transaction data tests of the mixture of distributions hypothesis

Transaction Data Tests of the Mixture of Distributions HypothesisAuthor(s): Lawrence HarrisSource: The Journal of Financial and Quantitative Analysis, Vol. 22, No. 2 (Jun., 1987), pp.127-141Published by: Cambridge University Press on behalf of the University of Washington School ofBusiness AdministrationStable URL: http://www.jstor.org/stable/2330708 .

Accessed: 04/09/2014 08:10

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

Cambridge University Press and University of Washington School of Business Administration are collaboratingwith JSTOR to digitize, preserve and extend access to The Journal of Financial and Quantitative Analysis.

http://www.jstor.org

This content downloaded from 109.135.15.50 on Thu, 4 Sep 2014 08:10:33 AMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=cup

http://www.jstor.org/action/showPublisher?publisherCode=uwash

http://www.jstor.org/action/showPublisher?publisherCode=uwash

http://www.jstor.org/stable/2330708?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


JOURNAL OF FINANCIAL AND QUANTITATIVE ANALYSIS VOL. 22, NO. 2, JUNE 1987

Transaction Data Tests of the Mixture of Distributions

Hypothesis

Lawrence Harris*

Abstract

This paper presents new tests of the mixture of distributions hypothesis. Previous tests examined security prices and volume measured only at daily intervals. Here, differential implications ofthe hypothesis for transaction data are derived and tested. The new predic? tions emanate from the assumption that prices and volume evolve at uniform rates in trans? action time. The results support this assumption and the mixture of distributions hypothe? sis in general. In addition, the tests suggest that the daily transaction-count may be a useful instrumental variable for estimating unobserved realizations of stochastic price variances.

I. Introduction

Clark [3] proposes that a mixture of normal distributions be used to model the distribution of daily security price changes. His model assumes that events

important to the pricing of a security occur at a random, not uniform, rate

through time. Using the same assumption, Harris [6] and Tauchen and Pitts [11] show that the joint distribution of daily price changes and volume also can be modeled by a mixture of bivariate normal distributions.

Empirical implications of a random rate of flow of information (the mixture of distributions hypothesis) for the distributions of daily price changes and vol? ume are examined in detail by Clark [3], Morgan [9], Westerfield [12], Harris

[6], [7], Tauchen and Pitts [11], and others. They show that the hypothesis can

explain why the distribution of daily price changes is kurtotic relative to the nor? mal distribution, why the distribution of the daily volume of trade is positively skewed, and why squared (or absolute) daily price changes are positively corre? lated with trading volume. These predictions are all confirmed in studies of daily data.

Although the empirical results in these studies strongly support the mixture

hypothesis, they are subject to criticism. It may be that the above-mentioned

properties of the daily data are simply a consequence of similar properties found

* School of Business Administration, University of Southern California, Los Angeles, CA 90089-1421. The author would like to thank Fisher Black, Tim Campbell, Marc Reinganum, and Mark Weinstein, and especially the anonymous JFQA referees for their helpful comments. Financial support was provided through a University of Southern California School of Business Summer Re? search Grant.

127



128 Journal of Financial and Quantitative Analysis

in transaction data. In particular, it is known that transaction price changes are

kurtotic, that transaction volumes are skewed, and that squared transaction price changes are correlated with transaction volumes.l Perhaps these properties are

preserved when transaction data are aggregated to obtain daily data. This article

investigates this question by studying new predictions of the mixture of distribu? tions hypothesis for transaction data. The empirical results support the mixture of distributions interpretation ofthe daily data.

The topic is important for methodological as well as scientific reasons.

Many financial studies conduct tests of economic hypotheses using daily data. These tests (for example, difference in means tests in event studies) are often made under the assumption that daily price changes are normally distributed and that they are independently and identically distributed over time. Departures from these assumptions may cause unreliable inferences. It is therefore important to determine the extent to which these assumptions apply, and whether depar? tures from them seriously affect the test results. For example, Brown and Warner

[2] show that while the shape of the distribution of daily returns is not generally critical in event studies, heteroskedasticity and autocorrelation can seriously af? fect the specification and power of some tests. Since the mixture of distributions

hypothesis can explain how heteroskedasticity and autocorrelation enter return

series, a better understanding of this theory may lead to improved testing meth?

odologies. The topic also has implications for option pricing models. Option pricing

theory depends crucially on the variance of the underlying price process. Under the mixture of distributions hypothesis, this variance is modeled as a stochastic variable. This article contributes to our understanding of that stochastic variable

by exploring its effects on price changes and also on volumes. In this context, the results of the new mixture of distributions tests are particularly interesting. Ap? parently, the daily transaction-count is a useful instrumental variable for estimat?

ing unobserved realizations of stochastic price variances. The remainder of the article is divided into three sections. Section II pre?

sents the mixture of distributions theory and its predictions for daily data and transaction data. The predictions are tested in Section III and the results are sum? marized in Section IV.

II. The Mixture of Distributions Theory

The mixture of distributions theory must be well understood before its im?

plications for transaction data can be discussed. Subsections A and B review the

theory and its implications for daily data, as developed in Harris [6] and Tauchen and Pitts [11]. The new predictions for transaction data are then introduced in Subsections C and D.

A. A Model for Daily Data

A simple mixture of distributions model for the joint distribution of daily price changes and trading volume can be derived from two assumptions. Suppose

See [13] and [4].



Harris 129

that a series of events takes place, each of which generates information important to the pricing of some security. After each event, price changes occur as traders

adjust their portfolios in response to the new information. The first assumption is that these post-event price changes and volumes of trade are jointly indepen? dently and identically distributed with finite variance.2 The second assumption is that the number of events occuring each day varies. Let nt count the number of events on day t.

The total price change and volume of trade recorded for a given day t repre? sent the cumulation of price changes and volumes that occur as a result of nt information events. The joint distribution of the daily price change and volume of trade will therefore be approximately bivariate normal conditional on nv This follows from the first assumption and from the Central Limit Theorem, provided n,islarge.

The mean and covariance matrices of this conditional distribution are both

proportional to the conditioning variable nt. In the Harris and Tauchen and Pitts

models, the conditional covariance of the daily price change AP, with the daily volume VOL, is equal to zero. This result is used here.3 Both studies also assume that the mean post-event price change is zero so that the conditional mean of AP, is zero. This assumption is relaxed in this study.

The unconditional distribution of AP, and VOL, differs from the conditional distribution because the number of information arrivals, nt, is assumed to vary across days. The unconditional distribution is therefore a mixed distribution. Its

density function is the expectation over the distribution of n, ofthe distribution of

AP, and VOL, given nt. It is said to be mixed because the expectation is a

weighted average ofthe different conditional distributions. This mixture of distributions model can be symbolically represented as

AP, ~

N(ant,bnt\n^),

VOL, ~

N{cnt,dnt\n^j, and

Cov(AP,,VOL,|/!,) = 0 ,

where a, b, c, and d are positive parameters. Variation in the mixing variable nt may be random, deterministic, and/or seasonal.

2 The post-event price change and trading volume are both related to the information generated by the event. The exact relation depends on the market microstructure, on the information dissemina? tion process, and on how trades process information. For the purpose of analyzing the distributional implications of a random flow of information, it is unnecessary to investigate these issues. The as? sumption that post-event price changes and the volume of trade both are realizations of random vari? ables that are independently and identically distributed for all events can be justified in any trading model in which only unexpected information is important.

3 Whether the post-event variables are correlated or not depends on the market microstructure. If the distribution of the unexpected information is symmetric, then some trading models predict that the post-event price change and the volume of trade will be uncorrelated. See, for example, Harris [6] and Tauchen and Pitts [11]. Their prediction is obtained by assuming that the traders are rational and that they use the market price to form their expectations. Under these assumptions, the a priori distri? bution of their trades will be symmetric with zero mean and orthogonal to the market price change, which also will have a symmetric distribution. Since total volume is one-half the summed absolute value of all trades, symmetries cause the expected cross product of AP, with VOL, to be zero. There? fore the correlation between these two variables will be zero.




B. Predictions for Daily Price Changes and Volume

The first set of predictions concerns distributional shapes:

1. The marginal unconditional distribution of AP, is kurtotic relative to the normal distribution [3].

2. The marginal unconditional distribution of AP, is slightly skewed to the

right [8].

3. The marginal unconditional distribution of VOL, is skewed to the right [6].

These well-known properties are a consequence of the mixing process. Kurtosis in the daily price changes results from the fact that the conditional price change variance is proportional to the conditioning variable nt. The unconditional den?

sity is fat-tailed and spiked because it is an average of diffuse (large nt) and com-

pact (small nt) conditional densities. The skewness of volume results from the fact that both the mean and variance of the conditional distribution of volume are

proportional to nt. The unconditional density is right-skewed because it is an

average of spiked densities centered near zero (small nt) and diffuse densities centered far from zero (large nt). Although prices will be right-skewed for the same reason, the price skewness will be slight relative to the volume skewness. The difference lies in the fact that for price changes, the conditional mean is small relative to the conditional variance for all nt. These predictions are all rigo- rously proven in Corollary 1A of Appendix A.

The next two predictions concern daily price-volume relations:

4. AP,2 is positively correlated with VOL, [6], [11].

5. AP, and VOL, are slightly correlated.

These correlations are due to the fact that the first two moments of both the price change and the volume conditional distributions are proportional to the same

conditioning variable, namely the number of information events, nt. The vari? ables AP 2 and AP are correlated with VOL because they are all likely to be large when nt is large, and small when nt is small. The correlation of AP with VOL, however, will be slight because variation in AP due to variation in the condi? tional mean is small relative to that due to variation about the mean. These two

predictions are proven in Corollary 2A of Appendix A. The next set of predictions concerns the time-series behavior of daily price

changes and volumes:

6. Price variance estimates are nonstationary if the level of nt varies through time.

7. Time-series of AP,, AP,2, and VOL, are autocorrelated if, and only if, nt is autocorrelated.

8. Autocorrelation should be strongest for the VOL, series, second strongest for the AP,2 series, and weakest for the AP, series.

Prediction 6 follows from the proportionality of the conditional variance of AP, to nt. The next prediction results from the fact that nt is the only source of auto? correlation in the model (the price changes and volume that follow each event are assumed to be serially independent). The last prediction is obtained by noting



Harris 131

that the extent to which autocorrelation is propagated from the mixing variable nt to a given observed series depends on the fraction of variation in the observed series that is due to variation in nv If this fraction is large, then the time-structure in nt will be transmitted into the observed series. For VOL,, the fraction is large because the conditional mean of the volume distribution is large relative to the conditional variance for all nv The fraction is smaller for AP} because the condi? tional price change distribution has substantial density near zero for all values of

nt. Thus AP} is often small when nt is large, and vice versa. The fraction is smallest for AP, because, for all nv the conditional mean of AP, is small relative to variation about that mean. Formulae that relate autocorrelation in AP,, AP,2, and VOL, to autocorrelation in nt are derived in Corollary 3A of Appendix A.

C. Predictions Concerning the Daily Number of Transactions

Additional predictions of the mixture model are derived by assuming that transactions occur at a uniform rate in event time. The empirical value of this

assumption ultimately depends on the predictive power of the resulting theory. However, some theoretical considerations motivate its use. The number of trans?

actions, Nt, will be proportional to the number of information events, n? in any model in which a fixed number of agents all trade in response to new informa? tion.

The assumed proportionality of Nt to nt yields several sets of new predic? tions for daily data. The first set concerns correlations:

9. Nt is correlated with AP? AP,2, and VOL,. 10. The correlation coefficients should be largest for VOL,, second largest for

AP,2, and smallest for AP,.

The correlations are caused by the proportionality of the conditional means of these variables to nv The strength of the correlation depends on the fraction of the variation in each variable that is due to variation in nv Formulae for these correlations are derived in Corollary 2A of Appendix A.

The next prediction concerns autocorrelations:

11. Autocorrelation in the Nt time-series should be stronger than that found in

any other daily series.

By assumption, all the variation in Nt is due to variation in nv Hence Nt should be as autocorrelated as nv The other daily series are less autocorrelated because

although variation in nt is their only source of autocorrelation, it is not their only source of variation. This prediction is proven in Corollary 3A of Appendix A.

The final set of predictions in this subsection concerns the distribution of the

daily price change divided by the square root of the daily number of trades. This

adjusted price change is distributed with the following mixed distribution,

f, \ r "*

where k is the constant of proportionality between Nt and nv This adjustment is

interesting because it eliminates daily variation due to nt in the conditional price




change variance and because it attenuates the daily variation in the conditional mean. The adjusted price change should therefore be more normally distributed than the unadjusted price change and it should be less affected by time-series

properties ofthe directing variable. In particular:

12. APt /jNt should be less kurtotic and more symmetric than the AP,.

13. Time series of adjusted price changes should be less heteroskedastic than are unadjusted series.

14. The adjusted price change and the squared adjusted price change should not be as autocorrelated as their unadjusted counterparts.

D. Predictions for Transaction Interval Price Changes and Volumes

A second implication of the assumption that transactions occur at a uniform rate in event time is that the number of information arrivals within different trans? action intervals of some fixed length should be constant. This implication yields several predictions for transaction data:

15. Price changes and volumes measured over fixed transaction intervals should be more normally distributed, the longer the transaction interval. In particular, kurtosis and skewness in these variables should decrease as the measurement interval increases.

16. Transaction interval price changes and squared price changes should not be correlated with transaction interval volumes.

17. Transaction interval price changes and volumes should not be autocorre? lated.

Prediction 15 is a consequence of the Central Limit Theorem. It was first tested and confirmed (for prices only) by Brada, Ernst, and Van Tassel [1]. Predictions 16 and 17 are based on the assumption that post-event price changes and volume are independently distributed and serially independent.

III. Empirical Results

The analysis is conducted on a sample of 50 New York Stock Exchange common stocks that traded between December 1, 1981, and January 31, 1983. For each security, price changes and volume are computed over fixed intervals of

1, 10, 50, and 100 transactions and over daily time intervals. A full description of the data set is presented in Appendix B.

As shown in Table 1, all previously tested predictions of the mixture of distributions hypothesis for daily data are familiar and reconfirmed in this sam?

ple: The distribution of AP, is kurtotic and skewed (Predictions 1 and 2), the distribution of VOL, is skewed (Prediction 3), and both AP,2 and AP, are corre? lated with VOL, (Predictions 4 and 5). These results are all statistically signifi? cant.

Before presenting and interpreting the new results, it is instructive to exam? ine the extent to which the daily transaction counts vary and whether they are autocorrelated. These properties are necessary for powerful tests of the new pre-



Harris 133

TABLE 1

Cross-Security Medians of Summary Statistics that Characterize Daily Price Changes and Volumes, Price Changes and Volumes Measured over Transaction Intervals8, and Adjusted

Daily Price Changes.* 50 NYSE Common Stocks, December 1,1981 -January 31,1983

* The adjusted daily price changes are obtained by dividing the daily price change by the square root of the daily number of trades (A/). All but one of these summary statistics would have an expected value of zero if there were no mixing process. The expected value of the heteroskedasticity statistic would be 31 percent. Daily data medians that are more than 2 standard deviations from these expected values are marked with an asterisk. a Price changes over an interval of k transactions are computed by differencing every /cth transaction price. Transaction interval volume is the sum of all shares traded in the k transactions. b Third moment skewness coefficient. c Fourth moment measure of excess kurtosis. d All correlations presented in this table are Spearman correlations. e Spearman first autocorrelation coefficient. f Coefficient of variation among 14 monthly estimates of price variance. dictions of the mixture of distributions hypothesis. Both properties are present in the data. The median among the 50 securities ofthe daily transaction-count coef? ficient of variation is 64 percent while the median first-order autocorrelation coefficient is0.583.4

Consider now tests of the new predictions of the mixture of distributions

hypothesis: the variables VOL,, AP,2, and AP, are all significantly correlated with Nt (Prediction 9) and for each security in the sample, the correlation coeffi? cient is greatest for VOLt and least for AP, (Prediction 10). The median coeffi? cients are 0.740, 0.364, and 0.117, respectively. For all 50 securities, the VOL,

4 All of the correlation and autocorrelation statistics reported in this study are Spearman rank correlations. The Spearman measure is used because it is robust and invariant to monotonic transformations ofthe data. Robustness is desirable because the sample distributions of many ofthe series are highly nonnormal. Invariance to monotonic transformations is important because there is no uniquely proper way to measure the size of a price change (two possibilities are absolute distance and squared distance). Despite these considerations, results using Pearson correlations are qualitatively and, in most cases, quantitatively the same.




and the AP2 correlations are both positive and significantly different from zero at the 5-percent level. The AP, correlation is positive for all but one of the securities and is significantly different from zero for 26 of the 50 securities. These results are fully consistent with the assumption that Nt can proxy for the unobserved realization ofnt.

Autocorrelation among the various daily series is strongest for the Nt series, second strongest for the VOL, series, third strongest for the AP,2 series, and weakest for the AP, series (Predictions 8 and 11). The median autocorrelation coefficients are 0.586, 0.347, 0.177, and 0.045, respectively. For every security but one, the Nt autocorrelation coefficient is greater than the VOL, coefficient. For all but four securities, the VOL, coefficient is greater than the AP,2 coeffi? cient. And for 34 of the 50 securities, the AP,2 coefficient is greater than the AP, coefficient. None of the exceptions are notable. Although these results cannot

prove that autocorrelation in the daily rate is due only to autocorrelation in nv they are completely consistent with this hypothesis.

The adjusted daily price changes, APtljNv are less kurtotic, less skewed, less heteroskedastic, and less autocorrelated than are the unadjusted price changes (Predictions 12-14). The median excess kurtosis for the adjusted price changes is 0.65 versus 2.92 for the unadjusted changes, the median third moment skewness is 0.106 versus 0.437, the median coefficient of variation of the 14

monthly sample price variances is 48.0 percent versus 79.1 percent,5 and the median first-order autocorrelation coefficient is 0.031 versus 0.045 for the

unsquared adjusted and unadjusted price changes, and 0.073 versus 0.177 for

squared adjusted and unadjusted price changes. Sample excess kurtosis is re? duced by the adjustment for every security but 1, skewness is reduced for all but 10 securities, heteroskedasticity is reduced for every security but 1, and first- order autocorrelation in the squared data is reduced for all securities. These re? sults would be extremely unlikely if the prior probability of reduction were as? sumed to be equal to J4.6

Price changes and volumes measured over fixed-length transaction intervals

appear to be more normally distributed the longer the measurement interval (Pre? diction 15). Although sample measures of price change kurtosis and of volume skewness and kurtosis are all large for short transaction intervals, they all de? crease toward zero as the interval length increases. (Sample price change skew? ness is small for all interval lengths.7) The results concerning prices reconfirm those found by Brada, Ernst, and Van Tassel [1]. Together with the new results for volume, these findings confirm that the asymptotic conditions of the Central Limit Theorem are approached as the measurement interval length increases.

Moreover, they demonstate that it is unlikely that the skewness and kurtosis ob? served in daily price changes and volumes is entirely due to similar properties

5 The expected coefficient of variation of the sample monthly variances can be shown to be approximately 31 percent under the assumption that the daily price changes are normally distributed.

6 If the true probability of reduction is one-half, the probability of observing no increase is less than 10~15 ; of observing one or fewer increases is less than 10~13 ; and of observing ten or fewer increases is less than 10-5 .

7 The high transaction price change kurtosis is probably due to the predominance of zero price changes and to the fact that prices are quoted only at discrete levels (see [5]). Skewness and kurtosis in the volume data are probably due to large blocks trades.



Harris 135

found in the transaction data. Skewness and kurtosis are both similar when mea? sured over long transaction intervals rather than over daily intervals.

Squared price changes are less correlated with volume when measured over fixed transaction intervals than when measured over daily intervals (Prediction 16). The median correlation coefficients range between 0.072 and 0.120 for the fixed transaction interval data, while the median for the daily data is 0.318. For

only 13 ofthe 50 securities are any ofthe transaction interval coefficients greater than the daily coefficient. These results suggest that the correlation of squared price changes with volume observed in the daily data is caused, at least in part, by a common dependence on the mixing variable, nt.

The correlation of unsquared transaction interval price changes with volume is also small (Prediction 16). The median coefficients for the various transaction intervals range between 0.033 and 0.113 versus 0.108 for the daily data. It is

interesting to note, however, that they tend to increase as the length ofthe trans? action interval increases.

Squared price changes and volumes are less autocorrelated when measured at transaction intervals than when measured over daily intervals (Prediction 17). For the squared price changes, the median first-order autocorrelation coefficients for the various transaction intervals range between 0.038 and 0.134 versus 0.177 for the daily data. For the volume data, they range between 0.093 and 0.228

compared to 0.347 for the daily data.8 These results suggest that autocorrelation in the daily data is not entirely caused by autocorrelation in transaction data se? ries. Instead, these daily series are probably autocorrelated because they are de?

pendent on the daily number of information events, nt, which is autocorrelated.

IV. Conclusion

This study conducts new tests of the mixture of distributions hypothesis us?

ing transaction data. These tests assume that transactions take place at a uniform rate in event time. The results do not contradict this assumption or the hypothe? sis. In particular, the results show that the following properties ofthe joint distri? bution of daily price changes and trade volumes are consistent with a bivariate normal mixture of distributions model in which means and variances, conditional on the daily number of transactions, are proportional to the conditioning variable:

? kurtosis, skewness, and heteroskedasticity in daily price changes; ? skewness (and, trivially, kurtosis) in daily volumes; ? positive correlation of squared daily price changes with volumes and weak

positive correlation of actual daily price changes with volumes; and

? autocorrelation in daily squared price changes and in daily volumes.

An analysis of price changes and volumes measured over transaction intervals shows that it is unlikely that these properties are caused by similar properties found in the transaction data.

8 Among the unsquared price changes, only those series measured over a single transaction dis? play any significant autocorrlation (median ?0.157). The negative autocorrelation, which has been observed by previous authors, is probably due to the bid-ask spread.




The results suggest that the daily number of transactions may be a good estimate (up to a constant of proportionality) of a time-varying information evolution rate. This conclusion is based on indirect evidence, since the information evolution rate is not observable. Predicted effects of a time-varying information evolution rate are, however, observed in daily data, and they are not observed in

daily data adjusted by the daily transaction count or in series of price changes measured over fixed transaction intervals. It is therefore likely that the unobserved information evolution rate is closely related to the process that generates transactions.

There is, however, another interpretation that is completely consistent with the data. It may be that trading is self-generating as Roll and French [10] suggest. If so, the daily transaction count would actually be the conditioning variable in the mixture model rather than merely a proxy for an unobserved conditioning variable interpreted as the information evolution rate. To discriminate between these two interpretations, new theories must be developed with differential pre? dictions.

Appendix A

Properties of Mean- and Variance-Subordinated Normal Mixture Distributions

This appendix provides formal derivations ofthe less obvious predictions of the mixture of distributions theory. The first lemma develops some preliminary results that simplify later derivations. Lemma 1. Univariate moments.

Let X be a random variable whose distribution is given by the following mean- and variance-subordinated normal distribution

X ~ N(an,bn\n) a ^ 0, b > 0 ,

where n is the directing random variable with mean (jl and central moments m2, m3, and m4. The unconditional mean, second, third, and fourth central moments of X are given by:

2 9 E(X ? a\L) = b\k + a m2

3 3 E(X-a\k) =

3abm2 + a m3

E(X-a\L)A = 3b2 \l + 3b2 m? + 6a2 bm. + 6a2b^m7 + a4mA

and the mean variance of X 2 (for a = 0) are given by:

EX2 = b\L

E(x2-biif = 3b2 m2 + 2?2|x2.



Harris 137

Praof. Let En be the expectations operator over the distribution of n, and let Ex\n be

the expectations operator over the conditional distribution of X, given n. The various moments are derived using the law of iterated expectations. The proofs ofthe first two moments are

EX = E E | X = aE n = aix n x\n n ^

E(X-a^f = EnExln[(X-an + an-aii)2]

= EnEx\n(X-an^2 +

EnEx\n^X~aH^an-a^

x\n 2

+ E E i (an ? aii) n x\nK ^J

= Enbn + 0 + Ena2(n-\xf

= b\k + a m2.

The proofs ofthe other moments are analogous. Corollary 1 A. Moment Skewness and Excess Kurtosis (Predictions 1,2, and 3)

ECX-aurf 3abm0 + a m~ Skew(X) = t{-X

a?\a = -1-?

[E(X-a\L?] (b? + a2m2) = 0, if a = 0 .

?m_ ?(X-a|x)4 . 3b \l +3b m2 + 6a bm3 + 6a b\im2 + a m4 K(X)-- ? 3 =---3

(E(X-ay.)2) (b[L + a2m2) 3/2

3b2m2 + 6a2b(m2) Skew(r) + a4m22K(r)

= 3-2 if a = 0 .

For a > 0 and m3 ̂ 0, skewness is an increasing function of a (Predictions 2 and 3). Excess kurtosis always positive if a = 0 or if the distribution of n is

right-skewed and kurtotic (Prediction 1). Lemma 2. Covariances

Let X and Y be random variables whose distributions are given by the fol?

lowing mean- and variance-subordinated normal distributions

X ~ N(an,bn\n) a ^ 0, b > 0 Y ~ N(cn,dn\n) c ^ 0, d > 0 with Cov(X,Y|rt) = 0 ,




where n is a directing random variable with mean (jl and central moments m2, m3, and m4. Covariances among X, Y, n, andX2 are given by

Cov(X,?) = am2

Cov(x2,n) = bm2 + tf2(m3 + 2|Jim2)

Cov(X,F) = acm2

Cov(X2,y) = bcm2 + az2(m3 + 2|Jim2)

.

Proof Let En be the expectation operator over the distribution of n, and let Ex\n and

Ey\n be the expectation operators over the conditional distributions of X and Y,

given n. Each cross-moment is derived using the law of iterated expectations

Cov(X,n) = ?[(X-flM')("-M')] =

EnEx\n[(X-an + an-a\L)(n-\k)]

= En[(an-a\L)(n-\k)~]

= am2.

The proof of Cov(X2,?) is analogous

Cov(X,7) = E[(X-a\L)(Y-wy\ =

En[E^(X-an + an-aii)E^(Y-cn

+ cn-ciL)] =

En[(an-a\k)(cn-c\Ly] =

acm2.

The proof ofCov(X2,y) is analogous. Corollary 2.A. Correlations (Predictions 4, 5, 9, 10).

-1/2 ami I h LL

Corr(X,?) = \ x : = sign(n) U ^+1|

^(z?lx + ?2w2)w2

The correlation between X and ? is an increasing function of a2/b, which is a

measure of the variation in X that is due to the mean-subordination. The correla?

tion approaches 0 as a 2/b ?> 0 and it approaches sign(<z) as a 2/b ?> ?.

-1/2

Ignoring the signs of <z and c, the correlation between X and y depends on the



Harris 139

ratios a2/b andcVd. The correlation is an increasing function of both ratios, and

equal to 0 if either ratio is equal to 0. For a = 0,

/ ? \ bm7 , 7 x-i/2

l(3b2m2 + 2b2iL2)m2 2

Corr(x2,y) = f2

/(3& m2 + 2b (jl ju/|ji + c w2j

(3m2 + 2^2)|4^ +

m2)

-1/2 /_ ~ 2\/</ \ = m? ?2

It is interesting to note that for a = 0, these two correlations do not depend on b. The first depends only on the coefficient of variation in n, while the second de?

pends on that coefficient of variation, and the ratio c 2/d. Lemma 3. Serial Covariances.

Let Xt be a random variable whose distribution is given by the following mean- and variance-subordinated normal distribution

Xt ~

N(anvbnt\nt) a^0,b>0, where nt is a directing random variable with mean (jl and variance m2, and with serial covariation represented by SCov(w). Assume that conditional on the series

(nt), the series (Xt) is serially independent. Then the serial covariances of Xt and

X,2are SCov(X) = a2SCow(n),

and for a = 0, SCov(X2) = b2SCov(n).

Proof The results are derived using the law of iterated expectations.

SCov(X) = E^-a^X^^a^j]

= EnEx\n [(Xt

- mt + ant

- a?) (Xt-l-ant-l+ant-l- W)]

= En[(mt-a*)(ant-\~a*)\

= a2SCov(n) .

Fora = 0,

SCov(x2) =

E(X2X2_{) -

(?X2)2

= ^A|,(^-i)"^V2

= *>V>",-i)-*V

= b2En[(nt~lx' + lL)(nt-\-ll+v)

- *W

= b2[SCow(n) + |x2 - |x2] = b2SCow(n) .




Corollary 3.A. Autocorrelations (Predictions 7, 8, 11).

ACorr(X) = ^ov(n) = ACorr(n)

b\L + a m2 ?_ JL + i a2 m2

The autocorrelation in Xt depends on the ratio a 2lb. It approaches 0 as a 2lb ?> oo

and it approaches ACorr(n) as a2/b ?> 0. In particular, note that ACorr(X) ^

ACorr(n). Fora = 0,

ACorrfx2) = ^Cov(n) = 1 AQMr(|l) v J 3b2 m2

3

Appendix B

Description of the Data

The security sample includes the first 50 New York Stock Exchange com? mon stocks, sorted by ticker symbol, that traded continuously between Decem? ber 1, 1981, and January 31, 1983. The prices and shares traded for each transac? tion in this period were collected from the Fitch NYSE Tapes. All trades on the floor ofthe NYSE are included in the sample.

The prices and volume are adjusted for the effects of dividends, stock splits, and stock dividends. The effect ofthe price adjustment is to produce a series with values equal to the current market value of one share purchased on December 1, 1981, assuming that cash dividends are reinvested. The effect ofthe volume ad?

justment is to produce a series with trade volume reported in units of stock out?

standing on December 1, 1981. The adjusted prices were filtered to identify bad data. An observation passed

the filter if it were within 10 percent of the previous price, 10 percent of the

following price, or 10 percent of the average of the preceding and following prices. Only 6 of 434,270 transaction prices failed to pass this filter. Of these, 5 were found to have been incorrectly coded.

The daily price change series was computed from closing prices. On days when a security did not trade, the price change is assumed to be zero. When

trading resumed, the price change was computed as the difference between the

closing price and the price on the last trade previous to that day. This procedure differs from that used by the Center for Research in Security Prices to compute the daily return of an untraded stock for its Daily Returns File. When available, CRSP uses the average of the closing bid and ask prices. The bid and ask prices are not included in the Fitch data.



Harris 141

References

[i

[2.

[3

[4;

[5

[6;

[7

[8

[9;

[io;

[ii

[12;

[13

Brada, J.; H. Ernst; and J. Van Tassel. "The Distribution of Stock Price Differences: Gaussian after All?" Operations Research, 14 (March-April 1966), 334-340.

Brown, S. J., and J. B. Warner. "Using Daily Stock Returns: The Case of Event Studies." Journal of Financial Economies, 14 (March 1985), 3-32.

Clark, P. K. "A Subordinated Stochastic Process Model with Finite Variance for Speculative Prices." Eeonometriea, 41 (Jan. 1973), 135-155.

Epps, T. W., and M. L. Epps. "The Stochastic Dependence of Security Price Changes and Transaction Volumes: Implications for the Mixture-of-Distributions Hypothesis." Eeonome? triea, 44 (March 1976), 305-321.

Gottlieb, G., and A. Kalay. "Implications of the Discreteness of Observed Stock Prices." Journal of Finance, 40 (March 1985), 135-153.

Harris, L. "A Theoretical and Empirical Analysis ofthe Distribution of Speculative Prices and ofthe Relation between Absolute Price Change and Volume." Unpubl. Ph.D. diss., Univ. of Chicago (1982). _"Cross-security Tests of the Mixture of Distribution Hypothesis." Journal of Financial and Quantitative Analysis, 21 (March 1986), 39-46.

Kon, S. "Models of Stock Returns?A Comparison." Journal of Finance, 39 (March 1984), 147-165.

Morgan, I. G. "Stock Prices and Heteroskedasticity." Journal of Business, 49 (Oct. 1976), 496-508.

Roll, R., and K. French. "Is Trading Self-Generating?" Working Paper, Grad. School of Management, UCLA (1984). Tauchen, G., and M. Pitts. "The Price Variability-Volume Relationship on Speculative Mar? kets." Eeonometriea, 51 (March 1983), 485-505.

Westerfield, R. "The Distribution of Common Stock Price Changes: An Application of Trans? actions Time and Subordinated Stochastic Models." Journal of Financial and Quantitative Analysis, 12 (Dec. 1977), 743-765.

Wood, R. A.; T. Mclnish; and J. K. Ord. "An Investigation of Transactions Data for NYSE Stocks." Journal of Finance, 15 (July 1985), 723-739.



transaction data tests of the mixture of distributions hypothesis

Documents