intra-day futures price volatility: information effects and variance persistence

Intra-Day Futures Price Volatility: Information Effects and Variance PersistenceAuthor(s): P. R. Locke and C. L. SayersSource: Journal of Applied Econometrics, Vol. 8, No. 1 (Jan. - Mar., 1993), pp. 15-30Published by: WileyStable URL: http://www.jstor.org/stable/2285108 .

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JOURNAL OF APPLIED ECONOMETRICS, VOL. 8, 15-30 (1993)

INTRA-DAY FUTURES PRICE VOLATILITY: INFORMATION EF'ECTS AND VARIANCE PERSISTENCE

P. R. LOCKE Commodity Futures Trading Commission, 2033 K Street NW, Washington, DC 20581, USA

AND

C. L. SAYERS University of Houston, Houston, TX 77204-5882, USA

SUMMARY

This paper examines the role of the rate of information arrival proxy variables, as they relate to persistence in the variance structure of minute-by-minute S&P 500 Index Futures returns series. The role of contract volume, floor transactions, the number of price changes, executed order imbalance, and an information composite in reducing variance persistence is examined. All proxy variables are found to explain a significant amount of returns variance. While the characteristics of returns data vary daily, some evidence of remaining variance persistence is found, regardless of the definition of the rate of information arrival variable. Our results suggest that utilization of a pure ARCH-type model for high- frequency returns data implies a mis-specification.

1. INTRODUCTION

The fact that empirical distributions of price changes usually appear too leptokurtic to be consistent with Gaussian populations has been long documented (see, for example, Granger and Morgenstern, 1970; Fama, 1965 and Mandelbrot, 1963). An explanation for this observation is provided in the mixture of distributions hypothesis discussed in, for example, Clark (1973), Epps and Epps (1976), Tauchen and Pitts (1983) and Westerfield (1977). A subordinated stochastic process approach (Mandelbrot and Taylor, 1967) is utilized in which price changes are subordinated to the directing process, where the directing process is defined to be the speed of evolution of the price change process. Given that the price change process evolves at different rates over different periods of time, the volume of trade is most commonly used to proxy for the rate of information arrival. The majority of existing literature on this topic has utilized daily data on price changes and volume. A major implication of the subordinated stochastic process approach is that the distribution of returns appears normal when conditioned on its variance. The variance formulation is then hypothesized to be a function of the rate of information arrival, as proxied by volume. It is this information-based variance structure which will be investigated in this paper by examining the effectiveness of several information proxy variables towards explaining return variance structure.

A parallel explanation for the existence of empirical price change distributions which display fat tails and spiked peaks lies in the auto-regressive-conditional-heteroscedasticity (ARCH)

0883-7252/93/010015-16$13.00 Received May 1991 © 1993 by John Wiley & Sons, Ltd. Revised February 1992



P. R. LOCKE AND C. L. SAYERS

(Engle, 1982) literature. The ARCH specification allows persistence in the variance structure, and provides a good approximation to many return series where returns of similar magnitude cluster together in chronological time. Lamoureux and Lastrapes (1990) investigate the viability of the ARCH specification, given that the daily rate of information arrival may be viewed as a heteroscedastic mixing variable in the variance structure of the GARCH(1,1) specification:

rt = ,t-i + Et Et |(t, Et- I, Et2,...) - N(0, ht) (1) ht = co + Oae2-1 + Ci2ht-l + 03 Vt

where rt = rate of return, A^t-I is the mean of rt conditional upon past information, and Vt = trading volume. Here, daily trading volume is used as a proxy variable for the mixing variable, where the mixing variable is defined to be the number of intra-day price changes. Utilizing daily data on a sample of 20 common stocks, little evidence of variance persistence is found, after controlling for volume. Given that the existence of ARCH-type processes reflects an uneven, yet persistent flow of information to the market, the authors conclude that volume explains much of the non-normality of the unconditional return distributions.

While it is appealing to search for information proxies to explain the variance of asset returns, care should be taken to compare this analysis with an underlying price formation process. If the underlying price formation process is directed by a serially correlated information process, the serial correlation may result in a significant correlation between the price formation process and trading variables. However, insofar as the process is independent of trading, and yet generates new prices, trading per se need not explain persistence in returns volatility.

The purpose of this paper is to investigate the explanatory role of information on variance persistence within intra-day price series. The basic premise (see for example, Diebold, 1986 and Stock, 1987, 1988) is that the rate of economic evolution may differ from the rate at which observations are recorded, such as on a quarterly, monthly or daily basis. Given that financial markets display high speeds of adjustment, studies based upon daily observations may fail to capture information contained in intra-day market movements. Thus, this study utilizes minute-by-minute data on the S&P 500 Index Futures. In addition, characteristics of this futures market differ over days of observation (see Kuserk, Locke, and Sayers 1992). This study utilizes minute-by-minute data on a per-day basis for a sample month.

We investigate the role of the rate of information arrival variable, as it relates to the futures price variance structure. In particular, we evaluate the effectiveness of contract volume, versus alternative information measures, in explaining the structure of price change variance in the S&P 500 Index Futures. Given that the existence of ARCH-type persistence in the variance structure may signal model mis-specification, our primary objective is to specify the rate of information arrival proxy variable structure so as to eliminate evidence of ARCH-type persistence in the variance structure. Given the hypothesis that price changes are related to an underlying information-based price process, we examine the role of the rate of information arrival proxy variable in explaining price change variance. While the correlation between price volatility and volume is well documented in the literature, (see Karpoff, 1987 for a survey) it is the hypothesis of this paper that other variables may yield valuable information on market mechanisms.

This paper is organized as follows. Section 2 discusses the data set and our variable formulation. Empirical results are presented in section 3, and section 4 contains concluding remarks.

16



INTRA-DAY FUTURES PRICE VOLATILITY

2. DATA FORMULATION

The data reflect trading in the Chicago Mercantile Exchange's S&P SOO Index Futures contract. The relatively event-free month of April 1990 was chosen arbitrarily, and we examine trading in the June 1990 expiration. This month yields 20 trading days of intra-day observations. The sampling frequency is 1 minute. We sample from 9:30 a.m. to 3:59 p.m Eastern time, which yields 390 observations per day. Two data sources are employed: the Time and Sales record (T&S)2 and the computerized trade reconstruction (CTR)3 record.

Prices are recorded into the T&S conditional on a price change. We select the first recorded price each minute on the T&S and retain the previous price for minutes with no price change events. From the T&S record we obtain the percentage change in the futures price on a per- minute basis. In addition, we record the number of price changes which occurred within the minute time interval. The average number of price changes per minute of trading in our sample is 5 9. From the T&S, as reported from the floor of the exchange, we select only those prices representative of transactions occurring at new prices. Thus, repeated prices, and prices with the qualifier 'bid' or 'ask' attached are eliminated. Bids and asks are entered onto the T&S as a signal of a broker's attempt to trade a customer order.

From the CTR record we obtain three potential rate of information arrival variables. Our volume variable is defined to be contract volume, equal to the number of S&P 500 futures contracts traded within a minute time interval, excluding error trades, spread trades, and exchanges for physical. This variable corresponds to the volume variable used by Lamoureux and Lastrapes (1990). In addition, the number of floor transactions is utilized in order to capture price volatility effects not necessarily represented by contract volume. On the futures exchange, customer orders may be broken up into smaller increments (to a minimum size of one contract) at the discretion of the floor broker. Thus, if Broker A is holding a customer order to buy 10 contracts at the market price, and Broker B is offering to sell six contracts, then Broker A will buy six from Broker B, and then perhaps buy four more from four separate floor traders trading for their own account, either at the same price or not. This transaction would appear as five records on the CTR, which are considered to be five floor transactions in our study. Transactions, rather than volume per se, may be more indicative of information flows.

The third variable obtained from the CTR record is order imbalance, defined to be the absolute value of the difference between all contracts bought and sold by floor traders for their own account within a minute. Floor traders trading for their own account are generally assumed to add liquidity to the market, by serving as market makers in the sense that they bid and offer, trade frequently, and attempt to keep their exposure to a minimum. Their aggregate position change over a time interval is a measure of an equal and opposite imbalance in

'Trading in the S&P 500 index futures is allowed from 9:30 a.m. until 4:15 p.m. Eastern time. The last 15 minutes of trading the index futures are eliminated because the market changes significantly when the New York Stock Exchange shuts down at 4:00 p.m. Eastern time. 2The T&S is considered to be an excellent source of intra-day time and price data, and is input by a pit recorder at the Chicago Mercantile Exchange with assistance from several pit observers. The time and sales record is used and checked by floor traders, customers and brokers to validate trade times and prices. 3The CTR contains a record of every S&P 500 futures transaction, where each transaction is a trade between two floor traders, acting either as principal or agent. Using a paper trail, these transactions are algorithmically placed chronologically (to the nearest minute) within the trading day by the exchange. The algorithm matches trades with prices recorded in the T&S record.

17



18 P. R. LOCKE AND C. L. SAYERS

customer trading. Thus, the degree of order imbalance,4 the aggregate change in floor traders' positions, serves as a proxy for information arrival to the market. Increased executed order imbalance is a sign that the present futures price is not consistent with off-floor price expectations.

While the CTR is expectationally correct up to the inute, there is room for error in the trade reconstruction algorithm. The primary use of the CTR is for trade sequence and clearing purposes, not exact reconstructed trade times.5 However, rather than aggregate to a lower frequency with the hope of improved CTR accuracy, we utilize high-frequency sampling in spite of possible data errors. We consider information flows to be of a high frequency, and believe that aggregation over lower frequencies would mask the relationships we wish to investigate in this study.

3. EMPIRICAL RESULTS

Figure 1 displays a time-series plot of the S&P 500 Index Futures price level by minute. The series appears relatively free from structural breaks when examined on a daily basis, as in this

0 6 A p R 9 0

0 9 A P R 9 0

I I 0 1 A A P P R R 9 9 0 0

6 A P R 9 0

1 7 A P R 9 0

I 8 A P R 9 0

2 3 A P R 9 0

2 4 A P R 9 0

DAYTIUE

Figure 1. S&P 500 Futures price by minute, April 1990

4 The use of the term order imbalance is not to be confused with the unexecuted order imbalance on stock exchanges. Our variable is an ex post executed imbalance signalling a realized change in market maker inventory. 5CTR records with customer types of own account and/or customer are believed to have the highest timing accuracy due to the nature of the record-keeping involved. Floor trader's trades for his own account are entered sequentially onto trading cards which are themselves sequentially numbered Thus, each floor trader's trade for his own account will be in the proper chronological order in the CTR tape. Each floor trader's trade for a customer has a corresponding printed order which is time-stamped some time near its arrival time. All trading cards are collected and time-stamped frequently, putting an upper bound on all execution time windows.

FUTURE 360 -

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INTRA-DAY FUTURES PRICE VOLATILITY 19

LOFUT 0. 00266 -

0.00246 0 00226 0.00206- 0 .0186 - 0 00 1 6 - 0, 01 4 6

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p p P P P P P P P P P P P P A R R R R R R R R R R R R R R R R R R R f t T 0 9 9 9 9 9 9 9 9 9 9 9 9 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

OAYTI C

Figure 2. S&P 500 Futures log price change by minute, April 1990

study. Let our price change variable hereafter be denoted by returns = rt = ln(ft/ft-), where ft = futures price at time t.6 Figure 2 provides a time-series plot of the returns series. Taken as a whole, the series is relatively uncorrelated, with a first order autocorrelation coefficient of - 0 04. Remaining autocorrelation coefficients are quite small. The returns series appears relatively free from outliers, with the exception of a few days of trading. Volatility appears to increase primarily on 6, 20 and 27 April, which are Fridays.7 The issue of increased volatility on Fridays and the ability of our rate of information arrival variables to account for this higher volatility will be addressed later in the paper.

Descriptive statistics and sample correlations for returns and rate of information arrival variables are displayed in Table I. Frequency distributions re displayed y ti ain Figures 3 through 7. Normality is rejected for returns and for all information arrival variables at the 1 per cent level of significance. Table II contains the means of squared returns and the information arrival variables on a daily basis. Note that the daily means change substantially over trading days, and that high daily mean values of contract volume tend to be associated with relatively high daily mean values of floor transactions and price changes.

Results of whiteness and normality tests for returns are displayed in Table III. Note that all daily series appear to be uncorrelated, as demonstrated by the Ljung-Box statistic, with the exception of 9 April.8 The 12 April and 24 April series were prefiltered by fitting an AR(2). The skewness and kurtosis coefficients are inconsistent with normality for many, though not

6Technically speaking, returns are not defined for changes in futures prices. The term returns is adopted out of convenience. 7 Data for 13 April, 1990 are missing from this study of trading days due to the Good Friday holiday. 8The 9 April series is consistent with whiteness at 18 lags, with a Ljung-Box statistic equal to 23-96 versus the 5 per cent critical value X2(18) = 28-87.




Table I. (a) Descriptive statistics, total series, April 1990, 7800 minute-by-minute observations

Standard Variable Mean deviation Maximum Minimum Skewness Kurtosis

Returns* -4-67 x 10-6 3-71 x 10-4 2-49 x 10-3 -1-94 x 10-3 -0-06 1-67 Contract volume* 94-24 93 75 1245 0 2-76 13-47 FIoor transactions* 35-27 29-06 266 0 2-06 6-05 No. of price changes* 5-92 2-97 20 0 0-59 0-44 Order imbalance* 17-90 19-79 184 0 2-48 8-74 Squared returns 1-38 x 10-7 2-64 x 10-7 61-78 x 10-7 0

* Normality is rejected at the 1 per cent level of significance, using the Kolmogorov-Smirnov D statistic.

Skewness = 0 and Kurtosis = 0 for a normal (Gaussian) distribution. Approximate error bars, assuming normality are: ask = 0-028 and rku = 0-056.

(b) Sample correlations, total series, April 1990

Contract Floor Order No. of price Squared volume tansactions imbalance changes returns

Contract volume 1 0-92 0*56 0.54 0*28 Floor transactions 1 0-47 0-56 0-28 Order imbalance 1 0-31 0-16 No. of price changes 1 0-26 Squared returns 1

R e , I 11 o ii 0 l7

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y I

I-

I - 3- 2' I-

t 0I -0. -0 1 .0 4 -0.1 0,0 0 2 0.4

Value o Pr ice Change

Figure 3. Frequency distribution of price changes, S&P 500 Index Futures, April 1990

20

I

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0 5 0 1 00 1 50 2 00 2 50 3 00 3 50

CO l t a ets r ade d

Figure 4. Frequency distribution of volume by minute, S&P 500 Index Futures, April 1990

25 50 75 100 125 ISO 17

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Figure 5. Frequency distribution of transactions by minute, S&P 500 Index Futures, April 1990

21

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22 P. R. LOCKE AND C. L. SAYERS

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7 1 0 0 1 2 3 4 5 6 7 8 9 10 I 1 12 13 14 15 16 17 18 19 20 21 22

Pr ice Changes

Figure 6. Frequency distribution of number of price changes by minute, S&P 500 Index Futures, April 1990

3-

2-

q

-100 -75 -50 -25 0 25 50 75 100

Order Inbolance

Figure 7. Frequency distribution of order imbalance by minute, S&P 500 Index Futures, April 1990

I I I I I I I I




Table II. Descriptive statistics; daily means of variables, April 1990

Squared return Contract Floor No. of price Order Date (x 107) volume transactions changes imbalance

2 April 1.74 96.63 34.90 7.31 19.28 3 April 1.35 108.56 38.39 6.99 21.13 4 April 1.65 103.40 37.84 7.18 16.99 5 April 1.59 84.36 35.71 7.02 15.91 6 April 1.73 87.28 33.26 6.21 18.18 9 April 0.84 60.23 24.45 4.86 13.89 10 April 0.82 65.07 27.37 5.10 14.05 11 April 0.97 71.37 30.79 4.93 13.55 12 April 0.70 62.64 24.47 4.03 14.50 16 April 1.03 98.47 34.04 5.01 17.07 17 April 1.02 89.68 34.11 4.74 17.31 18 April 1.22 110.21 41.20 5.93 19.95 19 April 1.85 99.52 37.16 5.89 19.23 20 April 2.10 126.42 41.67 6.84 21.29 23 April 1.48 111.49 39.58 5.96 20.29 24 April 1.12 95.06 36.93 5.45 18.75 25 April 1.14 89.60 34.99 5.58 17.62 26 April 1.59 110.39 41.60 6.20 20.57 27 April 2.31 117.24 41.70 7.04 19.90 30 April 1.31 97.16 35.33 6.07 18.64

all, of the sample distributions.

days. The positive kurtosis coefficients are indicative of leptokurtic

The bispectral based test proposed by Hinich (1982) is applied in order to test the null hypothesis of Gaussianity, in which the bispectrum will be equal to zero at all frequencies. Should the null hypothesis not be rejected, the data may be consistent with a zero-bispectrum, yet non-Gaussian. As an additional diagnostic, the Hinich (1982) linearity test is applied. This test is based upon the constancy of skewness over different frequencies, given a linear process. An ARCH-type process such as (1), which is linear in lagged values of e2 and h, should yield a constant bispectrum. Ashley, Patterson, and Hinich (1986) provide results which demonstrate that the bispectral tests have good power versus seven alternative nonlinear time- series models, none of which are of the ARCH class. The majority of sample days appear inconsistent with the null hypothesis of Gaussianity. Most, but not all, sample days pass the linearity test.

The Lin-Mudholkar (1980) univariate test for normality is based on the characterization of a Gaussian distribution by the independence of the sample mean and the sample variance. The test has high power against asymmetric alternatives, and against long-tailed symmetric distributions. Under the null hypothesis of a Gaussian series, the Lin-Mudholkar test statistic is asymptotically Gaussian with zero mean and unit variance. The majority of sample days appear normal by this test.

The BDS test discussed in Brock, Dechert, Scheinkman, and LeBaron (BDSL) (1991) provides a test formulated under the null hypothesis of independent and identically distributed (i.i.d.) data series. Based on the correlation integral, the test statistic follows an asymptotic N(0,1) distribution under the null. Rejection of the null implies evidence against the null

hypothesis of i.i.d. BDSL discuss the power of the BDS test versus various nonlinear

23




Table III. Normality and whiteness tests; minute-by-minute returns, April 1990

Date SK KU G LIN L-M L-Box BDS Engle M-L

2 April 0-16 1-16 4-06 4-93 - 104 28-27 3-12 20-64 39-03 3 April 0-28 1-33 9-28 4-63 - 176 21-40 0-89 45-71 56-67 4 April -0-23 1-69 9-15 6-24 1-38 25-95 4-35 56-75 100-67 5 April -0-23 1.00 3-52 2-43 1-54 17-59 1-43 22-04 22-06 6 April 0-39 4-99 8-10 8-00 - 171 12-26 0-70 30-36 29-24 9 April 0.00 0-06 -0-14 0-59 -0-01 37-17 1-59 51-30 80-90 10 April -0-17 1-08 1-92 0-29 1-09 19-87 3-08 13-76 15-96 11 April -0-11 0-63 -0-88 0-29 0-78 15-85 1-52 37-42 38-91 12 April 0.11 0-38 0-84 0-62 -0-84 30-07 -0-80 7-26 12-44 16 April -0-49 2-24 4-02 0.09 2-76 11-68 4-89 26-31 34-77 17 April 0-34 1-35 3-01 0.05 -2-14 14-38 3-64 31-24 37 04 18 April -0-17 0-37 3-18 1-82 1-27 23-20 1-62 39-60 48-85 19 April -0-09 1-18 2-96 1-51 0-59 20-30 1-98 31-39 37-07 20 April -0-35 0-71 4-00 0-71 2-43 23-66 1-83 34-67 44-55 23 April -0-40 1-04 1-49 0-98 2-65 19-02 4-21 24-99 27-28 24 April -0-11 0-54 1-17 0-66 0-81 23-06 1-65 9-57 18-76 25 April 0.00 0-33 1-54 -0-13 0.00 15-65 3-48 43-20 81-23 26 April 0.10 0-50 0.19 1-88 -0-70 20 82 0-74 15-71 17-13 27 April 0-09 0-77 1-58 1-75 -0-66 28-46 1-58 18-58 19-51 30 April 0-25 0-78 4-16 1-92 -1-73 9-31 0-12 45-71 46-09

Number of daily observations = 389 (387 for April 12 and 24); SK = coefficient of skewness, KU = coefficient of kurtosis. Approximate standard errors, assuming a normal distribution are ask= 0*124 and aku= 0*248. G = Hinich-Gaussian - N(0,l)approx; LIN = Hinich-linearity - N(0,l)approx; L-M = Lin-Mudholkar - N(0,l)asy, L-Box = Ljung-Box -X2(20) with 5 per cent critical value = 31 41; BDS = Brock-Dechert-Scheinkman - N(0,1)asy, embedding dimension = 3, epsilon = std. dev./spread; critical values based on 5000 replications of normal random variables from Brock, Hsieh and LeBaron (1991); N= 250: upper 2-5 per cent = 2 37, lower 2-5 per cent = -2-17; N= 500: upper 2 5 per cent = 2 26, lower 2 5 per cent = - 2*02; Engle = Engle test - x2(20) with 5 per cent critical value = 31 41; M-L = McLeod-Li - 2(20) with 5 per cent critical value = 31 *41.

alternatives, including ARCH and GARCH. The finite sample results presented in Brock, Hsieh, and LeBaron (1991) are utilized in order to determine approximate critical values for the BDS statistic. The BDS test rejects the null hypothesis on seven of the sample days.

The existence of persistence in the variance structure will be tested by applying the ARCH specification test of Engle (1982) to rt. In particular, estimate:

p r2= a + Fir2-i + t (2)

i=1

=t = error, and test Tx R2 as a x2, where T= number of observations and R2 represents the coefficient of determination from (2). In addition, results of the McLeod and Li (1983) test for whiteness are presented. This portmanteau statistic based upon the sample autocorrelations of the squared time-series is believed to have value when testing linearity against ARCH-type alternatives. The McLeod-Li test statistic is asymptotically equivalent to the Engle test statistic under the null hypothesis of no ARCH effects. See Luukkonen, Saikkonen, and Terasvirta (1988) for a discussion of the performance of various Lagrange multiplier tests.

Evidence of ARCH-type effects in our intra-day series appears mixed. Twelve out of 20 sample days display little evidence of variance persistence, as demonstrated by the Engle test statistic. The McLeod-Li test statistic rejects the null hypothesis of whiteness more frequently, on 12 of 20 trading days. Note that many of the days which demonstrate persistent variance

24




structure also appear inconsistent with normality. Given that the existence of ARCH-type processes may serve as a signal of model mis-specification, our goal is to identify the sources of variance persistence in our returns series.

A major implication of the subordinated stochastic process approach is that the distribution of returns appears normal when conditioned on its variance. The variance formulation is then hypothesized to be a function of the rate of information arrival, as proxied by volume. It is this information-based variance structure which will be investigated in this paper by examining the effectiveness of several information proxy variables towards explaining return variance structure. Equations of the general form:

20

rt2= a + yIt + firt-i + et (3) i=l

will be estimated, where et = error, and It represents information arrival. Given the lack of serial correlation exhibited in the returns series, we assume an expected return of zero for the futures price. Thus, r is used as a representative for var(rt). Given the importance of volume as an information arrival variable, as demonstrated by Lamoureux and Lastrapes (1990), we investigate whether evidence of variance persistence is removed in intra-day S&P 500 Index Futures, after controlling for volume. Alternative definitions of It are the number of floor transactions within the minute, the number of price changes within the minute, and absolute order imbalance over the minute time interval.

As a fifth information arrival proxy we perform principal-components analysis on the four variables contract volume, floor transactions, number of price changes, and order imbalance. We interpret the first principal component as an information composite. The first principal component eigenvectors (by date) are presented in Table IV. The reported eigenvectors are

Table IV. Principal-components analysis; first principal eigenvector elements by days, April 1990

No. of price Date Contract volume Floor transactions Order imbalance changes

2 April 0-590051 0-578384 0.381774 0.414199 3 April 0-563759 0-545602 0.414319 0.461296 4 April 0.561277 0-559377 0.397386 0.462763 5 April 0-581497 0-572435 0.380796 0.434942 6 April 0.562058 0-558926 0.449702 0.411656 9 April 0-563856 0-553312 0.418895 0.447704 10 April 0-565951 0-564581 0.414152 0.435231 11 April 0.567171 0-560154 0.400900 0.451468 12 April 0.556613 0-546788 0.440038 0.444490 16 April 0-575491 0-566540 0.355527 0-470577 17 April 0.551492 0-546760 0.458745 0.431814 18 April 0.571869 0-560365 0.390655 0.454253 19 April 0-585579 0-575998 0.413265 0.393110 20 April 0-577848 0-576179 0-370484 0-443679 23 April 0-574897 0-563981 0.411628 0.426593 24 April 0-567735 0-551297 0.428479 0.436066 25 April 0-584910 0-572744 0.390267 0.421350 26 April 0.584298 0-566394 0.438563 0-381386 27 April 0-558748 0-557535 0.420277 0.447574 30 April 0-569592 0-551841 0.419592 0-441564

Eigenvectors were formed from the correlation matrix to avoid scaling problems.

25




formed from the correlation matrix to avoid scaling problems. From the table, the near- equivalent weights for contract volume and floor transactions indicate that these two variables contribute approximately equally to the first component. Order imbalance has the smallest element, or weight, and thus contributes the least to the first component. The information composite is mainly influenced by floor transactions and contract volume. For our analysis we utilize the first two principal components, which generally explain around 85 per cent of the total variation. Utilizing the information composite as It in (3), we test for evidence of remaining persistence in the returns variance.

Results of the variance persistence tests are shown in Table V. In general, the coefficients of the information variables were highly significant. However, the information variables appear insufficient to control for variance structure persistence. On trading days where evidence of ARCH-type effects appeared strong, as indicated by large Engle and McLeod-Li test statistics, evidence of significant variance persistence remains, after controlling for rate of information arrival variables.9 Variance persistence remains when lagged values of It are included in (3).

In contrast to the results of Lamoureux and Lastrapes (1990), we find significant variance

Table V. Information arrival results; minute-by-minute S&P 500 Index Futures return variance, April 1990; variance persistence F-test results

Contract Floor No. of price Order Information Date volume transactions changes imbalance composite

2 April 1.01 0-97 0-80 1-12 0-94 3 April 216** 2.39** 217** 2-35 220** 4 April 2-55** 2-69** 1.59* 2-86** 205** 5 April 0-96 1-02 0-81 1.09 0-87 6 April 1-43 1-45 1-47 1-61* 1-53 9 April 2-16** 2-15** 230** 2-59** 212* 10 April 0-60 0-68 0-48 0-67 0-56 11 April 1-93** 1-98** 1.74** 2-00** 182** 12 April 0-62 0-58 0-45 0-65 0-67 16 April 1-07 0-95 1.01 1-22 0-94 17 April 1-15 1-13 1-30 1.58* 1-21 18 April 1.91** 189* 78** 2-07* 193* 19 April 1-53 1-57* 1-40 1.61* 140 20 April 1-54 1.66* 1-54 1-86** 153 23 April 1-04 0-97 1-13 1-30 1-06 24 April 1-16 1-22 1-02 1-23 1-24 25 April 2-22** 2-25** 2.08** 2-43** 219** 26 April 0-73 0-79 0-72 0-78 0-72 27 April 0-84 0-83 0-86 0-92 0-83 30 April 2-29** 2-36** 2-25** 2-27** 2-25*

Equation estimated: 20

rt2= aY + t + irt-i + e, i=1

where et = error, and It represents definition of information arrival. Ho = Insignificant variance persistence; 5 per cent critical value = 1I57; 1 per cent critical value = 1 71.

920 April provides an exception, where we fail to reject the null hypothesis of insignificant variance persistence in the three instances where I, is defined to be contract volume, number of price changes, and the information composite.

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persistence after controlling for contract volume. In general, the performance of floor transactions and order imbalance as information arrival proxy variables appears less satisfactory than the performance of contract volume, number of price changes, and the information composite. The F-test appears unable to differentiate between the quality of the information composite and the use of contract volume or the number of price changes in controlling for variance persistence.

Finally, we test the subordinated stochastic process implication that the distribution of returns, conditional upon information arrival, appears normal. By utilizing the predicted value of r2 from (4):

rt = al + yIt + et (4)

which we will denote by at, weighted returns were calculated and their distributional qualities evaluated. An evaluation of It is based upon the specification of (4) which renders the weighted return series consistent with Gaussianity. Letting weighted returns be denoted by Zt = rt/at, where at represents the estimated standard deviation from (4), we evaluate the distribution of [z} by a series of tests for Gaussianity.

Results of BDS statistics on weighted returns are shown in Table VI, as well as comparable

Table VI. BDS statistics, minute-by-minute returns, S&P 500 Index Futures

Original Contract No. of price Information Date series volume changes composite

2 April 3.12 0-22 0.22 -0-06 3 April 0-89 - 179 - 124 - 146 4 April 4-35 1-13 -0-06 0.12 5 April 1.43 -0-22 0-03 -0-20 6 April 0-70 0-59 0-15 0-56 9 April 1-59 - 029 - 129 - 087 10 April 3-08 0-65 0.18 -0-41 11 April 1-52 - 085 - 168 - 149 12 April -0-80 0-94 0-41 0.93 16 April 4.89 1.08 1-07 -0-17 17 April 3-64 1.10 -0-26 - 105 18 April 1-62 -0-16 -0-52 -0-05 19 April 1-98 1.13 -0-48 -0-35 20 April 1.83 0-44 -1-16 -0-77 23 April 4-21 1.88 1-38 1-57 24 April 1-65 -0.83 -0-50 - 109 25 April 3-48 0-51 0-51 0-52 26 April 0-74 0-75 0.30 0.18 27 April 1-58 1-04 0-65 0-75 30 April 0-12 - 135 -0-91 - 104

Number of observations = 389 (387 for 12 and 24 April). Variable tested is the return or weighted return. Weights are estimated from regressions of squared returns on contract volume, number of price changes, or the first two principal components (from information variables contract volume, floor transactions, no. of price changes and absolute order imbalance). BDS = Brock-Dechert- Scheinkman -N(0,1)asy, embedding dimension =3, epsilon= std./spread. Critical values from Brock, Hsieh, and LeBaron (1991): N= 250: upper 2 5 per cent = 2-37, lower 2 5 per cent = -2 17; N= 500: upper 2 5 per cent= 2.26, lower 2 5 per cent = -2.02.

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Table VII. Hinich test for Gaussianity

Original Contract No. of price Information Date series volume changes composite

2 April 4-06 1-69 1-23 2-26 3 April 9-28 -0-31 0-88 -0-58 4 April 9-15 5-05 2-69 14-87 5 April 3-52 -0-43 1-24 0-07 6 April 8-10 2-16 10-44 2-95 9 April -0-14 -0-28 -0-73 -0-76 10 April 1-92 0-08 - 104 - 048 11 April -0-88 - 139 -0-41 - 112 12 April 0-84 -0-09 -0-77 0.05 16 April 4-02 0-56 1-08 1-16 17 April 3-01 -0-18 5-97 9-39 18 April 3-18 0-88 1-18 2-11 19 April 2-96 0-68 -0-43 -0-53 20 April 4-00 1-06 0-80 0-72 23 April 1-49 0-33 -1-35 0.01 24 April 1-17 0.00 0-68 0.01 25 April 1-54 1-80 1-04 0-96 26 April 0.19 0-76 0-27 0-49 27 April 1-58 0-50 0-61 0-40 30 April 4-16 0-48 1-56 -0-32

Number of observations = 389 (387 for 12 and 24 April). Variable tested is the return or adjusted return. Weights are estimated from regressions of squared returns on contract volume, number of price changes, or the first two principal components (from an information composite containing contract volume, floor transactions, number of price changes and absolute order imbalance). Hinich Gaussianity test - N(O,l)approx.

BDS statistics on the original returns series. We fail to reject the null hypothesis of i.i.d. for weighted returns series formed by defining It in (4) to be contract volume, the number of price changes, and the information composite. Results of the Hinich test for Gaussianity on the weighted returns are shown in Table VII. As should be expected, many of the weighted series appear more consistent with the hypothesis of normality. However, a large number of returns series remain strongly leptokurtic.

Note that weighted returns for the relatively volatile Fridays, dated 20 and 27 April, appear consistent with Gaussianity. In contrast, weighted returns for Friday, 6 April, the most volatile day in our sample, still appear inconsistent with Gaussianity. Results for 4 and 17 April could easily be attributed to the existence of a small number of outliers, as the Hinich test appears relatively sensitive to the presence of outliers, as compared to the BDS test. When equation (3) was utilized in order to generate the weights, the result was less satisfactory. In particular, while the BDS test was unable to differentiate weighted returns from the null of i.i.d., the Hinich test often strongly rejected the null of Gaussianity. The magnitude and frequency of outliers in the weighted returns series constructed by utilizing equation (3) was considerable.

4. CONCLUSION

This paper investigates the role of the information arrival variable, as it relates to persistence in the variance structure of minute-by-minute S&P 500 Index Futures returns series. The role

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of contract volume, floor transactions, the number of price changes, and executed order imbalance as proxy variables for information flows in reducing variance persistence are examined. With the methods utilized we are unable to differentiate between the performance of the rate of information arrival proxy variables contract volume, number of price changes, and an information composite in eliminating variance persistence. Contract volume and the number of price changes appear to be equally successful information arrival proxies as evidenced by the F-tests in Table V.

In addition, individual weighted returns formulated by utilizing predicted variances which are functions of contract volume, number of price changes, and the information composite, appear consistent with the hypothesis of i.i.d. All information proxy variables are found to explain a significant amount of returns variance. While the characteristics of returns data vary daily, some evidence of remaining variance persistence is found, regardless of the definition of the rate of information arrival variable. Thus, adjusting the S&P 500 returns series by contract volume, price changes, or the information composite alone does not eliminate serial dependence in the second moment on all days. The results suggest that utilization of a pure ARCH-type model for high-frequency returns implies a mis-specification.

As would be expected, the adjusted returns series appear more consistent with normality than the original series. However, many of the adjusted series remain strongly leptokurtic. This could be attributed to the presence of outliers. In particular, the contract volume variable contains a small number of large observations. These outlier observations could possibly be attributed to an alternative pricing mechanism. In addition, the power of our asymptotic tests may be limited in finite samples.

Finally, price volatility may be attributed to a market information signal which is distinct from our rate of information arrival proxy variables. For example, a favourable public announcement could cause a relatively large increase in price with a relatively small number of price changes, and without a similar increase in volume. If the underlying price formation process is directed by a serially correlated information process, the serial correlation may result in a relationship between the price formation process and trading variables. In other words, trading per se will not explain persistence in returns volatility.

ACKNOWLEDGEMENTS

This paper was prepared for presentation at the Conference on Nonlinear Dynamics and Econometrics, UCLA, 5-6 April, 1991, under the title 'Information and chronological time effects in intra-day futures price volatility.' The authors are grateful to participants of the Nonlinear Dynamics Conference and to Leigh Riddick, George Wang, and two anonymous referees for helpful comments. The BDS program was provided by W. D. Dechert and the BISPEC program was provided by Doug Patterson. The views stated within are those of the authors, and do not necessarily reflect those of the Commodity Futures Trading Commission or its staff.

REFERENCES

Ashley, R. A., D. M. Patterson, and M. J. Hinich (1986), 'A diagnostic test for nonlinear serial dependence in time series fitting errors', Journal of Time Series Analysis, 7, 165-178.

Brock, W. A., D. A. Hsieh, and B. LeBaron (1991), A Test of Nonlinear Dynamics, Chaos and Instability, MIT Press, Cambridge, MA.

Brock, W. A., W. D. Dechert, J. A. Scheinkman, and B. LeBaron (1991), 'A test for independence based upon the correlation dimension', Department of Economics, University of Wisconsin-Madison.

29




Clark, P. K. (1973), 'A subordinated stochastic process model with finite variance for speculative prices', Econometrica, 41, 135-159.

Diebold, F. X. (1986), 'Comments on modelling the persistence of conditional variance', Econometric Reviews, 5, 51-56.

Engle, R. E. (1982), 'Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inflation', Econometrica, 41, 135-156.

Epps, T. W., and M. L. Epps (1976), 'The, stochastic dependence of security price changes and transaction volumes: implications for the mixture-of-distributions hypothesis', Econometrica, 44, 305-321.

Fama, E. F. (1965), 'The behavior of stock market prices', Journal of Business, 38, 34-105. Granger, C. W. J. and 0. Morgenstern (1970), Predictability of Stock Market Prices, D. C. Heath,

Lexington, MA. Hinich, M. J. (1982), 'Testing for Gaussianity and linearity of a stationary time series', Journal of Time

Series Analysis, 3, 169-176. Karpoff, J. M. (1987), 'The relationship between price changes and trading volume: A survey', Journal

of Financial and Quantitative Analysis, 22, 109-126. Kuserk, G. J., P. R. Locke, and C. L. Sayers (1992), 'The effects of amendments to rule 80A on liquidity,

volatility and price efficiency in the S&P 500 Futures', Journal of Futures Markets, 12, 383-409. Lamoureux, C. G., and W. D. Lastrapes (1990), 'Heteroskedasticity in stock return data: volume versus

GARCH effects,' Journal of Finance, 45, 221-229. Lin, C-C., and G. S. Mudholkar (1980), 'A simple test for normality against asymmetric alternatives',

Biometrika, 67, 455-461. Luukkonen, R., P. Saikkonen, and T. Terasvirta (1988), 'Testing linearity in univariate time series

models', Scandinavian Journal of Statistics, 15, 161-175. Mandelbrot, B. (1963), 'The variation of certain speculative prices,' Journal of Business, 36, 394-419. Mandelbrot, B. and H. M. Taylor (1967), 'On the distribution of stock price differences', Operations

Research, 15, 1057-1062. McLeod, A. I., and W. K. Li (1983), 'Diagnostic checking ARMA time series models using squared-

residual autocorrelations', Journal of Time Series Analysis, 4, 269-273. Stock, J. H. (1987), 'Measuring business cycle time', Journal of Political Economy, 95, 1240-1261. Stock, J. H. (1988), 'Estimating continuous-time processes subject to time deformation: an application

to postwar U.S. GNP', Journal of the American Statistical Association, 83, 77-85. Tauchen, G. E., and M. Pitts (1983), 'The price variability-volume relationship on speculative markets',

Econometrica, 51, 485-505. Westerfield, R. (1977), 'The distribution of common stock price changes: an application of transactions

time and subordinated stochastic models', Journal of Financial and Quantitative Analysis, 12, 743-765.

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intra-day futures price volatility: information effects and variance persistence

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