forecasting the nikkei spot index with fractional cointegration

$: Forecasting the Nikkei spot index with fractional cointegration$
Forecasting the Nikkei Spot Index withFractional Cointegration

DONALD LIEN1* and YIU KUEN TSE2

1University of Kansas, USA2National University of Singapore, Singapore

ABSTRACT

We investigate the forecast performance of the fractionally integrated errorcorrection model against several competing models for the prediction of theNikkei stock average index. The competing models include the martingalemodel, the vector autoregressive model and the conventional error correc-tion model. We consider models with and without conditional hetero-scedasticity. For forecast horizons of over twenty days, the best forecastingperformance is obtained for the model when fractional cointegration iscombined with conditional heteroscedasticity. Our results reinforce thenotion that cointegration and fractional cointegration are important forlong-horizon prediction. Copyright # 1999 John Wiley & Sons, Ltd.

KEY WORDS conditional heteroscedasticity; error correction model;fractionally integrated error correction model; martingale;Nikkei stock average index; vector autoregression

INTRODUCTION

Future markets are commonly recognized to serve three purposes: price discovery, risk transferand transaction-cost reduction. In terms of price discovery, futures price provides signals for thespot price in the future. Thus, a statistical model incorporating past futures prices is expected toimprove the forecasting performance.

A method to incorporate the e�ects of the futures prices on the spot prices is the VectorAutoregression (VAR) model. This model postulates that the past spot and futures prices a�ectthe current spot and futures prices. As both spot and futures prices are often found to have a unitroot (that is, they are integrated of order one and are non-stationary), VAR models are usuallyconstructed for the di�erenced prices rather than the price levels.

When modelling spot and futures prices, a restricted VAR model is usually preferred. Therestriction arises from the no-arbitrage relationship: the future price should equal the spot priceplus the carrying charge, which is the total cost associated with purchasing and holding the assetunderlying the futures contract. While this equilibrium relationship does not hold all the time, the

CCC 0277±6693/99/040259±15$17.50 Received January 1998Copyright # 1999 John Wiley & Sons, Ltd. Revised July 1998

Accepted November 1998

Journal of Forecasting

J. Forecast. 18, 259±273 (1999)

* Correspondence to: Donald Lien, Department of Economics, College of Liberal Arts and Sciences, University ofKansas, 213 Summer®eld Hall, Lawrence, KS 66045-2113, USA.

spot and futures prices would respond to the deviations as arbitrage operations take place.The responses of the spot and futures prices to the deviations, a case of `error correction', imposerestrictions on the parameter of the VAR model. Thus, one may consider a model in which thecurrent prices are a�ected by the past prices and the error correction term. The resulting model iscalled the error correction model (ECM). Forecasting with ECM is discussed in Engle and Yoo(1987).

The ECM is related to the notion of cointegration. If a linear combination of two variablesthat are individually integrated of order one is stationary, the two variables are said to becointegrated. The stationary linear combination of integrated variables is called the cointegrationresidual. The Engle±Granger (1987) representation theorem states that if two variables arecointegrated, there exists an ECM.

While earlier research focused on cointegrated residuals that are integrated of order zero, recentresearch examined the more general case in which the cointegration residuals follow a fractionallyintegrated process. An important feature of a fractionally integrated process is that its auto-correlation function dies down in a hyperbolic manner which is characteristic of a long memoryprocess. When the cointegration residual of two (individually) integrated series is fractionallyintegrated, the two series are said to be fractionally cointegrated. As a generalization of the Engle±Granger representation theorem, when two series are fractionally cointegrated, there exists afractionally integrated ECM (FIECM). These extensions can be found in Granger (1986).

Fractional cointegration has been found empirically in the literature. In examining thepurchasing power parity, Cheung and Lai (1993) found that the deviation from the parity has along memory and may be described by a fractionally integrated process. Similar ®ndings wereestablished by Baillie and Bollerslev (1994) and Masih and Masih (1996) for exchange-rate data.Also, Dueker and Startz (1994) found that the three-month and one-year Treasury bill rates arefractionally cointegrated.

The importance of modelling the cointegration relationship by a fractional process lies in itsincorporation of the e�ects of long memory. For example, in the spot-futures framework, theECM models allow only the ®rst-order lag of the cointegration residual to a�ect the spot andfutures prices. In contrast, the FIECM incorporates a long history of past cointegration residuals.Thus, the FIECM nests the ECM. When fractional cointegration prevails the ECM is mis-speci®ed due to the omitted variables. Because the FIECM accounts for the long memory of thecointegration relationship, it should provide better forecasts (especially over long forecasthorizons) if fractional cointegration indeed exists. Similarly, one would expect the univariateARFIMA models to outperform the univariate AR models in long-range forecasting. None-theless, Crato and Ray (1996) found that the AR models provide compatible performance exceptfor long forecasting horizons with a large number of observations.

In this paper we consider the forecast of the Nikkei Stock Average (NSA) spot index. We ®ndthat the spot index and the futures price are integrated of order one but the basis (the di�erencebetween the futures prices and the spot index) is fractionally integrated. Thus, the FIECM isadopted for forecasting. We compare the FIECM against the EC and VAR models. As recentresearch supports the prevalence of time-varying second moments in many ®nancial time series(see Lien and Tse 1997 for some results on the NSA index), we incorporate the generalizedautoregressive conditional heteroscedasticity (GARCH) structure into ourmodels. To examine theperformance of themodels in long-horizon prediction, we consider the forecasts of theNikkei spotindex up to 40 days ahead. Based upon the mean squared error and mean absolute percentageerror, we ®nd that the FIECM±GARCH model provides the best forecast, the ECM±GARCH

Copyright # 1999 John Wiley & Sons, Ltd. J. Forecast. 18, 259±273 (1999)

260 D. Lien and Y. K. Tse

and VAR models are close second best. As expected, the superiority of the FIECM±GARCHmodel is more evident when the forecast horizon is long. These results indicate the signi®cance ofincorporating the long memory structure of the basis in doing prediction.

FORECASTING MODELS

Let st denote the logarithm of the spot price and ft the logarithm of the futures price at time t.Given the sample data st and ft for t � 1, 2, . . . ,T, we consider the forecast of sT�h , i.e. the h-step-ahead forecast. If we ignore the futures price data, a univariate ARFIMA(p, d, q) model can beapplied to the spot prices as follows:

f�B��1 ÿ B�dst � y�B�et �1�where B is the backward-shift operator; f(.) and y(.) are polynomials of orders p and q,respectively, and et is a white noise. We assume that all roots of f(B) and y(B) lie outside the unitcircle. The parameter d is not necessarily an integer, so that fractional di�erencing is permitted.Whenÿ0.55 d5 0.5, the process st is stationary. However, when d4 0.5, st has in®nite varianceand is thus not covariance stationary. It is noted that

�1 ÿ B�d �X1j�0

G�j ÿ d�Bj

G�ÿd�G�j � 1� �2�

and when d is an integer, we have the usual Box±Jenkins type of ARIMA model.To incorporate the futures price information into forecasting the spot index a VAR model may

be speci®ed as:

st � a0 �Xmi�1

aistÿi �Xmj�1

bjftÿj � est �3�

This is the spot equation in the VAR model with a lag order of m. A corresponding equation forthe futures price can be similarly constructed. When the spot index and the futures price are bothI(1), i.e. integrated of order one, st and ft will be replaced by the di�erenced variables Dst and Dft ,respectively. For example, for the spot-index equation we have

Dst � a0 �Xmi�1

aiDstÿi �Xmj�1

bjDftÿj � est �4�

Suppose both st and ft are I(k), but there is a linear combination of st and ft , denoted by zt , thatis of a lower integration order r, where both k and r non-negative integers with r5 k. Then wehave the case of a cointegration relationship. Engle and Granger (1987) showed that anappropriate model for st is the following:

Dst � f0 �Xpi�1

fiDstÿi �Xqj�1

jjDftÿj � gztÿ1 � est �5�

Many studies have found that k � 1, r � 0 and that zt may be approximated by the basis de®nedas zt � ftÿ st.


Forecasting the Nikkei Spot Index 261

Whenm � p � q, the VARmodel in equation (4) is nested in the ECM in equation (5) (see, forexample, LeSage, 1990, and Ghosh and Lien, 1997). In case thatm5 p and/orm5 q with at leastone strict inequality, the two models are non-nested. To ®t the VAR model the lag order m mustbe speci®ed. In this paper we determine the lag order based on the minimization of the AkaikeInformation Criterion (AIC).

The integration orders k and r can be extended to include the cases when these values are notintegers. In the literature, the case that k � 1 and r4 0 is referred to as `fractional cointegration'.Granger (1986) characterized a representation for such a process as follows:

Dst � o0 �XIi�1

oiDstÿ1 �XJj�1

tjDftÿj � d��1 ÿ B�r ÿ �1 ÿ B��zt � est �6�

When r � 0, equation (6) reduces to equation (5). Otherwise, the long history of the pastcointegration residuals will a�ect the spot index. As a consequence, when the basis is fractionallyintegrated, the error correction model is a misspeci®cation of the true fractional cointegrationmodel due to the omitted variables ztÿj for j4 1. Upon expanding the polynomial within thesquare brackets on the right-hand side of equation (6), it can be found that the coe�cient of ztÿjis dG�j ÿ r�=fG�ÿr�G�j � 1�g. Thus, while an in®nite history of zt is applied to determine thespot price, coe�cients of the lagged terms are determined by d, r and j only.

To date, univariate ARFIMA models are commonly applied in the economics and ®nanceliterature. FIECM is a restricted multivariate ARFIMA model. Ray and Tsay (1997) discussedanother type of restricted multivariate ARFIMAmodel. Ravishanker and Ray (1997) provided aBayesian analysis for general unrestricted multivariate ARFIMA models.

All the above models leave the second moments of the spot and futures prices unspeci®ed. Asimple method would be to assume that the variances and covariances are constant over time.Recent research, however, suggests the existence of conditional heteroscedasticity. Thus, wemodel the variance of est using the following GARCH process:

s2st � a0 �XMi�1

ais2s;tÿi �

XNj�1

bje2s;tÿj �7�

where s2st is the conditional variance of est . The possibilities of constant variance as well as time-varying variances are considered when comparing the forecast performance.

DATA DESCRIPTION

Our data set consists of 2106 daily observations of the spot index and the futures price of theNikkei Stock Average 225 (NSA), covering the period from January 1989 through August 1997.Daily closing values of the spot index and the settlement price of the futures contracts are used.The regular futures contract mature in March, June, September, and December. The contractsexpire on the third Wednesday of the contract month. For the futures price series, we use thenearest regular contract ( for example, to extract the futures price series in February the prices ofthe march contract are used) and roll over to the next contract around the tenth of the contractmonth. This is to ensure su�cient trading volume for the futures contracts and to avoid excessivevolatility.



To examine the post-sample forecasts of various models, we use the data from January 1989through August 1996 (a total of 1861 observations) for the in-sample analysis and estimation,and the data from September 1996 through August 1997 (a total of 245 observations) for thepost-sample forecast comparisons.

The spot index and the futures price are plotted in Figure 1. The index lost about 53% of itsvalue from the beginning of 1989 to August 1997, Figure 2 presents the logarithmic di�erences ofthe spot and future prices. The clustering of the logarithmic di�erences suggest that conditionalheteroscedasticity prevails. Thus, it would be appropriate to model the time-varying secondmoments of the series.

ESTIMATION RESULTS

To estimate the ARFIMA±GARCH model, we employ the Conditional Sum of Squares (CSS)method as in Baillie, Chung, and Tieslau (1995) (all estimations and graphics performed in thispaper were programmed using the GAUSS matrix language with the module MAXLIK). Lingand Li (1997) proved the consistency and asymptotic normality of the CSS estimators in thepresence of conditional heteroscedasticity. The method, however, depend upon the (theoretically)in®nite past history of the series. In empirical estimation, the pre-sample observations are trunc-ated at a ®nite lag value. Unlike Ling and Li (1997), who impose the pre-sample observations tobe zeroes, we use the actual data as pre-sample values. This method reduces the e�ective sample

Figure 1. Nikkei stock average, 1989/1 through 1997/8



size of the estimation. As long memory processes are characterized by hyperbolically decliningexpansions, long lags are required. After some experimentation, we truncate the pre-sampleobservations at lag 500. Thus, the e�ective sample size of the estimated models is 1361.

To choose the appropriate orders for the ARFIMA±GARCH speci®cations, one may considera set of reasonable orders and then apply model selection criteria to the estimated results. We donot follow this approach. Instead, we adopt the Box±Jenkins approach. That is, we ®rst estimatethe lower-order (parsimonious) models and then go to higher-order models only when thediagnostics reveal inadequacy. This parsimony principle is particularly important when fore-casting with non-linear models as higher-order non-linear models tend to over®t and forecastpoorly. Lower order models, on the other hand, provide better forecasts as long as modeldiagnostics are acceptable.

Following the parsimony principle, we consider low-order ARFIMA±GARCH models. Thefollowing ARFIMA(1, d, 1)±GARCH(1, 1) model for the spot return, futures return and thebasis is adopted:

�1 ÿ fB��1 ÿ B�d�xt ÿ m� � �1 ÿ yB�et �8�s2t � o � ae2tÿ1 � bs2tÿ1 �9�

where etjFtÿ1 � IID N(0, s2t ); y, d, m, f, o, a and b are parameters and Ft is the information set attime t. The series xt represents generically Dst, Dft or zt . The results are shown in Table I. For the

Figure 2. Nikkei stock average (di�erenced log), 1989/1 through 1997/8



spot and futures returns, the estimated order of the fractional integration, d, is not signi®cantlydi�erent from zero. We have experimented with dropping the insigni®cant parameters from theequations, and the results remain similar. Thus, both spot and futures prices are I(1). We furtherconducted the Dickey±Fuller (DF) test to ascertain this conclusion. The t-ratios of the DFregressions of the spot and futures prices were found to be ÿ1.536 and ÿ1.568, respectively,which are higher than the approximate critical value of ÿ 2.865 at the 5% level. thus, thestationarity of the spot and futures series is rejected.

On the other hand, the estimated fractional integration order of the basis is statisticallydi�erent from zero at the 1% level. We conclude that the NSA markets exhibit fractionalcointegration. Figure 3 displays the autocorrelation coe�cients of the NSA basis. The diagram isstrikingly similar to that of exchange rates displayed in Baillie and Bollerslev (1994) but withmore prominent periodicity. The persistence of the coe�cients supports the prevalence offractional cointegration. To determine the fractional integration order of the basis, the restrictedmodel in which the insigni®cant parameter y is deleted is re-estimated. This result is displayed inthe last column of Table I. The diagnostic statistics Q1 and Q2 , which test for serial correlation inthe residuals and squared residuals, respectively, do not detect any misspeci®cations. Thus, zt canbe taken as a series of I(0.3774).

All the parameters in the variance equation are statistically signi®cant, indicating the presenceof conditional heretoscedasticity. Moreover, in each case we have 05 a, b and a � b5 1,satisfying the necessary conditions for stationarity.

Gives the existence of fractional cointegration, we follow Granger (1986) and model the spotand futures price movements as follows (now xt is either Dst or Dft):

xt � m � y1Dstÿ1 � y2Dftÿ1 � d��1 ÿ B�0:3774 ÿ �1 ÿ B��zt � et �10�

Table I. Estimation results of the ARFIMA(1, d, 1)±GARCH(1, 1) models

Parameter Futures return Spot return Basis Basis

m ÿ0.0067 0.0141 0.1405** 0.1395(0.0329) (0.0318) (0.0513) (0.1078)

f ÿ0.5397* ÿ0.5283 ÿ0.1332** ÿ0.1198**(0.2344) (0.6767) (0.0513) (0.0405)

y ÿ0.4784* ÿ0.5152 ÿ0.0147(0.2497) (0.6866) (0.0578)

d 0.0010 ÿ0.0011 0.3763** 0.3774**(0.0278) (0.0295) (0.0262) (0.0308)

o 0.0509** 0.0573** 0.0021* 0.0021*(0.0165) (0.0201) (0.0009) (0.0009)

a 0.0661** 0.0841** 0.1032** 0.1032**(0.0139) (0.0197) (0.0192) (0.0216)

b 0.9095** 0.8873** 0.8852** 0.8852**(0.0191) (0.0265) (0.0207) (0.0234)

Q1(24) 25.38 23.60 21.85 21.82Q2(24) 24.07 29.64 29.42 29.37

Here and in Table II the numbers in parentheses are the corresponding standard errors of the estimates. An asteriskindicates statistical signi®cance at the 5% level whereas two asterisks indicate statistical signi®cance at the 1% level.Q1(24) is the Box±Pearce statistic based on the correlation coe�cients of the residuals up to lag order 24. Similarly Q2(24)is based on the squared residuals.



while the conditional variance is again modelled by the GARCH(1,1) process in equation (9).Equation (10) is then estimated by CSS, with or without conditional heteroscedasticity. Note thatin the estimation we have followed a two-step procedure in which the di�erencing parameter r aswell as the cointegration residuals are treated as known in the second stage. This two-stepprocedure was ®rst proposed by Engle andGranger (1987), and has been commonly applied in theliterature. While Engle and Granger provided justi®cations of the procedure for the ECM, weshould point out that the e�ects of taking r as known in the second stage on the limiting distri-bution of the two-step estimator are unknown. In particular, the standard errors of the coe�cientsin the second-step equation, i.e. equation (10), are calculated with the assumption that thefractional integration order is known. When taking into account that the order is an estimatedvalue, the standard errors have to be corrected. However, the corrections are very complicated andare not pursued here.

The estimation results are summarized in Table II. For fractional cointegration models to bevalid, d must be statistically signi®cant for at least one of the spot or futures equations. From the®rst two columns, we ®nd that the spot return indeed responds to the basis (and nothing else).On the other hand, the futures return is a�ected by the lagged futures returns but is not a�ectedby the basis. The third and fourth columns present the models with the statistically insigni®cantparameters deleted.

Figure 3. Autocorrelation function of the basis of the NSA



In addition to the FIECM±GARCH model, several competitive models are considered. Inparticular, we consider the ECM and VARmodels, with and without GARCH. As a benchmark,we also examine the martingale model in which the optimal forecast of future returns is thein-sample average return. For the VAR model, the lag order m is determined as 2 based on theAIC. The ECM and VAR models without GARCH are estimated using ordinary least squares(OLS), while their counterparts with GARCH are estimated using the quasi maximum likelihoodestimation (QMLE) method. The various estimated models are summarized below (the returnsand basis are measured in percentages).

FIECM

Dst � ÿ0:0267�0:0375�

� 0:7929�0:1479�

�1 ÿ B�0:3774 ÿ �1 ÿ B��zt

Dft � ÿ0:0257�0:0393�

ÿ 0:0441�0:0271�

Dftÿ1

ECM±GARCH

Dst � 0:0591�0:0433�

� 0:1476�0:0516�

ztÿ1

Dft � 0:0013�0:0309�

ÿ 0:0620�0:0296�

Dftÿ1

s2st � 0:1216�0:0269�

� 0:1259�0:0229�

e2s;tÿ1 � 0:8152�0:0295�

s2s;tÿ1

s2ft � 0:1094�0:0251�

� 0:0975�0:0179�

e2f;tÿ1 � 0:8524�0:0246�

s2f;tÿ1

Table II. Estimation results of the FIECM±GARCH models

Parameter Futures return Spot return Futures return Spot return

m ÿ0.002 0.0071 0.0013 0.0027(0.0336) (0.0326) (0.0309) (0.0344)

y1 0.0862 ÿ0.1537(0.0444) (0.1059)

y2 ÿ0.1411** 0.1307 ÿ0.0620**(0.0431) (0.1049) (0.0296)

d 0.0217 0.5837** 0.7069**(0.0704) (0.1786) (0.1394)

o 0.1084** 0.1256** 0.1094** 0.1219**(0.0220) (0.0279) (0.0251) (0.0268)

a 0.0983** 0.1227** 0.0975** 0.1236**(0.0164) (0.0224) (0.0179) (0.0223)

b 0.8522** 0.8145** 0.8524** 0.8161**(0.0215) (0.0298) (0.0246) (0.0291)

Q1(24) 27.40 24.85 27.58 25.41Q2(24) 27.22 33.04 36.85 34.01



ECM

Dst � ÿ0:9072�0:0496�

� 0:1473�0:0584�

ztÿ1

Dft � ÿ0:0257�0:0393�

ÿ 0:0441�0:0271�

Dftÿ1

VAR±GARCH

Dst � 0:0324�0:0514�

� 0:5179�0:0250�

Dftÿ1 � 0:2606�0:0293�

Dftÿ2 ÿ 0:5523�0:0265�

Dstÿ1 ÿ 0:2143�0:0282�

Dstÿ2

Dft � 0:0074�0:0557�

ÿ 0:1103�0:0272�

Dftÿ1 ÿ 0:0301�0:0348�

Dftÿ2 � 0:0440�0:0282�

Dstÿ1 � 0:0763�0:0325�

Dstÿ2

s2st � 0:0966�0:0150�

� 0:0684�0:0078�

e2s;tÿ1 � 0:8758�0:0142�

s2s;tÿ1

s2ft � 0:0876�0:0119�

� 0:0591�0:0060�

e2f;tÿ1 � 0:8951�0:0101�

s2f;tÿ1

r � 0:9662�0:0013�

VAR

Dst � 0:0018�0:0375�

� 0:5879�0:0997�

Dftÿ1 � 0:2738�0:0998�

Dftÿ2 ÿ 0:6008�0:1047�

Dstÿ1 ÿ 0:2698�0:1009�

Dstÿ2

Dft � ÿ0:0261�0:0395�ÿ 0:0322�0:1050�

Dftÿ1 ÿ 0:0140�0:1051�

Dftÿ2 ÿ 0:0129�0:1103�

Dstÿ1 � 0:0142�0:1062�

Dstÿ2

For the models with time-varying variances, GARCH(1,1) models are assumed. It is notedthat for one-step forecast, we do not need the futures equation. This brings up the issue ofwhether the spot and futures equations should be estimated jointly. We considered both casesand found that jointly estimated models produce inferior results. Therefore we adopt the single-equation estimation results presented above. Note that the coe�cient of Dftÿ1 in the futuresequation of the FIECM or ECM models is statistically insigni®cant. We have also considered amodel by removing this variable (i.e. the future price follows a random walk), but the forecastresults are similar. These results are not reported here for the purpose of conserving space.

FORECAST PERFORMANCE

In the studies by LeSage (1990) and Shoesmith (1992), the ECM was found to outperform theVAR model in forecast accuracy, particularly over long horizons. As the underlying equilibriumrelationship in the EC model is expected to hold only in the long run, better performance shouldbe more evident in the long run (see Stock, 1995). Engle and Yoo (1987) demonstrated that theECM outperforms the VAR model (in levels) in forecasting. Christo�ersen and Diebold (1996)suggested that the VARmodel performs poorly because it fails to impose integration. They arguedthat the VAR model in di�erences will perform as well as the ECM. Moreover, they noticed thatsimilar results were obtained in Reinsel and Ahn (1992) and Clements and Hendry (1995).



We compare the forecast performance of the models discussed in the previous section with stepsizes ranging from one to forty. Four criteria are employed: root mean squared errors (RMSE),mean absolute errors (MAE), mean absolute percentage errors (MAPE), and percentage ofcorrect direction predicted (CDIR). CDIR summarizes the percentage of cases where thedirection of movement of the spot index (up or down) is correctly predicted. Christo�ersen andDiebold (1996) and Clements and Hendry (1995) pointed out that, due to the non-stationarity ofthe forecasted variable, ECM will not outperform VAR model in the long run based uponconventional criteria such as the MSE. They proposed di�erent criteria that account for the long-run equilibrium relationship. In this paper, however, we adhere to the simpler conventionalmeasures, with the caveat that these measures might be biased against the ECM and FIECMmodels. The results are summarized in Tables III±VIII.

Table III. One-step forecast performance

Model RMSE MAE MAPE(%) CDIR(%)

Martingale 250.7412 195.3180 1.0075 50.6122VAR 253.5640 194.0731 1.0009 53.0612VAR±GARCH 253.0693 193.7710 0.9998 56.3265ECM 250.6327 195.2439 1.0067 50.6122ECM±GARCH 250.5194 195.0721 1.0061 49.3878FIECM 251.6255 194.6264 1.0040 51.0204FIECM±GARCH 251.5253 194.7638 1.0049 51.4286

Here and in Tables IV±VIII RMSE � root mean square error, MAE � mean absolute error, MAPE � mean absolutepercentage error and CDIR � correct direction prediction.

Table IV. Five-step forecast performance



Table V. Ten-step forecast performance





For one-step forecast, Table III indicates that the ECM±GARCH model provides the bestperformance in terms of RMSE, followed by the ECM and martingale models. Similar tothe results in Tse (1995), the EC model outperforms the martingale model marginally. TheVAR±GARCH model gives the best performance based on MAE, MAPE and CDIR. Theimportance of fractional cointegration in modelling and prediction, however, is not apparentbased on the one-step prediction.

When the step size increases to ®ve, martingale model dominates others in terms of RMSE.Also, VAR becomes the best model by any of the other three criteria, followed by FIECM±GARCH. The same conclusions apply to the case of ten-step forecasts. However, when the stepsize is larger than twenty, fractional cointegration plays an important role. In Tables VI±VIII

Table VI. Twenty-step forecast performance



Table VII. Thirty-step forecast performance



Table VIII. Forty-step forecast performance





FIECM±GARCH dominates other models in RMSE, MAE and MAPE. Also, the relativeperformance of ECM±GARCH improves as the step size increases. For twenty-step forecasts,ECM±GARCH is outperformed by VAR in every criterion. However, ECM±GARCH performsbetter than VAR in RMSE when the step size is thirty and in every criterion when the step sizeis forty. That is, for the forty-step forecast, the best model is FIECM±GARCH, followed byECM±GARCH.

Several conclusions can be drawn from these results. First, for long-horizon forecasts such asup to forty periods ahead, incorporating cointegration or fractional cointegration as in the ECMand FIECM does lead to better performance. Second, GARCH e�ects help improve forecastperformance for the ECM and FIECM models. On the other hand, imposing GARCH e�ectson the VAR models leads to worse forecast performance. Third, combining the GARCH e�ectswith cointegration or fractional cointegration provides the best forecast performance for thelong-horizon forecasts. In fact, the improvement over other competitive models increases as thestep size increases. Without the GARCH e�ects, both the martingale and VAR (in di�erences)models perform consistently well across di�erent forecast horizons. On the other hand, theperformances of the ECM and FIECM models quickly decline as the forecast horizon increases.It appears that misspeci®cations of the second moments have less adverse e�ects on simplermodels.

CONCLUSIONS

In this paper we have investigated the forecast performance of FIECM against several com-petitive models using daily Nikkei spot index and futures price data. We found that, whenignoring conditional heteroscedasticity, the fractional cointegration does not improve forecastperformance. In fact, the VAR model provides the best performance. When conditional hetero-scedasticity is correctly incorporated into the model together with fractional cointegration, weobtain the best forecasting performance for step sizes larger than twenty. When the step size isforty, the ECM±GARCH model is the second best. The results are consistent with the notionthat cointegration or fractional cointegration is important only for long-run predictions. We alsoadd that incorporating fractional cointegration in an EC model improves the forecastingperformance over conventional EC models.

It should be noted that our results are based on the NSA index, and generalization to othertime series must be taken with care. Nonetheless, the results show that the use of the FIECMcoupled with the recognition of time-varying variances may produce relatively good forecastsover long forecasting horizons. Whether the results can be further substantiated with other datais a topic for future research.

ACKNOWLEDGEMENTS

The authors wish to acknowledge De-Min Wu for helpful discussions. Robert Shumway (theeditor) and two anonymous referees provided many useful comments and suggestions. The dataused in this paper were kindly provided by the Singapore International Monetary Exchange(SIMEX).



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Authors' Biographies :Donald Lien is a professor with the Department of Economics and School of Business, the Universityof Kansas. He holds a PhD in social sciences from California Institute of Technology and is an editorialboard member for Journal of Futures Markets . He has published articles in Advance in Futures and OptionsResearch, Communications in Statistics, Econometric Review, Journal of Econometrics, Journal ofForecasting, and Journal of Futures Markets.

Yiu Kuen Tse is an associate professor with the Department of Economics, National University ofSingapore. He holds a PhD in econometrics from the London School of Economics and is a Fellow of theSociety of Actuaries. He has published articles in Journal of Econometrics, Journal of Business and EconomicStatistics, Journal of Applied Econometrics, Journal of Financial and Quantitative Analysis, Review ofEconomics and Statistics, Journal of Futures Markets and Journal of Forecasting.

Authors' addresses :Donald Lien, Department of Economics, College of Liberal Arts and Sciences, University of Kansas,213 Summer®eld Hall, Lawrence, KS 66045-2113, USA.

Yiu Kuen Tse, Department of Economics, National University of Singapore, Singapore.



forecasting the nikkei spot index with fractional cointegration

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