Forecasting exchange rates using cointegration models and intra-day data

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  • Journal of ForecastingJ. Forecast. 21, 151166 (2002)Published online 20 March 2002 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/for.822

    Forecasting Exchange Rates usingCointegration Models and Intra-day Data

    ADRIAN TRAPLETTI,1 ALOIS GEYER1* ANDFRIEDRICH LEISCH21 Department of Operations Research, Vienna University ofEconomics, Austria2 Department of Statistics and Probability Theory, ViennaUniversity of Technology, Austria

    ABSTRACTWe present a cointegration analysis on the triangle (USDDEM, USDJPY,DEMJPY) of foreign exchange rates using intra-day data. A vector autore-gressive model is estimated and evaluated in terms of out-of-sample forecastaccuracy measures. Its economic value is measured on the basis of tradingstrategies that account for transaction costs. We show that the typical sea-sonal volatility in high-frequency data can be accounted for by transformingthe underlying time scale. Results are presented for the original and themodified time scales. We find that utilizing the cointegration relation amongthe exchange rates and the time scale transformation improves forecastingresults. Copyright 2002 John Wiley & Sons, Ltd.

    KEY WORDS cointegration; exchange rates; high-frequency data; tradingstrategies

    INTRODUCTION

    Forecasting foreign exchange (FX) rates has been generally found to be a difficult task. Since theimportant paper by Meese and Rogoff (1983) numerous studies have shown that out-of-sampleforecasts of a variety of models are roughly equivalent to or only slightly better than forecasts froma simple random walk (see, for instance, Diebold et al., 1994; Satchell and Timmermann, 1995;Brooks, 1997). This supports the view that FX markets process available information in a highlyefficient way. Therefore, constructing any forecasting model requires us to utilize every potentialsource of information. The present paper attempts to do so in a number of ways.

    First, we make use of the fact that FX rates are traded almost continuously on a worldwidebasis. In order to account for this we use high-frequency intra-day data. We assume that at leastsome (possibly very short) time period is required to integrate new information into the differentmarkets, in particular, when considering how diverse the interpretation of information can be. When

    * Correspondence to: Alois Geyer, Department of Operations Research, Vienna University of Economics, Augasse 2-6,A-1090 Vienna, Austria. E-mail: Alois.Geyer@wu-wien.ac.at

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  • 152 A. Trapletti, A. Geyer and F. Leisch

    using intra-day data we expect that this provides the opportunity to detect (short-term) dependenciesintroduced by price adjustments. Provided that such adjustments occur very quickly this may explainwhy daily data is not suitable for building forecasting models.

    Second, we take into account that the trading activity on FX markets is not uniformly distributedover time. If activity is high, forecasts may have to be produced more frequently than duringquiet periods. Looking at the FX markets on a worldwide basis Dacorogna et al. (1996) haveshown distinct patterns in the daily trading activity. We confine ourselves to model the (almost)deterministic seasonality of FX market volatility that can be found during a day and during aweek. We use a deseasonalization procedure similar to the method proposed by Dacorogna et al.(1993) and apply a volatility-based time scale to the price generating process. We do not attemptto account for endogenously induced changes in the volatility of the markets.1 We rather rely onlyon the seasonality introduced by the pattern of opening times and intra-day activities of differentmarkets which can be assumed to stay rather constant over time.

    Third, we consider a triangle of FX rates, namely USDDEM, USDJPY and DEMJPY rates.We attempt to use the so-called triangular identity, which states that the ratio of two FX rates, calledthe main rates, must be equal to the cross rate. In principle, direct trading of the cross rate should beequivalent to carrying out the trade through the main rates. For daily data this condition is almostthe definition of the cross rate in terms of the main rates. However, looking at higher frequencies,the triangular identity does not continuously hold. In practical terms, it is necessary to allow forsome time to elapse such that new information can be processed by the market and be incorporatedinto prices. This relationship forces the three FX rates to be cointegrated, provided that each rateis integrated. Based on the work of Johansen (1991) we estimate a vector autoregressive (VAR)model, analyse the resulting cointegration term, and use the VAR model to generate forecasts. Forcomparison we also present forecasting results obtained from univariate models.

    We compute out-of-sample forecasts for several horizons and evaluate the performance of thehigh-frequency time series models relative to that of a simple random walk. The ultimate test forany forecasting model can be considered to be its economic value (cf. Satchell and Timmermann,1995). We therefore evaluate the profitability of the forecasts in terms of two trading strategies thataccount for transaction costs and differ with respect to the frequency of trades. Since the changein the time scale is a major element of our approach, we present results for models based on the(original) physical time scales and on the modified time scale.

    In the next section we introduce high-frequency FX rates and briefly summarize the deseason-alization procedure and its effects. The results of the VAR analysis appear in the third section. Inthe fourth section we outline the experimental design and present the results of the forecasting andtrading experiments. The final section offers a summary and conclusions.

    HIGH-FREQUENCY EXCHANGE RATES

    DataThe basic data for this study are the exchange rate quotes for USDDEM, USDJPY, and thecross rate DEMJPY. The data set covers the period from 1 October 1992 to 30 September 1993on a tick-by-tick basis and contains 1,472,241 (USDDEM), 570,813 (USDJPY), and 158,978

    1 The corresponding time scale is called intrinsic by Dacorogna et al., (1996).

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  • Forecasting Intra-day Exchange Rates using Cointegration Models 153

    (DEMJPY) data records. Each record consists of the time the record was collected, the bid andask price, the identification of the reporting institution, and a validation flag.

    The price values are computed as the average of the log of bid and ask prices. The returns are theprice changes over a one-hour time interval, and the volatility is the absolute value of the returns.The one-hour interval was chosen in order to keep the intra-day character of the time series, but toavoid a bidask spread of the same order as the returns. To construct an equally spaced time seriesfrom the original tick-by-tick series, we take the most recent valid price record as a proxy for thecurrent price record. We have also tried a linear interpolation between the two neighbouring pricesbut the forecasting and trading results were not strongly affected. We decided to the use the mostrecent record because it conforms more closely to an out-of-sample forecast situation.

    Time scaleOne of the major characteristics of high-frequency data is the strong intra-week and intra-dayseasonal behaviour of the volatility. This is shown, for instance, by the autocorrelation function ofvolatility in physical time for the USDDEM rate in Figure 1.

    A data generation process with strong seasonal distribution patterns is not stationary. Thereforeit is necessary to control for these seasonalities before fitting any time series model. However, ourprimary goal is not only to improve the efficiency of the parameter estimates of the model but alsoto utilize the underlying regularities in the trading activities to obtain better forecasts and resultsof the trading strategies.

    A very promising approach to filter these seasonal patterns is to apply a new time scale to theprice generating process similar to the one suggested by Dacorogna et al. (1993). In probabilitytheory this is formally equivalent to subordinated process modelling. In this model the returnsfollow a subordinated process in physical time and are non-stationary. However, they follow thestationary parent process on another time scale which we call operational.

    0 100 200 300 400

    0.1

    0.0

    0.1

    0.2

    0.3

    0.4

    Lag

    ACF

    Figure 1. Correlogram for USDDEM returns and volatilities for the period from 1 October 1992 to 30September 1993. Autocorrelations of returns are mainly inside the asymptotic 95% confidence bounds forwhite noise, whereas correlations of volatility exhibit a strong seasonality

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  • 154 A. Trapletti, A. Geyer and F. Leisch

    The first step towards obtaining such a time scale is to omit the weekends. Thereby the original(physical) time scale is turned into a so-called business time. The business time stops while physicaltime runs through the weekend. It therefore still contains intra-weekly and intra-daily seasonalities(mainly the hour-of-the-day effects). To control for the remaining seasonal patterns, we applyanother time scale to the business time scale which is computed by stretching highly volatilemarket periods and shortening less volatile periods. Based on the transformation procedure explainedin Appendix A a one-hour time interval in operational time corresponds to about 15 minutes inphysical time when the volatility in the market is high (for instance, in the afternoon according toGreenwich Mean Time). On average, however, time intervals on both scales are equally long.

    The three FX series are constructed by sampling equally spaced in operational time with asampling frequency corresponding to one hour time intervals. During a highly volatile period thisimplies, for instance, that four consecutive hourly observations on the operational time scale aretaken from a period of about one physical hour in the original database. As noted above, the mostrecent valid price record is taken as the current price.

    The effect of changing the time scale can be seen in Figure 2.2 Although conditional heteroscedas-ticity is still present, most of the seasonal effects have been removed. We find a significantly negativeautocorrelation at lag one. Bollerslev and Domowitz (1993) find negative autocorrelations in five-minute returns of USDDEM quotes and attribute these to the nonsynchronous construction of theprice series. Zhou (1996) explains the negative autocorrelation by the noise in prices. He arguesthat the returns of high-frequency data are effectively the difference between two noise series,with an first-order autocorrelation approaching 0.5 as the observation frequency increases. Fortick-by-tick data of the USDDEM returns Zhou reports a correlation of 0.45. For the one-hour

    0 100 200 300 400

    0.1

    0.0

    0.1

    0.2

    0.3

    0.4

    Lag

    ACF

    Figure 2. Correlogram for USDDEM returns and volatilities in operational time. Autocorrelations of returnsare mainly inside the asymptotic 95% confidence bounds for white noise, whereas correlations of volatilityare mainly outside the bounds

    2 Since the results for the two other rates are very similar to those of the USDDEM rate, they are not reported here.

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  • Forecasting Intra-day Exchange Rates using Cointegration Models 155

    interval we find a value of only about 0.05 (0.06 and 0.02 for USDJPY and DEMJPY,respectively).

    Based on the Akaike information criterion and using only the first half of the sample (n D 4387)we have selected univariate AR(2) models for the returns3 to compute forecasts, although weobtained very similar results (not reported below) with AR(1), MA(1) and MA(2) models.

    COINTEGRATION ANALYSIS

    The cointegration analysis is done for the first half of the sample (n D 4387). Based on augmentedDickeyFuller tests we find that all series are I1 which is hardly surprising. We then estimate akth-order VAR of the three series by the maximum likelihood procedure following Johansen (1991).The ordering of the variables is (USDDEM, USDJPY, DEMJPY). Allowing for linear trendsand for a varying number of independent unit roots the VAR system can be written as

    Xt Dk1iD10iXti C5Xtk C m C et 1

    where et are NID.0,3/ with the 3 3 covariance matrix 3. Further parameters are the 3 1 vectorm, the 3 3 matrices 0i, and the 3 3 matrix 5. The rank of the latter is equal to the number ofcointegrating vectors.

    We select the lag length in (1) using the Akaike information and final prediction error criteria(AIC and FPE). The unrestricted model (1) with Rank5 D 3 is estimated using the first half ofthe sample. AIC and FPE both yield an optimal lag length of Ok D 4.

    On the basis of the plots of the series (see Figure 3) a model without a linear trend was assumed.The results of the trace and maximum eigenvalue test statistics are given in Table I (cf. Johansenand Juselius, 1990). The support for the existence of exactly one cointegrating vector is strong.Moreover, assuming the absence of a linear trend is data consistent. The likelihood ratio test statisticis LR D 5.05 which is asymptotically 22 and, thus, not significant.

    Table II presents the normalized estimate of the cointegrating vector b D b0,b000 where 5 Dab0.a and b are 3 r matrices, and m D ab00.b determines the error correction mechanism of themodel. It is natural to assume that all rates of the FX triangle enter this mechanism with the sameweight. Given the chosen normalization we consider the hypothesis

    b D 1, 1, 1, 00 j 2

    where j is a real-valued parameter. The likelihood ratio test is given by LR D 6.89 which shouldbe compared with the quantiles of the 23 distribution. It is not significant and, therefore, hypoth-esis (2) is accepted with Oj D 2.612 103. We therefore maintain the model (1) and hypothesis (2)using Ok D 4 and Or D 1. The estimates of the other parameters of the model are presented inAppendix B.

    The autocorrelation function for the error correction term 1, 1, 1Xtk is plotted in Figure 4.While the process is apparently I0, it is immediately clear that this process is not uncorrelated.

    3 These are equivalent to ARIMA(2,1,0) models in the rates.

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    0.35

    0.40

    0.45

    0.50

    0.55

    USD

    DEM

    4.65

    4.70

    4.75

    4.80

    USD

    JPY

    4.1

    4.2

    4.3

    4.4

    DEM

    JP

    Y

    0 2000 4000 6000 8000Figure 3. The price series sampled equally spaced in operational time with a sampling frequency correspondingto one hour time intervals for the period from 1 October 1992 to 30 September 1993

    Table I. Trace and maximum eigenvalue test statistics for variousvalues of the cointegration rank r with critical values for a 5%significance level

    Trace Critical value max Critical value

    r 2 3.35 9.09 3.35 9.09r 1 14.01 20.17 10.66 15.75r D 0 608.98 35.07 594.97 21.89

    Any information about a disequilibrium in the market that is less than 10 operational hours oldhas a significant effect on the current rates. This implies that non-trivial forecasts of the FX ratesshould be possible. The relatively rapid decay of the autocorrelation coefficients indicates the purelyintra-day character of this phenomenon.

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  • Forecasting Intra-day Exchange Rates using Cointegration Models 157

    Table II. The estimated cointegrating vector b0

    normalizedon USDDEM

    USDDEM USDJPY DEMJPY Constant

    1.000 0.999 1.000 0.005

    0 5 10 15 20 25 30

    0.0

    0.1

    0.2

    0.3

    0.4

    Lag

    ACF

    Figure 4. Correlogram for the error correction term 1, 1, 1Xtk with asymptotic 95% confidence boundsfor white noise estimated from observations 1 to 4387

    The average speed of adjustment towards equilibrium is determined by the corresponding estimateof the adjustment parameters

    Oa D 0.164, 1.764 104, 0.3430

    The estimated coefficients indicate a much faster reaction for the DEMJPY cross rate and for theUSDDEM rate than for the USDJPY rate. This suggests the conclusion that arbitrage trades aretypically not carried out using the USDJPY rate.

    The results of the cointegration analysis indicate that the constraint on the FX rates introducedby the triangular relationship implies a single cointegrating vector that controls the levels of theFX rates. The adjustment of the rates induced by the cointegration term suggests a potential fornon-trivial short-term forecasts. We finally note that fitting an unrestricted VAR in the rates pro-vides forecasting results (not reported) that can hardly be distinguished from the (restricted) errorcorrection model estimated above.

    PREDICTION

    Experimental designOut-of-sample forecasts are performed on the second half of the sample. We start with predictingobservation 4388 while fixing the parameters at their estimated values. At each time period one-,

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    two-, four-, and eight-step-ahead forecasts are computed by iterating forward through time. Thehorizons correspond to about 15 minutes, 30 minutes, 1 hour and 3 hours in physical time if theseasonal volatility in the market is high. The average forecasting horizon in operational time,however, is the same as in physical time.

    We evaluate the forecasts in terms of the mean squared error (MSE) and the mean absolute error(MAE). We report the performance of the AR and VAR models relative to the random walk. Theseaccuracy measures are, however, parametric in the sense that they rely on the desirable propertiesof mean and variance. Therefore we also apply a distribution free procedure. A measure havingthis desirable property is the percentage of forecasts in the right direction which will be calledthe direction quality (DQ). In conjunction with the DQ the signal correlation (SC) between theforecasting signal and the actual price signal is computed.

    In order to avoid decisions based on point estimates of the forecast accuracy, we test for theequality of two forecasts as proposed by Diebold and Mariano (1995). We give a brief explanationof how this test works.

    Consider two scalar forecasts producing the errors fe1t, e2tg, t D 1, . . . , T. The null hypothesis ofthe equality of the expected forecast performance is E[ge1t] D E[ge2t], or E[gdt] D 0, wheredt D ge1t ge2t and g is some loss function. It is natural to base a test on the sample meand D T1TtD1 dt of the loss-differential series. If the loss-differential series is weakly stationaryand short memory, we have the asymptotic result d

    d! N, 2, where D E[gdt], 2 DT1

    1hD1 h, and h is the autocovariance function of fdtg. The obvious test statistic for

    testing the null is then S D d/ O, where O is a consistent estimator of . We also computed amodified version of the above test as suggested by Harvey et al. (1997). Since the results are verysimilar they are not reported here.

    Choosing g as the square function or as the absolute value function results in the MSE or theMAE, respectively. To evaluate the DQ we set e1t to one if the forecast is in the right direction,and to zero otherwise. Zero changes either of the forecast or of the actual price are omitted.The DQ of the AR and VAR models is compared to a coin flip with probabilities 1/2 for eachdirection. Hence, e2t D 0.5 and g is chosen as the identity function. For the SC the forecast errorsare chosen as e1t D XtXt/

    TtD1Xt2

    TtD1Xt2 and e2t D 0 for the benchmark of no

    signal correlation. Xt and Xt are observed and predicted returns, respectively.These measures of forecasting performance are primarily of academic interest. However, they

    neither provide necessary nor sufficient conditions for forecast value in economic terms, i.e. a prof-itable trading strategy yielding positive returns (cf. Satchell and Timmermann, 1995). To evaluatethe forecast value also in that sense we implement two simple trading strategies. In the first strat-egy a trader takes a long position if the forecast direction is up and a short position otherwise.However, this trader is very busy and changes the position frequently. Therefore, we implement amore realistic second strategy which accounts for transaction costs. Since actual transaction costsare not available we used the quoted bidask spreads for simplicity.

    The motivation of the second strategy is to combine the direction forecasts with a technicaltrading rule in order to reduce the trading frequency. If the current position is neutral, then thedirection forecast is used to take a position. We assume that a position can be taken at the currentlyobserved rate (i.e. the rate at the time when the forecasts are made). In reality it takes some timeuntil the trade is actually executed. Thus, one would not obtain the current price but a price thatprevails later. Given the high liquidity of markets and provided that the trading signals are quickly

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  • Forecasting Intra-day Exchange Rates using Cointegration Models 159

    converted to place orders we expect that it takes not more than a minute (or even less) to executethe trade. Thus, the current price may be considered to be an acceptable proxy for the actual price.

    A position is held until a so called stop-loss rule is violated. In an ad-hoc way we usedtwice the transaction costs as a limit. If cumulative negative returns exceed this limit the positionis immediately cleared. Next, the direction forecast is used again and so on. Transaction costsare subtracted from returns whenever a position is changed. For this strategy, winning runs areunlimited. On the other hand, long negative runs are not possible, since they are stopped out.

    The trading strategies are carried out in operational time. This implies that depending on the fixed,seasonal stretching and contracting of the time scale the trader is relatively more or less frequentlyacting (or ready to act) than if the forecasts were generated in physical time. According to themapping of the time scales, traders are more frequently evaluating the trading rule on, for instance,Friday afternoon (15 : 30 GMT) than on Friday morning (10 : 00 GMT). Using this design raisesthe question whether trading (and modeling) in business time yields different results. Therefore wehave also estimated the models and executed the trading strategies in business time.

    ResultsThe results of the out-of-sample forecasting exercise are reported in Tables III and IV. Measuringthe forecasting performance in terms of direction quality (DQ) and signal correlation (SC) theVAR model is significantly better than the random walk for the one- and two-hour horizons. ForUSDJPY and DEMJPY we find significant improvements even four- and eight hours ahead. TheAR(2) model performs clearly worse than the VAR model. Only in a few cases is the AR(2) modelsignificantly better than the random walk, mainly for the one-hour horizon.

    For the VAR model the percentage of forecasts in the right direction is between 53.0% for theUSDDEM and 55.9% for the DEMJPY for the one-hour horizon. For the two-hour forecasts theDQ is between 51.6% (USDDEM) and 54.1% (USDJPY). The corresponding SC varies from6% to 15% (one hour) and from 2.7% and 13.3% (two hours).

    Table III. Out-of-sample forecasting results of univariate AR(2) models for theobservations 4388 to 8774. The results are given in terms of the MSE, the MAE, theDQ, and the SC relative to the random walk

    FXrate

    Forecasthorizon(hours)

    MSE(108

    MAE(106

    DQ(102

    SC(102

    USDDEM 1 0.47 1.90 2.70 6.992 0.05 0.95 0.43 3.194 0.06 0.93 0.43 1.838 0.06 0.92 0.43 3.87

    USDJPY 1 0.72 4.42 5.09 5.672 0.18 0.42 0.82 2.414 0.03 0.32 0.79 0.058 0.03 0.32 0.79 0.11

    DEMJPY 1 0.55 1.43 0.49 2.102 0.66 1.94 0.13 3.744 0.00 2.22 0.39 1.588 0.01 2.24 0.39 3.53

    Significant at the 5% level. Significant at the 10% level.

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  • 160 A. Trapletti, A. Geyer and F. Leisch

    Table IV. Out-of-sample forecasting results of the VAR(4) model for the observations 4388 to 8774. Theresults are given in terms of the MSE, the MAE, the DQ, and the SC relative to the random walk

    FXrate

    Forecasthorizon(hours)

    MSE(108)

    MAE(106)

    DQ(102)

    SC(102)

    USDDEM 1 0.31 2.15 3.03 6.012 0.58 4.96 1.58 2.914 3.58 11.02 0.24 0.108 9.53 17.89 1.32 1.22

    USDJPY 1 1.55 6.22 5.22 8.512 0.60 2.92 4.13 2.744 2.25 0.54 1.73 0.508 1.26 1.90 1.08 1.25

    DEMJPY 1 4.78 0.79 5.88 15.522 6.40 1.68 3.80 13.324 5.55 1.26 2.52 9.068 1.42 8.59 1.38 9.06

    Significant at the 5% level. Significant at the 10% level.

    To a certain extent the superior performance for short horizons may be due to the negativeautocorrelations noted above. In particular since the quality of the forecasts decreases continuouslyas the forecasting horizon increases. Nevertheless, the DQ for an 8-hour forecast of the cross-rateDEMJPY is 51.4%. Furthermore, the ranking of the rates from lower to higher predictability usingthe VAR model is USDDEM, USDJPY, and DEMJPY.

    Concerning the MSE and the MAE, the results are not that clear. For the shorter horizons thereseems to be some predictability, but not for all rates and not for both models. For the longerhorizons the random walk significantly outperforms the VAR in case of the USDDEM rate. Incontrast, for the DEMJPY rate the long-horizon forecasts of the VAR are better than the randomwalk.

    We have two explanations for these results. First, the underlying distributions of the returns havefat tails. This means that the MSE and the MAE performances depend very much on a few largeobservations, making the results very unstable. Second, the MSE and the MAE loss series have longmemory. For instance, for one of the MSE series, we estimated the order of fractional integration tobe 0.22. When making l-step predictions the situation becomes even worse. Thus, the asymptoticsfor the sample mean could change dramatically as shown in Taqqu (1975). Consequently, oneshould not over-interpret the results from the MSE and the MAE significance tests. Concerning theDQ and the SC the situation is less critical since the magnitude of the forecasts is irrelevant.

    We now turn to the economic significance of the forecasts. For that purpose we present results onlyfor the VAR model.4 The cumulative return of the first trading strategy without taking transactioncosts into account is shown in Figure 5. The growth path mainly reflects the forecasting power ofthe VAR model. However, frequent trading is necessary to achieve this return. Taking transactioncosts into account yields a negative overall net-return (not shown). The second trading strategy ismore successful as shown in Figure 6. Although we have not optimized the stop-loss rule in anysense, the cumulated net-returns are positive. However, Table V shows that the average net return

    4 Results for the AR(2) model are available upon request.

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  • Forecasting Intra-day Exchange Rates using Cointegration Models 161

    is significantly positive only for the DEMJPY rate. On the other hand, it is a well-known factamong of FX traders that the actual transaction costs are much smaller than the quoted bidaskspreads. Therefore, our estimates of transaction costs can be considered conservative estimates ofthe actual costs and potential profits may consequently be even higher.

    We have emphasized the importance to account for the seasonal volatility patterns in the data.This raises the question to what extent the trading results are affected by using the operationaltime scale. For that purpose we have estimated the VAR model and executed the second tradingstrategy in business time, too. Table V shows that net returns obtained in business time are loweron average except for the USDJPY rate. None of the returns is significantly positive, however.

    CONCLUSIONS

    We have presented a cointegration analysis on the triangle (USDDEM, USDJPY, DEMJPY)of foreign exchange (FX) rates using high-frequency data. We have evaluated the out-of-sampleforecasting power of a VAR model in both statistical and economic terms. We find that focusingon the FX triangle provides additional information derived from the cointegration relation betweenthe rates that allows for improving short-term forecasts. Based on a simple trading strategy thataccounts for transaction costs we show that these forecasts may even be economically significant.We also found that changing the time scale such that it accounts for the typical seasonal volatilityin high-frequency data improves the results of the trading strategy.

    APPENDIX A TIME SCALE TRANSFORMATION

    The basis for the computation of the operational time scale is the empirical scaling power lawfound by Muller et al. (1990). It expresses the average volatility V over some time interval t asa function of the length of the time interval t

    V D c te A1

    Since we focus on multivariate modelling, V is computed as the volatility of a weighted averageof the underlying time series. c and e are estimated from a regression of logV on logt anddepend on the underlying time series and the averaging weights. The results of the regression arereported in Table AI.

    Table V. Out-of-sample results on testing for positive average hourly net returns ofthe second trading strategy. Results are presented for estimated VAR models andtrading strategies in operational and business time

    Mean (105) t-value P>toper. busin. oper. busin. oper. busin.

    USDDEM 1.89 0.52 0.953 0.268 0.170 0.395USDJPY 2.08 2.28 0.819 0.907 0.206 0.183DEMJPY 5.14 3.27 2.097 1.373 0.018 0.085

    Note: oper. D operational time scale; busin. D business time scale.

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    0.00

    0.05

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    0.30

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    DEM

    0.0

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    0.6

    0.7

    DEM

    JP

    Y

    5000 6000 7000 8000Figure 5. Cumulative returns (ignoring transaction costs) of the first trading strategy for the out-of-sampleobservations 4388 to 8774

    To express time as a function of the volatility we invert the scaling power law as suggested inSchnidrig and Wurtz (1995). Application to each hour of an average week gives the length of thishour in operational time

    topi D(

    Vti

    c

    )1/ei D 1, . . . , 168 A2

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  • Forecasting Intra-day Exchange Rates using Cointegration Models 163

    0.00

    0.05

    0.10

    USD

    DEM

    0.00

    0.02

    0.04

    0.06

    0.08

    0.10

    0.12

    0.14

    USD

    JPY

    0.00

    0.05

    0.10

    0.15

    0.20

    DEM

    JP

    Y

    5000 6000 7000 8000Figure 6. Cumulative net-returns of the second trading strategy for the out-of-sample observations 4388 to8774

    where Vti is the average volatility over the ith hour of the week. topi is the resulting lengthof the ith hour in operational time.

    The complete time mapping is found by interpolating between these 168 tabulated points topi ,ti and synchronizing the two time scales each week. We have used a cubic spline interpolationprocedure as described in Press et al. (1995). Note that business time instead of physical time wasused as the underlying time scale for these computations. The time mapping from business time tophysical time is shown in Figure AI.

    Copyright 2002 John Wiley & Sons, Ltd. J. Forecast. 21, 151166 (2002)

  • 164 A. Trapletti, A. Geyer and F. Leisch

    Table AI. Parameter estimates forthe scaling power law. Thenumbers in parentheses areestimated standard errors. t ismeasured in seconds

    logc Oe R2 fit11.703 0.521 0.998

    (0.055) (0.006)

    0 50 100 150

    0

    50

    100

    150

    Operational Time

    Busi

    ness

    Tim

    e

    Figure AI. Time mapping from operational time to business time

    The averaging weights are chosen in such a way that the sum of the seasonal fluctuations inoperational time over all time series becomes minimal. We use the root mean square error ofthe average volatility around the mean (RMSEV) as a measure of the seasonal fluctuations (seeDacorogna et al., 1993). Table AII shows the averaging weights which minimize the RMSEVcriterion. Furthermore, the corresponding RMSEV in operational time are reported as percentagesof the RMSEV in physical time.

    Table AII. The averaging weights which minimize the sum of theseasonal fluctuations in operational time over all time series. TheRMSEV-ratios are computed as the ratio of the RMSEV inoperational time to the RMSEV in physical time

    USDDEM USDJPY DEMJPY

    Weights 0.68 0.13 0.19RMSEV-ratio 0.29 0.33 0.34

    Copyright 2002 John Wiley & Sons, Ltd. J. Forecast. 21, 151166 (2002)

  • Forecasting Intra-day Exchange Rates using Cointegration Models 165

    APPENDIX B VAR PARAMETER ESTIMATES

    The maximum likelihood procedure gives the following estimates:

    O01 D

    0.129 0.089 0.0960.040 0.043 0.0430.156 0.209 0.169

    0.029 0.031 0.0310.440 0.420 0.4520.035 0.037 0.037

    O02 D

    0.050 0.070 0.0340.050 0.052 0.0510.073 0.095 0.072

    0.036 0.038 0.0380.460 0.410 0.4720.043 0.045 0.045

    and

    O03 D

    0.096 0.102 0.1230.055 0.056 0.0560.086 0.087 0.073

    0.040 0.041 0.0410.356 0.337 0.3350.048 0.049 0.049

    where the numbers in parentheses are estimated standard errors. Furthermore,

    O5 D

    0.164 0.164 0.1640.037 0.037 0.037

    1.764 104 1.764 104 1.764 1040.027 0.027 0.0270.343 0.343 0.3430.032 0.032 0.032

    and

    O3 D[ 1.940 0.653 1.121

    0.653 1.035 0.3061.121 0.306 1.465

    ] 106

    ACKNOWLEDGEMENTS

    The data for this study was kindly distributed by Olsen & Associates, Research Institute in Zurich,Switzerland, for the First International Conference on High Frequency Data in Finance.

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    Authors biographies:Adrian Trapletti is research consultant for Erste Bank, Vienna. He holds a PhD in Econometrics fromthe Vienna University of Technology. His research interests include non-linear time series analysis and thedevelopment of financial forecasting and trading models.

    Alois Geyer is associate professor at the Vienna University of Economics, Department of Operations Research.One of his major research areas is time series analysis. He is mainly interested in empirical analyses requiredfor decision making in financial applications.

    Friedrich Leisch is assistant professor at the Vienna University of Technology, Department of Statistics andProbability Theory. His main research interests are statistical computing, biostatistics, econometrics, clusteranalysis and time series analysis. A special focus are software development and statistical applications ineconomics, management science and medical research.

    Authors addresses:Adrian Trapletti, Department of Operations Research, Vienna University of Economics, Augasse 2-6, A-1090Vienna, Austria.

    Alois Geyer, Department of Operations Research, Vienna University of Economics, Augasse 2-6, A-1090Vienna, Austria.

    Friedrich Leisch, Department of Statistics and Probability Theory, Vienna University of Technology, WiednerHauptstrasse 8-10, A-1040 Vienna, Austria.

    Copyright 2002 John Wiley & Sons, Ltd. J. Forecast. 21, 151166 (2002)

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