linear and non-linear (non-)forecastability of high-frequency exchange rates

Linear and Non-linear(Non-)Forecastabilityof High-frequencyExchange Rates

CHRIS BROOKS

University of Reading, UK

ABSTRACT

This paper forecasts Daily Sterling exchange rate returns using variousnaive, linear and non-linear univariate time-series models. The accuracy ofthe forecasts is evaluated using mean squared error and sign predictioncriteria. These show only a very modest improvement over forecasts gener-ated by a random walk model. The Pesaran±Timmerman test and a com-parison with forecasts generated arti®cially shows that even the best modelshave no evidence of market timing ability.

KEY WORDS nonlinear; time series; forecasting; exchange rates

INTRODUCTION

``The ultimate test of an economic model . . . comes with checking its predictions'' (Christ, 1951).Two seminal papers by Meese and Rogo� (1983, 1986) have highlighted the poor out-of-sampleforecasting performance of structural models of exchange rates, such as the Dornbusch (1976)sticky price monetary model, and the Hooper±Morton (1982) model. Speci®cally, Meese andRogo� showed that structural post-sample forecasts could be bettered by a simple random walkmodel of exchange rate movements. This seemingly paradoxical result created a great interest intesting re®ned structural models for di�erent currencies over di�erent time periods, since itappeared that a fundamental raison d'eÃtre of structural modelling had disappeared. In fact, asimilar result had been observed previously by Wallis (1982), who argued that model-basedforecasts may have larger MSE than time-series forecasts, in contrast to the common perceptionat the time.More recent studies have yielded a mixed set of results. Boughton (1987), for example,®nds that while monetary models perform poorly, better results are obtained using portfoliobalance models which are able to outperform the random walk. Finn (1986) and Boothe andGlassman (1987) ®nd a similar result for the late 1970s and early 1980s using RMSE criteria,although the structural equations yielded higher pro®ts than the random walk or forward ratepredictors.

There is some evidence, however, that if structural models are generalized to include laggedadjustment mechanisms, their forecasts can be somewhat improved (Edison, 1991; Somanth,

CCC 0277±6693/97/020125±21 Received June 1996# 1997 by John Wiley & Sons, Ltd. Revised October 1996

Journal of Forecasting, Vol. 16, 125±145 (1997)

1986). There is also some suggestion that structural models may be improved by allowing theirparameters to vary over time (De Arcangelis, 1992; Schinasi and Swamy, 1989). Furthermore,while time-series models may be superior in the short run, structural and error correction modelsoften have appeal for prediction over longer time periods (Hogan, 1986; Chinn and Meese, 1995;Kim and Mo, 1995). Kim and Mo cite this as evidence that foreign exchange dealers may beusing `chartist' techniques for prediction in the short run, and analysing `the fundamentals' in thelong run. Fundamentals, such as money supply and real incomes, do appear to have morepredictive power in the longer term (see for example Mark, 1995).

Linear time-series models (such as the Box and Jenkins (1976) ARIMA models or Brown's(1959) exponential smoothing model) have not proved particularly useful for forecasting®nancial asset returns either (see, for example, Kuan and Lim, 1994). A natural question whichfollows, therefore, is whether (more complex) non-linear models can do better. All the entries in arecent forecasting competition, which included ®nancial and other types of data, were allessentially non-linear in nature (see Weigend, 1994), and a recent paper using the same data asthat used here (Brooks, 1996) showed very strong evidence of non-linear dependence.1 Theseobservations provide strong motivation for assessing the forecastability of exchange rate returnsusing both linear and various non-linear time series models with a common set of objectivecriteria, which is the central theme of this paper.

The outline of the rest of the paper is as follows. The next section gives a brief description ofthe data. A methodological description of each of the models used for forecasting and how theymight be estimated is given in the third section. The fourth section discusses di�erent metricswhich may be used to evaluate the accuracy of the forecasts, and the ®fth section highlights themost important features of the results. The ®nal section presents the conclusions and tentativelysuggests some directions for future research.

THE DATA

The analysis presented here is based on just over twenty years of daily mid-price spot exchangerate data, denominated in sterling. The sample period taken covers the whole of the post-BrettonWoods era until the present day from 2 January 1974 until 1 July 1994 inclusive. Three currenciesare analysed, namely the French franc/pound, the German mark/pound, and the US dollar/pound. The raw exchange rates were transformed into log-returns giving a series of 5191observations.2 Some summary statistics and the results of non-linearity tests are presented inTable I.

It is evident from Table I that all three series show evidence of leptokurtosisÐthat is, they aremore peaked at the mean and with fatter tails than a corresponding normal distribution with thesame mean and variance, although none of the series is highly skewed. Hence the excess kurtosisleads the Bera±Jarque normality test to extreme rejections in all three cases. An augmentedDickey±Fuller test shows that the log returns do not posses a further unit root. Also given inTable I are a small sample of the non-linearity tests used in Brooks (1996), which show verystrong evidence of non-linearity, particularly in the cases of the German mark and US dollar.

126 Journal of Forecasting Vol. 16, Iss. No. 2

1 An extremely small subset of these results is presented in Table I.2 The data were tested for the presence of unit root non-stationary using the Dickey-Fuller (Dickey and Fuller, 1979;Fuller, 1976) Phillips±Perron (Phillips, 1987; Perron and Phillips, 1987; Phillips and Perron, 1988) and Sargan±Bhargava (Sargan and Bhargava, 1983, Bhargava, 1986) tests. The levels data and the log-levels data were found in allcases to be strongly I(1), but there was no evidence of non-stationarity in the returns series.

MODELS FOR FORECASTING AND CALCULATION OF FORECASTS

The selection of models for consideration and comparison is somewhat arbitrary and necessarilynot exhaustive. In each case, models and forecasting equations are given for the log-returns, asopposed to the levels or the log-levels. While forecasts for the latter are probably of considerablymore use, the issue is greatly complicated by the issue of possible non-stationarity and non-invertibility of models in the levels. Furthermore, it should be possible to obtain forecasts for thelevels based on those values predicted for the returns, and previous values of the levels. It is alsopossible to construct intervals for the predictions, but these can only be computed by assumingsome underlying distribution (such as asymptotic normality) for the forecast errors. But thisassumption is likely to be untenable for most ®nancial series (see, for example, the Bera±Jarquestatistics in Table I), and hence only point forecasts are calculated here.

The models are estimated over the ®rst 4800 observations with the remainder (391) being leftentirely for post-sample forecast evaluation. The modelling and forecasting procedure is notcarried out recursively, that is either by adding an observation and then recalculating theparameters of the model, or by modelling using a moving window. One reason for not using thesetechniques is that estimation of non-linear models still remains a relatively CPU-intensiveexercise which renders recursive estimation infeasible. In any case, adding another 10 or 50or even a 100 observations is hardly likely to signi®cantly alter the model parameters when wealready have 4800. The notation used in this study is as follows. The total number of observ-ations is denoted as T, and the number of observations used for in-sample modelling as T1 . ThenT±T1 observations are retained as a hold-out sample. Let the actual value of the series at timet � n, and an n-step-ahead forecast of the value of that series made at time t be written as xt�nand ft,n respectively. We can also write

f t;n � E�xt � njOt� �1�

Chris Brooks High-frequency Exchange Rates 127

Table I. Summary statistics and non-linearity tests for the French franc, German mark and US dollarreturns

Series French franc German mark US dollar

Mean ÿ0.0058 ÿ0.0181 ÿ0.0079Variance 0.2645 0.244 0.4190Skewness ÿ0.0346 ÿ0.6079 ÿ0.0693Kurtosis 7.7472 6.1814 3.5162Normality 12982c 8584c 2678c

LB QX(5) 19.5c 17.3c 38.0c

ADF(5) ÿ29.63c ÿ68.80c ÿ66.81cARCH(5) 196.70c 215.67c 219.70c

BDS 1.076 1.217 5.062c

Tsay 0.0057 7.4116c 5.2613b

RESET 2.119a 8.9523c 1.8252

Notes: Kurtosis quoted is excess kurtosis; ADF(5) is an augmented Dickey±Fuller unit root test with ®ve lags to soak upautocorrelation; Normality is the Bera±Jarque (1981) normality test which is asymptotically distributed as a w2 (2); LBQX(5) is the Ljung±Box (1978) portmanteau test for autocorrelation of order up to ®ve, and is asymptotically distributedas a w2 (5); ARCH(5) is Engle's LM test for ARCH; BDS is the test of Brock, Dechert and Scheinkman (1987) withm � 5, e=s � 1; Tsay is Tsay's (1986) test for non-linearity; RESET is Ramsey's (1969) Regression Error Speci®cationtest for misspeci®cation of functional form. All tests for non-linearity are carried out on the residuals of a pre-whiteningautoregressive model (order chosen by AIC) with daily and bank holiday dummies; a;b;csigni®cance at the 10%, 5% and1% levels respectively.

so that the n-step-ahead forecast of the series made at time t is the expected value of the series nperiods in the future given all information available at time t. Since the forecasting models usedhere are all univariate,

Ot � x1; x2; . . . ; xt �2�where t � T1; . . . ; T±n and n � 1; . . . ; 20. Although only one-, ®ve-, ten- and twenty-steps-ahead forecasts are evaluated, it is also, of course, necessary to produce nineteen-, eighteen-, . . .steps-ahead forecasts in order to calculate the twenty-steps-ahead forecasts. The forecast horizonis ®xed at n steps ahead, and the starting point t is varied.

The random walk

If a series is postulated to contain an exact unit root in the (log) levels, then the optimal forecastfor the n-period ahead returns will be zero. The random walk model for the log-levels is given by

st�1 � st � et�1 �3�where st denotes the log of the spot exchange rate at time t. For the returns,

xt � et �4�If a constant is included in the estimated regression, the expected value of the error, et , denotedE[et�n], is zero, so that the best forecast of future errors is also zero:

f t;n � 0 8n �5�

Unfortunately, in this case, the sign prediction criterion, which will be discussed below, cannotbe used since the prediction is always zero, and therefore the predicted sign will always beincorrect (apart from in the rare case when the return is exactly zero).

Single exponential smoothing

If forecasts are required relatively quickly or at low cost, and if a series ¯uctuates about somebase level, one forecasting method which may prove useful is simple or single exponentialsmoothing, originally suggested by Brown (1959). Denoting St as the smoothed average of aseries at time t conditional upon previous realizations of x up to and including xt , we can de®nethe exponential smoothing relationship as

St � axt � �1ÿ a�Stÿ1 �6�where 0 < a < 1 is a smoothing constant determined by minimizing mean square in-sampleerror. The n-steps-ahead forecast is given by

f t;n � St �7�

Denoting the forecast error as et,n , we can write

St � Stÿ1 � aet;1 �8�for a one-step-ahead forecast. Thus the smoothed estimate, which is the future forecast, isupdated based upon the current error.


Modi®cations to the original algorithm have been suggested to incorporate trend and season-ality (Holt's and Winters' methods, respectively), but these are not considered here for theobvious reason of lack of trend or constant-period seasonality in the type of series analysed.Furthermore, Bartolomei and Sweet (1989) ®nd that Winters' method gives better forecaststhan simple exponential smoothing only 55% of the time using 47 seasonal series from theM-competition.

ARMA modelsThe ARMA class of models, popularized a quarter of a century ago (Box and Jenkins, 1976), stillremain arguably the most popular set of models for economic application, and this may be aconsequence of their relative ease of application, low computational cost, and their ability toproduce at least reasonable forecasts across a diverse set of data types. An ARMA(p,q)3 modelfor the returns may be written

f�L�xt � y�L�et �9�where f(L) and y(L) are polynomials in the lag or backshift operator. Model order can bedetermined using Akaike's (1974) or Schwarz's Bayesian (1978) information criteria (hereafterdenoted AIC and SBIC, respectively). Both are used here, although the former lead to poorerforecasts for reasons discussed below, and hence only results using the latter are shown.

An n-steps-ahead forecast from an estimated ARMA(p,q) model may be written

f t;n �Xpi�1

ai f t;nÿ i �Xqj�1

bjet�nÿ j �10�

where

f t;k � xt�k; k4 0 et�k � 0; k > 0

� et�k; k4 0

The class is often restricted to autoregressive models for simplicity and since the moving averagecomponent will disappear in forecasts with longer lead times. Furthermore, Makridakis andHibon (1995) state that the post-sample forecasting performance of pure AR models usually atleast matches those of more complex ARMAmodels. This is the position taken in this study. Theautoregressive models may also be augmented by the addition of daily and holiday seasonaldummies, which will appear in the forecasting model as `innovations' to use the Box±Jenkinsterminology. It is thus possible to see whether we can gain any forecasting power by simplyaccounting for day-of-the-week or holiday e�ects.

GARCH models and variantsTime-varying models of conditional heteroscedasticity have become extremely popular in the®nance literature (see, for example, Bollerslev et al., 1992, for an excellent survey). The focus ofthis study is in forecasting the mean of exchange rate returns, as opposed to the volatility, andhence these models are only of importance insofar as inappropriately modelling the conditionalvariance as being ®xed over time will lead to incorrect estimates of the parameters in the mean


3 The returns exhibit no evidence of non-stationarity, and thus the class of integrated ARMA, that is, ARIMA, modelsneed not to be considered for the log-returns.

equation. Bera and Higgins (1995) note that the parameters in the mean equation do changesigni®cantly when a GARCH-(1,1) model is formulated rather than assuming homoscedasticity.The equations for model estimation are given by

xt � b0 �Xpi�1

bixtÿ i � et; et � N�0; ht� �11�

ht � a0 � a1htÿ1 � a2e2tÿ1 �12�

Forecasts for the dependent variable in a GARCHmodel are computed in an identical fashion tothose of an autoregressive model, since it is not necessary to forecast the variance in order tobe able to forecast the mean. It is also possible to model the errors as being drawn from at-distribution rather than a Normal, which may account for more of the unconditional lepto-kurtosis in the underlying series than by assuming Gaussianity. This task was undertaken, butthe estimated parameters of the mean equation were altered by such small orders of magnitudethat forecasting performance would not be noticeably altered. Hence the results for modellingusing a GARCH-t are not shown.

A useful extension due to Engle et al. (1987) is to allow the conditional variance to enter themean equation (a GARCH-in-the-mean, or GARCH-M model), and thus a proxy for risk isallowed to in¯uence the return directly. In this case the forecasting equation (10) becomes

f t;n � b0 �Xpj�1

bj f t;nÿ j � g��f t;n�ht�

q�13�

under the GARCH-M formulation, where f t;k � xt�k; k4 0 and ft,n(ht) is an n-steps-aheadforecast of the variance given by equation (12).

The bilinear modelA potentially useful model which has generally not become popular in ®nancial econometrics isthe bilinear model. Poskit and Tremayne (1986) show that bilinear models may be useful formodelling non-Gaussian time series. A general bilinear ARMA model may be written

f�L�xt � a00 � y�L�et �Xri�1

Xsj�1

aijxtÿ ietÿ j �14�

with et distributed as a standard normal variate. Initial exploratory work was undertaken byGranger and Andersen (1978) and a detailed survey of the detection of bilinearity usingbispectral methods and estimation of bilinear models can be found in Subba Rao and Gabr(1980). Denoting the bilinear-ARMA model with autoregressive order p and moving averageorder q (with r and s as in equation (14) above) as BL(p,q,r,s), it is advantageous to limit the classof models to BL(1,0,1,1) and BL(0,0,1,1) following Brunner and Hess (1995). Invertibility andstationarity conditions for these models have been developed and their properties studiedelsewhere (Sesay and Subba Rao, 1988), but only for these simple bilinear models; invertibility isnecessary for multi-step-ahead forecasts to be computed using exact methods (De Gooijer andKumar, 1992) and the properties of more complex bilinear models is not well understood.Bilinear AR, bilinear MA, and BL-ARMA models were also estimated, but yielded no improve-ment in forecasting over the simple BL model, and hence those results are not shown.


Weiss (1986) gives the conditional log likelihood function for a bilinear model:

LT � ÿ 1

2log�s2e � ÿ

1

2s2T

Xe2t �15�

Forecasts can easily be generated in a manner similar to equation (10) and, as in the case of theARMA model, for the BL(0,0,1,1) model, the forecast component from the moving average willbe optimally zero for lead times greater than one period. Hence a ®rst-order bilinear model isuseful only for one-period-ahead forecasts. Forecasts using the bilinear model can thus be madeusing

f t;n � a00 � a11xtÿ1etÿ1

if n � 1, and f t;n � 0 otherwise. Granger and TeraÈ svirta (1993) argue that bilinear models may beparticularly useful if the prediction period contains ``occasional strong perturbances''.

Threshold modelsA simple relaxation of a standard linear autoregression is to allow a locally linear approximationover a number of states. This model is known as a threshold autoregression, and is globally non-linear, although each component is piece-wise linear. A general threshold autoregressive model(TAR) can be written

xt �XJj�1

I� j �t f� j �0 �

Xpji�1

f�j�i xtÿ i � e� j �t

!; rjÿ1 4 ztÿd 4 rj �16�

where I� j �t is an indicator function for the jth regime taking the value one if the underlying

variable is in state j and zero otherwise. ztÿd is an observed variable determining the switchingpoint and e� j �t is a zero-mean independently and identically distributed error process. If theregime changes are driven by own lags of the underlying variable, xt (i.e. ztÿd � xtÿd), then themodel is a self-exciting TAR (SETAR). The SETAR is associated primarily with Tong (Tongand Lim, 1980; Tong, 1983, 1990; Chan and Tong, 1986; Tsay, 1989).

Estimation of the model parameters (fi , rj , d, pj) is considerably more di�cult than for astandard autoregressive process, since in general they cannot be determined simultaneously, andthe values chosen for one parameter are likely to in¯uence estimates of the others. Tong (1983,1990) suggests a complex non-parametric lag regression procedure to estimate the values of thethresholds (rj) and the delay parameter (d ). Estimation of the autoregressive coe�cients can thenbe achieved using non-linear least squares (NLS), and the order of the piece-wise linear com-ponents (pj) by Akaike's (as Tong, 1990, suggests), or some other information criterion.

Ideally, it may be preferable to endogenously estimate the values of the threshold(s) as part ofthe NLS4 optimization procedure, but this is not feasible. The underlying functional relationshipbetween the variables of interest is discontinuous in the thresholds, such that the only method ofestimating the thresholds in this manner is to evaluate the function and to optimize with respectto the other parameters for every feasible value of the thresholds to some given level of accuracy.Thus the procedure adopted here is as follows. The delay parameter, d, is set to one on theoreticalgrounds in common with KraÈ ger and Kugler (1993): in the context of a ®nancial market such as


4 Subba Rao and Gabr (1980) use maximum likelihood estimation rather than NLS. This was also used here, but wascomputationally much slower, although the coe�cient estimates were very similar, so only the results for NLS estimationare shown.

that analysed here, it is most likely that the previous day's return would be the one to determinethe current state, rather than the return two days, two weeks or two months ago. An initialAR(2) is selected for all states, and either one or two thresholds estimated (depending on thepostulated model) using a grid-search procedure based on the fractiles of the distribution of thereturns.5 Given value(s) for the threshold(s), the order of the autoregressive models is determinedusing SBIC and AIC, with the model having the lowest value of the information criteria beingselected for forecasting.

Neural networksArti®cial neural networks (ANNs) are a class of non-linear regression models inspired by the waycomputation is performed by the brain. Their primary advantage over more conventionaleconometric techniques lies in their ability to model complex, possibly non-linear processeswithout assuming any prior knowledge about the underlying data-generating process. The fully¯exible functional form makes them particularly suited to a ®nancial (and, moreover, a foreignexchange) application where non-linear patterns are clearly present but an adequate structuralmodel is conspicuously absent.

Whereas the superior performance of neural networks for classi®cation relative to alternativetechniques is fairly evident (see, for example, Cillins et al., 1993; or Dutta and Shekhar, 1993),their success in point prediction of ®nancial variables (i.e. pattern recognition in time series) is byno means clear. In a particularly useful study, Kuan and Lim (1994) forecast ®ve daily dollar-denominated exchange rates based on around ®ve years of daily data. Using feedforward andrecurrent networks, they ®nd the network models generally perform signi®cantly better thanARMA models, although there were considerable di�erences between the series. In some cases,the ARMA model performed particularly poorly, and was unable to predict even 50% of signchanges. Haefke and Helmenstein (1994) ®nd that an ANN can outperform an AR(2) in pre-dicting the Austrian stock index. Tsibouris and Zeidenberg (1995) ®nd up to 60% correct signpredictions for four US stocks using a neural network with nine inputs and ®ve hidden layers.LeBaron and Weigend (1994), however, ®nd no signi®cant improvement over linear predictors.They argue that they ``. . . can be fairly con®dent that this is a fairly general result for the timeseries and models we considered . . . there is probably little hope of ®ne tuning the networks weused''. Episcopos and Davis (1995) compare the forecasting performance of EGARCH-M andneural network models for predicting daily US dollar foreign exchange series. They ®nd thatboth outperform the random walk, but neither is consistently better than the other.

By far the most popular type of model, and the one studied here, is known as a single hiddenlayer feedforward neural network. The model can be speci®ed as follows. The structure consistsof three layers: the inputs (akin to regressors in a linear regression model), which are connectedto the output(s) (the regressand) via a hidden or intermediate layer. From an econometricperspective, the problem reduces to one of estimating the synaptic weights or connectionstrengths between the layers. Formally the network model can be written

x̂N;m�x; b;w; b� �XNj�1

bjfXmi�1

wijZi � bj

!�17�

where the number of hidden units in the intermediate layer is N. The inputs were selected as ownlagged values of the series from t ÿ 1 to t ÿ m, where m is the number of inputs. xÃ is a vector of


5 I am grateful to Tom Maycock at Estima for initially suggesting this methodology and for helping to transform it intoworkable RATS code.

®tted values, Z is the input, b represents the hidden to output weights, and w and b represent theinput to hidden weights. Let

xmt � �xt�mÿ1; xt�mÿ2; . . . ; xt� �18�

The multivariate non-linear least squares minimization problem is then given by

minb;w;b

XT ÿmÿ1

t�0

�xt�m ÿ x̂N;m�xmt ; b;w; b��2 �19�

Non-linear least squares (NLS) estimates are computed using an application of the Levenberg±Marquardt algorithm (see Marquardt, 1963). The activation function for the hidden layer is thesigmoid

f�p� � 1

1 � exp�ÿp� �20�

The number of inputs was varied from 1 to 6, and the number of hidden layers from 1 to 10. The`best ®t' from among all combinations of alternative models can be chosen using SIC or byminimizing the in-sample mean square prediction error. Hornik et al. (1989) have shown that aneural network model with one hidden layer and a su�cient number of hidden nodes canapproximate any continuous function to an arbitrary degree of accuracy. Hence it is unlikely thatany additional hidden layer would add to predictive power, and is likely to represent anoverparameterization.

FORECAST EVALUATION CRITERIA

In order to assess the accuracy of diverse time-series forecasting methods, some kind of objectivecriterion which can be applied in di�erent situations and across di�erent forecasting horizons isrequired. In a recent survey of fourteen forecast evaluation measures, Makridakis and Hibon(1995) recommend the use of mean square error (MSE) and adjusted mean absolute percentageerror (AMAPE) on the grounds of the former's ability to discriminate well between good andbad forecasts, and the latter's intuitiveness and reliability. MSE provides a quadratic lossfunction, and so may be particularly useful in situations where large forecast errors aredisproportionately more serious than smaller errors. This may, however, also be viewed as adisadvantage if large errors are not disproportionately more serious than small ones, althoughthe same critique could also, of course, be applied to the whole least squares methodology.Indeed, Dieleman (1986) goes as far as to say that when there are outliers present, least absolutevalues should be used to determine model parameters rather than least squares. Makridakis(1993, p.528) argues that MAPE is `a relative measure that incorporates the best characteristicsamong the various accuracy criteria.' The mean square error can be de®ned as

MSE � 1

T ÿ T1

XTt�T1

�xt�n ÿ f t;n�2 �21�

where T is the total number of observations available and (T ÿ T1) are unused in the modellingprocedure but are retained as a hold-out sample purely for forecasting. AMAPE or symmetric


MAPE corrects for the problem of asymmetry between the actual and forecast values:

AMAPE � 1

T ÿ T1

XTt�T1

xt�n ÿ f t;nxt�n � f t;n

�� 22�

The symmetry in equation (22) arises since the forecast error is divided by twice the average ofthe actual and forecast values, whereas in the standard MAPE formula, the denominator issimply xtn , so that whether xtn or ft,n is larger will a�ect the result:

MAPE � 1

T ÿ T1

XTt�T1

xt�n ÿ f t;nxt�n

�� 23�

AMAPE also has the attractive additional property that it can be interpreted as a percentageerror, and furthermore, its value is bounded between 0 and 100, and that AMAPE will take thevalue one for a random walk in the levels (i.e. a zero forecast for the returns), so that if a modelhas an AMAPE < 1, it is superior to the random walk.

Unfortunately, it is not possible to use either MAPE or AMAPE when the actual realization ofthe series can take on values which are very much smaller than the corresponding forecast value.Consider the following example. Say we forecast a value of f t;n � 3, but the out-turn is thatxt�n � 0:0001. The addition to total MSE from this one observation is given by

1

391��0:0001ÿ 3�2 � 0:0230

This value for the forecast is large, but perfectly feasible since it is well within the range of thereturns data. But the addition to total MAPE from just this single observation is given by

1

391

0:0001ÿ 3

0:0001

�� 76:7

Thus it is clear that the use of these measures could potentially be misleading since they may beseriously increased by one or two large forecast error outliers.

It has, however, also recently been shown (Gerlow et al., 1993) that the accuracy of forecastsaccording to traditional statistical criteria may give little guide to the potential pro®tability ofemploying those forecasts in a market trading strategy, so that models which perform poorly onstatistical grounds may still yield a pro®t if used for trading, and vice versa. Models which canaccurately forecast the sign of future returns, or can predict turning points in a series have beenfound to be more pro®table (Leitch and Tanner, 1991). A possible indicator of the ability of amodel to predict direction changes irrespective of their magnitude is that suggested by Pesaranand Timmerman (1990). The relevant formula to compute this measure is

% correct sign predictions � 1

T ÿ T1

XTt�T1

zt�n �24�

where

zt�n � 1 if �xt�n:f t;n� > 0

zt�n � 0 otherwise


The Pesaran±Timmerman non-parametric test of predictive performancePesaran and Timmerman (1992, 1994), hereafter PT, have recently suggested a non-parametrictest for market timing ability which is generalised from the Henriksson±Merton (Henrikssonand Merton, 1981; Merton, 1981) test for independence between forecast and realized values.Using the same notation as above (which di�ers slightly from that of PT), let Px � Pr�xt�n > 0�;Pf � Pr�f t;n > 0� and

P̂ � 1

T ÿ T1

XTt�T1

zt�n

Denoting the ex ante probability that the sign will correctly be predicted as, P�, then

P� � Pr�zt�n � 1� � Pr�xt�nf t;n > 0� � PxPf � �1ÿ Px��1ÿ Pf � �25�

It can be shown that

Sn � P̂ÿ P�fcvar�P̂� ÿ cvar�P̂��g1=2 �26�

is a standardized test statistic for predictive performance which is asymptotically distributed as astandard normal variate under the null hypothesis of independence between correspondingactual and forecast values, where PÃ� denotes estimated values,

cvar�P̂� � 1

T ÿ T1P̂��1ÿ P̂��

and

cvar�P̂�� 1

T ÿ T1�2P̂x ÿ 1�2P̂f �1ÿ P̂f � � 1

T ÿ T1�2P̂f ÿ 1�2P̂x�1ÿ P̂x�

� 1

�T ÿ T1�2P̂xP̂f �1ÿ P̂x��1ÿ P̂f �

26643775

The PT statistic is not easily applied to multi-step-ahead forecasts from any kind ofautoregressive model since, by de®nition, if the modulus of the estimated coe�cients is verysmall (which they are here), then the forecast value should tend to the value of the constant termas the forecast horizon is increased if the autoregressive coe�cient tends to zero. Hence the testcannot be reliably computed for multi-step-ahead forecasts using these models as all the forecastswill take on the sign of the coe�cient on the constant term. Neither can the test be readily appliedto a GARCH-M model, since

��htp

will always be positive. Then it is possible that all theforecasts, while changing considerably in magnitude, might not change sign. Whether all theforecasts are positive, all are negative, or a combination will depend on the relative magnitudesof the coe�cients on the constant and volatility term. If one is considerably greater than theother, then the forecast will not change sign and hence the statistic cannot be computed. It is alsoonly worth applying the test to those models which yield the highest proportion of correct signpredictions, since this is a necessary but by no means su�cient condition for market-timingability. The test is therefore only utilized on the neural network and autoregressive modelsforecast one period ahead.


Comparisons with arti®cially generated dataMotivated by bootstrapping (see Efron, 1979), forecasts are also generated using random drawsfrom a normal distribution with the same mean and variance as the original in-sample returnsdata. A total of 391 forecasts are generated in this manner, and the proportion of correct signpredictions and the PT statistics are calculated. This procedure is repeated 5000 times for eachseries, and should help to decide whether any perceived improvement in forecasting the next signis real or simply the result of a statistical ¯uke.

FORECASTING RESULTS AND ANALYSIS

The MSE and proportion of correct sign predictions for each of the models outlined earlier aregiven in Tables II to V.6 Also given in Table II is the typical number of parameters to beestimated for each model, so that one may compare the relative forecasting performance ofparsimonious versus more complex models. The random walk, exponential smoothing and linearautoregressive models fall into the former category, with more complex models and the neuralnetwork with order chosen by in-sample MSE clearly falling into the latter. The remainingmodels rest somewhere in between. An important argument that is often advanced againstpro¯igate models is that they are sample-speci®c and thus result in poor generalization andtherefore poor out-of-sample performance (see, for example, Clements and Hendry, 1995).

Table V gives the overall rankings for each model and criterion, for forecasts made 1, 5, 10,and 20 steps ahead, where a ranking of `1' indicates the best model, and `� ' denotes two modelsyielding identical performance to three decimal places.

An obvious and important ®rst comparison to make is one between the simple random walkbenchmark and all other models, for this will determine whether time-series forecasting isworthwhile or not. The random walk is only the best model for the German mark, and, onaverage, ranks around four or ®ve. But it is usually a fairly safe bet, performing badly only forthe French franc.

Exponential smoothing is a technique originally formulated for forecasting periodic, seasonaldata, so it was not envisaged at the outset that it would perform particularly well for high-frequency ®nancial time series. In particular, single exponential smoothing implies that thereturns are generated by an ARIMA (0,1,1) process with a moving average parameter equal toone minus the smoothing coe�cient, a. Given that the returns have been found to be stationary,this must imply an overdi�erencing, and hence the smoothing parameter must be close to zero sothat the unit root is cancelled out to leave a model very similar to the random walk.7 The resultswere, however, surprisingly not unfavourable. Exponential smoothing was not at all useful onMSE grounds, and indeed it was the worst model overall, but the story using the sign predictioncriterion is rather di�erent; for the German mark, it is the best model, and generally seems betterat forecasting over longer horizons, relative to its competitors.

Linear models fare little better than the random walk, perhaps surprisingly, since preliminaryexploratory data analysis (for example, as shown by the Ljung±Box Q-statistic in Table I)highlights some evidence of linear dependence in the series.8 Predictions based additionally on aday-of-the-week and bank holiday e�ects (`AR�dummies') were notably bad.


6 The estimated models are not shown due to lack of space, although all are available from the author in an appendixupon request.7 I am grateful to the anonymous referee for pointing this out.8 The Ljung±Box Q-statistics are just signi®cant at the 1% level.

GARCH-M performed only marginally better than the GARCH model,9 and indeed theseproved overall to be the most satisfactory. Glosten et al. (1993) ®nd a signi®cant negativerelationship between the conditional mean and conditional variance of excess returns on a CRSPindex of stocks, indicating the importance of incorporating a volatility term in the meanequation, although it appears of slightly less use here in the foreign exchange markets, since the`leverage' e�ect which provides the motivation for GARCH-M does not apply.


Table II. Forecast performance of various models for the French franc

Number of steps forecast ahead1 5 10 20

Model #param. MSE Sign MSE Sign MSE Sign MSE Sign

r.w. in levels 0 2.190 Ð 2.220 Ð 2.253 Ð 2.220 ÐExp. smooth. 1 2.107 0.489 2.113 0.497 2.098 0.501 2.011 0.457AR-SBIC 1-2 2.184 0.491 2.222 0.475 2.226 0.468 2.197 0.454AR�dummies 7-8 2.199 0.460 2.237 0.462 2.272 0.463 2.214 0.443GARCH 4-5 2.004 0.492 2.031 0.494 2.060 0.489 1.983 0.470GARCH-M 5 2.183 0.483 2.031 0.475 2.060 0.468 1.982 0.454BL 2 2.188 0.485 Ð Ð Ð Ð Ð ÐSETAR-1-SBIC 4 2.072 0.459 2.089 0.457 2.101 0.450 1.992 0.432SETAR-1-AIC 3-5 2.072 0.454 2.088 0.444 2.101 0.437 1.995 0.422SETAR-2-SBIC 7-8 2.076 0.492 2.093 0.481 2.105 0.466 1.998 0.459SETAR-2-AIC 9-12 2.071 0.459 2.087 0.455 2.098 0.447 1.995 0.435NN-SBIC 2-6 1.972 0.450 1.988 0.452 2.008 0.463 1.917 0.486NN-MSE 60-66 2.092 0.414 2.013 0.517 2.045 0.493 2.069 0.457

Notes: MSE shown is the true MSE multiplied by a factor of 10 so that small di�erences in MSE across models can bediscerned. # param. denotes the typical number of parameters estimated for each model. This is shown only for theFrench franc since it is approximately the same for all series, although obviously the actual number depends on the orderof the model selected. Lag lengths of linear models are chosen by SBIC.

Table III. Forecast performance of various models for the German mark


Model MSE Sign MSE Sign MSE Sign MSE Sign

r.w. in levels 2.000 Ð 2.032 Ð 2.061 Ð 1.982 ÐExp. smooth 2.191 0.497 2.207 0.503 2.192 0.487 2.124 0.473AR-SBIC 2.074 0.458 2.111 0.447 2.144 0.439 2.068 0.427AR�dummies 2.085 0.481 2.121 0.468 2.153 0.461 2.078 0.446GARCH 2.075 0.459 2.106 0.460 2.138 0.453 2.067 0.435GARCH-M 2.069 0.455 2.106 0.447 2.138 0.439 2.064 0.427BL 2.082 0.454 Ð Ð Ð Ð Ð ÐSETAR-1-SBIC 2.139 0.459 2.162 0.455 2.182 0.447 2.079 0.435SETAR-1-AIC 2.134 0.462 2.157 0.457 2.177 0.450 2.080 0.438SETAR-2-SBIC 2.137 0.454 2.160 0.447 2.179 0.442 2.078 0.432SETAR-2-AIC 2.129 0.482 2.153 0.475 2.174 0.461 2.078 0.448NN-SBIC 2.034 0.424 2.055 0.426 2.084 0.434 2.007 0.457NN-MSE 2.087 0.414 2.121 0.460 2.197 0.450 2.158 0.462

Notes: As for Table II.

9 The GARCH model is not nested within GARCH-M in this case, since the former includes a number of lags of thedependent variable in the mean equation (with order chosen by SBIC), while the latter does not.

The bilinear and threshold models performed moderately. The former, as previously stated,was not expected to produce accurate forecasts, since the bispectrum test shown in Brooks (1996)has high power against bilinear processes but did not indicate the presence of non-linearity in anyof the series. Bilinear models perform well when there are a small number of large shocks to aseries, which is not the case here (see Granger and Andersen, 1978). Threshold models with threestates (i.e. two thresholds) are far superior to those with two, although in both cases for a givennumber of states, relatively parsimonious models were more accurate than complex ones.

Finally, turning to the neural network models, the highly overparameterized network withorder chosen by minimizing in-sample MSE performed very poorly judged by both criteria. Thisis in stark contrast to the network chosen using the stringent Schwarz's Bayesian informationcriterion, which gives excellent performance on MSE grounds but worse than averageperformance on sign prediction grounds, particularly with shorter forecast horizons. This


Table IV. Forecast performance of various models for the US dollar


Model MSE Sign MSE Sign MSE Sign MSE Sign

r.w. in levels 4.269 Ð 4.302 Ð 4.277 Ð 4.162 ÐExp. smooth. 4.540 0.462 4.559 0.470 4.371 0.492 4.264 0.470AR-SBIC 4.269 0.491 4.302 0.509 4.277 0.500 4.162 0.486AR�dummies 4.323 0.463 4.359 0.449 4.328 0.450 4.212 0.448GARCH 4.281 0.500 4.344 0.494 4.316 0.489 4.193 0.484GARCH-M 4.236 0.529 4.299 0.525 4.274 0.518 4.160 0.503BL 4.239 0.523 Ð Ð Ð Ð Ð ÐSETAR-1-SBIC 4.292 0.482 4.354 0.475 4.345 0.471 4.210 0.457SETAR-1-AIC 4.302 0.482 4.363 0.475 4.354 0.474 4.222 0.459SETAR-2-SBIC 4.270 0.526 4.330 0.519 4.321 0.516 4.190 0.500SETAR-2-AIC 4.287 0.544 4.348 0.543 4.336 0.542 4.208 0.522NN-SBIC 4.177 0.491 4.199 0.499 4.178 0.463 4.054 0.484NN-MSE 4.758 0.440 4.310 0.501 4.113 0.518 4.097 0.495

Notes: As for Table II.

Table V. Rankings of various models averaged across all three currencies

Number of steps forecast ahead1 5 10 20 Average

Model MSE Sign MSE Sign MSE Sign MSE Sign MSE Sign

r.w. in levels 3 Ð 3� Ð 5 Ð 4 Ð 4 ÐExp. smooth. 13 4� 12 3 11 1� 12 3 12 3AR-SBIC 4 4� 6 7 6 6 5 8 5 8AR�dummies 12 10 11 8 8� 8 11 9 11 9GARCH 2 1 3� 4� 3 4� 3 6 3 1GARCH-M 6� 4� 2 4� 2 4� 2 7 2 5BL 5 7 Ð Ð Ð Ð Ð Ð Ð ÐSETAR-1-SBIC 10 8� 10 9 10 10 8� 11 9 12SETAR-1-AIC 9 8� 9 10 8� 9 10 10 10 11SETAR-2-SBIC 8 3 8 4� 12 7 6� 4� 8 4SETAR-2-AIC 6� 2 7 2 7 3 6� 4� 7 2NN-SBIC 1 11 1 11 1 11 1 2 1 10NN-MSE 11 12 5 1 4 1� 8� 1 6 6

Note: `Average' is the average ranking over all three series and over all the forecast horizons.

serves to highlight the importance of guarding against over®tting with neural network models. Avery similar result is observed by Swanston and White (1995) using currency futures data.

The results of the Pesaran±Timmerman test of market timing ability are presented in Table VI.As they clearly show, neither the autoregressive nor the neural network models have any market-timing ability. In all cases except for the neural network model for the US dollar, the proportionof correct sign predictions �P̂� is less than one would expect if the forecasts and actual values werecompletely independent �P̂��. The results of the autoregressive model for the German mark arethe most startling: they show a signi®cant negative value of the test statistic, indicating thatpredictions made using this model are signi®cantly worse than if there were no relationshipbetween corresponding actual and forecast values. The 5000 sets of forecasts generated normallywith the same mean and variance show a similar picture in Table VII.

In fact, the results show that the forecasts generated using time-series models are much worsethan they ®rst appear. The median proportion of correction sign predictions for the random dataover the 5000 replications is not much lower than the best of the models. For none of the threeexchange rate series analysed here is the best forecasting model able to do better than the 95thpercentile of the randomly generated forecasts. To put this another way, a forecast generatedpurely randomly is better than all 13 of the time series models at least 5% of the time for all threeof the currencies investigated. The same is true of the PT statistic. The models were never able tooutperform the top 5% of the randomly generated forecasts. In some cases, the test statistic waslower (i.e. more negative) for the models than for the median arti®cial forecast. This must surelybe a damning indictment of the usefulness of time-series forecasting in this context.

CONCLUSIONS

The random walk model is able to produce reasonably accurate forecasts of exchange ratereturns, although it cannot, by de®nition, produce sign predictions. The use of time-series


Table VI. The Pesaran±Timmerman test of market timing ability for one-step-ahead forecasts usingautoregressive and neural network models with order selected using SBIC

Autoregressive model Neural network modelCurrency PÄ P* Sn PÄ P* Sn

French franc 0.491 0.500 ÿ0.689 0.463 0.483 ÿ0.811German mark 0.458 0.511 ÿ3.220 0.455 0.455 ÿ0.001US dollar 0.491 0.504 ÿ0.572 0.512 0.514 ÿ0.101

Table VII. Proportion of correct sign predictions and Pesaran±Timmerman test of market timing abilityfor one-step-ahead forecasts generated using 5000 normally distributed series with the same mean andvariance as the original in-sample returns data

Proportion of correct sign Pesaran±TimmermanPredictions Statistics, Sn

Currency 5%ile Median 95%ile 5%ile Median 95%ile

French franc 0.449 0.492 0.531 ÿ2.026 ÿ0.304 1.218German mark 0.429 0.469 0.510 ÿ2.812 ÿ1.201 0.408US dollar 0.451 0.492 0.533 ÿ1.904 ÿ0.295 1.336

Notes: 5% ile and 95% ile denote the 5th and 95th percentile of the distribution, respectively. These fractiles are shownrather than the mean since Sn values either side of zero will cancel each other out.

forecasting models does seem worthwhile from this perspective, and furthermore, the additionalcomplexity and the loss of any kind of diagnostic ability when one switches from a linear to anon-linear model also seems a sacri®ce worth making. However, the magnitude of theimprovement over the naive benchmark is extremely small. Moreover, the typical di�erencebetween the best and the worst model for each series is typically only of the order of 5%, and thismasks an even closer similarity between many of the `moderately' performing models which haveaccuracies that are virtually indistinguishable. Thus, although no formal statistical tests havebeen undertaken,10 it is considered highly unlikely that the improvement in forecastingperformance over the random walk is signi®cant.

Parsimonious models seemed to perform marginally better than those with more parameters,as one might expect, `. . . the accuracy of forecasts aggregated over all horizons is positivelyrelated to parsimony' (Beveridge and Oickle, 1994). One reason for this may be the tendency ofcomplex, over®tted models to mould to sample-speci®c noise as well as signal, leading to poorout-of-sample generalisation. This point has been made on numerous occasions regarding neuralnetworks, but was observed nearly forty years ago by Bassie (1958):

The more a function is complicated by additional variables or by nonlinear relationships, thesurer it is to make a good ®t with past data and the surer it is to go wrong at some time in thefuture (p. 81).

A survey of forecasting `experts' by Armstrong (1978), however, revealed that experts thoughtthat complex models would provide better forecasts although Armstrong's examination of theliterature revealed that this was not the case.

There was a considerable di�erence between the forecastabilities of the series. During theperiod studied, there was some forecast improvement over random or naive forecasts for theFrench franc and US dollar, although these were modest and would probably be outweighed bytransactions costs if they were utilized by an active trading strategy. We would have had virtuallyno hope of forecasting the German mark, and for this currency, almost certainly the best forecastfor the exchange rate for any given date in the future would be the exchange rate prevailing today.

A comparison of a number of di�erent time-series forecasting methods showed modestadvantages for non-linear models over random walk and autoregressive models. In particular,parsimonious neural network and GARCH-type models (as Nachane and Ray, 1993, ®nd), aree�ective over a range of series and forecast horizons. The simple martingale di�erence model wasnot the best model in this study, although it always performed reasonably well. This is in starkcontrast to Dimson and Marsh's (1990) volatility forecasts which showed the random walk invariance to be the worst of ®ve methods. They attribute this failure to the random walk beingunable to capture the mean-reverting property of volatility. There was some considerabledivergence between the usefulness of each model between the series and also between criteriaused to evaluate the models. To quote Makridakis et al. (1982) `. . . there is no such thing as thebest approach or method, just as there is no such thing as the best car or hi-® system (p.112).Future studies may determine whether these results can be viewed as general, or if they are data-and time-speci®c.

An issue which has received surprisingly little attention in applied ®nancial econometrics is theextent to which it is desirable to estimate models over extremely long periods of time. Taylor(1986), for example, states that, if possible, at least eight years of daily data should be analysed.When building an empirical model, one implicitly assumes that the coe�cients of the model are


10 This would require some distributional assumption for the forecast errors, such as asymptotic normality, which ishighly unlikely to hold.

constant throughout the entire sample period (see Makridakis, 1981, for a cursory treatmentof this issue).

There are at least two obvious solutions to this problem. The ®rst is to use higher-frequencydata: this yields large numbers of observations in a relatively short period of time, and thussatis®es any data requirements; at the same time, strict stationarity is much more likely to holdover short intervals of calendar time. The second method would be to relax the restriction thatthe model coe�cients must be ®xed over time (i.e. to estimate a time-varying coe�cients model).Riddlington (1993) ®nds in 21 applications that time-varying coe�cients models have muchlower RMS errors than ARMA, random walk or ®xed coe�cient structural models. From aforecasting perspective, it may be preferable to simply dispose of early data altogether, and toestimate the model over the latter part of the sample period since any exploitable relationshipwhich held at the beginning of a long sample period may have disappeared by the end. But howmuch data should we use, and how much should we discard? The answer may be obvious if therehas been some important structural change which will have altered the dynamics, but a moresophisticated approach may be to use all available data in model construction, but to give pastobservations relatively lower weights in parameter estimation. In any case, many econome-tricians may be worried by what appears to be discarding information.

This issue has been completely ignored in a number recent scholarly papers which haveattempted to produce models over very long series of observations spanning many years (forexample, Akgiray, 1989; West and Cho, 1995; Pesaran and Timmerman, 1995). The worsto�enders in this regard are perhaps Ding et al. (1993) and Hentschel (1995), both of whomestimate GARCH-variants using daily data from the 1920s to the 1990s, a period spanning aworld war and numerous other important structural changes. Although neither of these papersare speci®cally interested in producing forecasts, the fact remains that they both estimate modelsover very long periods of time without giving much consideration to the issue of parameterstructural stability, a problem which may only become apparent if they had attempted to use themodels for forecasting. The only paper that su�ciently addresses this issue to the knowledge ofthe author is Satchell and Timmerman (1995), who use a recursive window encompassing onlythe 1000 most recent observations to construct models and therefore forecasts. The over-whelming conclusion of this paper is that one cannot produce very accurate forecasts using high-frequency time-series models over a number of years. This must surely also cast doubt upon thee�cacy of constructing such models in the ®rst place.

ACKNOWLEDGEMENTS

I am grateful for extensive comments on earlier versions of this paper by Simon Burke, SethGreenblatt, Tim Harrington, Olan Henry, and an anonymous referee. All remaining errors are,of course, mine alone.

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Author's biography:Chris Brooks is a Lecturer in Finance at the ISMA Centre, University of Reading, where he also obtainedhis PhD. His research interests are in the ®eld of ®nancial econometrics, particularly in time-series modellingand forecasting.

Author's address:Chris Brooks, ISMA Centre, Department of Economics, Faculty of Letters and Social Sciences, Universityof Reading, PO Box 218, Whiteknights, Reading RG6 6AA, UK.


linear and non-linear (non-)forecastability of high-frequency exchange rates

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