Transactions Costs andCorrelations in a Large FirmIndexBlake LeBaron

SFI WORKING PAPER: 1991-11-047

SFI Working Papers contain accounts of scientific work of the author(s) and do not necessarily represent theviews of the Santa Fe Institute. We accept papers intended for publication in peer-reviewed journals or proceedings volumes, but not papers that have already appeared in print. Except for papers by our externalfaculty, papers must be based on work done at SFI, inspired by an invited visit to or collaboration at SFI, orfunded by an SFI grant.©NOTICE: This working paper is included by permission of the contributing author(s) as a means to ensuretimely distribution of the scholarly and technical work on a non-commercial basis. Copyright and all rightstherein are maintained by the author(s). It is understood that all persons copying this information willadhere to the terms and constraints invoked by each author's copyright. These works may be reposted onlywith the explicit permission of the copyright holder.www.santafe.edu

SANTA FE INSTITUTE

TRANSACTIONS COST AND CORRELATIONS

IN A LARGE FIRM INDEX

Blake LeBaron

University of Wisconsin-Madison

November 1991

Abstract

This paper presents results showing an inverse relation between volatility and autocorrelation in an indexof large firms. Evidence is given showing that index autocorrelation is also related to the level of tradingvolume. Tests are then performed on the individual fims that show much of this phenomenon is coming fromcross firm correlations. Simple models with transactions costs are then simulated to see if they can replicatethis phenomenon.

Acknowledgments

This research is part of the Economics Research Program at the Santa Fe Institute which is funded by grantsfrom Citicorp/Citibank and the Russell Sage Foundation and by grants to SFI from the John D. and Cather­ine T. MacArthur Foundation, the National Science Foundation (PHY-8714918), the U.S. Department ofEnergy (ER-FG05-88ER25054), and the Alex C. Walker Educational and Charitable Foundation-PittsburghNational Bank. The author is grateful t Rochelle Antoniewicz for collecting the bid-ask data used in thisstudy.

I. Introduction

The positive autocorrelation of stock return indices is a generally consistent phenomenon across

many portfolios. This effect has been noted since Fisher(1966) documented it and gave an initial

explanation due to nontrading. It is therefore sometimes referred to as the "Fisher Effect." Recent

papers have reopened the debate on what the cause is. Atchison et. al. (1987) and Lo and

Mackinlay(1990), using parametric models, find that the observed autocorrelations in market indices

are large relative to reasonable correlations generated from nontrading models. Also, several papers

note correlations, both negative and positive at longer horizons.' Several other authors have

presented evidence showing that these correlations are probably not constant over time and are

moving inversely with the current level of market volatility and trading volume.2 This result is

qualitatively consistent with nontrading and correlations for stock return indices.3

This paper extends these results by examining a very homogenous index of large firms. It is

shown that this large firm index displays similar dynamic properties to those found in the broader

S&P 500 composite. It is also demonstrated that trading volume does playa role in this effect,

although the exact volume-volatility connection will not be explored. Some tests will be performed

showing most of the changing autocorrelations coming from cross correlations as opposed to own

firm correlations.

A simple model of differential transactions costs is presented which is capable of generating the

phenomenon seen in the stock return series. It is shown that for reasonable values of transaction

costs the model does not generate results close to those from the actual data.

Section II will present the empirical results in the paper in various subsections covering, volatil-

ity, volume, and cross correlations respectively. The simple transactions cost model is presented in

section III with its simulations. Section IV concludes.

1 Among these are Conrad and Kaul(1989),Jegadeesh (1990) Lehman(1990), and La and MackinlaY(1988).

2 Some of the papers documenting this fact are Campbell et. a1.(1991), LeBaron(1992), and Sentana andWadhwani(1990) for stock returns. Also, Bilson(1990), Kim(1989), and LeBaron(1992) have reported a similareffect in foreign exchange markets. An early description of this effect with some other conjectures as to its causescan be found in Taylor(19S6). In more distantly related work, Hinich and Patterson(1985) and Scheinkman andLeBaron(1989} found evidence for nonlinearities in some of these series.

3 Given the time series result that volume and volatility are correlated over time, autocorrelations should behigher when volume (volatility) is lower corresponding to periods of more nontrading. See Karpov(1987) for a surveyof trading volume effects.

1

II. Empirical Results

A. Data Description

One purpose of this study is to analyze some previous results for very large firms which

are relatively cheap to trade, and for which trade occurs very frequently. To do this a portfolio

was constructed using the CRSP daily returns files from 1963-1988 to closely replicate the Dow

Industrials. The portfolio was chosen not to exactly replicate the Dow, but to have a strong overlap

with the Dow for most of the sample period.·

The firms that were initially chosen are given in Table 1. This list includes many of the largest

and most frequently traded stocks on the NYSE. Some idea of the relative costs of trading these

firms is given in the first column of table 1 where the percentage bid-ask spread is given. These

spreads were sampled on 9/18/91 at 9:45. It will be assumed that these spread characteristics are

fairly representative of the previously sampled data. These spreads will be used in simulations as

an estimate of the relative costs of trading large firms.

A plot of these spreads is given in figure 1. This figure shows that for 21 of the firms selected

the spreads are below 0.6% with an increase over the range starting at 0.1 %. It is clear that several

of the firms initially selected do not currently exhibit costs of trading similar to the other firms.

Some of these firms have been dropped from the actual Dow index many years ago, and some of

them are having serious financial problems. Since the goal of this study is to concentrate on an

index of large frequently traded firms most of the results will be examined on the set of 21 firms

with smallest spreads. A line is drawn in table 1 to show the portfolio of firms used.5

Table 2 presents some summary statistics for equally weighted indices formed from these two

portfolios, the entire group of 27, and the smaller group of 21. The two portfolios exhibit the general

properties of most stock portfolios of little skewness and large kurtosis.6 The numbers Pi show the

autocorrelations of the series at lag i. This table shows a large positive autocorrelation at lag 1

with little correlation after that. This fact is generally consistent with nontrading explanations,

4 The sample period begins in 1963 rather than 1962 to allow for moving average adjustment of the volumeseries.

5 Out of this smaller group only two firms, Inca and American Brands, are not currently in the Dow Industrials.

6 To clarify this fact these numbers are included with the October 1987 crash removed.

2

but it is somewhat puzzling that the correlation is present for an index of large firms. Many of the

tests run here will be concerned with the exact nature of this autocorrelation. The final row in this

table shows the daily correlation with the actual Dow Jones Industrial index.

B. Correlations and Volatility

This section replicates the results of LeBaron(1992) which found a connection between corre­

lations and volatility for the S&P 500 index for the larger firm indices described in the previous

section.

A local estimate of conditional variance is constructed using the past 10 lags of squared re-

turns.7

A large table is then constructed in the following form.

This table is then sorted on 0-1, the estimate of the conditional volatility at time t. This allows the

examination of the correlation of Tt and Tt+l for any given range of volatility. Specifically, a band

of length 500 will be moved across the sorted sample and the correlations (adjusting for means

within the band) will be estimated in each range.B

A plot of these correlations is shown in figure 2 for equally weighted portfolios on the original

series and the sample of 21 firms with smallest spreads. The y-axis in this figure plots the conditional

correlation for each band, while the x-axis plots the volatility quantile order (0.5 corresponds to the

median volatility).9 This figure shows a strong inverse relation between volatility and conditional

correlation for both series. Figure 3 checks the sensitivity of these results to the lag length of the

7 The 10 day length was used to make these results comparable with those in LeBaron(1992). For this largerfirm index, many of the results seen here are stronger and more reliable when a 5 day volatility index is used.

8 This paper differs slightly from the technique used in LeBaron(1992). The estimated correlations are actually

the estimated AR(!) coefficient, Ertr~±l, where the returns are mean adjusted. In normal time series circumstancesEr,

this is identical to the correlation, but in this case it is not since estimation is done conditionally on different setsof rt. The results are not greatly affected by this change, but it was felt that this measure made the results moreconsistent with the regression results in future sections. Also, some initial simulations have shown this to be a morereliable measure for several different processes.

9 This technique was inspired by the diagnostics used in Tsay(1989).

3

volatility index. For small changes in the length of this index (5 days) very little change is seen in

the downward pattern of the graph.

This relation is consistent with the previous results for the S&P 500 and could be explained as

a consequence of varying levels of nontrading. This explanation depends on the positive correlation

between volume and volatility. When volume is low volatility is low, and the level of nontrading is

high. This would cause larger correlations in a broad index. This explanation seems less appropriate

for this small group of large, frequently traded firms.

The statistical significance of this result is difficult to analyze since there are many dependencies

going from volatility to correlation estimates. Simulations will be presented to demonstrate the

significance of these results. Several types of simulations will be presented in this paper. In this

section several parametric models will be estimated and then simulated to compare the results with

the original data. The first model used will be a simple AR(l) model estimated on the index returns.

This model will be simulated using scrambled estimated residuals as suggested in Friedman and

Peters(1984). The second model used is a simple parametric model designed to capture some of

the movements in volalility, a GARCH(l,l).'O This model also includes an MA(l) term to capture

the 1 day correlations. Simulations for the GARCH will also use scrambled estimated residuals."

Parameter estimates for the GARCH(l,l) are given in table 3. They show the strong persistence

in volatility in the series as evident from the large value of fl. Also, note that the estimate of the

moving average parameter, b, is significantly positive.

Comparisons of the simulations with the actual series are performed in table 4. The graphical

information contained in the figures is coarsely summarized as the difference between estimated

autocorrelations for high and low volatility ranges. The columns of the table are labeled, indicating

which differences are used. For example, the first column labeled 0-25, represents the difference

between the autocorrelation estimated over the lowest 25 percent of volatility, and the autocorre-

lation measured over the highest 25 percent. The difference that is presented for the 27 Dow firms

in the first row of the table indicates a difference of 0.207 between the highest and lowest volatility

10 See Bollerslev(1986) and Engle(1982) for the early development of these models, and Bollerslev(1990) for arecent survey of litterature.

11 This again follows Freedman and Peters(1984) in method, but the properties of the parametric bootstrap undernonlinear models such as the GARCH have not been derived. See Brock et. a1.(1990) for another application of thistechnique.

4

firms. Similar comparisons are done for 3 other partitions. The first, 2-27, moves the initial range

over to eliminate volatility outliers at either extreme. The third column, labeled 0-50, divides the

series in half and compares the correlations over each half, and the final column, 25-50, looks at

the interquartile range.

For both Dow portfolios the differences are all positive with the smallest difference coming from

the interquartile range. These results are all consistent with those presented in the previous plots.

These same numbers are now computed for each of 500 replications of representative simulated

series for comparison with the actual series. The results shown in parenthesis show the fraction of

these simulation runs which generate values greater than those measured for the actual series.

For both the 27 firm portfolio and the smaller 21 firm portfolio the results from the actual

time series are difficult to replicate in the simulations from either the AR(I) or GARCH-MA. This

is shown in simulated p-values for all the tests being close to zero except for the interquartile range

test which is 6.2 percent.

The next section of table 4 performs subsample comparisons by splitting the data into two

halves. Concentrating on the Dow 21 portfolio, the subsamples show some reduction in the esti­

mated differences. However, these differences are still significant at the 10 percent level with the

exception of the 25-50 range for the first subsample which gives a difference of 0.00. For the sub­

samples each of the bins is getting substantially smaller and the ability to see detailed differences

is probably diminishing. For this reason attention should be concentrated on the 0-50 differences

which show a consistent pattern across the two subsamples. Also, the result appears to be getting

stronger in the later half of the data with a 0-50 difference of 0.097. This is interesting since this

period is closest to the bid-ask spread measurement date.

C. Correlations and Volume

The previous section showed some connections between return correlations and volatility. This

section extends some of these results by using trading volume as the conditioning variable for

determining local correlations. The previous procedures are followed exactly except now volatility

is replaced by a local estimate of trading volume.

The volume series used here is the total number of shares traded on the Dow Industrials. This

series does not overlap exactly with the indices used here, but it should serve as a good proxy for

5

general market activity. The series is clearly nonstationary and needs to be normalized in some

way. This is done by dividing today's volume by a 100 day moving average of past volume.12 This

normalized series is now used as the information for estimating conditional correlations. This study

uses only the current volume at time t in estimating the conditional correlations from t to t + 1.

Table 5 presents the correlation differences for both the Dow 27 and Dow 21 portfolios. Both

exhibit correlation differences similar to those estimated using volatility conditioning information

with large positive differences for the 0-25, 2-27, 0-50 bands, and a very small difference for the

25-50 band.

Simulation comparisons are made much more difficult now that two series are being used.

A very simple comparison will be made which accounts for the contemporaneous correlations in

volume and volatility, but does not account for any of the relations across time. In this case a

scramble of volume and volatility is performed maintaining the cross-sectional structure of the

series. In other words, vectors of the form (x" Yt) are scrambled over time as a connected unit,

replicating cross sectional dependencies in the joint series. The fraction of values from 500 of these

simulations which generate values larger than those from the original series are given in parenthesis.

They show that none of the joint simulations generated values as large as those for the original

series.

The previous simulations accounted for dependencies between contemporaneous observations in

the series, but were unable to recreate any dependencies over time. The next simulations make some

progress in this direction using an m-dependent bootstrap technique described in Carlstein(1986)

and Kunsh(1989). The procedure resamples the time series in m length blocks as opposed to one

entry at a time. This method attempts to nonparametricly recreate time dependencies in the data.

The technique is still very new, and other competing methods are on the horizon.'"

Three m-dependent bootstraps are run. Each represents a different block length in the re-

sampling procedure. The first, labeled MDEP2, draws block lengths of 2. Since we are looking at

correlations from t to t+1 it is natural to see how likely it is that a two dependent process might

12 This procedure follows that used in Campbell et. al.(1991). However, they use loggged turnover ratios insteadof volume.

13 See Leger et. a1. (1991) for an excellent survey.

6

have generated the results that are seen here. The simulated p-values in table 5 show that this is

very unlikely. For each of the tests on 0-25, 2-27, 0-50 they are below 5 percent.

The second use for these m-dependent simnlations will be to test the reliability of the estimates

under a dependence which is long enough to give a good representation of what is going on in the

time series. In this test simnlations are done for sampling blocks of 15 and 20. The table presents

the means and standard deviations for these tests. The numbers presented give a picture of how

reliable the original differences were for these series. For example, for the 0-25 difference in the Dow

27 series, the 15 m-dependent simulations give a mean of 0.184 and a standard deviation of 0.48.

This indicates two things, the original estimate of 0.201 was reliably positive, and a 15 dependent

process does a good job of replicating this result. The results are basically consistent across for

the two different portfolios and simulations. These results should still be viewed with some caution

since this type of resampling is still relatively new and its properties are still somewhat unknown.

Obviously, increasing the block size will start to give tighter and tighter standard errors on the

sample estimate since the simulations are approaching the original sample.

The results of this section and the previous one are summarized in figure 4. This figure plots

estimates of the return correlation conditioned on the two pieces of information used, volatility

and volume for the 21 firm index. Each of the two pieces of conditioning information are mapped

into distribution fractiles. Estimation of the correlation of T, to r,+, is then performed using a

nonparametric smoother, weighting observations based on volume and volatility using a Gaussian

kernel, with bandwidth of 0.2. While no statistical analysis of this figure will be given it gives a

good graphical summary consistent with previous results. There is a general downward pattern

in the estimated correlations which is connected to both volume and volatility. There are some

other interesting aspects to figure 4 which will not be tested in this paper, but may give some

information on what is going on. Note from the figure that both volume and volatility appear to

be useful pieces of information. Knowledge of one does not eliminate the usefulness of the other.

This indicates that one is not a noisey proxy for the other in terms of this relation. However, both

may still be related to some common factor. Also, it should be kept in mind that both pieces of

information may be very imprecise estimates.

7

D. Cross Correlations

In the previous two sections results were presented demonstrating changing correlation patterns

in an index of large firms. This section will explore where this changing correlation is coming from.

Correlation in any index constructed from individual firms may be coming from the firms

themselves or from correlations across firms and time.14 The correlation patterns presented so far

could be coming from either type of effect. Either the level of cross correlation is changing over

time, or the level of own correlation, or both.

To directly test this a simple regression will be estimated conditionally over the different

volatility ranges. '5 The following equation will be estimated for the set of 21 firms.

ri,t = CKi + f3f rm-i,t-1 +Pi ri,t-1 + 'i,t

Where ri,t is the return of firm j at time t, and rm-i,t is the return on the equal weighted index of

the 21 Dow firms with firm j removed.

The coefficients are estimated over the different volatility ranges using the same 10 day volatil­

ity index used in the first section. Since there is no control for outliers in this experiment the crash

in 1987 is removed from the series. The estimated values are then averaged across firms. Results

are presented in figure 5. The figure shows very little change in the estimated conditional means,

CKi' However, the downward pattern is repeated for the two correlation measures. The pattern

appears much more dramatic for the cross correlation term than the own correlation.

This pattern is confirmed in table 6 which reports the coefficient differences from the largest to

the smallest half of volatility. The table reports a difference of 0.100 for the f3L difference and 0.032

for the P difference. The CK difference is close to zero. The significance of these results are tested by

performing a joint bootstrap for all the firms in the portfolio. As was done previously with volume,

the simulations scramble the vector of firms redrawing each vector of firms at a certain time. This

again maintains the contemporaneous correlations across firms, but eliminates any structure over

time. The simulated p-values from 500 simulations are given in parenthesis and show the fraction

of simulation runs generating values greater than those estimated. The simple joint bootstrap

14 See Fisher(1966), Cohen et. a1.(1986), Lo and Mackinlay(1990), and Mech(1990).

15 This test is related to some of the tests performed in Wiggins(1990) using volume data.

8

simulation is clearly not able to generate differences as large as those in the data for either f3 or p.

The m-dependent boostrap is again used, and the simulated p-values are reported for the 2 block

length case. The p-values are slightly larger than the independent case, but they basically repeat

the previous results of significant difference for the cross correlations for both volume and volatility,

and significant difference for the own coefficient for volatility ouly.

These results are further strengthened by the reliability tests using the large block sampling

procedures. Again 15, and 20 length blocks will be used. f3L and p appear to both be reliably

positive for the volatility conditioning tests, and f3L only appears to be reliably positive for the

volume conditioning tests.

Table 7 also reports results for the two halves of the series. For volatility the f3L differences

drop, while the p differences remain close to their values for the entire sample.!6 For volume the p

differences remain close to zero, and the f3L differences are both positive, with the first half giving

a larger difference.

The results presented in this section indicate that much of the changing correlation pattern

is coming from cross correlations suggesting that models of differing transaction costs across firms

may be a useful place to look for explanations. However, this may not be the entire story. There

still remains a puzzling large amount of own correlation difference when using the volatility index.

III. Differing Transaction Costs

In this section some explanations for the previous results based on differing transactions costs

will be tested. The large positive cross correlation observed during periods of low volatility could

be the effect of transactions costs. Consider a one factor model for the movements of the individual

firms. If there is an upward shock to the market factor the prices of stocks will react to this

shock as long as the shock is larger than their proportionate transaction cost. In a market with

different levels of costs across firms the firms which are cheaper to trade will react immediately to

all information and those that are more expensive to trade will react with delay. This could be

responsible for the observed cross correlations seen here. If the conditional variance of the market

16 As mentioned previously a 5 day volatility index was more appropriate for this portfolio. Using this index theresults for f3L where much more stable across the subperiods.

9

factor was changing over time, then during periods of low volatility the impact of the differential

transactions costs would be greater causing more autocorrelation.

A simple simulation will be performed here. The model simulated here is very similar to the

one tested in Mech(1990). Let the return of each firm be driven by a common factor and some

idiosynchratic noise,

rj,t = At +<Ij€j,t

where At is the common market factor. It is uncorrelated over time and assumed to follow a

GARCH(l,l) process with parameters from table 3, and €j,t is N(O,l).17 €j,t is N(O,l).

Returns of each firm at time t will reflect the increment to the market At plus <Ij €j,t if this

amount is larger than the firm's transaction costs. If the amount is smaller, it is added to the true

value of the firm, but the current observed return is zero. The firm will reflect the new information

as soon as its true return is greater than the transaction cost. This mechanism can clearly generate

autocorrelation in an index since firms may respond to a market factor move on different days.

A rough estimate of the transaction cost for each firm is made using one half the percentage

bid-ask spread plus a fixed commission component. The estimates of the bid-ask spreads for

this simulation come from those in table 1. It is assumed that this distribution of spreads is

representative of the spreads of a set of large firms. For the fixed component a value of 0.15 percent

is used. This value was obtained from Chan and Lakonishok(1991) who report a mean commission

cost of 0.15 from the largest firm trades in their data set on institutional trading. The stability of

this number over time is clearly an important issue for this research. Some experiments have been

performed with larger fixed costs up to 0.5 percent. Increasing the fixed cost component did not

have a big impact on the results seen here. The effect of the fixed cost component of the transaction

cost on the correlations cannot be signed since it increases the probability of prices not reflecting

new information for all firms. Therefore it is unclear how it affects the autocorrelation which is is

driven by differences in transaction costs.

Table 7 reports some results of simulations of this transaction cost model using the parameters

given above. In the first set of simulations individual firm risk is set to zero. The set of firms ouly

17 The MA component is eliminated in these simulations since it is desired to generate the correlation from themodel itself.

10

react to the common factor. This is done for two reasons. First, this sets up the most likely case

for persistent correlations to occur. Second, when the individual components are simulated it will

never be quite clear whether they are being modeled correctly, so it is a good idea to view the

model with these components turned off.

The first row repeats the correlation differences from table 3 for the 21 large firm portfolio. All

results in this table use the 10 day volatility conditioning information. The final column, labeled

"Autocorrelation", is the unconditional autocorrelation of the series. The second and third rows

present the mean and standard deviations from 500 simulations using the fixed cost plus one half

the percentage bid-ask spread. Given the parameter estimates are correct, and that traders are

paying the spread, this should be close to the "true cost" .'8 Individual firm risk is not used, (Jj = 0,

and the common factor follows a GARCH process as in table 3 with no MA component. None ofthe

means from the simulation differences are positive and the standard deviations show that few of the

simulations were able to come close to the results from the actual series. Also, the unconditional

autocorrelation is low with a value of only 0.005 as compared with 0.104 for the actual series.

The bid-ask spread numbers used here may not be very representative of the time period

under study. For that reason, and to see how reasonable a mechanism this model is for this

phenomenon, further tests are performed on other spreads. The next two rows double all the

spreads used previously. The correlation differences are starting to rise, but they are still small.

The mean simulation difference at 0-25 is 0.081 as compared with 0.217 for the actual series.

The unconditional correlations still remain low at 0.008. From the standpoint of the correlation

differences the results are starting to look closer to the actual data. The next row boosts the bid-ask

spreads to 4 times their original levels. At this level the model starts to match the data very well.

For the 0-25 difference the simulation mean is 0.211 which is very close to the value seen in the

actual series of 0.217. For the 0-50 difference, the simulation generates a mean value of 0.118, with

a standard deviation of 0.041. This compares with the value of 0.143 for the actual series. While

the simulation value is a little low, it is well within two standard deviations of the actual series.

Interestingly, while these numbers appear to be lining up very well with the data, the unconditional

autocorrelation remains too low, with a simulation mean of 0.025.

18 It is probably an overestimate since not all traders end up paying the observed spread.

11

Given the crudeness of these simulations and the fact that no attempt has been made to model

the own correlations of firms these results start to look like a success for this model. However, two

important cautions are in order. The success was reached at spread levels that were 4 times those

seen at the sampling point in 1991, and no individual firm noise has been added. This second

point will now be addressed. To get a rough estimate of the level of individual firm noise the

common factor model will be used. Let a2 be the variance of the individual firm component, ri' be

the unconditional variance of the market factor .x, and n the number of firms. Ignoring the serial

correlations in the series the time series variance for the index is therefore

var(xm ,,) = ri' +(lin )a2,

and the variance for each firm is

var(x;,,) = r/ +a2•

Estimating the two terms of the left from the actual series allows the calculation of a rough estimate

for the ratio a 2 1'1)2. This ratio for the 21 firm portfolio was 1.87.

Noise of the appropriate magnitude is then added to each individual firm's true return using

normal random numbers. This will obviously make the daily return for each firm more likely to

fall outside the transaction costs bands, and the price more likely to reflect current information.

This can be seen in the last four rows of table 7. In the first experiment the original parameters

are used again with the added noise. No correlation differences close to those for the actual firms

are seen. In the second experiment the level of transactions costs that was capable of generating

these differences, 4 times the actual levels, was used. Here, the addition of individual firm noise

eliminates the correlation differences that were simulated in the previous experiments. For example,

for the 0-25 difference the mean simulation value is now only 0.016 as compared to 0.211 from the

simulation of the model without individual firm noise, and 0.217 from the original data.

In the previous experiments it was assumed that the distribution of the idiosyncratic component

for firm returns was independent of the common factor. This may not be a very realistic assumption.

One very simple modification will be tested here. The conditional variance of the individual firm

components will be assumed to move in proportion to the variance of the common market factor.

Specifically, if the conditional variance of the market factor is h, then for each individual firm the

12

conditional vari3Jlce for its disturb3Jlce, 'j,t will be 1.87ht • This dependence clearly could influence

the correlation difference results, in that periods of small index volatility will now also be periods

small idiosyncratic volatility. This proportional link assumed here is surely unrealistic, but it is

probably most favorable in terms of generating correlation differences.

The last four lines of table 7 present results for this experiment. Using the actual spread

numbers the results are very similar to those for independent noise case. The correlation differences

are small, and the unconditional autocorrelation is also close to zero. When the spreads are boosted

to 4 times their observed values the results show some increase in the correlation differences, but the

increases are still pretty small compared with the actual data. For example, for the 0-50 differences

the mean simulation value was 0.039, as compared with 0.143 for the actual data. The standard

deviation of 0.41 shows that few of the simulations generated a value as large as that in the actual

series. Also, the autocorrelation remains unusually low with a mean of 0.012.

The simulations presented here show that this very simple model roughly calibrated is incapable

of generating the results seen in the data. In the first group of simulations the actual data had

to be multiplied by a factor of 4 to obtain results consistent with the data. At this level bid-ask

spread differences were large enough to generate results close to the observed time series. However,

this would imply spreads for some large firms of close to 2 percent, which seems unreasonable for

these large firms. The second experiments also showed that adding individual firm noise moved the

correlation differences close to zero even for the expanded spreads.

There are still many issues to be addressed here, 3Jld these simulations do not rule out dif­

ferential transactions costs in general as an explanation for these findings. Further explorations

into the distribution of spreads need to be performed. The question of the representativeness of

the distribution given in figure 1 is an import3Jlt one since scaled versions of this distribution are

used in all the simulations. Maybe there were periods when the distribution appeared closer to

two distinct groups as opposed to a continuous progression. Also, the fraction of trading that goes

on within the spread needs to be considered as well. The empirical reality of the simple model

simulated here needs to be considered further. The response of individual firms to the common

market factor needs to be examined more closely. The model used here implicitly assumes a market

factor beta of one for each firm. Also, the model used here for the movements of individual firm

13

volatility is probably not very realistic.!9 However, in defense of the model used here, it is not clear

whether adding these complications will change the results much. Lastly, other aspects of trading

costs need to be considered. It is possible that the missing component here is execution cost, or the

costs imposed on trading from trade related price movements. If execution costs are higher when

trading volume (and therefore volatility) is lower then it could be argued that some of the cost

differences across firms might widen during these periods. It is difficult to estimate execution costs

without high frequency data, but simulations such as these could try to estimate the magnitude of

these costs to see if they were reasonable.

IV. Conclusions

This paper has presented results documenting an inverse relation between serial correlations

and volatility for an index of large firms. These results extend those presented in LeBaron(1992)

which demonstrated this phenomenon for a broad index, the S&P 500. This connection is shown

to extend to trading volume, and much, but not all, of this changing correlation is coming from

cross firm effects. It is also important that the effects are still seen in the much narrower and

homogenous index used here.

Most of these findings can be classified as being broadly consistent with the wide range of mar­

ket microstructure issues described in Cohen et. al. (1980,1986). The changing cross correlation,

or "intervalling effect" could be caused by changing levels of nontrading and conditional variances

of new information. Possibly these models blended with the ideas of economic versus clock time

contained in Clark(1973) and Stock(1988) could yield explanations.

If nontrading as such is the cause a solid economic reason for this nontrading should be given.

For this reason a simple model of differing transactions costs is simulated to see how well it can

replicate the results from the actual series. It is clear how differing transactions costs will generate

positive unconditional correlations in an index with firms which are more or less reactive to news

depending on the cost of trade. If the volatility in the market is changing over time then the

differences in transaction costs causing autocorrelation would loom larger during the less volatile

periods. Models of this form are roughly calibrated to the actual data and then simulated. The

19 Conrad et. a1. (1990) present evidence on lead lag patterns in volatility adding further complications to thevolatility story.

14

results show that these models are not a good representation of what is going on. It is still possible

that either the parameters used are not representative of the entire sample, or the very simplified

models do not adequately represent the trading process, or transactions costs are not the entire

story.

These negative results immediately suggest several modifications that might change these find­

ings. First, better modeling of the movement of individual firms in the time-series cross-section

is needed. The model simulated here has a single factor and the relation of firms to this factor is

constant over firms and across time. The addition of more realistic dynamics for the reaction of

individual firms to an aggregate factor would certainly be useful, although it is not clear how this

will affect the results. Related to this point is the question of the conditional variances of the indi­

vidual firm shocks and how they move with the common market variables used. Clearly, a better

understanding of these joint dynamics would be useful. Second, other components of transactions

costs not considered here may be able to explain some of these results. Some of the effects presented

here could be consistent with more complicated models where traders are concerned about both

the cost they incur from their own price impact, and the opportunity cost of missing a trade at a

given price and time. Both these types of costs should certainly change over time with volatility

and volume. Also the effects of specialist behavior and inventory adjustment may cause some of

the effects observed for individual firms?" It is possible that the inventory adjustment behavior of

specialists becomes more important during periods of higher volume and volatility thus reducing

the observed autocorrelation.

Even though further micro structure explanations need to be studied it is not clear whether

they will provide the needed results. This study concentrated on a uniform sample designed to

minimize information and transaction costs differences over firms. There are still many other

explanations for this behavior which have yet to be explored. Many of these would involve models

of learning. For example, it is possible that learning might be reponsible for a delayed response of

some firms to movements in a common factor if the effect of this factor on their price is less well

understood by the market. This phenomenon could be reduced during periods of higher volatility

when the costs of delaying trade outweigh those of quickly interpreting new information.

20 See Ho and Stoll(1981) and Grossman and Miller(1988).

15

References

Atchison, M., K. Butler, and R. Simonds. 1987. Nonsynchronous Security Trading and MarketIndex Autocorrelation. Journal of Finance 42 (March) : 111-118.

Bollerslev, Tim. 1986. Generalized Autoregressive Conditional Heteroskedasticity. Journal ofEconometrics 31 (April) : 307-328.

Bollerslev, T., R. Y. Chou, N. Jayaraman, and K. F. Kroner. 1990. ARCH Modeling in Finance: ASelective Review of the Theory and Empirical Evidence, with Suggestions for Future Research.University of Arizona, Tuscon, Arizona.

Bilson, J. F. 1990. "Technical" Currency Trading. In The Currency-Hedging Debate. Edited byL. R. Thomas. London: IFR Publishing Ltd.

Brock, W. A., J. Lakonishok, and B. LeBaron. 1990. Simple Technical Trading Rules and theStochastic Properties of Stock Returns. University of Wisconsin - Madison, Madison, WI.

Campbell, John, Sanford J. Grossman, and Jiang Wang. 1991. Trading Volume and Serial Corre­lation in Stock Returns. Princeton University, Princeton, NJ.

Carlstein, Edward. 1986. The Use of Subseries Values For Estimating the Variance of a GeneralStatistic from a Stationary Process. Annals of Statistics 14: 1171-1179.

Chan, Louis K. C., and Josef Lakonishok. 1991. Institutional Trades and Intra-Day Stock PriceBehavior. College of Commerce, University of Illinois, Urbana-Champaign, II..

Clark, Peter K. 1973. A Subordinated Stochastic Process Model with Finite Variance for Specula­tive Prices. Econometrica 41 (January) : 135-55.

Cohen, K. J., G. A. Hawawini, S. F. Maier, R. A. Schwartz, and D. K. Whitcomb. 1980. Impli­cations of Microstructure Theory for Empirical Research on Stock Price Behavior. Journal ofFinance 35 (May) : 249-257.

Cohen, Kalman J., Steven F. Maier, Robert A. Schwartz, and David K. Whitcomb. 1986. TheMicrostructure of Securities Markets. Englewood Cliffs, New Jersey: Prentice-Hall.

Conrad, Jennifer, and Gautam Kaul. 1989. Mean Reversion in Short-Horizon Expected Returns.The Review of Financial Studies 2 (2) : 225-240.

Conrad, Jennifer, Mustafa Gultekin, and Gautam Kaul. 1990. Conditional Variances and Asym­metric Information Flow Across Securities. University of North Carolina, Chapel Hill, NC.

Engle, Robert F. 1982. Autoregressive Conditional Heteroskedasticity with Estimates of the Vari­ance of United Kingdom Inflation. Econometrica 50 (July) : 987-1007.

Fisher, L. 1966. Some New Stock-Market Indexes. Journal of Business 39 (January) : 191-225.

French, K. R. 1980. Stock Returns and the Weekend Effect. Journal of Financial Economics 8(March) : 55-69.

Freedman, D., and S. Peters. 1984. Bootstrapping a Regression Equation: Some Empirical Results.Journal of the Americal Statistical Society 79 (385) : 97-106.

Gallant, A. R., P. E. Rossi, and G. Tauchen. 1990. Stock Prices and Volume. North Carolina StateUniversity. Chapel Hill, NC.

16

Hinich, M. J., and D. M. Patterson. 1985. Evidence of Nonlinearity in Daily Stock Returns.Journal of Business and Economic Statistics 3 (January) : 69-77.

Ho, T. S. Y., and H.R. Stoll. 1981. Optimal Dealer Pricing Under Transactions and ReturnUncertainty. Journal of Financial Economics 9 : 47-73.

Jegadeesh, N. 1990. Evidence of Predictable Behavior of Securities Returns. Journal of Finance45 (3) : 881-898.

Karpov, Jonathan M. 1987. The Relation Between Price Changes and Trading Volume: A Survey.Journal of Financial and Quantitative Analysis 22 (March) : 109-126.

Kim, Chong Man. 1989. Volatility Effect on Time Series Behavior of Exchange Rate Changes.Korea Institute for International Economic Policy, Soul, Korea.

Kunsch, Hans R. 1989. The Jacknife and the Bootstrap for General Stationary Observations.Annals of Statistics 17 (3) : 1217-1241.

LeBaron, Blake. 1992 forthcoming. Improving Forecasts Using a Volatility Index. Journal ofApplied Econometrics.

LeBaron, B. 1992 forthcoming. Some Relations Between Volatility and Serial Correlations in StockMarket Returns. Journal of Business.

Leger, C., D. N. Politis, and J. P. Romano. 1991. Bootstrap Technology and Applications. StanfordUniversity, Dept. of Statistics, Stanford, CA.

Lehmann, Bruce N. 1990. Fads, Martingales, and Market Efficiency. Quarterly Journal of Eco­nomics 105 (February) : 1-28.

Lo, Andrew W., and A. Craig MacKinlay. 1988. Stock Prices Do Not Follow Random Walks:Evidence From a Simple Specification Test. Review of Financial Studies 1 : 41-66.

Lo, Andrew W., and A. Craig MacKinlay. 1990. An Econometric Analysis of NonsynchronousTrading. Journal of Econometrics 45 (July/August): 181-211.

Lo, Andrew W., and Craig A. MacKinlay. 1990. When Are Contrarian Profits Due To StockMarket Overreaction? Review of Financial Studies 3 (2) : 175-206.

Mech, Timothy S. 1991. What Causes Lagged Price Adjustment? Boston College, Chetnut Hill,MA.

Scheinkman, J. A., and B. LeBaron. 1989. Nonlinear Dynamics and Stock Returns. Journal ofBusiness 62 (July) : 311-38.

Sentana, Enrique, and Sushil Wadhwani. 1990. Feedback Traders and Stock Return Autocorrela.­tions. London School of Economics, London.

Stock, James H. 1987. Measuring Business Cycle Time. Journal of Political Economy 95 (Decem­ber) : 1240-61.

Taylor, Stephen J. 1986. Modelling Financial Time Series. Chichester: John Wiley.

Tsay, Ruey S. 1989. Testing and Modeling Threshold Autoregressive Processes. Journal of theAmerican Statistical Society 84 (March) : 231-240.

17

Table 1Firms: Sorted By Percentage Spread

Percentage Spread IName ITicker

0.001194 INTERNATIONAL BUSINESS MACHS IBM

0.001810 GENERAL ELEC CO GE0.001843 INTERNATIONAL PAPER CO IP0.001907 MERCK & CO INC MRK

0.002829 MINNESOTA MNG & MFG CO MMM0.002985 PROCTER & GAMBLE CO PG0.003328 AMERICAN TEL & TELEG CO T

0.003431 GENERAL MTRS CORP GM0.003899 INCO LTD N

0.003968 TEXACO INC TX

0.004228 EXXON CORP XON0.004415 WOOLWORTH F W CO Z0.005231 CHEVRON CORPORATION CHV0.005348 UNITED TECHNOLOGIES CORP UTX0.005495 DU PONT DD

0.005602 ALUMINUM CO AMER AA0.005634 WESTINGHOUSE ELEC CORP WX

0.005666 AMERICAN BRANDS INC AMB

0.005900 EASTMAN KODAK CO EK0.005917 GOODYEAR TIRE & RUBR CO GT0.006270 SEARS ROEBUCK & CO S

0.009050 U S X CORP X0.011561 UNION CARBIDE CORP UK0.011976 CHRYSLER CORP C

0.014599 BETHLEHEM STL CORP BS

0.037736 MANVILLE CORP MVL

0.042553 NAVISTAR INTL CORP NAV

Spread data is 2(a - b)/(b + a) for each firm sampled at 9:45 ET on September 9th, 1991.

Table 2Summary Statistics

Description Dow 27 Dow 21

N 6435 6435

Meau*lOO 0.0283 0.0358

Std.*lOO 0.9860 0.9657

Skewness -2.512 -2.644

(w/o crash) 0.012 0.035

Kurtosis 71.31 76.13

(w/o crash) 8.73 8.39

Pl 0.1126 0.1043

P2 -0.0277 -0.0301

Ps 0.0016 -0.0120

P4 -0.0217 -0.0286

Ps 0.0262 0.0248

P6 -0.0194 -0.0125

P7 0.0068 0.0059

Ps 0.0114 0.0077

P9 -0.0180 -0.0195

PlO -0.0192 -0.0220

(Std.) 0.0125 0.0125

Corr(Dow) 0.930 0.935

Summary statistics for the two large firm portfolios. Dow 21 is the group of firms with the smallest spreadsshowu as the firms above the line iu table 1. Pi is the autocorrelation at lag i, and (Std.) is the Bartlettstandard error. Corr(Dow) is the correlation with the actual Dow Iudustrial iudex.

Table 3GARCH(l,l) Parameter Estimates

Xt = a+b1ft_l +ft

z, ~ N(O, 1)

Dow

(Std)

"'0

1.078

(0.161)

(3

0.885

(0.004)

"'1

0.109

(0.029)

0.165

(0.013)

a

5.00

(1.07)

Estimation is by maximum likelihood, and standard numbers in parenthesis are asymptotic standard errors.

Table 4Correlation Differences

Daily Returns: Volatility Conditioning

0-25 2-27 0-50 25-50

Daily Dow27 0.207 0.190 0.139 0.068

(AR1) (0.000) (0.000) (0.000) (0.044)

(GARCH-MA) (0.000) (0.000) (0.002) (0.062)

Daily Dow21 0.217 0.175 0.143 0.070

(GARCH-MA) (0.000) (0.000) (0.002) (0.058)

Dow27 First Half 0.129 0.129 0.068 -0.013

(GARCH-MA) (0.036) (0.030) (0.082) (0.617)

Dow27 Second Half 0.092 0.102 0.103 0.054

(GARCH-MA) (0.078) (0.062) (0.014) (0.104)

Dow21 First Half 0.122 0.132 0.067 0.000

(GARCH-MA) (0.040) (0.028) (0.082) (0.515)

Dow21 Second Half 0.081 0.102 0.097 0.070

(GARCH-MA) (0.102) (0.062) (0.016) (0.096)

Numbers are autocorrelation differences between low and high levels of volatility. Fractions used are given atthe top of each column. Numbers in parenthesis are the fraction of 500 simulated AR(l) and GARCH(l,l)models with constant correlations that generate differences as large as those from the respective series.

Table 5Daily Returns: Volume Conditioning

0-25 2-27 0-50 25-50

Dow27 0.201 0.188 0.124 0.014

(Joint Scramble p) (0.000) (0.000) (0.000) (0.403)

(MDEP2 p) (0.032) (0.008) (0.034) (0.390)

MDEP15 (Mean) 0.184 0.175 0.107 0.005

[Std] [0.048] [0.045] [0.031] [0.043]

MDEP20 (Mean) 0.192 0.180 0.111 0.007

[Std] [0.045] [0.045] [0.029] [0.040]

Dow21 0.200 0.178 0.121 0.010

(Joint Scramble) (0.000) (0.000) (0.000) (0.452)

(MDEP2 p) (0.026) (0.028) (0.042) (0.424)MDEP15 (Mean) 0.183 0.165 0.104 -0.000

[Std] [0.045] [0.045] [0.031] [0.040]

MDEP20 (Mean) 0.189 0.172 0.107 0.001

[Std] [0.045] [0.045] [0.031] [0.037]Numbers are autocorrelation differences between low and high levels of volume. Fractions used are givenat the top of each column. Numbers in parenthesis are the fraction of 500 joint distribution simulatonsthat generate differences as large as those from the respective series. Numbers in brackets are the estimatedstandard deviations from 500 runs of the corresponding simulation.

Table 6Cross Correlations: 0 - 50 Differences

{3L P "Daily Dow - Volatility 0.100 0.032 -0.000

(Joint Scramble) (0.000) (0.000) (0.677)

(MDEP 2 p) (0.002) (0.008) (0.587)

MDEP15 0.072 0.031 0.000

[Std] [0.029] [0.008] [0.000]

MDEP20 0.082 0.031 -0.000

[Std] [0.028] [0.009] [0.000]

Volatility First Half 0.037 0.028 -0.000

Volatility Second Half 0.054 0.046 -0.000

Daily Dow - Volume 0.113 -0.005 -0.000

(Joint Scramble) (0.000) (0.796) (0.679)

(MDEP 2 p) (0.038) (0.667) (0.571)

MDEP15 0.102 -0.005 -0.000

[Stdj [0.031] [0.008] [0.000]

MDEP20 0.105 -0.005 -0.000

[Stdj [0.031] [0.008] [0.000]

Volume First Half 0.177 -0.009 -0.000

Volume Second Half 0.047 0.002 0.000..

Numbers are autocorrelation dIfferences between low and hIgh levels of volatIhty. For this table differencesare taken only between the lower and upper halves. Numbers in parenthesis are the fraction of 500 jointdistribution simulatons that generate differences as large as those from the respective series. Numbers inbrackets are the estimated standard deviations from 500 runs of the corresponding simulation.

Table 7Transaction Cost Simulations

0-25 2-27 0-50 25-50 Autocorrelation

Dow21 0.217 0.175 0.143 0.069 0.104

Ask-Bid (Mean) 0.025 0.023 0.011 0.003 0.005

[Std] [0.054] [0.046] [0.043] [0.038] [0.032]

2(Ask-Bid) (Mean) 0.081 0.071 0.041 0.017 0.008

[Std] [0.054] [0.047] [0.042] [0.041] [0.032]

4(Ask-Bid) (Mean) 0.211 0.191 0.118 0.053 0.025

[Std] [0.056] [0.051] [0.041] [0.040] [0.031]

(Ask-Bid)+Ind. Noise (Mean) 0.002 0.000 0.003 0.000 -0.001

[Std] [0.051] [0.043] [0.040] [0.040] [0.029]

4(Ask-Bid)+Ind. Noise (Mean) 0.016 0.015 0.011 0.001 0.008

[Std] [0.050] [0.043] [0.039] [0.040] [0.029]

(Ask-Bid)+GARCH Ind Noise(Mean) 0.009 0.008 0.005 -0.000 0.001

[Std] [0.050] [0.043] [0.038] [0.039] [0.030]

4(Ask-Bid)+GARCH Noise(Mean) 0.061 0.055 0.039 0.018 0.012

[Std] [0.055] [0.045] [0.041] [0.039] [0.032]SimulatIOn results for transactIOn cost model. Table rows present the means and standard deviatIons from500 simulations of the transaction cost model. Costs are set to 0.15 percent plus one half the indicatedmultiple of the bid-spread for 21 firms. The spreads used are the 21 smallest spreads from table 1. The firstthree tests used have no individual firm shocks, and the final two tests include these shocks.

Figure 1Proportionate Bid-Ask Spreads

27 Firms measured 9/18/91

I

~.I

------

0.045

0.04

0.035

0.03'"0~ 0.025Hg. 0.02

0.015

0.01

0.005

oo 5 10 15 20 25 30

Firm

Figure 2Daily Dow and Dow21: 63-88

Volatility = 10 Lags Squared

~o.j 0.35

«l l' ~!Qj O. 3-tII}11lH-1i--+-+-l--+-+-l--+-+--l

~ O. 25 -+--+I~:\lit---1IA~~r-+.~-+-f-f--+-+---1';;j O. 2 +-t--->'I-ll1-IIf-rv~~rt--+-J\-t-+1A-.[\,t--t-f

~ O. 15+-1--t-Wi-+-ll"¥-iJo--+h.----.I,Hlr'\ir.\,;---/t---1.... a. 1 +--+-t-+--+----'lfF-YJV4'-V

§ 0.05 Lu

o rl N M ~ ~ w ~ 00 ffi rl. ..o 0 0 0 0 0 0 0 0

Volatility Quantile Order

Dow27

Dow21

Conditional correlations using volatilityinformation. The graph is generated using 500day bands moved through a table sorted byvolatility.

Figure 3Daily Dow : 63-88

Volatility = 5, 10, 15 Lags Squared

l::o

-rl O. 35 -tJIc-++t-iI--+--+--+-t-+--+--+--1-WcooJ 0.3

"~ O. 2 5 -:ftl-v+--'Mrt-lIl----h--H---,---t-+--+-t-lu.-l O. 2 -HH---lco§ O. 15+--I--t---"'1 cl-----I--'-I-"H---'l--·rl

_~ O. 1 +-+---11-+-+--'1--+'d

§ o. 05 -1--+---1I--+--+--+-t--+-,--++u

o M N M ~ ~ w ~ 00 m M. . . . . . . . .o 0 0 0 0 0 0 0 0

Volatility Quantile Order

5 Day

10 Day

15 Day

Figure 4Daily Dow21 Correlations

10 Day Volatility1 Day Volume

0. 25

0. 2

0.0 5

O'<s

00<

Oo~s q00"~

~'i

Oo~ ~

~8

ooos

Conditional correlations using volume andvolatility information. Correlation is estimatedusing using a gaussian kernel weighting at eachpoint in volume-volatility space. Both volume andvolatility are initially mapped into theirdistribution quantile orders before thenonparametric procedure. Bandwidth is set to 0.2.

Figure 5Daily Dow21

Cross CorrelationsVolatility = 10 Lags Squared

0.25

0.2

0.15

0.1

0.05

o

-0.05

I;f~ ~

~

k

fv.1 1/~v '1fV

~ i ft.

-v'\1J" 1 ~' V1hJ

o rl N ~ ~ ~ ~ ~ 00 m rl

o 0 0 000 0 0 0

Volatility Quantile Order

Alpha

Beta Lag

Rho

Conditional regressions for firm j at time tregressed on a constant, Alpha, the lagged index,Beta Lag, and itself lagged, Rho. The graph showsthe average coefficient estimates across all 21firms.