368∆y 1.227 1.000 0.268All 2.727 4.000 0.605Note: Model 1 in 1st block, model 2 in 2nd.

but we still want to follow the investigation if the reduction to mod-els with variables that have a Granger-cause effect improves our

14

Page 16: Forecasting the Exchange Rate - Zentraler …homepage.univie.ac.at/robert.kunst/prog05bihawei.pdf · Forecasting the Exchange Rate A forecasting application with the exchange rate

forecasts, we increase the significance levels for rejection as dramat-ically as necessary. Every model should include at least one variable.The minimum of all probability values in table 9 is 0.268. Hence, weinclude all variables with probability smaller than or equal to thisvalue. Table 10 and table 11 give the new estimations.

Table 10: VAR Model set 1 (reduced)

∆e ∆mEU ∆sEU ∆pEU

L(∆e) 0.322 -0.033 0.269 0.015t-stat 5.59 -1.48 2.48 3.22L(∆mEU ) 0.293 -0.293 0.137 0.003t-stat 1.97 -5.11 0.49 0.27L(∆sEU ) -0.052 -0.040 0.066 0.004t-stat -1.64 -3.27 1.11 1.52L(∆pEU ) 1.159 0.119 1.874 0.564t-stat 2.03 0.54 1.74 11.82Const. -0.004 0.007 -0.012 0.001t-stat -1.74 7.50 -2.48 6.12

R2 0.151 0.134 0.046 0.400R2 0.139 0.121 0.032 0.391SSR 0.158 0.023 0.564 0.001AIC -4.582 -6.490 -3.311 -9.548SIC -4.516 -6.425 -3.245 -9.482Note: see table 7.

Table 11: VAR Model set 2 (reduced)

∆e ∆yL(∆e) 0.347 -0.068t-stat 6.10 -2.77L(∆y) -0.153 -0.271t-stat -1.14 -4.69Const. 0.001 -0.001t-stat 0.69 -1.45

R2 0.126 0.096R2 0.119 0.090SSR 0.163 0.030AIC -4.567 -6.255SIC -4.528 -6.215Note: see table 7.

Not only that in the two reduced models the t-statistics improveda lot – almost all of the variables are significant now – also the fit as

15

Page 17: Forecasting the Exchange Rate - Zentraler …homepage.univie.ac.at/robert.kunst/prog05bihawei.pdf · Forecasting the Exchange Rate A forecasting application with the exchange rate

measured by R2 is better compared to the models in table 7 and 10with all variables, as well as both AIC and SIC are smaller. Table 12shows the results for the tests for Granger-causality. Two variablesfrom the reduced model from set 1 are significantly related to theexchange rate at the five percent level and the third, short terminterest rate, misses the ten percent level by a hair.

Table 12: Granger causality test results for ∆e in reduced models

Variable Chi-sq df Prob.∆mEU 3.895 1.000 0.048∆sEU 2.683 1.000 0.101∆pEU 4.121 1.000 0.042All 9.403 3.000 0.024

Variable Chi-sq df Prob.∆y 1.294 1.000 0.255Note: Model 1 in 1st block, model 2 in 2nd.

Bottom Line Thus, for this section, we consider our modellingstrategy as successful in so far that we improved the R2, AIC andSIC, and found more significant test results for Granger causality.

5 Forecast Comparison

As already announced, we now want to make our last step consistingof the forecast comparison, which is intended to be the watershedof model quality. For this exercise, we left the twelve months ofdata from 2003 as a validation sample, which was never touchedfor the model selection. We carry this step out by three criteria:visual inspection and comparison, ranking of the correlations of theforecast per model with the actual model and the ranking of themean squared error of prediction.

We used all models that we estimated. A few comments seemworthwhile mentioning: SES (1) corresponds to the application ofsingle exponential smoothing to the differentiated series, i.e. thesmoothing is done with the differentiated series. The series called”Random Walk” is a random walk of the first differences.

5.1 Plotted Forecasts

Figure 3 shows all forecasts we produced, which amounts to a totalof 23. The straight line at the top corresponds to SES, which is

16

Page 18: Forecasting the Exchange Rate - Zentraler …homepage.univie.ac.at/robert.kunst/prog05bihawei.pdf · Forecasting the Exchange Rate A forecasting application with the exchange rate

unsurprisingly our worst forecast. Just below are the Holt-Wintersseasonal forecasts. They are well above every observation of the ex-change rate for the whole forecast period. The same holds for theHolt-Winter without a seasonal component, though we can yet un-doubtedly conclude from visual inspection that for this time periodit is clearly better than the seasonal versions. For the next straightline, which matches double exponential smoothing, the situation isless clear. On the one hand, it looks as if it captures the averagequite well around its trend line, except for the first quarter, but thereno forecast fits well. On the other hand, there is a band of serieswhich is potentially following the moves of the raw data reasonably.We have to rely on other criteria to draw a definite conclusion. Onelast feature that attracted our attention are the performances of theforecasts in June 2003: between April and May one can see clearlyhow the band of forecasts’ moves lag one period behind in capturingthe sharp decline, and, moreover, then overshoot in June – excepttwo series – the SES of the differentiated exchange rate and the dy-namic forecast of the single equation model with the first, whole setof variables.

Figure 3: Line Graph of Forecasts

0.80

0.84

0.88

0.92

0.96

1.00

03M01 03M03 03M05 03M07 03M09 03M11

5.2 Mean Squared Error of Predictions

To get a measure that allows for a clear ranking of the forecastsuccess of each model, we calculated their mean squared errors ofpredictions and their correlations with the exchange rate for the

17

Page 19: Forecasting the Exchange Rate - Zentraler …homepage.univie.ac.at/robert.kunst/prog05bihawei.pdf · Forecasting the Exchange Rate A forecasting application with the exchange rate

forecast period. In table 13 both are shown, together with the rankof each model according to PMSE and correlation. We made noseparate comparison for dynamic and static forecasts. The justifi-cation is the surprising finding mirrored in the table: the dynamicforecast of the AR(1)-model in first differences is better than manyof the static forecasts – especially better than all of the multivariatespecifications. But the evaluation of the forecasts suggests that ifone wants more accurate values for the future than obtained fromthe univariate models one should step back once more in model com-plexity and choose the simplest method of exponential smoothing:SES of the differentiated series beats the rest in terms of minimumPMSE. Still, it is important to emphasise that only the SES in dif-ferences yields a strong competitor for the top places in the forecastevaluation, not the one in levels, which finishes last in the table.

Table 13: Forecasts’ PMSEs and Correlations and their Ranks

Rank RankPMSE Model PMSE Correlations Corr

1 SES I(1) 0.00056677 0.81673695 142 S: ARI(1,1) 0.00066086 0.80755415 193 S: ARIMA(1,1,2) 0.00067266 0.81170252 184 DES 0.00067527 0.85477754 25 D: ARI(1,1) 0.00067984 0.81689973 136 S: VAR Set 1 reduced 0.00068673 0.82726105 57 S: VAR Set 2 reduced 0.00069132 0.80424002 208 S: SingleEq Set 1 0.00069206 0.82442980 69 S: SingleEq Set 2 0.00069786 0.80320240 21

10 D: ARIMA(1,1,2) 0.00070177 0.81804057 1111 Random Walk 0.00072259 0.81673695 1512 D: VAR Set 2 reduced 0.00072782 0.81657004 1613 D: SingleEq Set 2 0.00072809 0.81283845 1714 S: VAR Set 1 0.00072829 0.82138410 815 S: VAR Set 2 0.00073800 0.79476647 2216 D: VAR Set 2 0.00074119 0.81735970 1217 D: VAR Set 1 reduced 0.00075049 0.81846788 918 D: SingleEq Set 1 0.00075400 0.82878847 419 D: VAR Set 1 0.00077482 0.81828878 1020 HW no season 0.00152359 0.85477801 121 HW additive 0.00209887 0.82892547 322 HW multiplicative 0.00332722 0.82295698 723 SES 0.01038184 - 23

Note: S is static, D dynamic forecast. SES forecast is constant.

What we can also see from our results is that the restrictions weimpose on our models by demanding a certain level of significance

18

Page 20: Forecasting the Exchange Rate - Zentraler …homepage.univie.ac.at/robert.kunst/prog05bihawei.pdf · Forecasting the Exchange Rate A forecasting application with the exchange rate

for the variables to remain tends to improve our forecasts, whereasthe nonsample restriction of forming the difference of variable fromEU minus its counterpart from the US is in our case not helpful.

Interpretation of the correlation ranks has to be done with care.We have been called attention to the serious problem of spuriouscorrelation of time series. That type of correlation can easily bestrong if a pair of series exhibits the same trend, whereas the changesfrom period to period may not be strongly related. Thus, we let thePMSE have the last word on declaring the best model, especiallysince it does not contradict the impression we have got from thefigure.

6 Conclusion

Figure 4 summarises giving the four series that capture the mostinteresting results, in our view. The best simple forecasting methodgives the best forecasts. Our VAR model of set 1 reduced to Granger-causing variables is the best multivariate model, if it is forecastedstatically. But even this static forecast is worse then the best uni-variate dynamic forecast. Several models are better than the randomwalk in first differences, which is sometimes used as a benchmarkmodel. The best model of our three groups – model free, univariateand multivariate – always beats the random walk.1

Figure 4: Line Graph of Selected Forecasts

0.80

0.84

0.88

0.92

0.96

1.00

03M01 03M03 03M05 03M07 03M09 03M11

Exchange RateRandom WalkS: VAR Set 1 reducedD: ARI(1,1)SES (1)

1All references are uncited: Enders (1995), E-Views 5 (2004), Kunst (2005)

19

Page 21: Forecasting the Exchange Rate - Zentraler …homepage.univie.ac.at/robert.kunst/prog05bihawei.pdf · Forecasting the Exchange Rate A forecasting application with the exchange rate

References

Enders, W. (1995), Applied econometric time series, John Wiley andSons, New York [u.a.].

E-Views 5 (2004), Help File. Quantitative Micro Software.

Kunst, R. (2005), Econometric forecasting. Lecture Notes from acourse of the Vienna doctorate program.

20