Discussion Papers In Economics
And BusinessTrading volume and serial correlation in stock returns:
a threshold regression approach

Shoko Morimoto and Mototsugu Shintani
Discussion Paper 10-28

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPAN

Trading volume and serial correlation in stock returns:
a threshold regression approach

Shoko Morimoto and Mototsugu Shintani
Discussion Paper 10-28
December 2010

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPAN

Trading volume and serial correlation in stock returns:
a threshold regression approach
Shoko Morimoto∗and Mototsugu Shintani†‡

Abstract
We extend the analysis of Campbell et al. (1993) on the relationship between
the ﬁrst-order daily stock return autocorrelation and stock market trading volume
by allowing abrupt and smooth transition structures using lagged stock returns as atransition variable. Using U.S. stock market data, we ﬁnd the evidence supporting
the nonlinear relationship characterized by a stronger return reversal eﬀect on ahigh-volume day combined with low lagged stock returns.

JEL Classiﬁcation: C22, G12
Keywords: TAR, STAR, Stock return autocorrelation, Trading volume∗

Graduate School of Economics, Osaka University, Toyonaka, Osaka 560-0043 Japan
Department of Economics, Vanderbilt University, Nashville, TN 37235 USA‡
Corresponding author. E-mail: mototsugu.shintani@vanderbilt.edu
†

1

1

Introduction
The ﬁrst-order autocorrelation in daily stock returns tends to be lower when theaggregate stock market trading volume is higher. Using both the stock price index andindividual stock price series in the U.S., Campbell et al. (1993, CGW) ﬁnd a signiﬁcantly
negative eﬀect of volume on the autoregressive coeﬃcients of stock returns. They explain
their ﬁnding using a model in which risk-averse “market makers” accommodate sellingpressure from “liquidity” or “noninformational” traders in exchange for the reward of ahigher expected stock return. They argue that a stock price decline on a high-volume day
is more likely the result of exogenous selling pressure by noninformational traders, andwill be followed by price increases on subsequent days. In contrast, a stock price decline
on a low-volume day may be caused by the arrival of public information on lower future
cash ﬂows (or fundamentals) with the lower possibility of price reversals.
In this paper, we extend the empirical analysis of CGW by introducing an additional
nonlinear structure where the serial correlation of stock returns does not depend only
on the size of trading volume but also on the sign of lagged stock returns. To this end,
we consider variants of the threshold autoregressive (TAR) model and smooth transition
autoregressive (STAR) model using past stock returns as a transition variable. The former assumes abrupt transition, while the latter assumes smooth transitions between twoalternative eﬀects of volume on the serial correlation of stock returns. We use updated
U.S. data on stock price index returns and trading volume, and examine whether theﬁndings of CGW are robust to these extensions.
There are a number of reasons why we may expect that the relationship between
trading volume and serial correlation in stock returns depends on the sign of lagged stock
returns, which is conveniently described by TAR/STAR models. First, if both types ofinvestors hold stock for more than one period, their behavior in the face of liquidity shock
obviously depends on the past performance of the stock returns. Second, negative returns
increase the risk in the following period measured by volatility because of an increased
debt-to-equity ratio (leverage eﬀect). Since risk-averse market makers demand higher
2expected returns for riskier assets, the sign of lagged stock returns will have an eﬀect
on the serial correlation of stock returns. Third, when market declines are larger, there
is a greater likelihood that margin accounts will be liquidated. Thus, noninformational
traders are expected to be more active following the negative stock return periods thanafter the positive stock return periods.
It should be noted that our analysis is also related to some prior work that considers
the asymmetric impact of stock market shocks on stock returns. De Bondt and Thaler(1989) ﬁnds that stock market overreaction eﬀects among losers are much stronger thanamong winners. Koutmos (1998), Nam et al. (2001) and Chiang et al. (2007) ﬁnd thatstock indexes incorporate negative shocks faster than positive shocks in many advanced
nations.

2

Model
The stock return, rt , is often assumed to follow an autoregressive model because of apartial adjustment of past price to its market fundamentals. To capture the dependenceof serial correlation of stock returns on volume, CGW include the product of lagged
trading volume, vt−1 , and the lagged stock return, rt−1 , as an additional regressor in the
autoregression. The benchmark CGW regression takes the form of
rt = α + βrt−1 + γvt−1 rt−1 + εt ,and∑5rt = α + (i=1βi Dit )rt−1 + γvt−1 rt−1 + εt ,

(1)

(2)

where Dit ’s are ﬁve day-of-the-week dummies, and εt is an error term with mean zero anda ﬁnite variance. In our analysis, we extend the CGW regression models (1) and (2) tothe following TAR/STAR models,
rt = α + βrt−1 + γ1 vt−1 rt−1 + γ2 vt−1 rt−1 F (zt ) + εt ,andrt = α + (∑5i=1βi Dit )rt−1 + γ1 vt−1 rt−1 + γ2 vt−1 rt−1 F (zt ) + εt ,
3

(3)

(4)

where F (zt ) is a transition function, which takes a value between 0 and 1 depending on thetransition variable zt . Here, the transition variable, zt , represents the past performance ofthe stock returns, e.g., the moving averages of lagged stock returns. Our model reduces
to the benchmark CGW regression when γ2 = 0.
If the relationship between the serial correlation of stock returns and trading volume
is determined only by the sign of past stock market returns, an abrupt transition can be
introduced by employing a transition function of the form, F (zt ) = 1[zt > 0], where 1[A]is an indicator function that takes a value 1 if A is true and a value 0, otherwise. In sucha case, the coeﬃcient on the product of volume and the stock return, γ1 , represents thedependence of the ﬁrst-order return autocorrelation on the trading volume for the casewith negative past stock market returns. With positive past stock market returns, thecoeﬃcient on the product is represented by γ1 + γ2 .
We can further extend the model to allow for the smooth transition by employing thelogistic transition function, F (zt ) = [1 + exp(−δzt )]−1 , where δ(> 0) is a scale parameterthat controls the rate of the transition. Note that the logistic transition function nests
the indicator function since the former approaches the latter as δ → ∞. However, forsimplicity, we simply refer to the model with the indicator transition function as the TAR
model and the one with the logistic transition function as the STAR model.

3

Data
For the stock return series, we use daily log returns deﬁned as rt = 100 × log(Pt /Pt−1 )where Pt is a value-weighted index of stocks traded on the New York Stock Exchange
and American Stock Exchange (NYSE/ASE), from the Center for Research in Security
Prices (CRSP) at the University of Chicago. Our data covers the period from 7/1/1963
to 12/31/2009. In addition to the full sample period, we consider a shorter sample period
through 9/30/1987, which focuses on the period prior to the stock market crash of October1987. We refer to this subsample as the CGW sample, as it roughly corresponds to themain sample period used in the analysis of CGW (sample A in their notion). For the stock4

market trading volumes, we use the CRSP data on a value-weighted number of shares
traded daily on the NYSE/ASE. We follow CGW and use a triangular moving average of
growth rates, by subtracting a one-year backward moving averages from the log trading
volumes (multiplied by 100).1
For the transition variable, which represents the past stock market performance, we
consider both a simple lagged return, zt = rt−1 , and a lagged ﬁve-day moving average of∑stock returns, zt = 5j=1 rt−j /5. Descriptive statistics of returns, their ﬁve-day moving
averages, and detrended volumes are reported in Table 1. A comparison of the CGW
sample in panel A and full sample in panel B shows that the variation of returns is larger
with the full sample period, but no obvious diﬀerence is observed for trade volume series.

4

Results
We ﬁrst estimate the benchmark CGW regression model (1) and (2) using the ordinary

least squares (OLS). The estimation results of two speciﬁcations are presented in Table 2.
The panel A of the table shows results based on the CGW sample. The estimate of thecoeﬃcient γ is negative and signiﬁcant at the 1 percent signiﬁcance level for both withand without ﬁve day-of-the-week dummies. Our result is thus consistent with the result
reported in Table 2 of CGW (page 912). Inclusion of day-of-the-week dummies contributes
to a somewhat higher γ estimate in absolute value, an increase in the R2 statistic andreduction in the sum of squared residuals. The full sample results reported in panel B
show the lower estimate of γ in absolute value and the smaller R2 statistic, compared to
the CGW sample results. However, it is important to note that the signiﬁcantly negative
estimate of γ, the main ﬁnding by CGW, remains the same even if the sample period is
extended for more than twenty years.
Next, we estimate the TAR model with F (zt ) = 1[zt > 0] in (3) and (4). The model
can again be estimated by the OLS since the additional regressor, vt−1 rt−1 1[zt > 0], is
observable. The results are presented in Table 3 for two alternative transition variables

1

Thus our volume measure is the detrended log volume rather than the detrended log turnover.

5and for two sample periods. When zt = rt−1 is used for the transition (or threshold)variable, the original coeﬃcient on the product of volume and the stock return, γ1 , isnegative and statistically signiﬁcant at the 1 percent level. However, at the same time,
the coeﬃcient on the additional regressor, γ2 , turns out to be positive and statisticallysigniﬁcant. This result suggests that the price reversals on a higher trading volume day
are more evident when the past lagged returns are negative. This ﬁnding holds for both∑
the CGW sample and full sample. When ﬁve-day moving averages, zt = 5j=1 rt−j /5, areused for the transition variable, results are very similar to the case of zt = rt−1 , except for
the full sample case with day-of-the-week dummies where the estimate of γ2 is positive
but not statistically signiﬁcant.
Finally, we estimate the STAR model with F (zt ) = [1 + exp(−δzt )]−1 in (3) and (4).
Here, for each speciﬁcation, the transition variable is normalized to have a unit sample
variance for the purpose of the unit-free interpretation of the scale parameter δ (see van
Dijk et al., 2000). The model is estimated by the nonlinear least squares (NLS) method,
and the results are presented in Table 4 for two alternative transition variables and fortwo sample periods.2 Since the NLS estimate corresponds to the OLS for a ﬁxed value ofδ, both the R2 statistic and the sum of squared residuals are also reported in the table.
The results for γ1 and γ2 in the estimated STAR model are not distinguishable from thosein the estimated TAR model. While the standard error is very large, the estimate of thescale parameter δ is also large for all cases, suggesting that the shape of transition function
is similar to the abrupt transition function in the TAR model.3 However, reduction in thesum of the squared residuals suggests some improvement in terms of the model ﬁt over
the TAR model.
Unlike the case of TAR models with known threshold values (zero in our case), thelinear hypothesis cannot be tested by the signiﬁcance of γ2 in the case of STAR models,
because δ is not identiﬁed if γ2 is zero. To conduct a formal speciﬁcation test of our STAR
2

Initial values are ﬁrst obtained by running OLS regressions with the regressor vt−1 rt−1 [1 +exp(−δzt )]−1 for ﬁxed δ’s from 300 equally spaced grids. The Newton method is then employed tominimize in the least square criterion for each of 300 initial values to obtain the ﬁnal NLS estimate.
3
A large standard error with a large scale parameter estimate is commonly observed in the estimationof the STAR model. See van Dijk et al. (2000) for the reasoning.

6

model, we employ the test proposed by Teräsvirta (1994). The test without day-of-theweek dummies is based on an auxiliary regression of the form,[]
rt =α + βrt−1 + γvt−1 rt−1 + φ11 rt−1 + φ12 vt−1 rt−1 zt[][]+ φ21 rt−1 + φ22 vt−1 rt−1 zt2 + φ31 rt−1 + φ32 vt−1 rt−1 zt3 + εt .

(5)Under the null hypothesis of linearity against the STAR model, φij = 0 holds for alli = 1, ..., 3 and j = 1, 2. For the test with day-of-the-week dummies, βrt−1 in (5) is∑
replaced by ( 5i=1 βi Dit )rt−1 . The results of the F test are reported in Table 5. For allcases, the linearity is signiﬁcantly rejected which justiﬁes the use of the STAR model.

5Conclusion
We investigated the relationship between the ﬁrst-order daily stock return autocorre-

lation and stock market trading volume using threshold and smooth transition autoregressive models. We found that, consistent with the ﬁnding by Campbell et al. (1993), a stock
price decline on a high-volume day tends to be followed by a return reversal comparedto that on a low-volume day for the extended series. Furthermore, we found statisticallysigniﬁcant evidence of an additional nonlinear relationship where serial correlation structure also depends on past stock returns. In particular, we found stronger return reversal
eﬀect on a high-volume day if the lagged stock returns are negative.

Acknowledgements
We are grateful to Zi-Yi Guo, Keiich Hori, and seminar participants at Osaka University
for helpful comments and suggestions.

7

References[1] Campbell, J. Y., Grossman, S. J. and Wang, J. (1993) Trading volume and serial
correlation in stock returns, Quarterly Journal of Economics, 108, 905-39.
[2] Chiang, T. C., Chen, C. W. S. and So, M. K. P. (2007) Asymmetric return andvolatility responses to composite news from stock markets, Multinational Finance
Journal, 11, 179-210.
[3] De Bondt, W. F. M. and Thaler, R. H. (1989) A mean-reverting walk down WallStreet, Journal of Economic Perspectives, 3, 189-202.
[4] Koutmos, G. (1998) Asymmetries in the conditional mean and the conditional variance: evidence from nine stock markets, Journal of Economics and Business, 50,
277-290.
[5] Nam, K., Pyun, C. S. and Avard, S. L. (2001) Asymmetric reverting behavior of shorthorizon stock returns: an evidence of stock market overreaction, Journal of Banking
and Finance, 25, 807-824.
[6] Teräsvirta, T. (1994) Speciﬁcation, estimation, and evaluation of smooth transition
autoregressive models, Journal of the American Statistical Association, 89, 208-218.
[7] van Dijk, D., Teräsvirta, T. and Franses, P. H. (2002) Smooth transition autoregressivemodels: a survey of recent developments, Econometrics Reviews, 21, 1-47.

8

Table 1: Descriptive statistics
Mean

Median SD

A: 7/1/1963-9/30/1987
Return.04
Return (5-day MA).04
Volume.07
B: 7/1/1963-12/31/2009
Return.04
Return (5-day MA).04
Volume
.06

Min

Max

Obs
6,095
6,095
6,095

.06
.07
.06

.77
.41
.22

-4.44
-2.01
-1.10

5.16
2.53
1.04

.07
.07
.06

.98
.46
.22

-18.80-5.57-1.42

10.90 11,707
3.64 11,707
1.04 11,707

Notes: Return series are log stock returns expressed in percentage. ‘5-day MA’ represents the ﬁve-day moving averages of returns. Volume series are the log trading volumes
detrended by subtracting the one-year backward moving averages. ‘SD,’ ‘Min,’ ‘Max’and ‘Obs’ are the standard deviation, minimum, maximum and number of observations,
respectively.

Table 2: Benchmark modelDay-of-theβweek dummy
A: 7/1/1963-9/30/1987
No.285∗∗∗(.015)
Yes—
B: 7/1/1963-12/31/2009
No.106∗∗∗(.012)
Yes
—

γ

R2

SSR

−.291∗∗∗(.049)−.340∗∗∗(.049).061

3, 384.075

3, 330−.153∗∗∗(.032)−.136∗∗∗(.032).007 11, 214.017 11, 096

Notes: Numbers in parentheses are standard errors. Statistically signiﬁcant estimates at
the 1%, 5%, and 10% levels are shown with asterisks, ***, **, and *, respectively. ‘SSR’is the sum of squared residuals.

9

Table 3: TAR modelTransition
Day-of-thevariable (zt ) week dummy
A: 7/1/1963-9/30/1987
rt−1
No
rt−1
∑5

i=1 rt−i /5∑5i=1 rt−i /5

Yes
No
Yes

B: 7/1/1963-12/31/2009
rt−1
No
rt−1
∑5

i=1 rt−i /5∑5i=1 rt−i /5

Yes
No
Yes

β

γ1

γ2

R2

SSR

.280∗∗∗(.015)—−.545∗∗∗(.079)−.610∗∗∗(.079)−.458∗∗∗(.075)−.509∗∗∗(.075).377∗∗∗(.092).400∗∗∗(.091).257∗∗∗(.087).259∗∗∗(.087).063

3, 374.078

3, 320.062

3, 379.076

3, 325−.220∗∗∗(.039)−.185∗∗∗(.039)−.181∗∗∗(.035)−.150∗∗∗(.035).164∗∗∗(.053).117∗∗(.054).113∗(.058).055(.058).008 11, 205.284∗∗∗(.015)—.103∗∗∗(.012)—.104∗∗∗(.012)—.017 11, 092.007 11, 210.017 11, 096

Notes: F (zt ) = 1[zt > 0]. Numbers in parentheses are standard errors. Statisticallysigniﬁcant estimates at the 1%, 5%, and 10% levels are shown with asterisks, ***, **, and*, respectively. ‘SSR’ is the sum of squared residuals.

10

Table 4: STAR modelTransition
Day-of-thevariable (zt ) week dummy
A: 7/1/1963-9/30/1987
rt−1
No
rt−1
∑5

i=1 rt−i /5∑5i=1 rt−i /5

Yes
No
Yes

B: 7/1/1963-12/31/2009
rt−1
No
rt−1
∑5

i=1 rt−i /5∑5i=1 rt−i /5

Yes
No
Yes

β

γ1

γ2

δ

R2

SSR

.280∗∗∗(.015)—−.547∗∗∗(.079)−.612∗∗∗(.079)−.450∗∗∗(.075)−.498∗∗∗(.075).378∗∗∗(.092).401∗∗∗(.092).239∗∗∗(.087).236∗∗∗(.087)

41.34(1842.75)
42.23(1741.65)
161.26(1961.55)
161.26(1972.91).063

3, 370.078

3, 315.062

3, 374.076

3, 321−.220∗∗∗(.039)−.185∗∗∗(.039)−.180∗∗∗(.036)−.150∗∗∗(.036).164∗∗∗(.053).117∗∗(.054).108∗(.059).053(.059)

36.83(2956.47)
36.83(4124.93)
58.11(668.42)58.11(1354.30).008

11, 200.017

11, 088.007

11, 206.017

11, 092.284∗∗∗(.015)—.103∗∗∗(.012)—.104∗∗∗(.012)—

Notes: F (zt ) = [1 + exp(−δzt )]−1 . Numbers in parentheses are standard errors. Statistically signiﬁcant estimates at the 1%, 5%, and 10% levels are shown with asterisks, ***,**, and *, respectively. ‘SSR’ is the sum of squared residuals.

Table 5: Linearity test
Transition
Day-of-thevariable (zt ) week dummy
A: 7/1/1963-9/30/1987
rt−1
No
r
Yes
∑5 t−1
r
/5
No
t−i
∑i=1
5
Yes
i=1 rt−i /5
B: 7/1/1963-12/31/2009
rt−1
No
r
Yes
∑5 t−1rt−i /5
No
∑i=1
5
Yes
i=1 rt−i /5

F statistic p-values

52.91
23.83
54.34
25.40

0.00
0.00
0.00
0.00

27.00
17.60
35.65
17.73

0.00
0.00
0.00
0.00

Notes: Teräsvirta’s (1994) test for linearity against the logistic STAR model.
11