Directory UMM :Data Elmu:jurnal:I:International Review of Economics And Finance:Vol10.Issue1.2001:

International Review of Economics and Finance
10 (2001) 19 ± 40

Economic sources of asymmetric cross-correlation among
stock returns
Chih-Hsien Yua, Chunchi Wub*
a

b

National Chung-Cheng University, Chia-Yi, Taiwan
School of Management, Syracuse University, Syracuse, NY 13244-2130, USA

Received 10 June 1998; revised 23 November 1999; accepted 20 January 2000

Abstract
We suggest an alternative framework to explain the asymmetric return cross (serial)-correlation. We
identify two major sources of the asymmetric cross-correlation: (1) the difference in the sensitivity of
stock returns to economic factors, and (2) the differential quality of information between large and
small firms. We find that the difference in the response of stock prices to economic factors is an
important determinant of the first-order cross-correlation relative to firm-specific factors. Further

evidence suggests that the asymmetric cross-correlation is mainly attributed to differences in the
sensitivity of stock prices to market-wide information and the differential quality of cash flows
information between large and small firms. D 2001 Elsevier Science Inc. All rights reserved.
JEL classification: G10; G12
Keywords: Asymmetric cross-correlation; Economic factors; Risk premium; Return innovations

1. Introduction
Previous studies have documented intriguing time-series properties of stock returns.
Returns of market indices and size-related portfolios in the short horizon are found to be
positively serially correlated, whereas individual stock returns are negatively serially
correlated (see, for example, Cohen, Hawawini, Maier, Schwartz, & Whitcomb, 1980; Fama,
1965; French & Roll, 1986; Perry, 1985). Lo and MacKinlay (1990a) document a striking
* Corresponding author. Tel.: +1-315-443-3399; fax: +1-315-443-5457.
E-mail address: cwu@syr.edu (C. Wu).
1059-0560/01/$ ± see front matter D 2001 Elsevier Science Inc. All rights reserved.
PII: S 1 0 5 9 - 0 5 6 0 ( 0 0 ) 0 0 0 6 9 - 1

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C.-H. Yu, C. Wu / International Review of Economics and Finance 10 (2001) 19±40

phenomenon of asymmetric cross (serial)-correlation among stock returns.1 They find a
significant positive cross-correlation between the lagged (weekly) returns of large-firm stocks
and the current returns of small-firm stocks. On the other hand, the cross-correlation between
the lagged returns of small-firm stocks and the current returns of large-firm stocks is small
and statistically insignificant. They conclude that the lead±lag relationship among returns of
size-sorted portfolios is an importance source of contrarian profits. They also indicate that this
lead±lag phenomenon may imply a complex information transmission between large and
small firms.
While the significance of short-horizon cross-correlations is not refutable, the economic
sources of asymmetric cross-correlation are less clear. Potential sources cited in the recent
finance literature include market frictions, lagged information transmission and institutional
structures.2 Chan (1993) suggests that cross-correlations among stock returns occur when
market-makers do not have perfect information. When market-makers condition prices only
on their noisy signals, the pricing error of one stock will be correlated with signals of other
stocks. As market-makers adjust prices after true values of stocks are revealed, stock returns
will appear to be positively cross-autocorrelated. Also, since the quality of information
signals for large stocks is often better than that for small stocks, prices of large stocks tend to
adjust to market information faster than prices of small stocks. Consequently, returns of large
stocks may lead returns of small stocks.

Although there are conjectures about how information is assimilated, many contend that
the performance of large stocks conveys information about the future prospects of small
stocks. In general, if more and higher-quality information is available for large firms and the
information transmission is not instantaneous, there is always a possibility that current prices
of small stocks will be conditional on past prices of large stocks. Furthermore, this lead±lag
effect could be carried over to the level of stock return volatility (see Conrad, Gultekin, &
Kaul, 1991).
However, in addressing t]he issue of information transmission, most of previous studies
did not carefully separate the transmission effects of market-wide from firm-specific
information (see, for example, Chan, 1993). The lead±lag relationship across firms may be
due to differences in the response of stock returns to economic factors. Alternatively, it could

1

In this paper, we use the term ``cross-correlation (covariance)'' to represent cross-serial correlation (covariance).
2
For the discussion of market frictions, see Boudoukh, Richardson, and Whitelaw (1994). They demonstrated
that nonsynchronous trading can be an important determinant of cross-correlation. They showed that cross-serial
correlations can be explained by portfolios' own autocorrelation patterns coupled with high contemporaneous
correlations across portfolios, and that lagged returns on large-firm stocks are simply proxying the lagged returns

of small-firm stocks. Other related studies are Atchison, Butler, and Simonds (1987), Hameed (1992), Lo and
MacKinlay (1990b) and Roll (1984). For lagged response to information, see Jegadeesh and Titman (1995). For
discussion of the institutional factors, see Badrinath, Kale, and Noe (1995). They indicated that different levels of
institutional interest in equities induce cross-correlation in equity returns. Institutional investors have an incentive
to invest in stocks of large firms, which are more closely followed by informed traders. Because more information
is produced for large firms, prices of large-firm stocks may convey information regarding the prospects of smallfirm stocks. Finally, McQueen, Pinegar, and Thorley (1996) discovered a directional asymmetry that small stocks
respond more slowly to increases in large-stock returns than to decreases.

C.-H. Yu, C. Wu / International Review of Economics and Finance 10 (2001) 19±40

21

be the firm-specific information of large stocks that conveys the future prospects of small
stocks. In an imperfect-information market, both effects can coexist. It will be important to
know how and to what extent these factors have contributed to the asymmetric crosscorrelation among stock returns.
In this paper, we propose a framework to distinguish the effects of economic from firmspecific factors on the asymmetric cross-correlation. We consider a factor-based model for
decomposition of economic and firm-specific information effects. This approach allows us to
evaluate the relative importance of economic vs. firm-specific factors in the determination of
cross-correlations among returns of size-sorted portfolios. We show that return crosscorrelation can be induced by both the differential quality of information among small and
large stocks and differences in the response of stock returns to economics factors. The relative

importance of these effects is assessed. Furthermore, we analyze the components of economic
and firm-specific factors in stock returns to provide more detailed information about the
fundamental sources of cross-correlation, using the decomposition method suggested by
Campbell and Ammer (1993).
The results support the contention that cross-correlation is induced by both the differential
information quality of stocks and the sensitivity of stock price to economic factors. The
results show that the first-order cross-correlation is mainly caused by the differences in the
response to market-wide information between large and small stocks. On the other hand, the
differential quality of cash flow information appears to play a more significant role in the
asymmetric cross-correlations of higher orders.
The remainder of this paper is organized as follows. Section 2 presents an analytical
framework for decomposition of return cross-covariance. We first specify the structure of
common factors and the generating process of asset returns. Following this, the crosscovariance of returns is decomposed into aggregate economic and firm-specific components. Section 3 discusses data and empirical results. Finally, Section 4 summarizes our
major findings.

2. The model
We posit that the excess returns on N assets obey the following dynamic process:
et ˆ Etÿ1 …et † ‡ B…Xt ÿ Etÿ1 …Xt †† ‡ et ;

…1†

where: et = an N  1 vector of excess returns on assets, i.e., asset returns less short-term
interest rates; Xt = a K  1 vector of state factors; B = an N  K matrix of factor sensitivities;
et = an N  1 vector of asset-specific (idiosycratic) error terms; and Et ÿ 1 = an expectational
operator conditional on information at time t ÿ 1.
Similar to conventional factor models, the unsystematic (firm-specific) factor in Eq. (1)
is assumed to be uncorrelated with innovations of systematic (state) factors; that is,
Ä t + j,et) = 0, for j 6ˆ 0 where X
Ä t = Xt ÿ Et ÿ 1(Xt) is the vector of
Ä t,et) = 0 and Cov(X
Cov(X
unexpected factor shocks. The conditional expectation of stock returns is linear in state
factors and the beta coefficients are constant over time. The model is dynamic in the sense
that expected excess returns vary through time.

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C.-H. Yu, C. Wu / International Review of Economics and Finance 10 (2001) 19±40

Eq. (1) can be rewritten as:

~

…1a†
et ˆ Etÿ1 …et † ‡ BX t ‡ et ˆ Etÿ1 …et † ‡ e~ t ;
Ä
unexpected excess returns on assets.
where eÄt = BX
t is the vector
0 t + e1
0 of 1
elt
bl0
Let et ˆ @ A, and B ˆ @ A, where l and s denote large- and small-stock portfolios,
est
bs0
respectively, and bl and bs are K  1 vectors of sensitivities to the concurrent factor
innovations. Then, excess returns of large and small stocks are:
~

elt ˆ Etÿ1 …elt † ‡ bl0 X t ‡ elt ˆ Etÿ1 …elt † ‡ e~ lt

~

est ˆ Etÿ1 …est † ‡ bs0 X t ‡ est ˆ Etÿ1 …est † ‡ e~ st :

…2a†
…2b†

The asymmetric return cross-correlation documented by Lo and MacKinlay (1990a)
implies that the covariance between excess returns of small-stock portfolios and lagged
excess returns of large-stock portfolios is not zero, i.e., Cov(elt, est + 1) 6ˆ 0, and that
Cov(elt,est + 1) ÿ Cov(elt ÿ 1,est) > 0, that is, the cross-covariance of lagged large-stock returns
with small-stock returns dominates that of lagged small-stock returns with large-stock returns.
From Eqs. (2a) and (2b), Cov(elt,est + 1) can be expressed as:
Cov…elt ; est‡1 † ˆ Cov…Etÿ1 …elt †; Et …est‡1 †† ‡ Cov…Et …est‡1 †; e~ lt †
‡ Cov…Etÿ1 …elt †; e~ st‡1 † ‡ Cov…e~ lt ; e~ st‡1 †:

…3†

The vector of expected excess returns can be specified as the product of factor sensitivities
(i.e., beta risks) and prices of risk:

Etÿ1 …et † ˆ BFtÿ1 ;

…4†

Ftÿ1 ˆ CXtÿ1 ;

…5†

where Ft ÿ 1 is a K  1 vector of risk prices at time t ÿ 1. The information about state factors
Xt ÿ 1 is supposedly available at time t ÿ 1. As will be explained later, our asset-pricing model
is based on observable factors. Since conditional expectations of stock returns are linear in
factors, the vector of risk prices can be written as:
where C is a matrix of coefficients which define market price of risk (see also Campbell &
Hamao, 1992). Substituting Eq. (5) into Eq. (4) for Ft ÿ 1, we have:
Etÿ1 …et † ˆ …BC†Xtÿ1 ˆ AXtÿ1 ;
…6†
0 1
al0
@
A, and al and as are K  1 vectors of the sensitivity coefficients of expected

where A ˆ
as0
returns of large and small stocks to lagged economic factors. More specifically, the expected
excess returns of large and small stocks are:
Etÿ1 …elt † ˆ al0 Xtÿ1

Etÿ1 …est † ˆ as0 Xtÿ1 :

…6a†

…6b†

C.-H. Yu, C. Wu / International Review of Economics and Finance 10 (2001) 19±40

23

Eqs. (6), (6a) and (6b) imply that the risk premiums are time-varying and will be negative if
either A or Xt ÿ 1 is negative. Substituting Eqs. (6a) and (6b) into Eq. (3) for the expected
returns, we have:
Cov…elt ; est‡1 † ˆ al 0 Cov…Xtÿ1 ; Xt †as ‡ as 0 Cov…Xt ; e~ lt † ‡ al 0 Cov…Xtÿ1 ; e~ st‡1 †

‡ Cov…e~ lt ; e~ st‡1 †:

…7†

The above relationship involves the stochastic process of state factors. Following Campbell and Ammer (1993), we characterize the behavior of state factors by a vector
autoregressive (VAR) process (Eq. (8)):
~

…8†
Xt ˆ  Xtÿ1 ‡ X t ;
Ä t is the innovation in Xt, and the matrix  is the coefficient matrix of the VAR
where X
system. The VAR process need not be restricted to the first order. A higher-order VAR
structure can always be handled by augmenting the state vector and reinterpreting  as the
companion matrix of the system.
Ä t + 1) and Cov(Xt ÿ 1X
Ä t) are equal to zero, we can rewrite Eq. (7) as:
Ä t, X
Assuming that Cov(X
~

Cov…elt ; est‡1 † ˆ al 0 Cov…Xtÿ1 ; Xt †as ‡ as 0 Cov…X t ; elt † ‡ Cov…e~ lt ; e~ st‡1 †:

…9†

The value of the first term on the right side, al0Cov(Xt ÿ 1,Xt)as, depends on the sensitivities
of large- and small-stock returns to lagged economic factors, al and as, and the
Ä t,eÄl ), includes
autocovariance and cross-covariance of factors. The second term, as0Cov(X
t
the covariance between innovations of large-stock returns and concurrent economic variables
in addition to the lagged coefficient as. The covariance term captures the concurrent
correlation between unexpected economic conditions and the unexpected (excess) returns of
Ä t,eÄl ) is positive, then large-firm stocks experience unexpected
large-firm stocks. If Cov(X
t
positive performance whenever there is good news for the economy. The last term is the
covariation between return innovations of large and small stocks. This term essentially
captures the interactions between firm-specific factors of large and small stocks since
Cov(eÄlt,eÄst + 1) = Cov(elt,est + 1) from Eq. (1a).
Similarly, we can express the cross-covariance of large-stock returns (in time t + 1) with
small-stock returns (in time t) as:
~

Cov…elt‡1 ; est † ˆ al 0 Cov…Xtÿ1 ; Xt †as ‡ al 0 Cov…X t ; e~ st † ‡ Cov…e~ lt‡1 ; e~ st †:

…10†

Subtracting Eq. (10) from Eq. (9), we obtain the difference in the first-order cross-covariance
between excess returns of large- and small-stock portfolios:
~

~

Cov…elt ; est‡1 † ÿ Cov…elt‡1 ; est † ˆ ‰as 0 Cov…X t ; e~ lt † ÿ al 0 Cov…X t ; e~ st †Š
‡ ‰Cov…e~ lt ; e~ st‡1 † ÿ Cov…e~ lt‡1 ; e~ st †Š:

…11†

Ä t,eÄl ) ÿ al0Cov(X
Ä t,eÄs )], represents the contribution of the effect
The first term, [as0Cov(X
t
t
of economic factors to the asymmetric cross-correlation between large and small stocks.
The second term, Cov(eÄlt eÄst + 1) ÿ Cov(eÄlt + 1,eÄst), represents the contribution of the covariation between firm-specific return components (of large and small stocks) to the asymmetric cross-correlation.

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C.-H. Yu, C. Wu / International Review of Economics and Finance 10 (2001) 19±40

The effect of economic factors can be separated into two parts: the concurrent relationships
Ä t,eÄs ), and the
Ä t,eÄl ) and Cov(X
between factor innovations and return innovations, Cov(X
t
t
sensitivity of stock returns to lagged economic factors, as0 and al0. The higher the correlation
between the innovations of large-stock returns and concurrent economic factors, the greater
the cross-correlation between large- and small-stock returns. Thus, the differences in the
correlation of stocks returns with economic factors may induce an asymmetric crosscorrelation. More specifically, the asymmetric cross-correlation may be due to the fact that
both returns of large and small stocks are driven by the same economic factors (with no
difference in the timing of the effects of these forces), but the return of large stocks are more
closely associated with shocks of concurrent economic variables. Furthermore, as0 (al0) has a
positive (negative) effect on return cross-correlation. Conrad and Kaul (1988) find that smallstock returns are more sensitive to lagged economic factors than large-stock returns, i.e., as0 is
Ä t,eÄl ) and Cov(X
Ä t,eÄs ) on the asymmetry of return
larger than al0. Thus, the effect of Cov(X
t
t
cross-correlation would be reinforced by the differences in sensitivities of large- and smallstock returns to lagged economic variables.
The second term in Eq. (11) is equal to the differences between cross-covariances of
residual returns of large and small stocks. This term may not be trivial under imperfect
markets. For example, if large stocks are more closely followed by informed investors, the
prices of large stocks will impound more information than those of small stocks. But if
market frictions exist, information transmission from large stocks to small stocks may not be
instantaneous. If investors infer the values of small stocks conditional on past prices, the
information will be impounded in the prices of small stocks only after observing the past
prices of large stocks. Then, the lead±lag relation between large- and small-stock returns will
arise, and the value of the second term is most likely to be positive.3
We next turn to higher-order cross-correlations. Lo and MacKinlay (1990a) show that
cross-correlation remains at longer lags. For the return cross-covariance of higher lag orders
( q  2), we have:
~

~

Cov…elt ; est‡q † ÿ Cov…elt‡q ; est † ˆ ‰as 0 Cov…X t‡qÿ1 ; e~ lt † ÿ al 0 Cov…X t‡qÿ1 ; e~ st †Š
‡ ‰Cov…e~ lt ; e~ st‡q † ÿ Cov…e~ lt‡q ; e~ st †Š;

…12†

Cov…elt ; est‡q † ÿ Cov…elt‡q ; est † ˆ Cov…e~ lt ; e~ st‡q † ÿ Cov…e~ lt‡q ; e~ st †;

…13†

Ä t + q ÿ 1 are the innovations of future economic factors. Since factor innovations are
where X
Ä t + q ÿ 1,eÄs ) are both zero.
Ä t + q ÿ 1,eÄl ) and Cov(X
uncorrelated with idiosyncratic terms, Cov(X
t
t
Thus, Eq. (12) reduces to:
which suggests that a higher-order cross-correlation is mainly caused by the crosscorrelation between return innovations of large and small stocks. Furthermore, if
Ä t,eÄl ) ÿ al0 Cov(X
Ä t,eÄs )] in Eq. (11) is positive, then the magnitude of the first[as0 Cov(X
t
t
order return cross-covariance would be greater than that of higher-order return cross3
The fact that Cov(elt, est + 1) is not equal to zero does not rule out the own autocorrelation effect. It can be
shown that the cross correlation of the return shocks (et) is equal to the contemporaneous correlation times
autocorrelation of return shocks. The existence of autocorrelation of return shocks has been used as a main
argument for market overreations (e.g., Jegadeesh & Titman, 1995).

C.-H. Yu, C. Wu / International Review of Economics and Finance 10 (2001) 19±40

25

covariance. This may explain the finding that the asymmetric return cross-correlation decays
sharply over time (see Lo & MacKinlay, 1990a).
The analysis above shows that both economic and firm-specific factors contribute to the
asymmetry of return cross-correlation. However, it does not indicate the specific channels
through which the cross-correlation is generated. In the following, we decompose the effects
of economic and firm-specific factors to provide more detailed information on how the
underlying economic and firm variables affect the magnitude of cross-correlation among
stock returns.
2.1. The effects of economic factors
Asset returns are driven by unanticipated changes in business conditions, and some assets
may have greater exposure to certain economic events than do others. Economic variables
(proxies for economic shocks) may affect stock returns via cash flows or discount rates. For
example, both the real interest rate and long±short yield spread affect the discount rate, and
by changing the time value of expected future cash flows, they eventually affect stock returns.
Unexpected inflation affects real expected cash flows, discount rates and ultimately real
returns on stocks. Furthermore, dividend yields can predict stock returns (Campbell & Shiller,
1988; Fama & French, 1988). This is because stock prices tend to be low relative to dividends
when discount rates and expected returns are high.
To assess the relative importance of different economic variables to the return crosscorrelation, we further divide the covariance effect into K (factor) terms, each representing the
contribution of an economic variable:
~

~

as0 Cov…X t ; e~ lt † ÿ al0 Cov…X t ; e~ st † ˆ ‰as1 Cov…x~ 1t ; e~ lt † ÿ al1 Cov…x~ 1t ; e~ st †Š
‡ ‰as2 Cov…x~ 2t ; e~ lt † ÿ al2 Cov…x~ 2t ; e~ st †Š ‡ . . .
‡ ‰asK Cov…x~ Kt ; e~ lt † ÿ alK Cov…x~ Kt ; e~ st †Š

…14†

where K is the number of economic variables xÄ1t, . . ., xÄKt, . . ., as1, . . ., asK are the elements in
as, and al1, . . ., alK are the elements in al. After estimating the covariance term for each
economic variable, we can test the significance of its contribution to the asymmetric crosscorrelation of returns.4
2.2. Decomposition of the firm-specific effect
To provide more information about the effects of firm-specific factors on return crosscorrelation, we look into the fundamentals of stock returns. Obviously, many fundamental
variables can affect firm value. In the following, we adopt a return decomposition method
4

^

^

Let Cov(d) be a function
parameter vector d, and d be the estimator of d. If d has the following
p ^ of the
d
d
asymptotic distribution: T (dÿd† !
N …0; †, where T is the number of observations, and ``!
'' denotes the
^
convergence
in distribution, then the asymptotic distribution of Cov(d) can be shown as:
p

^
d
T ‰Cov…d† ÿ Cov…d†Š !
N ‰0; …@Cov…d†=@d0 †…@Cov…d†0 =@d†Š, where (@Cov(d)/@d0) is the Jacobian vector of
^
covariance evaluated at d. Each term in Eq.
p(14) can be seen as a function of parameters, i.e., Cov(d) above, and its
asymptotic standard error is simply …1= T †‰…@Cov…d†=@d0 †…@Cov…d†0 =@d†Š1=2.

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C.-H. Yu, C. Wu / International Review of Economics and Finance 10 (2001) 19±40

suggested by Campbell and Ammer (1993) to analyze the fundamental elements of returns
and link them to the cross-correlation of size-sorted stock portfolios. Campbell and Ammer
related unexpected excess stock returns to changes in rational expectations of future
dividends, real interest rates and future excess stock returns. Their model offers a parsimonious framework for summarizing the effects of news on stock returns. It also enables us to
trace the channels through which economic variables affect stock prices. For example,
unexpected inflation may affect expected future cash flows and discount rates. The Campbell±Ammer model allows us to assess the relative importance of these effects. Furthermore,
it is a dynamic accounting identity that imposes internal consistency on expectations, rather
than a behavior model that often requires certain strong assumptions on the underlying
variables and the validity of the model.
Denote ht + 1 as the log real return on a stock, dt + 1 the log real dividend, rt + 1 the
log real interest rate in period t + 1. By definition, the excess stock return in logarithm is
(Eq. (15)):
et‡1  ht‡1 ÿ rt‡1 :

…15†

Let eÄt + 1 be the innovation in excess return, et + 1 ÿ Etet + 1, where Et is an expectation formed
at the end of period t. Then, following Campbell and Ammer (1993), we have (Eq. (16)):5
~

e t‡1  et‡1 ÿ Et et‡1  …Et‡1 ÿ Et †

1
X
jˆ0

ÿ …Et‡1 ÿ Et †

j

r Ddt‡1‡j ÿ …Et‡1 ÿ Et †

1
X

r j et‡1‡j ;

1
X

r j rt‡1‡j

jˆ0

…16†

jˆ1

where D denotes a one-period backward difference and r is a constant discount factor. Or, in
more compact notations (Eq. (17)),
e~ t‡1 ˆ e~ d;t‡1 ÿ e~ r;t‡1 ÿ e~ e;t‡1 ;

…17†

where: eÄd,t + 1 = news about future cash flows (or dividends), eÄr,t + 1 = news about future real
interest rates, and eÄe,t + 1 = news about future excess returns.
As shown above, return innovations can be separated into three components, i.e., cash
flows, discount rates and future excess return growth. This decomposition allows us to assess
the relative importance of each return component in the determination of cross-covariance
among securities. For instance, we can examine the relative importance of each firm's return
component in the covariance of returns of large and small stocks with economic factors (Eqs.
(18) and (19)):
~

~

~

~

~

~

~

~

Cov…X t ; e~ l;t † ˆ Cov…X t ; e~ dl;t † ÿ Cov…X t ; e~ el;t † ÿ Cov…X t ; e~ r;t †

…18†

and
Cov…X t ; e~ s;t † ˆ Cov…X t ; e~ ds;t † ÿ Cov…X t ; e~ es;t † ÿ Cov…X t ; e~ r;t †:
5

…19†

Their decomposition method draws on Campbell and Shiller (1988). Campbell and Mei (1993) use a
similar method.

C.-H. Yu, C. Wu / International Review of Economics and Finance 10 (2001) 19±40

27

We can also examine the relative contribution of each firm-specific return component to the
cross-correlation between return innovations of large and small stocks:
Cov…e~ s;t‡q ; e~ l;t † ˆ Cov…e~ ds;t‡q ; e~ l;t † ÿ Cov…e~ es;t‡q ; e~ l;t † ÿ Cov…e~ r;t‡q ; e~ l;t †;

…20†

Cov…e~ s;t‡q ; e~ l;t † ˆ Cov…e~ s;t‡q ; e~ dl;t † ÿ Cov…e~ s;t‡q ; e~ el;t †
ÿCov…e~ s;t‡q ; e~ r;t †; …q  1†:

…21†

or

Each term on the right-hand side of Eqs. (20) or (21) measures the contribution of a return
component to the qth-order cross-covariance between large and small stocks.
We estimate the VAR model and the elements of covariance using the Generalized Method
of Moments (GMM). The GMM requires much weaker assumptions and offers heteroskedasticity-consistent estimates of the variance±covariance matrix (Hansen, 1982). Since the
model requires all variables to be stationary, we also conduct the Augmented Dickey±Fuller
(ADF) test on the stationarity of the variables (Dickey & Fuller, 1979).

3. Data and empirical results
We construct 10 equally weighted portfolios based on the value of outstanding equity at
the beginning of each year from January 2, 1981 to December 31, 1992. These size-based
stock portfolios include all stocks in the CRSP tape that have no missing return values during
the sample period. There are a total of 1168 stocks in the sample, and each portfolio has 116
or 117 stocks. Weekly returns are computed from Wednesday to Wednesday.6
The economic variables are selected primarily based on their ability to predict stock
returns. We use four important variables suggested by Campbell and Ammer (1993): the
market excess return, the short-term real interest rate, the dividend yield on the market
portfolio and the inflation rate. Also included in our model is the long±short yield spread
suggested by Chen, Roll, and Ross (1986). Campbell and Ammer offered an analytical
foundation to explain why these variables have strong forecasting power for excess stock
returns. Campbell (1987, 1991) provided evidence that these forecasting variables are
powerful. In a paper attempting to explain how betas are determined, Campbell and Mei
(1993) used similar variables for forecasting excess stock returns. To some extent, our study
is an extension of their analysis for own stock return covariance with market returns to the
cross-covariance between size portfolios.
We employ both the value-weighted and equally weighted indices from the CRSP tape as a
proxy for the market index. The weekly market returns are calculated by compounding the
daily returns. Nominal interest rates are obtained from the Federal Reserve Board in
Washington, DC. Short-term and long-term interest rates are based on yields of 3-month
Treasury bills and 10-year Treasury bonds, respectively. We convert the nominal interest rates
6

The weekly return is computed as the return from Wednesday's closing price to the following Wednesday's
closing price. If the following Wednesday's closing is missing, then Thursday's price is used. If Thursday's price is
also missing, then Tuesday's price is used.

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C.-H. Yu, C. Wu / International Review of Economics and Finance 10 (2001) 19±40

into real rates, by taking the difference between nominal interest and inflation rates. The
weekly inflation rates are interpolated from the monthly rates. Finally, dividend yields on the
market portfolio are calculated from the difference between the market indices with and
without dividends, both from the CRSP files. Real market dividend yields are obtained by
adjusting for inflation rates.
We conduct unit-root tests on all the variables in the VAR model and the excess return
equations. The results show that all variables are stationary. The ADF tests reject the unit-root
hypothesis at the 5% level or better for all variables.7 After assuring that all variables are
stationary, we perform GMM estimation for the VAR system.
3.1. The result of VAR estimation
The VAR system includes variables of excess market returns, real (3 months) Treasury bill
rates, inflation rates, yield spreads and real market dividend yields. Yield spreads are the
differences between 10-year Treasury bond and 3-month T-bill rates. Since the number of
parameters in the VAR system will increase rapidly as we lengthen the lag, there is some risk
of overfitting when a higher-order VAR is employed. The multivariate identification test on
the variables suggests that a second-order VAR system is appropriate for our sample.8
The results of VAR estimation are reported in Table 1. Note that although we specify the
VAR model in Section 2 as a first-order system, the second-order VAR can always be
rearranged in the first-order form (Sargent, 1979). The asymptotic heteroskedasticityconsistent standard errors are reported in the parentheses. Since the results are quite similar
for equally weighted and value-weighted indices, we only report the result for value-weighted
returns. Table 1 shows that inflation rates, yield spreads, real interest rates and dividend yields
have significant first-order autoregressive coefficients. On the other hand, market returns
behave much like white noise.
3.2. Excess return cross-correlation
Table 2 reports the own and cross-correlation matrices for the stock portfolios in the
three smallest and the three largest size deciles. The concurrent correlations among
portfolio returns are all significant. Similar to the finding of Lo and MacKinlay (1990a),
cross (serial)-correlations are asymmetric with returns of larger stock portfolios leading
those of smaller stocks. Similar patterns appear in higher-order autocorrelation matrices,
although the magnitude of cross-correlation is considerably smaller and less significant.
3.3. Return cross-covariance between large and small stocks
We next analyze the sources of the return cross-covariance. As shown earlier, the pattern of
cross-correlations may depend on the order structure. Although both economic and firm7

See Dickey and Fuller (1979) and Fuller (1976) for the details of the unit-root test procedure. Our finding is
similar to Campbell and Ammer (1993). The results of unit root tests are available upon request.
8
The order of the VAR system is determined by the Akaike Information Criterion (AIC, Akaike, 1973).

29

C.-H. Yu, C. Wu / International Review of Economics and Finance 10 (2001) 19±40
Table 1
Summary of VAR coefficient estimates
em,t ÿ 1
rt ÿ 1
pt ÿ 1
yt ÿ 1
dt ÿ 1
em,t ÿ 2
rt ÿ 2
pt ÿ 2
yt ÿ 2
dt ÿ 2

em,t

rt

pt

yt

dt

0.0270
(0.0442)
ÿ 1.4658
(3.8874)
0.5004
(4.3194)
ÿ 14.3555
(11.8491)
0.9125
(0.9863)

0.0001
(0.0003)
0.4236*
(0.0628)
ÿ 0.0675
(0.0620)
ÿ 0.0304
(0.1739)
ÿ 0.0134
(0.0132)

ÿ 0.0001
(0.0003)
ÿ 0.0238
(0.0654)
0.4879*
(0.0632)
ÿ 0.1798
(0.1720)
0.0084
(0.0118)

ÿ 0.00007
(0.00004)
ÿ 0.0726*
(0.0098)
ÿ 0.0682*
(0.0107)
0.5738*
(0.0321)
ÿ 0.0044*
(0.0020)

0.0004
(0.0006)
0.0264
(0.1358)
ÿ 0.4289*
(0.1467)
ÿ 0.0932
(0.3791)
0.1802*
(0.0279)

ÿ 0.0176
(0.0309)
2.8859
(3.8743)
ÿ 1.5066
(4.3633)
18.7246
(11.8441)
ÿ 1.3462
(1.1413)

0.0002
(0.0003)
0.0931
(0.0634)
0.0851
(0.0627)
0.0616
(0.1788)
0.0088
(0.0150)

ÿ 0.0001
(0.0003)
0.0965
(0.0650)
0.0893
(0.0633)
0.1594
(0.1757)
ÿ 0.0126
(0.0144)

0.00007
(0.00005)
0.0707*
(0.0096)
0.0732*
(0.0105)
0.0142
(0.0325)
0.0052*
(0.0021)

ÿ 0.0004
(0.0005)
0.1318
(0.1344)
0.2731
(0.1481)
0.3938
(0.3779)
ÿ 0.0084
(0.0328)

This table reports coefficient estimates for a VAR(2) of the weekly economic variables:
~
Xt ˆ l1 Xtÿ1 ‡ l2 Xtÿ2 ‡ X t
where Xt is the vector of the economic factors at time t, l1 and l2 are the lagged one and two coefficient matrices,
Ä t is the vector of factor innovations. There are five economic variables included, where em,t is
respectively, and X
the real excess market index return; rt is the real 3-month T-bill rate; pt is the inflation rate; yt is the 10-year and 3month yield spreads; and dt is the real market dividend yield.
The asymptotic standard errors are in parentheses.
* Significance at the 5% level.

specific factors play a significant role in the first-order cross-correlation, firm-specific return
innovations may be relatively important in the higher-order cross-correlations. We first
discuss the results for the first-order and then for higher-order cross-covariances. Most of our
results are based on weekly returns. However, the lead±lag relationship may also stem from
innovations in fairly slow-moving macro factors. To assess this potential effect, we also report
results of major empirical tests for monthly returns.
3.3.1. Results of the first-order cross-covariance
Table 3a reports the relative contributions of economic and firm-specific factors to the
asymmetry of the first-order return cross-covariance between large- and small-stock portfolios. The standard errors of the estimates are reported in parentheses.9 The figure in the first
row minus that in the second row represents the net contribution from economic factors. On
9

See footnote 4 for the estimation of standard errors.

30

C.-H. Yu, C. Wu / International Review of Economics and Finance 10 (2001) 19±40

Table 2
Autocorrelations and cross-correlations of weekly portfolio returns
R1t

R2t

R3t

R8t

R9t

R10t

R1t
R2t
R3t
R8t
R9t
R10t

1.00*
0.83*
0.82*
0.65*
0.60*
0.57*

1.00*
0.92*
0.80*
0.76*
0.73*

1.00*
0.86*
0.82*
0.79*

1.00*
0.97*
0.95*

1.00*
0.97*

R1t ÿ 1
R2t ÿ 1
R3t ÿ 1
R8t ÿ 1
R9t ÿ 1
R10t ÿ 1

0.31*
0.35*
0.31*
0.26*
0.25*
0.22*

0.24*
0.27*
0.28*
0.28*
0.27*
0.25*

0.18*
0.23*
0.22*
0.24*
0.24*
0.22*

0.02
0.06
0.07
0.10*
0.11*
0.10*

0.03
0.00
0.02
0.05
0.05
0.05

ÿ 0.05
ÿ 0.02
0.00
0.02
0.02
0.02

R1t ÿ 2
R2t ÿ 2
R3t ÿ 2
R8t ÿ 2
R9t ÿ 2
R10t ÿ 2

0.17*
0.15*
0.14*
0.11*
0.09*
0.09*

0.08
0.06
0.06
0.05
0.05
0.05

0.06
0.03
0.04
0.03
0.02
0.03

0.00
ÿ 0.02
ÿ 0.02
ÿ 0.03
ÿ 0.03
ÿ 0.01

0.00
ÿ 0.02
ÿ 0.02
ÿ 0.03
ÿ 0.04
ÿ 0.02

0.00
ÿ 0.02
ÿ 0.02
ÿ 0.03
ÿ 0.02
ÿ 0.02

R1t ÿ 3
R2t ÿ 3
R3t ÿ 3
R8t ÿ 3
R9t ÿ 3
R10t ÿ 3

0.16*
0.14*
0.14*
0.12*
0.11*
0.11*

0.12*
0.08*
0.09*
0.07
0.06
0.05

0.10*
0.09*
0.08*
0.07
0.06
0.06

0.04
0.04
0.05
0.04
0.04
0.03

0.05
0.04
0.06
0.05
0.04
0.04

0.04
0.05
0.05
0.03
0.03
0.03

R1t ÿ 4
R2t ÿ 4
R3t ÿ 4
R8t ÿ 4
R9t ÿ 4
R10t ÿ 4

0.10*
0.07
0.08*
0.09*
0.08*
0.07

0.09*
0.08*
0.08*
0.08*
0.08*
0.06

0.07
0.03
0.03
0.03
0.02
0.01

ÿ 0.04
ÿ 0.06
ÿ 0.06
ÿ 0.05
ÿ 0.05
ÿ 0.05

ÿ 0.04
ÿ 0.06
ÿ 0.07
ÿ 0.05
ÿ 0.05
ÿ 0.05

ÿ 0.05
ÿ 0.07
ÿ 0.07
ÿ 0.07
ÿ 0.06
ÿ 0.06

R1t ÿ 5
R2t ÿ 5
R3t ÿ 5
R8t ÿ 5
R9t ÿ 5
R10t ÿ 5

0.08*
0.10*
0.09*
0.06
0.05
0.04

0.03
0.06
0.05
0.02
0.02
0.01

0.03
0.06
0.04
0.01
0.01
0.00

0.00
0.00
ÿ 0.01
ÿ 0.04
ÿ 0.05
ÿ 0.06

0.00
0.00
ÿ 0.01
ÿ 0.04
ÿ 0.04
ÿ 0.06

0.01
0.01
0.00
ÿ 0.04
ÿ 0.04
ÿ 0.05

R1t ÿ 6
R2t ÿ 6
R3t ÿ 6
R8t ÿ 6

0.11*
0.12*
0.11*
0.08*

0.09
0.08*
0.09*
0.07

0.09*
0.08*
0.10*
0.07

0.04
0.05
0.05
0.04

1.00*

0.06
0.03
0.05
0.04
0.06
0.04
0.04
0.03
(continued on next page)

31

C.-H. Yu, C. Wu / International Review of Economics and Finance 10 (2001) 19±40
Table 2 (continued)
R1t

R2t

R3t

R8t

R9t

R10t

R9t ÿ 6
R10t ÿ 6

0.08*
0.07

0.07
0.06

0.07
0.06

0.04
0.03

0.04
0.03

0.03
0.02

R1t ÿ 7
R2t ÿ 7
R3t ÿ 7
R8t ÿ 7
R9t ÿ 7
R10t ÿ 7

0.08*
0.09*
0.11*
0.11*
0.11*
0.11*

0.06
0.06
0.08*
0.09*
0.09*
0.10*

0.03
0.03
0.06
0.09*
0.09*
0.09*

0.00
0.00
0.04
0.07
0.07
0.09*

0.01
0.02
0.05
0.08*
0.08*
0.09*

0.01
0.01
0.04
0.08*
0.08*
0.10*

R1t ÿ 8
R2t ÿ 8
R3t ÿ 8
R8t ÿ 8
R9t ÿ 8
R10t ÿ 8

0.03
0.02
ÿ 0.01
0.01
0.00
ÿ 0.01

ÿ 0.02
ÿ 0.02
ÿ 0.05
ÿ 0.03
ÿ 0.04
ÿ 0.05

ÿ 0.02
ÿ 0.02
ÿ 0.05
ÿ 0.04
ÿ 0.04
ÿ 0.04

ÿ 0.06
ÿ 0.05
ÿ 0.07
ÿ 0.06
ÿ 0.05
ÿ 0.04

ÿ 0.06
ÿ 0.05
ÿ 0.07
ÿ 0.06
ÿ 0.05
ÿ 0.05

ÿ 0.07
ÿ 0.05
ÿ 0.07
ÿ 0.06
ÿ 0.04
ÿ 0.04

Autocorrelation and cross-correlation matrices of the weekly portfolio return vector [R1t, R2t, R3t, R8t, R9t, R10t]0
where Rjt is the weekly return on the stock portfolio in the jth decile, j = 1, 2, 3, 8, 9, 10 (decile 1 represents the stocks
of the smallest market values and decile 10 includes the stocks of the largest market values). Each size-related
portfolio contains 116 stocks over the period from January 2, 1981 to December 31, 1992 (626 observations).
The asymptotic standard errors for the autocorrelations and cross-correlations under an i.i.d. null hypothesis are
p
given by 1= n ˆ 0:03997.
* Significance at the 5% level.

the other hand, the figure in the third row less that in the fourth is the net contribution from
the firm-specific return innovations. Results show that economic factors play a more
important role in explaining the first-order asymmetric return cross-correlation. The crosscovariance attributed to the response of returns to economic factors is significant. The first
column of Table 3a shows that results for the weekly returns. As shown, the contribution from
Ä t,eÄl ) ÿ al 0 Cov(X
Ä t,eÄs ), is 0.5057, whereas that from
economic factors, measured by as0 Cov(X
t
t
firm-specific return innovations, Cov(eÄlt,eÄst + 1) ÿ Cov(eÄlt + 1,eÄst), is equal to 0.4473. The last
row of the first column shows the sum of the effects of economic and firm-specific factors.
Note that the figures in the table are standardized by the cross-covariance term, Cov(elt,
est + 1) ÿ Cov(elt + 1,est). The fact that the total effect is close to one suggests that the assumption
regarding the independence between factor innovations and the idiosyncratic term is
reasonable for weekly returns.
We also estimate the contributions of the economic and firm-specific factors to the crosscovariance between large- and small-stock returns using the monthly data. The results are
reported in the second column of Table 3a. The results show that the contribution of the
economic factors to the lead±lag relation of stock returns tends to be larger as the return
horizon is increased. These results may reflect the nature of the slow-moving macro factors.
Another important concern is whether the results are stable over time. In particular, there
might be a possibility that the results are contingent on the overall market performance.
Recently, McQueen et al. (1996) have found that returns on small stocks are very sensitive to
down-market movements but change only slowly in response to up-market movements. To

32

C.-H. Yu, C. Wu / International Review of Economics and Finance 10 (2001) 19±40

Table 3
(a) The role of economic factors and firm-specific return innovations in the first-order cross-covariance between
large and small stocks
Ä t,eÄl )
as0Cov(X
t
Ä
al0Cov(Xt,eÄst)
Cov(eÄlt,eÄst + 1)
Cov(eÄlt + 1,eÄst)
Total

Weekly returns

Monthly returns

0.5245* (0.1844)
0.0188 (0.0917)
0.2984 (0.1957)
ÿ 0.1489 (0.0895)
0.9531* (0.0291)

0.8970* (0.3231)
0.1094 (0.2578)
0.4385 (0.3249)
0.4179 (0.2565)
0.8083* (0.1456)

(b) The first-order cross-covariance between large and small stocks: subperiod analysis
Ä t,eÄl )
as0Cov(X
t
Ä t,eÄs )
al0Cov(X
t
Cov(eÄlt,eÄst + 1)
Cov(eÄlt + 1,eÄst)
Total

1982 ± 1989 and 1991 ± 1992

1981 and 1990

0.5526* (0.1684)
0.0813 (0.1227)
0.2145 (0.2018)
ÿ 0.2388 (0.1596)
0.9255* (0.0168)

0.6125* (0.2546)
0.1034 (0.0658)
0.2642 (0.1857)
ÿ 0.1865 (0.1294)
0.9598* (0.0315)

The difference in the first-order cross-covariance between large- and small-firm stocks is:
~
~
Cov…elt ; est‡1 † ÿ Cov…elt‡1 ; est † ˆ ‰as 0 Cov…X t ; e~ lt † ÿ al 0 Cov…X t ; e~ st †Š ‡ ‰Cov…e~ lt ; e~ st‡1 † ÿ Cov…e~ lt‡1 ; e~ st †Š;
where elt and est are excess returns of large and small stocks, and eÄlt and eÄst are the innovations in the real excess
Ä t is the 5  1 vector of innovations of economic factors, and al and
returns of large and small stocks, respectively; X
as are, respectively, the regression coefficients of elt and elt on the lagged state variables, Xt ÿ 1 which include the
real excess market index return, the real 3-month T-bill rate, the inflation rate, the 10-year and 3-month yield
spread and the real market dividend yield.
Ä t,eÄl ) ÿ al 0 Cov(X
Ä t,eÄs )], represents the contribution of the economic factors to
The first component, [as0 Cov(X
t
t
the asymmetric cross-correlation, while the second component, [Cov(eÄlt,eÄst + 1) ÿ Cov(eÄlt + 1,eÄst)], represents the
contribution of firm-specific return innovations. The last row in the table is the total effect, which is
~ t ; e~st †Š ‡ ‰Cov…~
~ t ; e~lt † ÿ a10 Cov…X
‰as0 Cov…X
elt ; e~st‡1 † ÿ Cov…~
elt‡1 ; e~st †Š. Each component is divided by Cov(elt,
est + 1) ÿ Cov(elt + 1,est).
The asymptotic standard errors are reported in the parentheses.
* Significance at the 5% level.

address the issues of temporal stability and the directional asymmetry of small stocks'
response to bull and bear market, we divide the whole sample period into several periods: the
bull periods in 1982±1989 and 1991±1992, and the bear periods in 1981 and 1990. The bull
period includes the years with positive annual returns while the bear period includes the years
with negative annual returns.
The results for the bull and bear periods are reported in Table 3b. The overall results are
remarkably similar for both periods. For the bull period, the proportion of the crosscovariance attributed to the response of returns to economic factors is 0.4713 whereas the
contribution of firm-specific return innovations is 0.4542. For the bear periods, the
corresponding numbers are 0.5091 and 0.4507, respectively. The results indicate the average
proportions of the lead±lag return relation explained by economic and firm-specific factors
Ä t,eÄl )
are quite stable over time. On the other hand, there is an indication that the term as0Cov(X
t
in the bear periods (0.6125) is higher than that in the bull periods (0.5526). This phenomenon
is consistent with the finding of McQueen et al. (1996) that small stocks are more sensitive to

C.-H. Yu, C. Wu / International Review of Economics and Finance 10 (2001) 19±40

33

Table 4
The covariation between innovations of stock excess returns and economic factors
Ä 1t)
Cov(eÄlt,X
Ä 2t)
Cov(eÄlt,X
Ä 3t)
Cov(eÄlt,X
Ä 4t)
Cov(eÄlt,X
Ä 5t)
Cov(eÄlt,X
Ä 1t)
Cov(eÄst,X
Ä 2t)
Cov(eÄst,X
Ä 3t)
Cov(eÄst,X
Ä 4t)
Cov(eÄst,X
Ä 5t)
Cov(eÄst X

Weekly returns

Monthly returns

1.0376*
2.4730*
ÿ 4.3006*
ÿ 18.1884*
2.4627*

(0.0053)
(0.6792)
(0.7305)
(4.8565)
(0.2784)

1.0107* (0.0042)
2.2796* (0.3662)
ÿ 3.6556* (0.3735)
ÿ 2.3548 (3.3291)
3.9991* (0.4099)

0.7023* (0.0117)
0.1981 (0.6685)
ÿ 1.6892* (0.7340)
ÿ 4.2655 (5.7718)
1.7797* (0.2777)

0.9310* (0.0163)
2.9502* (0.5711)
ÿ 4.2933* (0.6492)
ÿ 0.2546 (5.5365)
3.9097* (0.5531)

This table reports the covariance between innovations of stock excess returns and each factor innovation. eÄlt and
Ä kt (k = 1, . . .,5) are the
eÄst are, respectively, the real excess return innovations of large and small stocks at time t, and X
innovations of real market returns, real short-term interest rates, inflation rates, long ± short yield spreads and real
market dividend yields, respectively. The numbers shown in the table are the covariances divided by the variances
Ä jt)/Var(X
Ä jt), where i = l (large firms) and s (small firms), j = 1, . . ., 5.
of the corresponding factors, i.e., Cov(eÄit,X
The asymptotic standard errors are reported in the parentheses.
* Significance at the 5% level.

market factors in the down-market. Their finding implies that the sensitivity coefficient (as) of
small-stock returns to market factors is larger in the down-market which, in turn, suggests that
Ä t,eÄl ) should have a larger value in the bear market. Our results appear to
the term as0Cov(X
t
support this contention.
We next examine the sensitivity of stock return innovations to each economic factor in
order to assess its contribution to the return cross-correlation. Table 4 reports the covariation
of return innovations to each concurrent factor innovation. Since the results for valueweighted and equally weighted indices are quite similar, for brevity, we only report the results
of the value-weighted market index in the remaining analysis. The upper and lower panels
report the results for the largest- and the smallest-stock portfolios, respectively. The
covariances of stock return with market returns, inflation rates and market dividend yields
are all significant for both the largest- and smallest-stock portfolios. Also, the large-stock
return innovation has a larger covariation with the concurrent market return innovation. For
example, the covariance between large-stock and market return innovations is 1.0376, while
the covariance between small-stock and market return innovations is 0.7023. The difference
(0.3353) is significant at the 1% level with the standard error equal to 0.0146.
We also estimate the covariation of return innovations to concurrent economic factor
innovations using monthly returns. The general pattern of covariance terms is quite similar to
that of weekly intervals. For large stocks, all covariance terms remain significant and of the
same sign as those of weekly returns. Except for the long±short yield spread (X4), the values
of covariance terms are fairly close to those of weekly returns. The differences between
weekly and monthly results are relatively larger for small stocks. Still, the signs of the
covariance terms are very consistent for both time intervals. Notice that the covariance
between small stock and market returns increases for monthly returns. For instance, the

34

C.-H. Yu, C. Wu / International Review of Economics and Finance 10 (2001) 19±40

Table 5
The role of economic factors and return innovations in higher-order cross-covariances between large and small
stocks
q
Ä t + q ÿ 1,eÄl )
as0 Cov(X
t
Ä t + q ÿ 1,eÄs )
al 0 Cov(X
t
Cov(eÄlt,eÄst + q)
Cov(eÄlt + q,eÄst)
Total

2

3

7

0.0736 (0.0663)
ÿ 0.0267 (0.0340)
0.8206* (0.0917)
ÿ 0.0577 (0.0338)
0.9786* (0.1146)

0.0179 (0.0833)
ÿ 0.0192 (0.0307)
1.7590* (0.0676)
0.7830* (0.0819)
1.0131* (0.1648)

0.0856 (0.0442)
0.0255 (0.0203)
0.9056* (0.0410)
0.0361 (0.0361)
0.9295* (0.0402)

The difference in higher-order return cross-covariances between large and small firm stocks is:
~
~
Cov…elt ; est‡q † ÿ Cov…elt‡q ; est † ˆ ‰as 0 Cov…X t‡qÿ1 ; e~ lt † ÿ al 0 Cov…X t‡qÿ1 ; e~ st †Š
‡ ‰Cov…e~ lt ; e~ st‡q † ÿ Cov…e~ lt‡q ; e~ st †Š;
where q = 2, 3 and 7, are the lagged orders at which return cross-correlations are significant; elt and est are real
excess returns of large and small stocks, and eÄlt and eÄst are the innovations in the real excess returns of large and
Ä t is the 5  1 vector of factor innovations, and al and as are, respectively, the
small stocks, respectively; X
regression coefficients of elt and est on the lagged factors, Xt ÿ 1 which include the real excess market return, the
real 3-month T-bill rate, the inflation rate, the 10-year and 3-month treasury yield spread and the real market
dividend yields.
Ä t + q ÿ 1,eÄl ) ÿ al0 Cov(X
Ä t + q ÿ 1,eÄs ], represents the contribution of economic
The first component, [as0 Cov(X
t
t
factors to the return lead ± lag effect, while the second part, [Cov(eÄlt,eÄst + q) ÿ Cov(eÄlxt + q,eÄst)], represents the
contribution of return innovations. The last row in the table is the total effect, which is
~ t ; e~st †Š ‡ ‰Cov…~
~ t ; e~lt † ÿ al0 Cov…X
‰as0 Cov…X
elt ; e~st‡1 † ÿ Cov…~
elt‡1 ; e~st †Š. Each component is divided by Cov(elt,
est + 1) ÿ Cov(elt + 1,est).
The asymptotic standard errors are reported in the parentheses.
* Significance at the 5% level.

covariance between small stock and market returns is 0.9310, which is only slightly smaller
than the covariance between large stock and market returns (1.0107).
The sign of the covariance terms determines the sign of the B coefficients in Eq. (1a),
which, in turn, affects the sign of A coefficients in Eq. (6) as well as the value of risk
premium. The results in Table 4 indicate that both B and A coefficients can be negative.
Therefore, risk premiums can be negative even if the economic factors are positive. Over
time, the values of economic factors will change and may become negative in certain periods.
Thus, risk premiums will be time-varying and may turn negative for some periods.
3.3.2. Results of higher-order cross-covariance
As shown in Eq. (13), the covariance between firm-specific return innovations may be
more responsible for higher-order return cross-covariances. Table 5 shows the relative
importance of firm-specific factors and economic factors in higher return cross-covariances.
In the interest of brevity, we only report results of weekly returns for selective orders ( q) at
which the return cross-correlations are found to be more prevalent. In contrast to the result in
Table 3a, the contribution from economic factors decreases dramatically, whereas the
covariances of firm-specific