Economic sources of asymmetric cross-correlation among
stock returns
Chih-Hsien Yu
a, Chunchi Wu
b*
a
National Chung-Cheng University, Chia-Yi, Taiwan b
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: [email protected] (C. Wu).
10 (2001) 19 ± 40
1059-0560/01/$ ± see front matterD2001 Elsevier Science Inc. All rights reserved.
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
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 small-firm 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.
1
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 cross-correlation among stock returns.
In this paper, we propose a framework to distinguish the effects of economic from firm-specific 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 correlations among returns of size-sorted portfolios. We show that return cross-correlation 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 cross-covariance of returns is decomposed into aggregate economic and firm-specific compo-nents. Section 3 discusses data and empirical results. Finally, Section 4 summarizes our major findings.
2. The model
We posit that the excess returns onNassets obey the following dynamic process:
et Etÿ1 et B XtÿEtÿ1 Xt et; 1
where: et= an N1 vector of excess returns on assets, i.e., asset returns less short-term
interest rates;Xt= a K1 vector of state factors; B = anNK matrix of factor sensitivities;
et= an N1 vector of asset-specific (idiosycratic) error terms; and Etÿ1= an expectational
operator conditional on information at timetÿ1.
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,
Cov(XÄt,et) = 0 and Cov(XÄt+j,et) = 0, for j60 where XÄt=XtÿEtÿ1(Xt) is the vector of
Eq. (1) can be rewritten as:
A, where l and s denote large- and small-stock portfolios,
respectively, and bl and bs are K1 vectors of sensitivities to the concurrent factor
innovations. Then, excess returns of large and small stocks are:
elt Etÿ1 elt bl0X
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)60, and that
Cov(elt,est+ 1)ÿCov(eltÿ1,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:
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 BFtÿ1; 4
whereFtÿ1is a K1 vector of risk prices at timetÿ1. The information about state factors
Xtÿ1is supposedly available at timetÿ1. 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:
Ftÿ1CXtÿ1; 5
whereCis a matrix of coefficients which define market price of risk (see also Campbell &
Hamao, 1992). Substituting Eq. (5) into Eq. (4) forFtÿ1, we have:
A, andalandasare K1 vectors of the sensitivity coefficients of expected
returns of large and small stocks to lagged economic factors. More specifically, the expected excess returns of large and small stocks are:
Etÿ1 elt al0Xtÿ1 6a
Eqs. (6), (6a) and (6b) imply that the risk premiums are time-varying and will be negative if
either Aor Xtÿ1 is negative. Substituting Eqs. (6a) and (6b) into Eq. (3) for the expected
returns, we have:
The above relationship involves the stochastic process of state factors. Following Camp-bell and Ammer (1993), we characterize the behavior of state factors by a vector autoregressive (VAR) process (Eq. (8)):
Xt Xtÿ1X ~
t; 8
where XÄt is the innovation in Xt, and the matrix is the coefficient matrix of the VAR
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.
of large- and small-stock returns to lagged economic factors, al and as, and the
autocovariance and cross-covariance of factors. The second term, as0Cov(XÄt,eÄl
t), includes
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
large-firm stocks. If Cov(XÄt,eÄlt) is positive, then large-firm stocks experience unexpected
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 timet) as:
Cov elt1;est al
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;est1 ÿCov elt1;est as
of economic factors to the asymmetric cross-correlation between large and small stocks.
The second term, Cov(eÄlt eÄst+ 1)ÿCov(eÄlt+ 1,eÄst), represents the contribution of the
covaria-tion between firm-specific return components (of large and small stocks) to the asym-metric cross-correlation.
The effect of economic factors can be separated into two parts: the concurrent relationships
between factor innovations and return innovations, Cov(XÄt,eÄlt) and Cov(XÄt,eÄst), and the
sensitivity of stock returns to lagged economic factors,as0andal0. 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 cross-correlation. 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
small-stock returns are more sensitive to lagged economic factors than large-small-stock returns, i.e.,as0is
larger than al0. Thus, the effect of Cov(XÄt,eÄl
t) and Cov(X
Ä
t,eÄst) on the asymmetry of return
cross-correlation would be reinforced by the differences in sensitivities of large- and small-stock 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
(q2), we have:
where XÄt+qÿ1 are the innovations of future economic factors. Since factor innovations are
uncorrelated with idiosyncratic terms, Cov(XÄt+qÿ1,eÄlt) and Cov(XÄt+qÿ1,eÄst) are both zero.
which suggests that a higher-order correlation is mainly caused by the cross-correlation between return innovations of large and small stocks. Furthermore, if [as0Cov(XÄt,eÄlt)ÿal
0Cov(XÄ
t,eÄst)] in Eq. (11) is positive, then the magnitude of the
first-order return covariance would be greater than that of higher-first-order return
cross-3
The fact that Cov(elt,est + 1) is not equal to zero does not rule out the own autocorrelation effect. It can be
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
cross-correlation, we further divide the covariance effect intoK(factor) terms, each representing the
contribution of an economic variable:
economic variable, we can test the significance of its contribution to the asymmetric
cross-correlation of returns.4
2.2. Decomposition of the firm-specific effect
To provide more information about the effects of firm-specific factors on return cross-correlation, 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
convergence in distribution, then the asymptotic distribution of Cov(d^) can be shown as:
T p
Cov d^ ÿCov d !dN0; @Cov d=@d0 @Cov d0=@d, where (@Cov(d)/@d0) is the Jacobian vector of covariance evaluated atd^. Each term in Eq. (14) can be seen as a function of parameters, i.e., Cov(d) above, and its asymptotic standard error is simply 1=
T p
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 parsimo-nious 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 Camp-bell±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)):
et1ht1ÿrt1: 15
LeteÄt+ 1be the innovation in excess return,et+ 1ÿEtet+ 1, whereEtis an expectation formed
at the end of periodt. Then, following Campbell and Ammer (1993), we have (Eq. (16)):5
e
whereDdenotes a one-period backward difference andris a constant discount factor. Or, in
more compact notations (Eq. (17)),
e
~
t1e~d;t1ÿe~r;t1ÿe~e;t1; 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)):
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;tq;e~l;t Cov e~ds;tq;e~l;t ÿCov e~es;tq;e~l;t ÿCov e~r;tq;e~l;t; 20
or
Cov e~
s;tq;e~l;t Cov e~s;tq;e~dl;t ÿCov e~s;tq;e~el;t
ÿCov e~s;tq;e~r;t; q1:
21
Each term on the right-hand side of Eqs. (20) or (21) measures the contribution of a return
component to theqth-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 heteroske-dasticity-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
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 heteroskedasticity-consistent 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
firm-7
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
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
portfo-lios. The standard errors of the estimates are reported in parentheses.9The figure in the first
row minus that in the second row represents the net contribution from economic factors. On
Table 1 This table reports coefficient estimates for a VAR(2) of the weekly economic variables:
Xtl1Xtÿ1l2Xtÿ2X~t
whereXtis the vector of the economic factors at timet,l1andl2are the lagged one and two coefficient matrices, respectively, and XÄtis the vector of factor innovations. There are five economic variables included, whereem,tis
the real excess market index return;rtis the real 3-month T-bill rate;ptis the inflation rate;ytis the 10-year and
3-month yield spreads; anddtis the real market dividend yield.
The asymptotic standard errors are in parentheses. * Significance at the 5% level.
9
(continued on next page) Table 2
Autocorrelations and cross-correlations of weekly portfolio returns
R1t R2t R3t R8t R9t R10t
R1t 1.00*
R2t 0.83* 1.00*
R3t 0.82* 0.92* 1.00*
R8t 0.65* 0.80* 0.86* 1.00*
R9t 0.60* 0.76* 0.82* 0.97* 1.00*
R10t 0.57* 0.73* 0.79* 0.95* 0.97* 1.00*
R1tÿ1 0.31* 0.24* 0.18* 0.02 0.03 ÿ0.05
R2tÿ1 0.35* 0.27* 0.23* 0.06 0.00 ÿ0.02
R3tÿ1 0.31* 0.28* 0.22* 0.07 0.02 0.00
R8tÿ1 0.26* 0.28* 0.24* 0.10* 0.05 0.02
R9tÿ1 0.25* 0.27* 0.24* 0.11* 0.05 0.02
R10tÿ1 0.22* 0.25* 0.22* 0.10* 0.05 0.02
R1tÿ2 0.17* 0.08 0.06 0.00 0.00 0.00
R2tÿ2 0.15* 0.06 0.03 ÿ0.02 ÿ0.02 ÿ0.02
R3tÿ2 0.14* 0.06 0.04 ÿ0.02 ÿ0.02 ÿ0.02
R8tÿ2 0.11* 0.05 0.03 ÿ0.03 ÿ0.03 ÿ0.03
R9tÿ2 0.09* 0.05 0.02 ÿ0.03 ÿ0.04 ÿ0.02
R10tÿ2 0.09* 0.05 0.03 ÿ0.01 ÿ0.02 ÿ0.02
R1tÿ3 0.16* 0.12* 0.10* 0.04 0.05 0.04
R2tÿ3 0.14* 0.08* 0.09* 0.04 0.04 0.05
R3tÿ3 0.14* 0.09* 0.08* 0.05 0.06 0.05
R8tÿ3 0.12* 0.07 0.07 0.04 0.05 0.03
R9tÿ3 0.11* 0.06 0.06 0.04 0.04 0.03
R10tÿ3 0.11* 0.05 0.06 0.03 0.04 0.03
R1tÿ4 0.10* 0.09* 0.07 ÿ0.04 ÿ0.04 ÿ0.05
R2tÿ4 0.07 0.08* 0.03 ÿ0.06 ÿ0.06 ÿ0.07
R3tÿ4 0.08* 0.08* 0.03 ÿ0.06 ÿ0.07 ÿ0.07
R8tÿ4 0.09* 0.08* 0.03 ÿ0.05 ÿ0.05 ÿ0.07
R9tÿ4 0.08* 0.08* 0.02 ÿ0.05 ÿ0.05 ÿ0.06
R10tÿ4 0.07 0.06 0.01 ÿ0.05 ÿ0.05 ÿ0.06
R1tÿ5 0.08* 0.03 0.03 0.00 0.00 0.01
R2tÿ5 0.10* 0.06 0.06 0.00 0.00 0.01
R3tÿ5 0.09* 0.05 0.04 ÿ0.01 ÿ0.01 0.00
R8tÿ5 0.06 0.02 0.01 ÿ0.04 ÿ0.04 ÿ0.04
R9tÿ5 0.05 0.02 0.01 ÿ0.05 ÿ0.04 ÿ0.04
R10tÿ5 0.04 0.01 0.00 ÿ0.06 ÿ0.06 ÿ0.05
R1tÿ6 0.11* 0.09 0.09* 0.04 0.06 0.03
R2tÿ6 0.12* 0.08* 0.08* 0.05 0.05 0.04
R3tÿ6 0.11* 0.09* 0.10* 0.05 0.06 0.04
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 correlation. The cross-covariance 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
economic factors, measured byas0Cov(XÄt,eÄlt)ÿal
0Cov(XÄ
t,eÄst), is 0.5057, whereas that from
firm-specific return innovations, Cov(eÄlt,eÄst+ 1)ÿCov(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)ÿCov(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 cross-covariance 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
R1t R2t R3t R8t R9t R10t
R9tÿ6 0.08* 0.07 0.07 0.04 0.04 0.03
R10tÿ6 0.07 0.06 0.06 0.03 0.03 0.02
R1tÿ7 0.08* 0.06 0.03 0.00 0.01 0.01
R2tÿ7 0.09* 0.06 0.03 0.00 0.02 0.01
R3tÿ7 0.11* 0.08* 0.06 0.04 0.05 0.04
R8tÿ7 0.11* 0.09* 0.09* 0.07 0.08* 0.08*
R9tÿ7 0.11* 0.09* 0.09* 0.07 0.08* 0.08*
R10tÿ7 0.11* 0.10* 0.09* 0.09* 0.09* 0.10*
R1tÿ8 0.03 ÿ0.02 ÿ0.02 ÿ0.06 ÿ0.06 ÿ0.07
R2tÿ8 0.02 ÿ0.02 ÿ0.02 ÿ0.05 ÿ0.05 ÿ0.05
R3tÿ8 ÿ0.01 ÿ0.05 ÿ0.05 ÿ0.07 ÿ0.07 ÿ0.07
R8tÿ8 0.01 ÿ0.03 ÿ0.04 ÿ0.06 ÿ0.06 ÿ0.06
R9tÿ8 0.00 ÿ0.04 ÿ0.04 ÿ0.05 ÿ0.05 ÿ0.04
R10tÿ8 ÿ0.01 ÿ0.05 ÿ0.04 ÿ0.04 ÿ0.05 ÿ0.04
Autocorrelation and cross-correlation matrices of the weekly portfolio return vector [R1t,R2t,R3t,R8t,R9t,R10t]0
whereRjtis the weekly return on the stock portfolio in thejth 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 given by 1=
n p
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 cross-covariance 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
are quite stable over time. On the other hand, there is an indication that the termas0Cov(XÄt,eÄl
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
Table 3
(a) The role of economic factors and firm-specific return innovations in the first-order cross-covariance between large and small stocks
(b) The first-order cross-covariance between large and small stocks: subperiod analysis
1982 ± 1989 and 1991 ± 1992 1981 and 1990
as0Cov(XÄt,eÄlt) 0.5526* (0.1684) 0.6125* (0.2546)
al0Cov(XÄt,eÄst) 0.0813 (0.1227) 0.1034 (0.0658)
Cov(eÄlt,eÄst+ 1) 0.2145 (0.2018) 0.2642 (0.1857)
Cov(eÄlt+ 1,eÄst) ÿ0.2388 (0.1596) ÿ0.1865 (0.1294)
Total 0.9255* (0.0168) 0.9598* (0.0315)
The difference in the first-order cross-covariance between large- and small-firm stocks is:
Cov elt;est1 ÿCov elt1;est as0Cov X
whereeltandestare excess returns of large and small stocks, andeÄltandeÄstare the innovations in the real excess
returns of large and small stocks, respectively; XÄtis the 51 vector of innovations of economic factors, andaland asare, respectively, the regression coefficients ofeltandelton the lagged state variables,Xtÿ1which 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.
The first component, [as0Cov(XÄt,eÄlt)ÿal 0Cov(XÄ
t,eÄst)], represents the contribution of the economic factors to
the asymmetric cross-correlation, while the second component, [Cov(eÄlt,eÄst+ 1)ÿCov(eÄlt+ 1,eÄst)], represents the
contribution of firm-specific return innovations. The last row in the table is the total effect, which is as0Cov X~t;~elt ÿa10Cov X~t;e~st Cov el~t;e~st1 ÿCov el~t1;~est. Each component is divided by Cov(elt,
est+ 1)ÿCov(elt+ 1,est).
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
the term as0Cov(XÄt,eÄlt) should have a larger value in the bear market. Our results appear to
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 value-weighted and equally value-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
Table 4
The covariation between innovations of stock excess returns and economic factors
Weekly returns Monthly returns
Cov(eÄlt,XÄ1t) 1.0376* (0.0053) 1.0107* (0.0042)
Cov(eÄlt,XÄ2t) 2.4730* (0.6792) 2.2796* (0.3662)
Cov(eÄlt,XÄ3t) ÿ4.3006* (0.7305) ÿ3.6556* (0.3735)
Cov(eÄlt,XÄ4t) ÿ18.1884* (4.8565) ÿ2.3548 (3.3291)
Cov(eÄlt,XÄ5t) 2.4627* (0.2784) 3.9991* (0.4099)
Cov(eÄst,XÄ1t) 0.7023* (0.0117) 0.9310* (0.0163)
Cov(eÄst,XÄ2t) 0.1981 (0.6685) 2.9502* (0.5711)
Cov(eÄst,XÄ3t) ÿ1.6892* (0.7340) ÿ4.2933* (0.6492)
Cov(eÄst,XÄ4t) ÿ4.2655 (5.7718) ÿ0.2546 (5.5365)
Cov(eÄstXÄ5t) 1.7797* (0.2777) 3.9097* (0.5531)
This table reports the covariance between innovations of stock excess returns and each factor innovation.eÄltand
eÄstare, respectively, the real excess return innovations of large and small stocks at timet, and XÄkt(k= 1,. . .,5) are the
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 of the corresponding factors, i.e., Cov(eÄit,XÄjt)/Var(XÄjt), wherei= l (large firms) and s (small firms),j= 1,. . ., 5.
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 return innovations become a major contributor to higher-order return cross-covariances. For example, for the first-order return cross-covariance in Table 3a, the contribution from economic factors is about 0.51. However, in Table 5, it decreases to
Table 5
The role of economic factors and return innovations in higher-order cross-covariances between large and small stocks
q
2 3 7
as0Cov(XÄt+qÿ1,eÄlt) 0.0736 (0.0663) 0.0179 (0.0833) 0.0856 (0.0442)
al0Cov(XÄt+qÿ1,eÄst) ÿ0.0267 (0.0340) ÿ0.0192 (0.0307) 0.0255 (0.0203)
Cov(eÄlt,eÄst+q) 0.8206* (0.0917) 1.7590* (0.0676) 0.9056* (0.0410)
Cov(eÄlt+q,eÄst) ÿ0.0577 (0.0338) 0.7830* (0.0819) 0.0361 (0.0361)
Total 0.9786* (0.1146) 1.0131* (0.1648) 0.9295* (0.0402) The difference in higher-order return cross-covariances between large and small firm stocks is:
Cov elt;estq ÿCov eltq;est as0Cov X
~
tqÿ1;e~lt ÿal0Cov X
~
tqÿ1;e~st
Cov e~lt;e
~
stq ÿCov e
~
ltq;e
~ st;
whereq= 2, 3 and 7, are the lagged orders at which return cross-correlations are significant;eltand estare real
excess returns of large and small stocks, andeÄltandeÄstare the innovations in the real excess returns of large and
small stocks, respectively; XÄt is the 51 vector of factor innovations, and al and as are, respectively, the regression coefficients ofeltandeston the lagged factors,Xtÿ1which 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.
The first component, [as0Cov(XÄt+qÿ1,eÄlt)ÿal 0Cov(XÄ
t+qÿ1,eÄst], represents the contribution of economic
factors to the return lead ± lag effect, while the second part, [Cov(eÄlt,eÄst+q)ÿCov(eÄlxt+q,eÄst)], represents the
contribution of return innovations. The last row in the table is the total effect, which is as0Cov X~t;~elt ÿal0Cov X~t;e~st Cov el~t;~est1 ÿCov el~t1;es~t. Each component is divided by Cov(elt,
est+ 1)ÿCov(elt+ 1,est).
0.09, 0.03 and 0.06 at lagged orders 2, 3 and 7, respectively and is statistically insignificant. On the other hand, the contribution from firm-specific return innovations rises from 0.45 in Table 3a, to 0.88, 0.98 and 0.87 in Table 5 at lagged orders 2, 3 and 7, respectively. These
results are generally consistent with the predictions of Eqs. (11) and (13).10
3.4. The roles of return components in the asymmetric return cross-covariance
We next decompose return innovations into components associated with cash flows, discount rates and expected future growth, and explore the role of each fundamental component in the determination of return cross-covariance. Fundamental return components
are estimated following the procedure suggested by Campbell and Ammer (1993).11In the
following, we first examine the role of each fundamental component in determining the covariances of stock returns to economic factors. Then, we report the relative importance of each fundamental component in the cross-covariance between stock return innovations.
3.4.1. Covariation of fundamental return components with economic factors
Table 6 shows the estimates of the covariances of each return component with factor innovations for both the largest and the smallest size portfolios. Recall that the return cross-covariance in Eq. (11) is affected by the concurrent cross-covariance of return innovations with factor innovations. A higher covariance between a particular fundamental return component with factor innovations implies a more important role for that component in the determination of return cross-covariance.
The results of weekly returns in the first panel of Table 6 show that unexpected increases in the market return and ex post real interest rate are associated with upward revisions in expected future cash flows of stocks. Positive innovations in the market dividend yield also increase expected future cash flows. On the other hand, both increases in inflation and yield spread are associated with a decrease in expected future cash flows, with the effect of inflation being much stronger.
The second panel of Table 6 shows that the future expected weekly return component is positively associated with innovations in the ex post real interest rate and this relationship is more significant for the large-stock portfolio. Positive innovations in yield spread increase future expected excess returns, but positive innovations in inflation decrease future returns. Other variables are only weakly associated with future expected returns. The third panel of Table 6 shows that the future real interest rate component increases with the ex post current interest rate and decreases with inflation rate. On the other hand, the innovations in market
10
In this study, the results are based on weekly returns. We choose this time period mainly because Lo and MacKinlay (1990a) found that the predictability of short-horizon returns is much stronger and more consistent at weekly intervals. A more complete time-series analysis would be required to determine the proper length of time intervals for the cross-correlation structure. This is however beyond the scope of this paper.
11
dividend yield and yield spread are positively associated with the component of future real interest rate.
An extension of the above analysis to monthly returns does not alter the basic conclusion. Although the size of coefficients varies somewhat, the direction of the correlation remains largely unchanged. All the covariance terms between the cash flow component and economic factor innovations remain quite significant. Also, similar to weekly returns, the covariance terms between the discount rate component and economic factors are all significant except for the first term (covariance with market returns). The only discernible difference is that the covariances between the future excess returns and economic factors are somewhat weakened for monthly returns.
Overall, the results in Table 6 indicate that the future cash flow component is more closely related to economic factor innovations than either the future excess return or the real discount rate component is. The results suggest that covariation of cash flows with economic factors is likely to be a more important source for the return cross-correlation between large and small stocks. The results also provide useful information about the channels through which economic variables affect stock prices. For example, unexpected inflation appears to have a greater effect on expected future cash flows than on future real interest rates and future expected excess returns. Furthermore, yield spread is positively
Table 6
Covariation of return components with economic factors
Weekly returns Monthly returns
Large stocks Small stocks Large stocks Small stocks
Cov(eÄd,t,XÄ1t) 1.0260* (0.0597) 0.9183* (0.1077) 0.9451* (0.0916) 1.1820* (0.0899)
Cov(eÄd,t,XÄ2t) 8.6541* (1.1916) 5.7499* (1.2903) 2.6332 (1.4128) 4.6523* (1.9854)
Cov(eÄd,t,XÄ3t) ÿ10.4196* (1.1223) ÿ7.4914* (1.2339) ÿ4.1805* (1.5187) ÿ7.0197* (2.0465)
Cov(eÄd,t,XÄ4t) ÿ3.2979* (1.4286) 9.5911* (1.8464) 1.5021 (1.9465) 7.1583* (2.8729)
Cov(eÄd,t,XÄ5t) 4.3802* (0.9869) 3.4298* (0.9472) 4.2771* (1.3944) 6.4075* (1.7741)
Cov(eÄe,t,XÄ1t) ÿ0.0133 (0.0617) 0.2143 (0.1138) ÿ0.0719 (0.0858) 0.2447* (0.0879)
Cov(eÄe,t,XÄ2t) 4.1939* (1.7884) 3.5648 (1.8838) ÿ0.9507 (1.4445) 0.3977 (2.1163)
Cov(eÄe,t,XÄ3t) ÿ4.3818* (1.7233) ÿ4.0653* (1.7776) 0.5661 (1.5199) ÿ1.6353 (2.0588)
Cov(eÄe,t,XÄ4t) 13.8034* (5.7731) 12.7699* (0.1803) 5.0419 (4.5485) 8.5977 (7.4190)
Cov(eÄe,t,XÄ5t) 1.2467 (1.1490) 0.9793 (1.0832) ÿ0.5201 (1.2490) 1.6997 (1.6042)
Cov(eÄr,t,XÄ1t) 0.0017 (0.1138) 0.0063 (0.0879)
Cov(eÄr,t,XÄ2t) 1.9874* (0.0654) 1.3044* (0.1043)
Cov(eÄr,t,XÄ3t) ÿ1.7374* (0.0544) ÿ1.0910* (0.0983)
Cov(eÄr,t,XÄ4t) 1.0866* (0.0675) ÿ1.1850* (0.1007)
Cov(eÄr,tXÄ5t) 0.6709* (0.0262) 0.7981* (0.0723)
This table reports the covariance of each fundamental component with economic factors.eÄd,tis the cash flow
news component, andeÄe,tis the excess return news component,eÄr,tis the discount rate news component in return
innovations. XÄkt,k= 1,2,. . .5, are the innovations of the real stock market return, the real short-term interest rate,
the inflation rate, the long ± short yield spread and the real market dividend yield, respectively. The numbers shown in the table are standardized by the variances of the corresponding factor innovations, Var(XÄjt),j= 1,. . ., 5.
associated with future real interest rates but is negatively associated with expected future cash flows.
3.4.2. Covariation of fundamental return components with return innovations
We now examine the covariances of the fundamental return components of a stock with return innovations of other stocks. Table 7 reports the estimates of the first-order cross-covariance between the fundamental components of large stocks and return innova-tions of small stocks, and vice versa. As shown, the results for both weekly and monthly returns show quite similar patterns. The results indicate that the cross-variances between cash flows and return innovations are relatively large and statistically significant. An
Table 7
The role of return components in the first-order cross-covariance between large and small stocks
Weekly returns Monthly returns Cov(eÄdl,t,eÄs,t+ 1) 0.9809* (0.0770) 0.9762* (0.1164)
Cov(eÄel,t,eÄs,t+ 1) ÿ0.0183 (0.0761) ÿ0.0206 (0.1142)
Cov(eÄr,t,eÄs,t+ 1) ÿ0.0009 (0.0081) ÿ0.0033 (0.0305)
Cov(eÄl,t,eÄs,t+ 1) 1.0395* (0.0609) 0.9980* (0.1575)
Cov(eÄl,t,eÄes,t+ 1) 0.0329 (0.0624) ÿ0.0147 (0.1480)
Cov(eÄl,t,eÄr,t+ 1) 0.0066 (0.0053) 0.0127 (0.0264)
This table reports the first-order cross-covariance of each return component between large and small stocks. eÄdl,tandeÄds,tare the cash flow news component, andeÄel,tandeÄes,tare the excess return news component of
large-and small-stock return innovations, respectively; large-and eÄr,tis the discount rate news component. The figures shown
in the table are standardized. Each covariance between return component and the excess return is divided by the covariance between the corresponding large- and small-stock excess returns.
The asymptotic standard errors are reported in the parentheses. * Significance at the 5% level.
Table 8
The role of return components in higher-order return cross-covariances between large and small stocks q
2 3 7
Cov(eÄdl,t,eÄs,t+q) 1.0086* (0.0822) 1.0432* (0.0699) 1.0140* (0.0562)
Cov(eÄel,t,eÄs,t+q) 0.0070 (0.0759) 0.0260 (0.0643) 0.0117 (0.0560)
Cov(eÄr,t,eÄs,t+q) 0.0016 (0.0076) 0.0173* (0.0042) 0.0023 (0.0030)
Cov(eÄl,t,eÄds,t+q) 0.9933* (0.0869) 1.0835* (0.0465) 1.1355* (0.1059)
Cov(eÄl,t,eÄes,t+q) ÿ0.0180 (0.0835) 0.0709 (0.0468) 0.1393 (0.1002)
Cov(eÄl,t,eÄr,t+q) 0.0114 (0.0066) 0.0125* (0.0012) ÿ0.0038 (0.0023)
This table reports higher-order cross-covariances of each return component between large and small stocks (where lagsq= 2, 3 and 7).eÄdl,tandeÄds,tare the cash flow news component, andeÄel,tandeÄes,tare the excess return
news component in the large- and small-stock return innovations, respectively, andeÄr,tis the discount rate news
component. The figures in the table are standardized by the covariance between the corresponding large- and small-stock excess returns.
increase in the cash flows of large stocks leads to an increase in the excess returns of small stocks in the next period. Also, larger returns of large stocks predict an increase in the future cash flows of small stocks. On the other hand, the cross-covariances between the remaining return components and excess return innovations are weak. The results support the contention that cash flows of large firms convey information for future returns of small-firm stocks.
Table 8 reports the estimates of higher-order cross-covariances. For brevity, we only report the covariances for those lagged orders with more prevalent weekly return cross-correlation. The results again show that cash flows remain the most important in determining the cross-covariance between return innovations of large and small stocks. Although the discount rate component is also significant in some cases, the magnitude of cross-covariance is much smaller compared to that of the cash flow component.
4. Summary
In this paper, we offer an alternative explanation for the phenomenon of asymmetric return cross-correlation documented first by Lo and MacKinlay (1990a). We identify two major sources of the asymmetric return cross-correlation among stock returns: (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 perform a cross-covariance decomposition to trace the channels of information transmission between large and small stocks. We show that both economic and firm-specific factors can contribute to the first-order cross-correlation. However, the firm-specific factor may play a more important role in higher-order cross-correlation. We further decompose returns into three fundamental components: cash flows, discount rates and expected future returns. We then link the behavior of stock returns to these components and assess the relative importance of each component in affecting the asym-metric return cross-correlation.
Empirical results are consistent with the contention that asymmetric cross-correlation is induced by both the difference in the response of stock prices to economic factors and the differential information impounded in stock return innovations. The impact of economic factors is more significant in the first-order return cross-covariance, whereas the influence of firm-specific factors is often carried over to higher-order cross-covariance. Further analysis shows that the asymmetric cross-correlation is mainly attributed to differences in the response of stock prices to market-wide information and the differential quality of cash flows information between large and small firms.
problems of market frictions such as transaction costs and nonsynchronous trading. Furthermore, our subperiod analysis shows that our empirical estimates are quite stable over time.
Acknowledgments
We thank Carl Chen, the Editor, two anonymous referees and Ravi Jagannathan for helpful comments, John Ammer for providing part of the data.
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