On Mean Reversion in Stock Index Futures Markets Shwu-Jane Shieh* National Cheng-Chi University

Abstract: The long-term dependent behavior in the closing prices and returns of the S&P 500, Nikkei 225, and Dow Jones index futures contracts are investigated by using the ARFIMA (p, d, q) model to estimate the order of the fractional integration parameters for a large range of sampling frequencies: from one- minute frequencies to monthly frequencies. The evidence shows that the closing prices and returns series exhibit mean-reverting properties for most of the sampling frequencies. This suggests that the contrarian’s trading strategies in relation to stock index futures markets have a positive value. Moreover, the empirical evidence indicates that the higher the frequency of the data, the stronger will be the degree of mean-reverting behavior in the close prices and returns series, particularly in the case of the S&P 500 and Dow Jones stock index futures contracts. JEL classification: C22, G13 Key words: mean reverting, contrarian strategy, ARFIMA (p, d, q)

*Associate Professor, Dept. of International Trade, College of Commerce, National Cheng-Chi University, Taipei, Taiwan, R.O.C., E-mail: [email protected].

I.

Introduction

The current literature on long memory in market prices and returns is largely derived from the pioneering work of Hurst (1951), who invented a new statistical method, the rescaled range analysis (R/S analysis), and studied the size of reservoir construction through its application. Mandelbrot’s (1972, 1975) studies on the use of Hurst’s rescaled range or R/S statistic for identifying the fractal behavior in a Brownian motion served as a stimulus for using the R/S statistic to analyze financial time series. A fractal Brownian motion process displays correlation that is persistent even during very large time lags and is distinct from the standard Brownian motion. The evidence that financial time series are examples of long memory processes is mixed. Greene and Fielitz (1977), Booth, Kaen, and Koveos (1982a, 1982b) and Aydogan and Booth (1988) by using the rescaled range statistic of Hurst (1951) arrive at the conclusion that some financial time series have long memory. In particular, Helms, et al. (1984) analyze price changes in futures contracts and claim that the returns of the series display long memory characteristics. Mills (1993) finds weak evidence of long memory in a sample of monthly U.S. stock returns by using the modified rescaled range statistic R/S and the semi-parametric approach of Geweke and Porter-Hudak (1983). Recently, Barkoulas, Labys, and Onochie (1997), conduct 1

the classical R/S analysis to reevaluate the memory of futures returns. They find evidence of persistent long memory in a group of futures contracts. Later, Barkoulas, Labys, and Onochie (1998) use the spectral regression method of Geweke and Porter-Hudak (1983) to estimate the fractionally integrated parameters and conclude that there is evidence of long memory behavior. By contrast, Lo (1991), using a modified rescaled range statistic, R/S, finds no evidence of long memory in a sample of U.S. stock returns. Many other researchers also document no evidence of long memory in financial returns (Fung and Lo, 1993; Cheung and Lai, 1993; Crato, 1994). Meanwhile, Lobato and Savin (1997) find no evidence of long memory in daily Standard and Poors 500 returns, but they find some evidence of long memory in the squared returns. Ding, Granger and Engle (1993) reach the same conclusions. Moreover, Monoyios and Sarno (2001) characterize the basis series of the S&P 500 and the FTSE 100 indices by means of nonlinearly mean-reverting models and conclude that the degree of mean reversion on this basis is a function of the size of the deviation from equilibrium. The rationalization behind the co-existence of such mixed conclusions regarding the persistent behavior of financial time series documented in empirical studies might arise for two reasons. First, the established empirical results might be influenced by the parameter estimation procedures adopted in the studies. Many

2

researchers point out the lack of robustness of the R/S statistic in the presence of short-term memory and heteroskedasticity (e.g., Mandelbrot (1972, 1975), Mandelbrot and Wallis (1968), Davies and Harte (1987), Aydogan and Booth (1988), and Lo (1991)). Fortunately, Sowell (1986, 1992) develops the exact maximum likelihood estimation procedure for the ARFIMA (p, d, q) process with unconditional normally distributed innovations. Such an approach is theoretically appealing according to Ballie (1996). This is also a reason why, armed with Sowell’s MLE estimation procedure, we investigate the mean reversion behavior of the stock index futures markets in this study. We perform the goodness-of- fit test for the model by diagnosing the randomness of the residuals. Second, much of the existing literature finds evidence of persistent behavior in financial time series by using series of returns or absolute returns for low frequency data (e.g., monthly, weekly, or daily). Whether or not the extent of the long memory in the series depends on the sampling frequency of the series is a challenging empirical problem. In this study, we investigate the stochastic memory behavior in regard to the Standard and Poors 500, Nikkei 225, and Dow Jones Industrial indices, futures prices and returns (but not absolute returns) both on an interdaily and intradaily basis. The sampling frequencies of the data set range from one minute, five minutes, daily and weekly to monthly.

3

The distinction between the data set studied here and that of other existing studies is threefold. First, we examine the long memory behavior of not only returns series but also closing price series. The reason for this is that in the case of the ARFIMA model and the related autocorrelation function (ACF), using the close prices does yield an interesting and practical result. While the other researchers focus on the return series in futures contracts, it might be a good complementary strategy to investigate the stochastic memory behavior for the close price series, given the assumption of the normally-distributed innovations. Secondly, we study not only the interday but also the intraday dynamic behavior of the returns and prices time series of futures contracts: from one- minute to monthly frequencies. Third, to avoid bias in the data set, we collect data not only from the main stock index futures prices in the U.S., but also from similar futures prices in Japan. Based on these three stock index futures series, the order of the fractionally integrated parameters in an ARFIMA (p, d, q) model is estimated by using Sowell’s (1992) exact maximum likelihood approach. From the empirical results, we find evidence that the S&P 500, Nikkei 225, and Dow Jones Industrial index futures exhibit mean-reverting properties in terms of both prices and returns series since most of the estimated values of the order of the fractional integration are negative. The results hold for all three stock index futures above and for a large range of sampling

4

frequencies: from one minute to monthly, except for the two low frequencies, weekly and monthly in the case of the Nikkei 225 index futures, which exhibit dynamically persistent behavior, or trend reinforcing series. The evidence that both the close prices and returns series of the Standard and Poor 500, Nikkei 225, and Dow Jones Industrial indexes exhibit mean-reverting properties lends strong support to the success of contrarian trading strategies, as documented first by DeBondt and Thaler (1985). The remainder of this paper is organized as follows. The Data and Methodology section describes the data sets used in this study and derives the stationarity test statistic and the ARFIMA (p, d, q) model. The empirical results obtained from the stationarity tests and the estimates of the fractionally integrated parameters are presented in the Empirical Results section. Some concluding remarks are discussed in the final section.

II.

Data and Methodology In this section, we describe the data sources and content used in this study

and the ARFIMA model estimated for closing prices in three stock index futures markets.

2.1 Data Our primary data set consists of tick-by-tick “raw” closing prices series and

5

returns in three stock index futures markets, the S&P 500 futures, Nikkei 225 futures, and Dow Jones Industrial index futures. Sampling frequencies, the number of observations (N), and the sampling periods are summarized in Table 1. The S&P 500 futures and Nikkei 225 futures are traded on the Chicago Mercantile Exchange (CME), but the Dow Jones Industrial index futures contracts are traded at the Chicago Board of Trade (CBT). The S&P 500 index futures contracts are traded from 8:30 in the morning to 3:15 in the afternoon. The Nikkei 225 index futures contracts are traded from 8:00 to 3:15, which means trading begins half an hour earlier than in the case of the S&P 500 index futures. The trading hours for the Dow Jones Industrial index futures contracts extend from 7:20 to 3:15. All of the data are obtained from the Tick Database. These futures contracts are chosen for two reasons. First, the closing prices of these futures contracts are available on a tick-by-tick basis. This enables us to construct intraday high- frequency data, including 1- min and 5-min frequencies for closing prices and returns. Second, all of these contracts are among the most heavily traded stock index futures contracts in the world. We automatically roll the futures contracts to determine the appropriate roll date for contracts based on total daily tick count. A roll of contracts will occur at the close of the day on which the next contract’s daily tick count exceeds that of the current contract.

6

However, to avoid confounding the evidence in view of the decidedly slower trading patterns over the trading periods, we also employ lower frequency data, such as daily, weekly, and monthly data. The price index series, yt , is differenced in log form to create a raw prices change series.

7

Table 1: Sampling Periods for S&P 500 Futures, Nikkei 225 Futures, and Dow Jones Industrial Index Futures. S&P 500 Futures

Nikkei 225 Futures

Dow Jones Futures

N

Sample period

N

Sample period

N

Sample period

1 min

65535

3/7/00~31/1/01

57274

1/11/97~31/1/01

65535

1/6/00~31/1/01

5 min

62729

1/2/98~31/1/01

63158

25/9/95~31/1/01

65535

1/4/98~31/1/01

Daily

4747

21/4/82~31/1/01

2442

25/9/90~31/1/01

838

6/10/97~31/1/01

Weekly

981

21/4/82~31/1/01

510

25/9/90~31/1/01

174

6/10/97~31/1/01

Monthly

226

21/4/82~31/1/01

119

25/9/90~31/1/01

40

6/10/97~31/1/01

*N is the number of observations in the sample.

8

2.2 The Autoregressive Fractionally Integrated Moving Average Process (ARFIMA (p,d,q) ) model To motivate the investigation of the fractional dynamics in the prices and returns series of the stock index futures contracts, we implement Lo ’s (1991) rescaled R/S range statistic to test the null of no long memory against the alternatives and Kwiatkowski, Phillips, Schmidt, and Shin’s (1992) stationarity test. The results, which can be requested by interested readers, lead to the conclusion that the close price series are neither stationary series nor long memory processes. That suggests that a fractionally integrated random process is adequate to describe the fractional dynamics of the stock index futures prices. The model of an autoregressive fractionally integrated moving average process of order parameters, (p,d,q), denoted by ARFIMA (p,d,q), with mean µ , may be written using backward shift lag operator notation as Φ ( L)(1 − L) d ( y t − µ ) = Θ( L )ε t ,

(9)

assuming ε t ~ N ( 0, σ ε2 ) , where yt is the prices or returns series, and L is the backward shift operator. Moreover,

Φ ( L) = 1 − φ1 L − φ 2 L2 − L − φ p L p ,

(10)

and

Θ( L ) = 1 + θ 1 L + θ 2 L2 + L + θ q Lq ,

9

(11)

where p, q are order parameters for AR and MA models. Furthermore, (1 − L) d is the fractional differencing operator defined by ∞

d Γ (k−d )L (1 − L ) = ∑ Γ( − d ) Γ( k +1) , k

(12)

k=0

where Γ (⋅) denotes the Gamma, or generalized factorial, function. The parameter d characterizes the persistence phenomenon in the prices and returns series. The stochastic process

yt is both stationary and invertible if all roots of

Φ (L ) and Θ(L) lie outside the unit circle and | d |≤ 0.5 . Assuming that | d |≤ 0.5 ,

and d ≠ 0 , Hosking (1981) shows that the autocovariance function, γ (⋅) ,of an ARFIMA process is defined as

γ (k ) =

Γ (1−d )Γ (k + d ) Γ ( k −d +1)Γ ( d )

~ Ck 2d −1 , k → ∞

(13)

where C = π1 Γ(1 − 2d ) sin πd is a constant and is described in detail by Resnick (1987), and γ (k ) is the autocovariance function with lag k . Helson and Sarason (1967) show that any process with d > −0.5 and an autocovariance function given by the above equation violates the strong mixing condition, and hence its long memory or long range dependence. Since we focus on the fractionally integrated parameter to describe the stochastic memory of the series, we estimate the ARFIMA (0, d, 0) model used here as a special case of the ARFIMA (p, d, q) model.

10

III. Empirical Results In this section, the empirical results obtained by detecting long-run dependence against stationarity alternatives are presented first. Based on the results of the stationarity tests, it is enough for us to fit the ARFIMA (p, d, q) models to detect the mean-reversion property in stock index futures markets. Due to space considerations, we only present the graphics of Autocorreoation Function (ACF) for the daily data for the close prices of the three stock index futures in Figure 1.

3.1 Estimating the ARFIMA (p,d,q) model for the close prices of the futures contracts

We estimate the order of the fractional integration parameter d based on the close prices of the S&P 500 index, Nikkei 225 index, and Dow Jones Industrial index. The empirical estimates are displayed in the first columns of Tables 4, 5, and 6, respectively. From the results, it is shown that the estimates of all close price series are negative except for the Nikkei 225 index of weekly and monthly frequency. The results are extremely interesting in that most of the close price series exhibit the mean-reverting characteristic that is statistically significant at the 5% level. Moreover, these series are not only mean-reverting in terms of the

11

1.0

ACF-close

0.5

0 1.0

20

40

60

80

100

120

140

160

180

200

220

240

40

60

80

100

120

140

160

180

200

220

240

40

60

80

100

120

140

160

180

200

220

240

ACF-close

0.5

0 1.0

20 ACF-close

0.5

0

20

Figure 1: ACF for Nikkei 225, S&P 500, and Dow Jones Index Futures Daily Close Prices (from top to bottom)

12

negative

differencing

parameter d ,

but

are

also

covariance

stationary

since − 0.5 < d < 0 . The implications of the mean-reverting evidence can be further explored. In particular, it indicates that the contrarian trading strategy has a positive value, which is documented widely in the finance literature. It is worth noting that in the case of the weekly and monthly Nikkei 225 index covering the sample period from September 25, 1995 to November 30, 2001, the series is characterized as exhibiting statistically significant long memory rather than being mean reverting, which contradicts the results of Lo’s test that are presented in Table 2. In addition, the last column of Tables 4, 5, and 6 shows the Akaike Information Criterion (1974) for each model. Not surprisingly, the values of AIC increase as the sample size gets larger. The results in Table 4 indicate that, for S&P 500 close prices, the estimates for parameter d range from -0.0466 to -0.1548, and the whole of the series exhibits the mean reversion characteristic at the 5% significance level. From Table 5, not surprisingly, it is confirmed that the Dow Jones Industrial index futures series exhibits mean reversion or price reversals since the estimates of the parameter d range from -0.0294 to -0.3840 for different sampling frequencies at the 5% significance level.

13

Table 2: Stationarity Tests for S&P 500 index, Nikkei 225 index, and Dow Jones Industrial Index Futures Closing Prices

S&P 500

Nikkei 225

Dow Jones

Lo

KPSS

Lo

KPSS

Lo

KPSS

1 min

12.04

14.7531**

14.5006

20.099**

8.2654

5.6318**

5 min

15.2021

23.2259**

12.7143

19.0158**

14.2159

20.1809**

Daily

7.0105

5.485**

2.9719

0.5054**

3.1558

1.0434**

Weekly

4.1080

1.8854**

1.838

0.1972

1.9271

0.3995*

Monthly

2.4684

0.6813**

1.3012

0.1016

1.2836

0.1721

*The critical value for Lo ’s test is 0.927 and that for the KPSS test is 0.347 at the 10% significance level. **The critical values at the 5% significance level are 0.861 and 0.463, respectively.

14

Table 3: Stationarity Tests for the S&P 500 index, Nikkei 225 index, and Dow Jones Industrial Index Futures Returns

S&P 500 Lo

KPSS

Nikkei 225 Lo

KPSS

Dow Jones Lo

KPSS

1 min

1.1064

0.0509

1.0289

0.0446

0.9539

0.0447

5 min

1.2237

0.0506

0.8297**

0.0228

1.3379

0.0903

Daily

1.089

0.049

1.3536

0.0668

1.025

0.0368

Weekly

1.1615

0.0565

1.3359

0.0722

1.0065

0.0410

Monthly

1.2198

0.0707

1.2031

0.0655

1.1943

0.0557

*The critical value for Lo ’s test is 0.927 and that for the KPSS test is 0.347 at the 10% significance level. **The critical values at the 5% significance level are 0.861 and 0.463, respectively.

15

Table 6 shows that the estimates of the parameter d for the Nikkei 225 index futures series for different frequencies range from -0.055 to 0.4985 at the 5% significance level. The usual phenomenon found here for Nikkei 225 stock index futures close prices is that the estimates of the parameter d for weekly and monthly data are greater than zero at the 5% significance level. This indicates that, for the same sampling period, intraday and daily Nikkei 225 index close prices exhibit the mean-reversion property, whereas long memory persistence is displayed for weekly and monthly data. To be more specific, the findings reveal inconsistencies in terms of the way in which the long memory in the close price series under different sampling frequencies is revealed. The findings also contradict the monotone relationship between the degree of the long memory and the sampling frequency, as later evidenced in the close prices for the other two stock index futures. Anderson and Bollerslev (1997) assert that “the degree of fractional integration in the absolute returns is invariant with respect to the sampling frequency”. For the sample sets that we have analyzed, the estimates for the fractional integration parameters in relation to the closing prices of the three stock index futures differ for different sampling frequencies, suggesting that the order of fractional integration is influenced by the sampling frequency of the data. The empirical evidence documented here is not consistent with the assertion made by Anderson and Bollerslev (1997). 16

Moreover, in most of the cases, there is a monotone relationship between the degree of the long memory and the sampling frequency. That is, the higher the frequency of the series, the stronger the degree of the fractional integration.

3.2 Estimating the ARFIMA (p,d,q) model for the returns of the futures contracts It should be noted that the assumption of normality underlying the formal statistical justification for the exact maximum likelihood estimates of the order of fractional integration is likely to be violated (especially for high frequency data) if we consider the closing prices of the stock index futures contracts. Thus, we examine the stochastic memory behavior of the returns series of these futures contracts with short term memory dynamics, and the estimates of the order of the fractional integration, d , are summarized in Tables 7, 8, and 9. The empirical evidence is consistent with that documented in the literature whereby the returns series of these futures contracts exhibit mean reverting properties. The results in Tables 7, 8, and 9 are not much different from those in Tables 4, 5 and 6. Table 7 shows that the estimates shown in Table 5 ranging from -0.0294 to -0.3840 for different sampling frequencies range from -0.0533 for the 1 minute frequency to -0.1448 for the monthly frequency and are statistically significant at the

17

5% level. It is interesting that, as the frequency gets higher, the value of the coefficient d becomes larger. From the last column of Table 7, it can be seen that all of the AIC values are negative, which are different from those used in estimating the closing price series. In Table 8, the Dow Jones Industrial index returns series and the Nikkei 225 index futures returns series display the same characteristics as those for the S&P 500 index return series. The only estimate that is not covariance stationary is the monthly Dow Jones Industrial index futures returns series since number of observations is only 40. As a result, it is not statistically significant. The results in Table 9 reveal an inconsistent relationship between the frequency of the data and the values of the parameter d .There does not exist a monotone relationship for the Nikkei 225 index futures returns series as in the case of the S&P 500 and Dow Jones Industrial index futures returns series. Except for the low frequency data in the Dow Jones Industrial and Nikkei 225 indices, the estimates for the order of the fractional integration are statistically significantly different from zero. It is evident that the log returns series of the stock index futures contracts exhibit the mean-reverting property. While the returns series of the stock index futures contracts do not exhibit long memory behavior, the empirical evidence lends support to the mean-reverting behavior, and anti-persistent or ergodic behavior. The results

18

found here are consistent with those documented by Crato and Ray (2000), who find that there is no long memory behavior in futures’ returns. The distinction between the finding here and that of Crato and Ray is that we find that the returns series exhibit mean reversion, but they conclude that the returns display no memory and exhibit a random- walk type behavior.

IV.

Conclusion By using estimations of the order of fractional integration from a ARFIMA (0,

d,0) model, we provide empirical evidence to show that the S&P 500 index, Dow Jones Industrial index, and Nikkei 225 index exhibit mean-reverting, or anti-persistent, behavior for a large range of sampling frequencies, ranging from intradaily one- minute frequencies to interdaily monthly frequencies. Most of the exact maximum likelihood estimators of d are statistically significant at the 5% level. In conducting further analysis, we show that the higher the frequency of the series, the stronger will be the degree of mean reversion in the series. This is true not only for the returns series but also for the close prices series.

19

Table 4: d Parameter Estimates for the Close Prices of S&P 500 Futures Coefficient d

Std. Error t-value

t-prob

AIC

1 min

0.0112*

0.0018

6.06

0.00

2.5306

5 min

-0.0466*

0.0041

-11.5

0.00

9.9592

Daily

-0.0735*

0.0120

-6.09

0.00

6.9873

Weekly

-0.0968*

0.0273

-3.54

0.00

8.9527

Monthly

-0.1548*

0.0748

-2.07

0.04

10.1424

*denotes 5% significance level.

Table 5: d Parameter Estimates for theClose Prices of Dow Jones Industrial Index Futures Coefficient d

Std. Error t-value

t-prob

AIC

1 min

-0.0294*

0.0006

-51.3

0.00

6.8003

5 min

-0.0489*

0.0027

-18.2

0.00

8.0769

Daily

-0.0679*

0.0298

-2.28

0.02

12.5175

Weekly

-0.1352*

0.0687

-1.97

0.05

13.9683

Monthly

-0.3840*

0.1699

-2.26

0.03

15.2674

*denotes 5% significance level.

20

Table 6: d Parameter Estimates for the Close Prices of Nikkei 225 Futures Coefficient d

Std. Error t-value

t-prob

AIC

1 min

-0.055*

0.003

-18.6

0.00

9.6126

5 min

-0.0362*

0.028

-12.9

0.00

10.0853

Daily

-0.0466*

0.0157

-2.97

0.00

14.0225

Weekly

0.4985*

0.0020

247

0.00

15.5255

Monthly

0.4937*

0.0088

56.1

0.00

17.411

*denotes 5% significance level.

Table 7: d Parameter Estimates for the Returns of S&P 500 Futures Coefficient d

Std. Error t-value

t-prob

AIC

1 min

-0.0533*

0.0054

-9.92

0.00

-11.8481

5 min

-0.0128*

0.0041

-2.66

0.00

-10.3132

Daily

-0.1224*

0.0176

-6.94

0.00

-7.0298

Weekly

-0.1217*

0.0596

-2.04

0.04

-4.5798

Monthly

-0.1448*

0.0722

-2.02

0.04

-4.0934

*denotes 5% significance level.

21

Table 8: d Parameter Estimates for the Returns of Dow Jones Industrial Index Futures Coefficient d

Std. Error t-value

t-prob

AIC

1 min

-0.0323*

0.0048

-6.75

0.00

-11.8418

5 min

-0.0222*

0.0046

-4.83

0.00

-10.4541

Daily

-0.1163*

0.0461

-2.52

0.01

-5.9829

Weekly

-0.1226*

0.059

-2.08

0.04

-4.5804

Monthly

-0.5158

0.4913

-1.05

0.3

-3.1695

*denotes 5% significance level.

Table 9: d Parameter Estimates for the Returns of Nikkei 225 Futures Coefficient d

Std. Error t-value

t-prob

AIC

1 min

-0.0611*

0.0047

-13

0.00

-9.7186

5 min

-0.0187*

0.0071

-2.63

0.01

-9.4343

Daily

-0.047*

0.0152

-3.09.

0.00

-5.6562

Weekly

-0.0202

0.035

-0.58

0.56

-4.2531

Monthly

-0.1001

0.0804

-1.25

0.21

-2.6321

*denotes 5% significance level. .

22

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