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The dynamic relation between returns and idiosyncratic volatility.

We claim that regressing excess returns on one-lagged volatility provides only a limited picture of the dynamic effect of idiosyncratic risk, which tends to be persistent over time. By correcting for the serial correlation in idiosyncratic volatility, we find that idiosyncratic volatility has a significant positive effect. This finding seems robust for various firm size portfolios, sample periods, and measures of idiosyncratic risk. Our findings suggest stock markets mis-price idiosyncratic risk. There may be some measurement problems with idiosyncratic risk that could be related to nondiversifiable risk.

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Recently, interest in idiosyncratic volatility has increased, and researchers are debating the relation between stock returns and idiosyncratic volatility. For example, Campbell, Lettau, Malkiel, and Xu (2001) find that firm-level volatility increased over the period 1962-1997, although the stock market as a whole has not become more volatile. Goyal and Santa-Clara (2003) examine the relation between stock returns and idiosyncratic volatility and find a significant positive relation between (equal-weighted) average stock variance, which is largely idiosyncratic risk, and the (value-weighted) portfolio returns on the NYSE/Amex/Nasdaq stocks for the sample period of 1963:8-1999:12. However, they find that the variance of the market returns does not forecast market returns. (1) Since asset pricing models in financial economics tend to predict that only systematic risk should affect returns, their finding seems rather unusual. (2)

In contrast, Bali, Cakici, Yan, and Zhang (2005) find that the Goyal and Santa-Clara (2003) result does not hold either for sample of 1963:8-2001:12 or for portfolios of stocks traded on the NYSE/Amex or NYSE. They claire that it is primarily due to small stocks traded on the Nasdaq and a liquidity premium associated with them.

Therefore, the empirical evidence on the relation between stock returns and idiosyncratic volatility remains mixed. Two fundamental questions remain. Does idiosyncratic risk really matter? And if it does, how do we understand the nature of the idiosyncratic risk that affects returns on the market? We address these questions in this article.

By regressing excess market returns on one-lagged values of idiosyncratic volatility, previous studies on the dynamic relation between stock returns and idiosyncratic risk focus on one-period forecasts. This one-period predictive regression approach is simple, intuitive, and widely used in the finance literature. However, inferences based on predictive regressions rely critically on the assumed stochastic properties of the posited explanatory variable, particular on whether it is highly persistent. These earlier studies show that a highly persistent explanatory variable may result in an inefficient estimate of predictive regressions for returns. (3)

In this article, we re-examine the dynamic relation between idiosyncratic volatility and returns on the market by addressing the problems associated with persistent volatility (see Ghysels, Santa-Clara, and Valkanov, 2005). Extensive studies of volatility that use various ARCH models find that stock return volatility tends to be persistent? Using only one lagged volatility in the excess return regression may provide only a limited picture of the dynamic relation between returns and idiosyncratic volatility. Including several lagged volatilities and examining autoregressive systems may also be misleading, because volatilities are strongly serially correlated.

Under a similar circumstance, Sims (1980a, 1980b, 1992) suggests that the best method is to analyze the response of returns to typical random volatility shocks (or innovations). (5) Pindyck (1984) and Poterba and Summers (1986) show that increases in volatility persistence, measured by serial correlation, will have a greater effect on the equity premium and thus stock prices. Then they examine the effect of volatility shocks, which can be viewed as innovations, on stock prices. In a similar vein, French, Schwert, and Stambaugh (1987) focus on the relation between excess market returns and return volatility shocks (unexpected volatility). They argue that the relation between realized excess return and the volatility shocks can be induced by the relation between ex-ante excess return and volatility. In a recent paper, Ghysels, Santa-Clara, and Valkanov (2005) find that what matters for the tests of the risk-return tradeoff is the persistence of the conditional variance process. That is, the window length plays a crucial role in forecasting variances and detecting the trade-off between risk and return.

Following these studies, we measure the dynamic (multiperiod) effects of volatility on excess returns by regressing excess returns on lagged innovations in volatility (i.e., pre-whitened volatility), which are white noise and thus not serially correlated. After correcting for the serial correlations in various measures of volatilities, we find significant positive effects of idiosyncratic volatility on stock returns, although the effects tend to be delayed. This positive effect is robust for both small and large firm portfolios, equal- and value-weighted volatilities, average volatility and idiosyncratic volatility, and for different sample periods.

In his seminal work on the intertemporal capital asset pricing model (ICAPM), Merton (1973) shows that the conditional excess market return is a linear function of its conditional variance (the risk component) and its covariance with investment opportunities (the hedging component). To account for the intertemporal hedging component induced by stochastic investment opportunities, Goyal and Santa-Clara (2003) and Bali et al. (2005) estimate the risk-return tradeoff with various control variables (see also Guo and Whitelaw, 2006). When we add a set of control variables to capture the state variables that determine changes in the investment opportunity set, we find that our results, which indicate a significant dynamic relation between returns and idiosyncratic risk, remain virtually unchanged. Further, we find that idiosyncratic risk is to some extent related to these state variables.

We find that idiosyncratic risk does affect market returns. This effect reflects the market's overreaction to idiosyncratic risk, which suggests mispricing of the stock market. However, because idiosyncratic volatility is related to various state variables or market fundamentals, there remain some measurement issues about the volatility measure used in recent studies, which may be related to investors' inability to hold well-diversified portfolios due to various reasons (Goyal et al., 2003 and Malkiel and Xu, 2003).

The article is organized as follows. In Section I, we describe our data and measures of market volatility, average stock volatility, and idiosyncratic volatility. In Section II, we provide some preliminary descriptive statistics. In Section III, we present estimation results of the dynamic relation between excess market returns and various measures of idiosyncratic volatility. In Section IV, we further explore the nature of idiosyncratic volatility and its economic value as a predictor of excess returns. Section V concludes.

I. Data and Measures of Idiosyncratic Volatility

We use CRSP daily return data for the sample period July 1962 to December 2002 to construct monthly market volatility, average stock volatility, and idiosyncratic volatility. Following French, Schwert, and Stambaugh (1987) and Goyal and Santa-Clara (2003), we construct equal-weighted market volatility (EWMV) and equal-weighted average volatility (EWAV) as:

(1) [EWMV.sub.t] = [[D.sub.t].summation over (d=1)] [r.sup.2.sub.ew,d] + 2 [[D.sub.t].summation over (d=2)] [r.sub.ew,d][r.sub.ew,d-1]

(2) [EWAV.sub.t] = 1/[N.sub.t] [[N.sub.t].summation over (i=1)][[[D.sub.t].summation over (d=1)] [r.sup.2.sub.i.d] 2 [[D.sub.t].summation over (d=2)][r.sub.i,d]r[r.sub.i,d-1]] (2)

respectively, where [r.sub.ew,d] is the equal-weighted stock return across the CRSP index on day d, [N.sub.t] is the number of stocks in month t, [D.sub.t] is the number of days in month t, and [r.sub.i,d] is the daily return of stock i on day d.

We construct value-weighted market volatility (VWMV) and average volatility (VWAV) as:

(3) [VWMV.sub.t] = [[D.sub.t].summation over (d=1)] [r.sup.2.sub.vw,d] + 2 [[D.sub.t].summation over (d=2)] [r.sub.vw,d][r.sub.vw,d-1]

(4) [VWAV.sub.t] = [[N.sub.t].summation over (i=1)] [CAP.sub.i,t-1]/ [CAP.sub.t-1] [[[D.sub.t].summation over (d=1)] [r.sup.2.sub.i.d] 2 [[D.sub.t].summation over (d=2)][r.sub.i,d]r[r.sub.i,d-1]]

respectively, where [r.sub.vw,d] is the value-weighted stock return across the market on day d, [N.sub.t] is the number of stocks in month t, [D.sub.t] is the number of days in month t, [r.sub.i,d] is the daily return of stock i on day d, [CAP.sub.i,t] is the market capitalization for stock i in month t, and [CAP.sub.t] is the market capitalization in month t.

In Equations (1) through (4), we add the second term to the right-hand-side to adjust for the autocorrelation in daily returns, again using the approach proposed by French, Schwert, and Stambaugh (1987) and Goyal and Santa-Clara (2003).

We construct idiosyncratic volatility based on both a constant mean return model for a high frequency measure, and a market model for a low frequency measure. We construct the constant mean return model-based equal-weighted (EWIV) and value-weighted (VWIV) idiosyncratic volatilities as:

(5) [EWIV.sub.t] = [EWAV.sub.t], - [EWMV.sub.t]

(6) [VWIV.sub.t] = [VWAV.sub.t] - [VWMV.sub.t]

We denote [e.sub.it] as the CAPM residual for security i at month t. We construct the CAPM model-based equal- ([EWIV.sup.m.sub.t]) and value-weighted ([VWIV.sup.m.sub.t]) idiosyncratic volatilities as: (6)

(7) [EWIV.sup.m.sub.t] = 1/[N.sub.t] [[N.sub.t].summation over (i=1)]VAR([e.sub.it])

[VWIV.sup.m.sub.t] = [[N.sub.t].summation over (i=1)] [CAP.sub.it-1]/[CAP.sub.t-1] VAR([e.sub.it])

However, using the one-factor CAPM market model-based residuals in Equations (7) and (8) may not provide an adequate adjustment for risk. Fama and French (1993, 1996) introduce size and book to market equity as two additional factors, and Carhart (1997) introduces one-year momentum as a fourth factor. To check the robustness of our results, we compute idiosyncratic volatility based on both Fama-French's three-factor model and Carhart's (1997) four-factor model to implement all out empirical procedures. We obtain the data on these factors from French's website. (7) We find that the results are similar to those of the CAPM model. Therefore, to save space we do not report the results of either the three-tactor or the four-factor models.

II. Preliminary Empirical Results: Descriptive Statistics

Table I presents descriptive statistics for CRSP returns and volatilities from July 1962 to December 2002. In Panel A, we observe that the means of equal-weighted market volatility, average volatility, and idiosyncratic volatility are 0.002, 0.032, and 0.030, respectively. This observation confirms that idiosyncratic risk represents a dominant component of average risk. We find a similar pattern for value-weighted volatilities. The standard deviations of value-weighted average volatility and idiosyncratic volatility are less than half of those of equal-weighted volatilities. The mean of value-weighted returns is much lower than that of equal-weighted returns. This finding implies that small firms tend to have higher returns with larger volatilities than do large firms.

Panel B shows that average volatility and idiosyncratic volatility are very persistent: their autocorrelation at lag one is about 0.8, and that autocorrelation decays very slowly as the number of lags increase. This finding suggests that using a one-lagged value of these persistent volatilities in examining the dynamic effect of these volatilities on returns can be quite misleading.

In contrast, market volatilities, which are frequently modeled as an ARCH (or GARCH) model to reflect their persistence, show a relatively low persistence: their autocorrelation at lag one is about 0.2, and the autocorrelation decays as the number of lags increase. Box-Pierce (1970) Q(12) statistics show that all measures of volatilities are serially correlated. Not surprisingly, market returns, particularly the value-weighted returns, show little autocorrelation. Goyal and Santa Clara (2003) show that the average stock variance has a lower measurement error than does market variance (e.g., equal-weighted portfolio variance). These authors note that they can measure the average stock variance more precisely in relation to its mean than its market variance. In Panel B, we observe that the average stock variance (e.g., EWAV, VWAV) is more persistent than market variance (e.g., EWMV, VWMV). The larger measurement error is likely to decrease the persistence in the realized series of the market variance.

Panel C shows a negative contemporaneous relation between market volatility and market returns. However, it shows a positive contemporaneous correlation between equal-weighted market returns and two measures of idiosyncratic volatility (average volatility and idiosyncratic volatility). The correlations of value-weighted market returns with two measures of idiosyncratic volatility are also positive except for the one with value-weighted average volatility, which is consistent with Duffee's (1995) finding.

III. Dynamic Effects of Idiosyncratic Risk on Market Returns

In this section, we propose and analyze two time-series approaches to proxy the expected idiosyncratic volatility. We examine the dynamic effects of idiosyncratic volatility controlling for the serial correlations in the volatility. We also provide the robustness checks for the dynamic effects.

A. Motivation

Goyal and Santa-Clara (2003) and Bali et al. (2005) provide new evidence on the risk-return relation by estimating a variant of Merton's ICAPM. They investigate the presence and significance of a relation between aggregate idiosyncratic volatility (IV) and the excess market return. Both studies are based on the following relation:

(9) E([R.sub.M,t] - [r.sub.f,t]|[[OMEGA].sub.t-1]) = [beta]E([IV.sub.t]| [[OMEGA].sub.t-1]),

which leads to:

(10) [R.sub.M,t] - [r.sub.f,t] = [alpha] + [beta]E([IV.sub.t]|[[OMEGA].sub.t-1]) + [[epsilon].sub.t]

where E([R.sub.M,t] - [r.sub.f,t]|[[OMEGA].sub.t-1]) and E([IV.sub.t]| [[OMEGA].sub.t-1]) are the conditional mean of excess market return and conditional variance of aggregate idiosyncratic risk, respectively.

Our question is how to obtain a reasonable proxy for E([IV.sub.t]| [[OMEGA].sub.t-1]). For any stationary stochastic process, it has a moving average representation (MA) by the Wold representation theorem, and it has an autoregressive representation (AR) as long as it is invertible, which we assume here. That is:

(11) [IV.sub.t] = A(L) [IV.sub.t-1] + [u.sub.t] = [[infinity].summation over (j=1)][a.sub.j][IV.sub.t-j] + [u.sub.t],

where A(L) is a polynomial in the lag operator (i.e., A(L) = [[infinity].summation over (j=1)][a.sub.j] [L.sup.j]) with L as a lag (or backshift) operator (i.e., [L.sup.n] [X.sub.t] [X.sub.t-n]). We can invert the AR of [IV.sub.t] to an MA of [IV.sub.t]:

[I - A(L)L] [IV.sub.t] = [u.sub.t],

or

[IV.sub.t] = [[I - A(L)L].sup.-1] [u.sub.t] B(L)[u.sub.t] = [[infinity].summation over (j=0)] [b.sub.j] [u.sub.t-j], (12)

where I is the identity matrix and B(L) = [[I - A(L)L].sup.-1].

Thus, we can obtain two reasonable proxies for E([IV.sub.t]| [[OMEGA].sub.t-1]) by taking conditional expectations of the above representations, one based on AR of [IV.sub.t] in Equation (11) and the other based on MA of [IV.sub.t] in Equation (12):

E([IV.sub.t]|[[OMEGA].sub.t-1]) = [[infinity].summation over (j=1)][a.sub.j] [IV.sub.t-j] (13)

E([IV.sub.t]|[[OMEGA].sub.t-1]) = [[infinity].summation over (j=1)][b.sub.j] [u.sub.t-j]. (14)

Equation (14) follows from the fact that E([u.sub.t]|[[OMEGA].sub.t-1]) = 0 and E([u.sub.t-j]|[[OMEGA].sub.t-1]) = [u.sub.t-j] for any j [greater than or equal to] 1. One way to understand the equivalence of these two representations is that the Hilbert space spanned by past [IV.sub.t]s and the one spanned by past innovations [u.sub.t]s are identical. Or we could use only one lagged [IV.sub.t-1] as a simpler proxy for E([IV.sub.t]|[[OMEGA].sub.t-1]) instead of many lagged [IV.sub.t]s, as in the above AR model Equation (13), which prior studies use.

Now we need to choose between the above two proxies, the one based on AR and the other based on MA, for E([IV.sub.t]|[[OMEGA].sub.t-1]). In general, both are legitimate proxies. However, for our study, we are concerned with a dynamic effect of IV on the excess market return and IV is quite persistent (i.e., IV is serially correlated), so we prefer to use the proxy based on MA, because the innovations in IV are serially uncorrelated by construction. Therefore, we can easily identify the dynamic effects of lagged IV on the excess market return. This is in the same spirit as Sims' VAR approach, which is adopted in this article.

B. Dynamic Effects after Corrections for Serial Correlations in Volatilities

Since Table I shows that idiosyncratic volatility is highly persistent and serially correlated, a one-period-ahead forecast regression approach may give us a partial understanding of the dynamic time-series relation between stock returns and idiosyncratic risk. (8) One way to examine a comprehensive dynamic effect of persistent volatility on returns would be to include several lagged values of volatilities as regressors and then use the autoregression approach to examine the significance of each coefficient of the lagged volatilities:

[r.sup.e.sub.t] = [alpha] + [m.summation over (q-1)] [[beta].sub.q][IV.sub.t-q] + [u.sub.t], (15)

where m is the number of lags.

As discussed above, the potential problem with this approach is that the coefficient [[beta].sub.k] in this regression might not measure the net effect of [IV.sub.t-k] on excess returns in k periods. This is because lagged volatilities are highly serially correlated so that a unit change in [IV.sub.t-k] affects not only excess return in k periods but also [IV.sub.t-j] for j < k, which in turn affects excess returns in j periods. That is, the coefficient of [IV.sub.t-k] in the regression does not identify the dynamic effects of each volatility on excess returns.

As Sims (1980a, 1980b, and 1992) suggests, an alternative is to use serially uncorrelated (i.e., orthogonalized) innovations in volatility as regressors in the regression and then use the moving average approach to examine the signif<cance ofeach coefficient. To implement this approach, our first step is to obtain innovations (or unexpected changes) in volatilities. To do so, we regress volatility on its own lagged values and take the residuals as the innovations. Then we regress excess returns on several lagged values of these innovations in volatility:

[r.sup.e.sub.t] = [alpha] + [m.summation over (q=1)] [[beta].sub.q][[??].sub.t-q] + [lambda][IV.sub.t-q-1] + [e.sub.t], (16)

where [[??].sub.t-q] is the innovation in [IV.sub.t-q].

In addition to several lagged innovations in volatility, we include the level of volatility at time t-q-l in estimation of Equation (16) to capture the remaining lagged innovations. By construction, the innovations [[??].sub.t-q] are serially uncorrelated, and thus the [coefficient.sub.q] measures the net effect of [IV.sub.t-q] on excess return in q periods. Using lagged innovations in volatilities and a lagged volatility is similar to the Mixed Data Sampling (MIDAS) estimator of the conditional variance used by Ghysels, Santa-Clara, and Valkanov (2005).

Panel D of Table I reports the serial correlations and contemporaneous correlations for these innovations in various measures of volatility for our sample. Compared to the autocorrelations in various volatilities in Panel B, the autocorrelations of innovations in Panel D are drastically lower for all our volatility measures, and the Q(6) statistics are nonsignificant for most measures of innovations. Contemporaneous correlations between innovations in various measures of idiosyncratic volatilities are relatively high.

Table II reports the estimation results for Equation (16). Considering both the Akaike (1974) Information Criterion and the Schwarz (1978) Bayesian Criterion, we choose six lags (i.e., [mu] = 6) in Equation (16). The table includes estimates of betas, lamda, the sure of betas, Newey-West (1987) adjusted t-statistics, and adjusted [R.sup.2].

To examine the significance of the overall dynamic relation between excess return and idiosyncratic volatility, we conduct two sets of tests on the coefficients of the regression. One is a test of the null hypothesis that the sure of the coefficients is zero, and the other is a test of the null hypothesis that each coefficient is zero. The former tests whether the net intertemporal effect of volatility on excess return is statistically significant, and the latter tests whether the effect of volatility as a group is statistically significant.

In Panel A, when we use equal-weighted average volatility, we find at least two significant betas for each sample period. In the first sample period, 1963-1999, [[beta].sub.2], [[beta].sub.3], and [[beta].sub.4] are significant. In the second sample period, 1963-2001, [[beta].sub.3] and [[beta].sub.4] are significant. In the extended sample period, 1963-2002, [[beta].sub.3] and [[beta].sub.4] are significant. These significant estimates of coefficients indicate that the null hypothesis that the overall dynamic effect of volatility is zero ([[beta].sub.t]= 0, for all t) is rejected at the conventional significance level for each sample period. Adjusted [R.sub.2] are 0.012, 0.008, and 0.011 for each sample period.

When we use value-weighted average volatility, we find similar results. Some coefficients are significant, and these estimates are all positive for each sample period (see Panel B). Adjusted [R.sub.2] are 0.014, 0.009, and 0.022 for each sample period. The null hypothesis that the overall dynamic effect of volatility is zero ([[beta].sub.t] = 0, for all t) is rejected at the conventional significance level for the first sample period, 1963-1999, and the extended sample period, 1963-2002.

We find a significant positive effect of lagged idiosyncratic volatility shocks on excess returns, and that this result holds for both equal- and value-weighted idiosyncratic volatilities. This finding implies that idiosyncratic risks are compensated by a higher excess return, which seems at variance with the conventional asset pricing models' predictions. (9)

C. Robustness of Dynamic Effects

To see whether the dynamic relation between excess return and idiosyncratic volatility is sensitive to various firm size portfolios, we run the regression in (16) using separate portfolios from the NYSE/Amex, NYSE, and Nasdaq. Table II presents the results when we use both equal- and value-weighted average volatilities. Panels C and D use a NYSE/Amex portfolio, Panels E and F use a NYSE portfolio, and Panels G and H use a Nasdaq portfolio.

When we use equal-weighted average volatility in Panels C, E, and G, some beta coefficients are significantly positive in each panel, and the two null hypotheses are rejected for each portfolio and for each sample period except for the second null hypothesis in panel C(NYSE/Amex) for the second sample period, 1963-2001.

When we use value-weighted average volatility in Panels D, F, and H, the relation appears to be somewhat weaker, but there still is at least one significant positive coefficient for the NYSE/Amex and NYSE portfolios. The two F-test results are somewhat sensitive to the sample periods, but weaker for the second sample period, 1963-2001. However, the two hypotheses are rejected for the extended sample period of 1963-2002 even for the NYSE portfolio (Panel E).

The results in Table II indicate that the positive effect of idiosyncratic volatility on excess returns is significant and robust even for portfolios excluding Nasdaq stocks, at least for the extended sample period, 1963-2002. When we estimate the dynamic relation for a sample period that starts from 1926 as in Goyal and Santa-Clara (2003), we find a similar result showing a significant dynamic effect based on volatility innovations.

In his seminal work on intertemporal capital asset pricing model (ICAPM), Merton (1973) shows that the conditional excess market return is a linear function of its conditional variance (the risk component) and its covariance with investment opportunities (the hedging component). Campbell (1996) and Scruggs (1998) point out that the approximate linear relationship between risk and return may be misspecified if the hedging term in the ICAPM is important. To account for the intertemporal hedging component induced by stochastic investment opportunities, Goyal and Santa-Clara (2003) and Bali et al. (2005) estimate the risk-return tradeoff with various control variables (see also Guo and Whitelaw, 2006).

To ensure that our results from estimating Equation (16) are robust, we re-estimate Equation (16). We add a set of control variables that have been used in other studies to capture the state variables that determine changes in the investment opportunity set. The state variables include term premium, relative T-bill rate, default spread, and dividend yield.

Table III presents the estimation results. Panels A and B show that at least two betas are positive and significant in each sample period regardless of whether we use equal- or value-weighted average volatility, and that some state variables are also significant. The two null hypotheses are rejected for the first sample period (1963-1999) and the extended sample period (1963-2002).

Panels C and D of Table III report estimation results we obtain by controlling for various state variables and for two measures of illiquidity due to Amihud (2002): expected and unexpected illiquidity. We find that at least one beta is positive and significant in each sample period regardless of whether we use equal- or value-weighted average volatility. Some state variables and unexpected illiquidity variable are also significant. The two null hypotheses are rejected for the extended sample period (1963-2002).

In these analyses, we use (innovations in) average volatility, which is mostly idiosyncratic volatility, as a measure of idiosyncratic risk. We do so to make our results directly comparable to previous studies (e.g., Goyal et al., 2003; and Bali et al., 2005). To see whether the positive dynamic relation between excess return and innovations in average volatility is robust for different measures of idiosyncratic risk, we examine the relation between excess return and innovations in alternative measures of idiosyncratic risk. These alternative measures are based on the constant mean model and the market model, which we define in Section I. Here, since prior studies find a relatively weak relation with value-weighted idiosyncratic volatilities, we focus on value-weighted idiosyncratic volatility.

Panel A of Table IV reports estimation results of a predictive regression when we measure value-weighted idiosyncratic volatility based on the constant mean model (using high frequency, daily, data). We find a significant positive relation between excess returns and innovations in value-weighted idiosyncratic volatility for the first sample period (1963-1999) and the extended sample period (1963-2002), but the positive relation is relatively weak for the second sample period (1963-2001). Therefore, we find that the null hypothesis that all betas are zero is rejected for the first and extended sample periods.

Panel B of Table IV reports the estimation results of a predictive regression when we measure value-weighted idiosyncratic volatility based on the market model in which we use low frequency, monthly data. Although we find a significant positive relation for the first sample period, the positive relation becomes less significant when we extend the sample period to 2001 or to 2002. We suspect that the weak dynamic relation in panel B may be in part due to the use of low frequency (i.e., monthly) data (see Schwert, 1989).

Panel C of Table IV reproduces the market model results of Panel B for the sample period starting from 1926. For this longer period, we find that the null hypothesis that all betas are zero is strongly rejected. However, the first null hypothesis that the sum of all betas is zero is not rejected. We also note that the adjusted [R.sup.2] for this longer period is substantially greater than that of the shorter period.

As mentioned in Section I, the one-factor CAPM market model-based residuals may not provide an adequate adjustment for risk. Thus, we implement our empirical procedures using idiosyncratic volatility constructed based on both Fama-French (1993, 1996) three-factor model and Carhart (1997) four-factor model. We find the results are similar to those of the CAPM model. For the multi-factor model, some high frequency daily data (e.g., book value) are not available.

IV. Idiosyncratic Volatility and State Variables

To clarify out understanding of the nature of idiosyncratic risk, we look at the relation between idiosyncratic risk and fundamental (state) variables. Following Goyal and Santa-Clara (2003), Bali et al. (2005), and Guo and Whitelaw (2006), we run the regression of idiosyncratic volatility on several fundamental (state) variables:

[IV.sub.t] = [alpha] [[beta].sub.1][dy.sub.t] + [[beta].sub.2][rtb.sub.t] + [[beta].sub.3][trem.sub.3] + [[beta].sub.4][def.sub.t] + [[beta].sub.5][cay.sub.t] + [e.sub.t], (17)

where [IV.sub.t] denotes idiosyncratic volatility; [dy.sub.t], [rtb.sub.t], [term.sub.t], [def.sub.t], and [cay.sub.t], denote dividend yield, relative T-bill, term spread, default spread, and consumption to wealth ratio at time t, respectively. Recently, Guo and Whitelow (2006) empirically show that the hedge component is important in the relation between return and risk. Following Guo and Whitelow (2006), we include [cay.sub.t], a consumption to wealth ratio, as a proxy for the hedge component, in addition to the detrended risk-free rate, [rtb.sub.t].

Table V reports the estimation results of Equation (17), when we use average volatility (Panel A), idiosyncratic volatility based on a constant mean return model (Panel B), and idiosyncratic volatility based on a market model (Panel C). In Panels A, B, and C, we report the estimates of coefficients with the t-statistics in parentheses. [R.sup.2] is the adjusted [R.sup.2].

We observe that most coefficient estimates, including the cay variable, are significant, and that the adjusted [R.sup.2]s are between 6% and 12% for equal-weighted average volatility and between 5% and 8% for value-weighted average volatility. The p-values indicate that the hypothesis that idiosyncratic volatility is not related to these fundamental state variables is strongly rejected.

Overall, we find that idiosyncratic volatility contains some elements of fundamentals including a hedge component of returns. Out finding is consistent with that of Brown and Ferreira (2005), that small-firm volatility bas the predictive power of big-firm returns, in part because it is a proxy for systematic volatility and consumption-wealth ratio.

Given out finding of in-sample dynamic relation between excess returns and idiosyncratic volatilities, an interesting question would be to examine out-of-sample evidence of the dynamic relation. Recently Goyal and Welch (2005) compare predictive regressions with historical average returns and find that historical average returns usually generate superior return forecasts. Based on this finding, they argue that in-sample correlations conceal a systematic failure of predictive variables out-of-sample forecasts. However, Campbell and Thompson (2005) argue that, once sensible restrictions are imposed on the signs of coefficients and return forecasts, forecasting variables with significant in-sample forecasting power generally have a better out-of-sample performance than a forecast based on the historical average return. Further, they provide incremental excess returns by using forecasting variables under reasonable assumptions.

When we implement usual out-of-sample forecasts using six lagged innovations in volatility following Goyal and Welch (2005) procedure, we obtain mixed results. For example, using equal-weighted idiosyncratic volatility, we find that the mean absolute error (MAE) from rolling out-of-sample errors from the OLS model is slightly smaller than that from the historical mean model. However, when we use the root mean squared error (RMSE), we find the opposite result. Our estimation shows that MAE = 0.00057 and RMSE = -0.00073.

When we follow Campbell and Thompson (2005) and impose restrictions on the regression in such a way that we consider only significant coefficients (in our case, only the third, fourth and sixth lagged innovations), we obtain improved [R.sup.2]s for the forecast models. Following the Campbell and Thompson procedure with the value of relative risk aversion of three, the estimated adjusted [R.sup.2], and the Sharpe ratio from out sample, we obtain an annualized incremental excess return of 6.4% when we use equal-weighted idiosyncratic volatility, and of 1.7% when we use value-weighted idiosyncratic volatility. This finding implies that if we are willing to impose sensible restrictions on the coefficients of forecast regressions, then idiosyncratic volatility may even provide an economic value in forecasting excess returns. (10)

V. Conclusion

Recently, researchers have hotly debated the dynamic relation between stock returns and idiosyncratic volatility. Given the persistent nature of idiosyncratic volatility, we argue that previous studies provide only a limited, partial picture on this dynamic relation. Using an alternative approach that corrects the autocorrelations in idiosyncratic volatility, we provide a more general picture of the intertemporal relation between stock returns and idiosyncratic volatility.

We find a significant dynamic relation but a delayed response of market returns to innovations in idiosyncratic volatility. This finding is not sensitive to different firm size portfolios. After we control for a set of variables used in previous studies to capture the state variables that determine changes in the investment opportunity set, we find a positive dynamic relation between the market returns and innovations in idiosyncratic volatility. Further, when we control for some nontraded risks associated with labor income returns and entrepreneur income returns, we still find that stock market returns are influenced by idiosyncratic volatility beyond its effect on revisions in future expected cash flows and discount factors. (11) This finding may provide evidence for the market's mispricing with respect to idiosyncratic risks.

We also find that idiosyncratic risks are related to macroeconomic state variables. This finding suggests that if we take the position that appropriately measured idiosyncratic risk should not be related to market fundamentals and even though we have considered some multi-factor models, there are still some measurement problems. Or perhaps this relation suggests that the idiosyncratic risks are not fully diversified.

A behavioral finance approach may provide an alternative interpretation for our findings. We find that the market responds positively to innovations in idiosyncratic volatility with an initial delay prior to its becoming nonsignificant after several periods, and that its effect is beyond what the efficient market hypothesis anticipates. This finding suggests that the market may be overreacting to idiosyncratic volatility. In contrast, we also find that the market at first responds negatively, but positively afterwards for several periods to innovations in market volatility. This finding suggests that the market may be underreacting briefly to market volatility at least in short horizons.

Recent behavioral hypotheses distinguish information between intangible and tangible with overreaction to the former and underreaction to the latter. If we regard idiosyncratic volatility as part of intangible information and market volatility as part of tangible information, our interpretation seems compatible with recent behavioral hypotheses (e.g., Daniel and Titman, 2005).

There are two common explanations for previous studies that find evidence for idiosyncratic volatility having forecast ability. One is that idiosyncratic volatility forecasts real stock market returns because it reflects the effect of non-traded human capital and entrepreneurial capital returns. The other explanation is that idiosyncratic volatility forecasts real stock market returns because investors are unable to hold well-diversified portfolios for various reasons, such as transactions costs, incomplete information, and institutional restrictions. Therefore, the investors demand a premium on idiosyncratic volatility.

To examine the effect of idiosyncratic risk on market returns, we incorporate the effect of nontraded human capital and entrepreneurial capital returns, and still find the dynamic effect of idiosyncratic risk. (12) It remains to be seen whether the dynamic relation is mainly due to investors holding nondiversified portfolios. Another way to extend this article would be to use more refined measures of innovations in volatility based on a multivariate framework (e.g., general VAR) in the return and volatility relation.

We would like to thank Amit Goyal, Hui Guo, Robert Savickas. Samuel Thompson, Tuomo Vuolteenaho, and seminar participants at the University of Northern Iowa, University of Hong Kong, 2004 Southern Finance Association meeting, 2005 American Finance Association meeting, and 2005 Financial Management Association meeting for their useful comments. We especially thank Lemma Senbet, Alexander Triantis (the Editors), and two anonymous referees for detailed and valuable comments. We also thank Sandra Sizer for editorial assistance. An earlier version of this paper received the best paper award in Investments at the 2004 Southern Finance Association meeting. Any errors are our own.

(1) They measure average stock risk as the cross-sectional average of the variances of all the stocks traded in that month, and then run predictive regressions of market returns on this variance measure as well as the variance of the market. Consistent with most previous studies, they find that market variance does not forecast market returns. However, they find a significant positive relation between lagged average stock variance and market returns.

(2) Guo and Savickas (2006) report that Goyal and Santa-Clara's (2003) finding is due to comovements of average stock volatility with stock market volatility. Although our paper focuses on the dynamic (time-series) effect of idiosyncratic risk on market returns, Malkiel and Xu (2002) and Ang, Hodrick, Xing, and Zhang (2005) examine the cross-sectional relationship between idiosyncratic volatility and expected returns. Malkiel and Xu (2002) find that idiosyncratic volatility helps explain the cross-sectional variation of stock returns after controlling for size, book-to-market, and liquidity effects. Ang, Hodrick, Xing, and Zhang (2005) find that stocks with high idiosyncratic volatility have low average returns. Their finding is in contrast to previous research that finds either a significant positive relation between idiosyncratic volatility and average returns or an insignificant relation. However, their result is consistent with the time-series results in Goyal and Santa-Clara under a Merton (1973) ICAPM interpretation. For example, firms that are exposed to the idiosyncratic risk factor provide a hedge against increases in that type of risk (and/or aggregate volatility risk), which lowers their average returns.

(3) For our sample, average volatility and idiosyncratic volatility are very persistent: their first-order autocorrelation is about 0.8, and it decays very slowly over time. For more details, see Table I and Section II. Several studies emphasize high persistence issues in small samples: French, Schwert, and Stambaugh (1987), Stambaugh (1999), Ang and Bekaert (2005), Campbell and Yogo (2006), Torous, Valkanov, and Yan, (2005), Amihud and Hurvich (2004), and Lewellen (2004). In particular, Lewellen (2004) states,

"The sample autocorrelation is strongly correlated with the slope estimate in the predictive regression, so any information conveyed by the autocorrelation helps produce a more powerful test of predictability. Incorporating this information into empirical tests has two effects: 1) the slope coefficient is often larger than Stambaugh's estimate, and 2) the standard error of the estimate is much lower. In combination, the two effects can substantially raise the power of empirical tests."

(4) See Bollerslev, Chou, and Kroner (1992) for an extensive survey of ARCH models in finance.

(5) For simplicity, we do not consider cross-equation feedbacks because we do not focus on the effect of excess returns on volatility.

(6) Bali et al. (2005) construct the equal and value-weighted idiosyncratic volatility in the same way.

(7) http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data library.html.

(8) Campbell and Yogo (2005), Torous, Valkanov, and Yan (2005), Ang and Bekaert (2005), and Lewellen (2004), among others, provide exploratory research on this issue.

(9) We compare the predictive power of the set of lagged innovations in average stock volatility with that of the value or equal-weighted average idiosyncratic volatility in the regression of the one-month-ahead conditional variance of excess market returns. We find that the set of lagged innovations in average stock volatility gives more accurate forecasts of the one-month-ahead conditional variance of excess market returns (e.g., the adjusted [R.sub.2] of 0.3341 for innovations in EWAV, and that of 0.4189 for innovations in VWAV) than does the value- or equal-weighted average idiosyncratic volatility (e.g., adjusted [R.sub.2] of 0.2684 lbr EWAIV, and that of 0.1559 for VWAIV).

(10) The details of these estimates are available from the authors.

(11) The details of these estimates are available from the authors.

(12) This result is not reported in the article, but is available from the authors.

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Xiaoquan Jiang and Bong-Soo Lee *

* Xiaoquan Jiang is an Assistant Professor of finance at the University of Northern Iowa in Cedar Falls, IA. Bong-Soo Lee is a Professor of Finance at the KAIST Graduate School of Finance, Seoul, Korea and Florida State University, Tallahassee, FL.
Table I. Descriptive Statistics of Returns and Idiosyncratic Volatility
Measures

This table presents descriptive and time-series properties of returns
and idiosyncratic volatility. EWRET and VWRET denote monthly equal- and
value-weighted CRSP index returns. EWMV, VWMV, EWAV, VWAV, EWIV,
and VWIV denote monthly CRSP index equal- and value-weighted average
market volatility, average stock volatility and idiosyncratic
volatility, respectively, which we calculate using daily data. EWAVS,
VWAVS, EWIVMS, VWIVMS, EWIVS, VWIVS, EWIVFFS, and VWIVFFS are the
innovations in monthly CRSP equal- and value-weighted average stock
volatility, and idiosyncratic volatility that we base on the market
and constant mean return models and the Fama-French three factor
model. Q(m) denotes the significant level of Box-Pierce (1970)
statistic for m lags of the autocorrelation function. The sample
period is from July 1962 to December 2002.

 Panel A. Summary Statistics

 Mean Std Dev Min

EWRET 0.018 0.058 -0.252
VWRET 0.009 0.044 -0.226
EWMV 0.002 0.003 0.000
VWMV 0.002 0.004 0.000
EWAV 0.032 0.019 0.010
VWAV 0.010 0.008 0.002
EWIV 0.030 0.018 0.009
VWIV 0.008 0.007 -0.060

 Panel A. Summary Statistics

 Max Skew Kurt

EWRET 0.303 -0.146 2.762
VWRET 0.157 -0.512 1.995
EWMV 0.051 8.882 115.618
VWMV 0.067 12.239 206.358
EWAV 0.123 1.796 4.298
VWAV 0.055 2.614 8.337
EWIV 0.117 1.620 3.406
VWIV 0.044 0.087 24.234

 Panel B. Autocorrelation at Lags

 1 2 3 4

EWRET 0.200 -0.037 -0.033 -0.026
VWRET 0.051 -0.049 0.002 -0.002
EWMV 0.168 0.119 0.105 0.042
VWMV 0.208 0.167 0.132 0.056
EWAV 0.806 0.736 0.699 0.636
VWAV 0.818 0.750 0.735 0.664
EWIV 0.859 0.791 0.758 0.704
VWIV 0.724 0.661 0.654 0.595

 Panel B. Autocorrelation at Lags

 5 6 7 8

EWRET 0.016 -0.005 -0.021 -0.119
VWRET 0.075 -0.023 -0.012 -0.063
EWMV 0.064 0.059 0.014 0.043
VWMV 0.079 0.103 0.065 0.048
EWAV 0.613 0.584 0.576 0.584
VWAV 0.651 0.674 0.621 0.614
EWIV 0.678 0.652 0.651 0.650
VWIV 0.581 0.556 0.556 0.508

 Panel B. Autocorrelation at Lags

 9 10 11 12 Q(12)

EWRET -0.001 0.054 0.070 0.101 37.7
VWRET 0.012 0.065 0.010 0.015 10.0
EWMV 0.081 0.069 0.031 0.040 38.7
VWMV 0.087 0.082 0.057 0.045 66.1
EWAV 0.622 0.631 0.640 0.665 2525.6
VWAV 0.653 0.609 0.553 0.546 2595.6
EWIV 0.690 0.705 0.715 0.736 3059.8
VWIV 0.521 0.501 0.471 0.463 1958.0

 Panel C. Cross Correlations

 EWRET VWRET EWMV

EWRET 1
VWRET 0.843 1
EWMV -0.262 -0.311 1
VWMV -0.301 -0.296 0.881
EWAV 0.136 0.021 0.429
VWAV 0.009 -0.019 0.355
EWIV 0.194 0.080 0.272
VWIV 0.167 0.134 -0.077

 Panel C. Cross Correlations

 VWMV EWAV VWAV EWIV VWIV

EWRET
VWRET
EWMV
VWMV 1
EWAV 0.460 1
VWAV 0.386 0.674 1
EWIV 0.327 0.986 0.653 1
VWIV -0.106 0.486 0.876 0.533 1

 Panel D. Autocorrelations of Innovations
 in Various Measures of Volatility

 1 2 3 4

EWAVS -0.019 -0.036 -0.040 -0.043
VWAVS -0.001 -0.013 -0.058 -0.003
EWIVMS -0.014 -0.003 -0.051 -0.075
VWIVMS -0.005 -0.029 -0.024 -0.037
EWIVS -0.017 -0.049 -0.037 -0.049
VWIVS -0.005 -0.002 -0.017 -0.002
EWIVFFS -0.011 -0.028 -0.045 -0.068
VWIVFFS -0.004 -0.021 -0.039 -0.009

 Panel D. Autocorrelations of Innovations
 in Various Measures of Volatility

 5 6 Q(6) p-value

EWAVS -0.057 -0.093 8.253 0.220
VWAVS 0.013 -0.058 3.844 0.698
EWIVMS -0.093 -0.112 14.816 0.022
VWIVMS -0.053 -0.063 4.661 0.588
EWIVS -0.046 -0.082 7.378 0.287
VWIVS -0.009 -0.023 0.633 0.996
EWIVFFS -0.091 -0.105 12.997 0.043
VWIVFFS -0.037 -0.094 5.979 0.426

 Panel E Cross Correlations of Innovations
 in Various Measures of Volatility

 EWAVS VWAVS EWIVMS VWIVMS

EWAVS 1
VWAVS 0.503 1
EWIVMS 0.457 0.169 1
VWIVMS 0.318 0.322 0.534 1
EWIVS 0.950 0.449 0.511 0.345
VWIVS -0.011 0.596 0.246 0.266
EWIVFFS 0.447 0.134 0.959 0.431
VWIVFFS 0.336 0.320 0.524 0.848

 Panel E Cross Correlations of Innovations
 in Various Measures of Volatility

 EWIVS VWIVS EWIVFFS VWIVFFS

EWAVS
VWAVS
EWIVMS
VWIVMS
EWIVS 1
VWIVS 0.130 1
EWIVFFS 0.512 0.235 1
VWIVFFS 0.351 0.272 0.520 1

Table II. Dynamic Effects of Innovations in Average Stock Volatility
on Excess Returns After Controlling for State Variables

This table contains the estimates of regression:

[r.sup.e.sub.t] = [alpha] + [6.summation over (q = 1)] [[beta].sub.q]
[[??].sub.t-q] + [lambda] [IV.sub.t-7] + [e.sub.t],

where [r.sup.e.sub.t] denotes the value-weighted portfolio excess
return, [[??].sub.t], is the estimated innovation in average
volatility, and [IV.sub.t-7] is the seven-period lagged average
volatility. We measure volatility as equal- and value-weighted average
variance from NYSE/Ame x/Nasdaq, NYSE/Amex, NYSE, and Nasdaq returns.
We report the Newey-West adjusted t-statistics in parentheses. We
conduct the tests of two hypotheses, [H.sub.1] and [H.sub.2], using
standard F-statistics and report their p- values in parentheses.
[R.sup.1] is the adjusted [R.sup.2].

 Panel A. Equal-weighted Average
 Volatility (NYSE/Amex/Nasdaq)

Period [[beta].sub.1] [[beta].sub.2] [[beta].sub.3]

1963:8- 0.133 0.449 0.483
1999:12 (0.417) (2.314) (2.843)
1963:8- 0.116 0.142 0.331
2001:12 (0.402) (0.566) (1.961)
1963:8- 0.025 0.088 0.422
2002:12 (0.093) (0.362) (2.504)

 Panel A. Equal-weighted Average
 Volatility (NYSE/Amex/Nasdaq)

Period [[beta].sub.4] [[beta].sub.5] [[beta].sub.6]

1963:8- 0.453 0.099 0.331
12.074
1999:12 (2.327) (0.517) (1.646)
-0.060
1963:8- 0.480 -0.086 0.265
8.247
2001:12 (2.811) (-0.468) (1.300)
-0.221
1963:8- 0.428 -0.082 0.316
12.573
2002:12 (2.547) (-0.452) (1.580)
-0.050

 Panel A. Equal-weighted Average
 Volatility (NYSE/Amex/Nasdaq)

 [summation]
Period [lambda] [[beta].sub.t] [R.sup.2]

1963:8- 0.224 1.949 0.012
1999:12 (1.852)
1963:8- 0.034 1.248 0.008
2001:12 (0.296)
1963:8- -0.009 1.198 0.011
2002:12 (-0.075)

 Panel A. Equal-weighted Average
 Volatility (NYSE/Amex/Nasdaq)

 [H.sub.1]:[summation] [H.sub.2]:[[beta].sub.t]=0
Period [[beta].sub.t]=0 for all t>0

1963:8- 10.258 19.088
1999:12 (0.001) (0.004)
1963:8- 3.759 11.315
2001:12 (0.053) (0.079)
1963:8- 3.703 13.740
2002:12 (0.054) (0.033)

 Panel B. Value-weighted Average
 Volatility (NYSE/Amex/Nasdaq)

Period [[beta].sub.1] [[beta].sub.2] [[beta].sub.3]

1963:8- -0.151 -0.213 0.849
1999:12 (-0.204) (-0.296) (1.249)
1963:8- -0.355 0.148 0.756
2001:12 (-0.570) (0.230) (1.258)
1963:8- -0.214 -0.342 1.002
2002:12 (-0.387) (-0.509) (1.864)

 Panel B. Value-weighted Average
 Volatility (NYSE/Amex/Nasdaq)

Period [[beta].sub.4] [[beta].sub.5] [[beta].sub.6]

1963:8- 1.534 0.451 1.577
1999:12 (2.633) (0.742) (2.458)
1963:8- 1.014 -0.141 1.216
2001:12 (1.629) (-0.246) (1.995)
1963:8- 1.099 -0.224 1.503
2002:12 (2.017) (-0.440) (2.557)

 Panel B. Value-weighted Average
 Volatility (NYSE/Amex/Nasdaq)

 [summation]
Period [lambda] [[beta].sub.t] [R.sup.2]

1963:8- 0.799 4.047 0.014
1999:12 (2.056)
1963:8- -0.157 2.636 0.009
2001:12 (-0.504)
1963:8- -0.211 2.824 0.022
2002:12 (-0.672)

 Panel B. Value-weighted Average
 Volatility (NYSE/Amex/Nasdaq)

 [H.sub.1]:[summation] [H.sub.2]:[[beta].sub.t]=0
Period [[beta].sub.t]=0 for all t>0

1963:8- 4.584 12.074
1999:12 (0.032) (0.060)
1963:8- 2.668 8.247
2001:12 (0.102) (0.221)
1963:8- 3.474 12.573
2002:12 (0.062) (0.050)

 Panel C. Equal-weighted Average
 Volatility (NYSE/Amex)

Period [[beta].sub.1] [[beta].sub.2] [[beta].sub.3]

1963:8- -0.058 0.480 0.507
1999:12 (-0.175) (2.442) (2.752)
1963:8- -0.084 0.373 0.483
2001:12 (-0.084) (1.628) (2.482)
1963:8- -0.092 0.243 0.543
2002:12 (-0.298) (0.944) (2.760)

 Panel C. Equal-weighted Average
 Volatility (NYSE/Amex)

Period [[beta].sub.4] [[beta].sub.5] [[beta].sub.6]

1963:8- 0.448 0.086 0.071
1999:12 (2.372) (0.456) (0.317)
1963:8- 0.387 0.031 0.108
2001:12 (1.867) (0.163) (0.504)
1963:8- 0.391 0.015 0.172
2002:12 (1.877) (0.078) (0.803)

 Panel C. Equal-weighted Average
 Volatility (NYSE/Amex)

 [summation]
Period [lambda] [[beta].sub.t] [R.sup.2]

1963:8- 0.200 1.533 0.006
1999:12 (1.045)
1963:8- 0.042 1.298 0.003
2001:12 (0.22)
1963:8- -0.028 1.271 0.004
2002:12 (0.145)

 Panel C. Equal-weighted Average
 Volatility (NYSE/Amex)

 [H.sub.1]:[summation] [H.sub.2]:[[beta].sub.t]=0
Period [[beta].sub.t]=0 for all t>0

1963:8- 4.709 16.355
1999:12 (0.030) (0.012)
1963:8- 3.084 9.959
2001:12 (0.079) (0.126)
1963:8- 2.968 10.649
2002:12 (0.085) (0.100)

 Panel D. Value-weighted Average
 Volatility (NYSE/Amex)

Period [[beta].sub.1] [[beta].sub.2] [[beta].sub.3]

1963:8- -0.285 -0.248 0.780
1999:12 (-0.317) (-0.281) (0.973)
1963:8- -0.552 -0.013 1.124
2001:12 (-0.679) (-0.017) (1.573)
1963:8- -0.366 -0.581 1.352
2002:12 (-0.537) (-0.720) (2.322)

 Panel D. Value-weighted Average
 Volatility (NYSE/Amex)

Period [[beta].sub.4] [[beta].sub.5] [[beta].sub.6]

1963:8- 1.318 0.614 1.636
1999:12 (1.985) (0.904) (2.263)
1963:8- 0.762 0.339 1.445
2001:12 (1.041) (0.505) (1.942)
1963:8- 0.951 0.053 1.734
2002:12 (1.562) (0.090) (2.430)

 Panel D. Value-weighted Average
 Volatility (NYSE/Amex)

 [summation]
Period [lambda] [[beta].sub.t] [R.sup.2]

1963:8- 0.858 3.815 0.008
1999:12 (1.703)
1963:8- 0.015 3.106 0.006
2001:12 (0.033)
1963:8- -0.089 3.142 0.019
2002:12 (-0.192)

 Panel D. Value-weighted Average
 Volatility (NYSE/Amex)

 [H.sub.1]:[summation] [H.sub.2]:[[beta].sub.t]=0
Period [[beta].sub.t]=0 for all t>0

1963:8- 2.799 8.440
1999:12 (0.094) (0.208)
1963:8- 2.243 7.116
2001:12 (0.134) (0.310)
1963:8- 2.774 12.873
2002:12 (0.096) (0.045)

 Panel E. Equal-weighted Average
 Volatility (NYSE)

Period [[beta].sub.1] [[beta].sub.2] [[beta].sub.3]

1963:8- -0.180 0.615 0.436
1999:12 (-0.515) (3.150) (2.152)
1963:8- -0.172 0.593 0.464
2001:12 (-0.503) (2.868) (2.283)
1963:8- -0.144 0.449 0.540
2002:12 (-0.425) (1.775) (2.636)

 Panel E. Equal-weighted Average
 Volatility (NYSE)

Period [[beta].sub.4] [[beta].sub.5] [[beta].sub.6]

1963:8- 0.536 0.207 0.058
1999:12 (2.717) (0.997) (0.228)
1963:8- 0.452 0.150 0.069
2001:12 (2.059) (0.720) (0.279)
1963:8- 0.494 0.110 0.136
2002:12 (2.272) (0.523) (0.557)

 Panel E. Equal-weighted Average
 Volatility (NYSE)

 [summation]
Period [lambda] [[beta].sub.t] [R.sup.2]

1963:8- 0.234 1.671 0.008
1999:12 (1.171)
1963:8- 0.100 1.556 0.005
2001:12 (0.495)
1963:8- 0.034 1.585 0.005
2002:12 (0.165)

 Panel E. Equal-weighted Average
 Volatility (NYSE)

 [H.sub.1]:[summation] [H.sub.2]:[[beta].sub.t]=0
Period [[beta].sub.t]=0 for all t>0

1963:8- 4.797 20.574
1999:12 (0.029) (0.002)
1963:8- 4.060 15.635
2001:12 (0.044) (0.016)
1963:8- 4.060 12.695
2002:12 (0.044) (0.048)

 Panel F. Value-weighted Average
 Volatility (NYSE)

Period [[beta].sub.1] [[beta].sub.2] [[beta].sub.3]

1963:8- -0.292 -0.279 0.754
1999:12 (0.314) (0.309) (0.908)
1963:8- -0.571 0.010 1.145
2001:12 (-0.685) (0.012) (1.558)
1963:8- -0.380 -0.585 1.373
2002:12 (0.717) (0.717) (2.319)

 Panel F. Value-weighted Average
 Volatility (NYSE)

Period [[beta].sub.4] [[beta].sub.5] [[beta].sub.6]

1963:8- 1.339 0.645 1.788
1999:12 (1.957) (0.935) (2.424)
1963:8- 0.715 0.374 1.537
2001:12 (0.943) (0.545) (1.995)
1963:8- 0.930 0.066 1.823
2002:12 (1.489) (0.110) (2.470)

 Panel F. Value-weighted Average
 Volatility (NYSE)

 [summation]
Period [lambda] [[beta].sub.t] [R.sup.2]

1963:8- 0.911 3.955 0.009
1999:12 (1.784)
1963:8- 0.033 3.210 0.006
2001:12 (0.071)
1963:8- -0.080 3.227 0.019
2002:12 (-0.173)

 Panel F. Value-weighted Average
 Volatility (NYSE)

 [H.sub.1]:[summation] [H.sub.2]:[[beta].sub.t]=0
Period [[beta].sub.t]=0 for all t>0

1963:8- 2.867 9.049
1999:12 (0.090) (0.171)
1963:8- 2.318 7.177
2001:12 (0.128) (0.305)
1963:8- 2.858 12.910
2002:12 (0.091) (0.044)

 Panel G. Equal-weighted Average
 Volatility (Nasdaq)

Period [[beta].sub.1] [[beta].sub.2] [[beta].sub.3]

1963:8- 0.276 0.324 0.340
1999:12 (1.077) (1.715) (2.015)
1963:8- 0.211 0.024 0.181
2001:12 (0.961) (0.114) (1.36)
1963:8- 0.093 0.002 0.270
2002:12 (0.422) (0.010) (2.011)

 Panel G. Equal-weighted Average
 Volatility (Nasdaq)

Period [[beta].sub.4] [[beta].sub.5] [[beta].sub.6]

1963:8- 0.322 0.083 0.442
1999:12 (1.608) (0.471) (2.371)
1963:8- 0.387 -0.096 0.263
2001:12 (2.678) (0.662) (1.377)
1963:8- 0.321 -0.096 0.293
2002:12 (2.270) (-0.645) (1.589)

 Panel G. Equal-weighted Average
 Volatility (Nasdaq)

 [summation]
Period [lambda] [[beta].sub.t] [R.sup.2]

1963:8- 0.124 1.786 0.012
1999:12 (1.033)
1963:8- -0.030 0.969 0.013
2001:12 (-0.283)
1963:8- -0.066 0.883 0.014
2002:12 (0.632)

 Panel G. Equal-weighted Average
 Volatility (Nasdaq)

 [H.sub.1]:[summation] [H.sub.2]:[[beta].sub.t]=0
Period [[beta].sub.t]=0 for all t>0

1963:8- 11.932 15.744
1999:12 (0.001) (0.015)
1963:8- 3.291 10.766
2001:12 (0.070) (0.096)
1963:8- 2.917 11.948
2002:12 (0.088) (0.063)

 Panel H. Value-weighted Average
 Volatility (Nasdaq)

Period [[beta].sub.1] [[beta].sub.2] [[beta].sub.3]

1963:8- 0.423 -0.192 0.696
1999:12 (1.138) (-0.564) (1.788)
1963:8- 0.302 0.079 0.201
2001:12 (1.106) (0.232) (0.667)
1963:8- 0.289 -0.099 0.365
2002:12 (1.068) (-0.297) (1.186)

 Panel H. Value-weighted Average
 Volatility (Nasdaq)

Period [[beta].sub.4] [[beta].sub.5] [[beta].sub.6]

1963:8- 1.239 -0.047 1.044
1999:12 (3.732) (-0.127) (2.947)
1963:8- 0.824 -0.523 0.510
2001:12 (2.83) (-1.775) (1.520)
1963:8- 0.733 -0.361 0.578
2002:12 (2.498) (-1.233) (1.816)

 Panel H. Value-weighted Average
 Volatility (Nasdaq)

 [summation]
Period [lambda] [[beta].sub.t] [R.sup.2]

1963:8- 0.268 3.163 0.030
1999:12 (1.326)
1963:8- -0.170 1.392 0.027
2001:12 (-1.018)
1963:8- -0.189 1.506 0.026
2002:12 (-1.141)

 Panel H. Value-weighted Average
 Volatility (Nasdaq)

 [H.sub.1]:[summation] [H.sub.2]:[[beta].sub.t]=0
Period [[beta].sub.t]=0 for all t>0

1963:8- 7.729 22.178
1999:12 (0.005) (0.001)
1963:8- 2.572 17.754
2001:12 (0.109) (0.007)
1963:8- 3.335 15.087
2002:12 (0.068) (0.020)

Table III. Dynamic Effects of Innovations in Average Stock Volatility
on Excess Returns after Controlling for State Variables and
Illiquidity Variables

This table contains the estimates of regressions:

[r.sup.e.sub.t] = [alpha] + [6.summation over (q=1)] [[beta].sub.q]
[[??].sub.t-q] + [[lambda]IV.sub.t-7] + [W.summation over (k=1)]
[[gamma].sub.k] [S.sub.k,t-1] +[e.sub.t],

[r.sup.e.sub.t] = [alpha] + [6.summation over (q=2)] [[beta].sub.q]
[[??].sub.t-q] + [[lambda]IV.sub.t-7] + [W.summation over (k=1)]
[[gamma].sub.k] [S.sub.k,t-1] + [[delta].sub.1][milliq.sub.t-1] +
[[delta].sub.2][umilliq.sub.t] +[esub.t],

where [r.sup.e.sub.t] denotes the value-weighted portfolio excess
return, [[??].sub.t] is the estimated innovation in average
volatility, and [IV.sub.t-7] is the seven-period lagged average
volatility. [S.sub.k] are state variables, such as term spread,
and dividend yield. Miliq and umilliq are the expected illiquidity
and unexpected illiquidity measure based on Amihud (2002). We
meaure volatility as equal- and value-weighted average variance
from NYSE/Amex/Nasdaq returns. We report the Newey-West adjusted
t-statistics in parentheses. We conduct the tests of two hypotheses,
[H.sub.1] and [H.sub.2], using standard F-statistics and report
their p-values in parentheses. [R.sup.2] is the adjusted [R.sup.2].

 Panel A. Equal-weighted Average Volatility (NYSE/Amex/Nasdaq)

Period [beta]1 [beta]2 [beta]3

1963:8- 0.144 0.425 0.438
1999:12 (0.463) (2.189) (2.598)
1963:8- 0.132 0.120 0.309
2001:12 (0.469) (0.482) (1.828)
1963:8- 0.035 0.067 0.403
2002:12 (0.129) (0.280) (2.411)

 Panel A. Equal-weighted Average Volatility (NYSE/Amex/Nasdaq)

Period [beta]4 [beta]5 [beta]6

1963:8- 0.388 0.058 0.296
1999:12 (1.918) (0.302) (1.486)
1963:8- 0.444 -0.118 0.243
2001:12 (2.570) (-0.653) (1.205)
1963:8- 0.393 -0.113 0.301
2002:12 (2.293) (-0.628) (1.524)

 Panel A. Equal-weighted Average Volatility (NYSE/Amex/Nasdaq)

Period [lambda] [gamma]1 [gamma]2

1963:8- 0.261 -0.451 -0.087
1999:12 (1.922) (-2.723) (0-040)
1963:8- 0.040 -0.370 -1.208
2001:12 (0.312) (-2.274) (-0.554)
1963:8- -0.016 -0.249 -1.246
2002:12 (-0.133) (-1.568) (-0.572)

 Panel A. Equal-weighted Average Volatility (NYSE/Amex/Nasdaq)

Period [gamma]3 [gamma]4 [summation]

1963:8- 2.426 -0.143 1.749
1999:12 (3.103) (0.112)
1963:8- 2.191 0.249 1.131
2001:12 (2.827) (0.194)
1963:8- 1.597 0.403 1.085
2002:12 (2.094) (0.315)

 Panel A. Equal-weighted Average Volatility (NYSE/Amex/Nasdaq)

Period [R.sup.2] [H.sub.1]: [H.sub.2]:
 [summation] [[beta].sub.t]=0
 [[beta].sub.t]=0 [for all]t>0

1963:8- 0.032 7.890 14.977
1999:12 (0.005) (0.020)
1963:8- 0.025 3.116 10.065
2001:12 (0.078) (0.122)
1963:8- 0.019 3.071 12.716
2002:12 (0.080) (0.048)

 Panel B. Value-weighted Average Volatility (NYSE/Amex/Nasdaq)

Period [beta]1 [beta]2 [beta]3

1963:8- 0.088 -0.143 0.855
1999:12 (0.115) (-0.203) (1.316)
1963:8- -0.140 0.239 0.794
2001:12 (-0.225) (0.385) (1.348)
1963:8- -0.069 -0.273 1.060
2002:12 (-0.126) (-0.399) (2.026)

 Panel B. Value-weighted Average Volatility (NYSE/Amex/Nasdaq)

Period [beta]4 [beta]5 [beta]6

1963:8- 1.449 0.370 1.503
1999:12 (2.561) (0.610) (2.422)
1963:8- 1.002 -0.118 1.235
2001:12 (1.667) (-0.207) (2.171)
1963:8- 1.081 -0.238 1.549
2002:12 (2.062) (-0.455) (2.795)

 Panel B. Value-weighted Average Volatility (NYSE/Amex/Nasdaq)

Period [lambda] [gamma]1 [gamma]2

1963:8- 0.495 -0.397 -1.667
1999:12 (1.198) (-2.547) (0.788)
1963:8- -0.352 -0.381 -2.209
2001:12 (-1.115) (-2.427) (-1.073)
1963:8- -0.370 -0.276 -2.212
2002:12 (-1.151) (-1.807) (-1.057)

 Panel B. Value-weighted Average Volatility (NYSE/Amex/Nasdaq)

Period [gamma]3 [gamma]4 [summation]

1963:8- 2.139 -0.078 4.122
1999:12 (2.848) (-0.061)
1963:8- 2.357 -0.426 3.013
2001:12 (3.124) (-0.336)
1963:8- 1.809 0.041 3.110
2002:12 (2.456) (0.032)

 Panel B. Value-weighted Average Volatility (NYSE/Amex/Nasdaq)

Period [R.sup.2] [H.sub.1]: [H.sub.2]:
 [summation] [[beta].sub.t]=0
 [[beta].sub.t]=0 [for all]t>0

1963:8- 0.034 4.632 11.559
1999:12 (0.031) (0.073)
1963:8- 0.033 3.548 8.761
2001:12 (0.060) (0.187)
1963:8- 0.037 4.262 13.880
2002:12 (0.039) (0.031)

 Panel C. Equal-Weighted Average Volatility (NYSE/Amex/Nasdaq)

Period [beta]1 [beta]2 [beta]3

1963:8- 0.131 0.319 0.58
1999:12 (0.511) (1.697) (3.089)
1963:8- 0.132 0.043 0.421
2001:12 (0.515) (0.174) (2.274)
1963:8- 0.023 0.034 0.547
2002:12 (0.093) (0.144) (3.050)

 Panel C. Equal-Weighted Average Volatility (NYSE/Amex/Nasdaq)

Period [beta]4 [beta]5 [beta]6

1963:8- 0.267 0.070 0.251
1999:12 (1.405) (0.360) (1.394)
1963:8- 0.313 -0.099 0.207
2001:12 (1.832) (-0.561) (1.064)
1963:8- 0.26 -0.083 0.277
2002:12 (1.528) (-0.466) (1.462)

 Panel C. Equal-Weighted Average Volatility (NYSE/Amex/Nasdaq)

Period [lambda] [gamma]1 [gamma]2

1963:8- 0.110 -0.388 0.598
1999:12 (0.810) (-2.318) (0.297)
1963:8- -0.072 -0.287 -0.490
2001:12 (-0.497) (-1.739) (-0.236)
1963:8- -0.070 -0.141 -0.670
2002:12 (-0.489) (-0.885) (-0.325)

 Panel C. Equal-Weighted Average Volatility (NYSE/Amex/Nasdaq)

Period [gamma]3 [gamma]4 [[delta].sub.1]

1963:8- 2.164 -0.879 -0.003
1999:12 (2.822) (-0.769) (-0.975)
1963:8- 1.871 -0.561 -0.002
2001:12 (2.446) (-0.487) (-0.492)
1963:8- 1.177 -0.710 0.001
2002:12 (1.565) (-0.613) (0.284)

 Panel C. Equal-Weighted Average Volatility (NYSE/Amex/Nasdaq)

Period [[delta].sub.2] [summation] [R.sup.2]
 [[beta].sub.t]

1963:8- -0.066 1.616 0.615
1999:12 (-7.482)
1963:8- -0.063 1.016 0.133
2001:12 (-7.016)
1963:8- -0.062 1.057 0.122
2002:12 (-6.918)

 Panel C. Equal-Weighted Average Volatility (NYSE/Amex/Nasdaq)

Period [H.sub.1]: [H.sub.2]:
 [summation] [[beta].sub.t]=0
 [[beta].sub.t]=0 [for all]t>0

1963:8- 6.926 13.623
1999:12 (0.008) (0.034)
1963:8- 2.401 8.879
2001:12 (0.121) (0.181)
1963:8- 2.821 13.682
2002:12 (0.093) (0.033)

 Panel D. Value-Weighted Average Volatility (NYSE/Amex/Nasdaq)

Period [beta]1 [beta]2 [beta]3

1963:8- 0.109 -0.521 1.231
1999:12 (0.149) (-0.693) (1.706)
1963:8- -0.057 -0.006 1.077
2001:12 (-0.092) (-0.009) (1.667)
1963:8- -0.013 -0.507 1.485
2002:12 (-0.024) (-0.730) (2.545)

 Panel D. Value-Weighted Average Volatility (NYSE/Amex/Nasdaq)

Period [beta]4 [beta]5 [beta]6

1963:8- 0.839 0.081 1.329
1999:12 (1.500) (0.135) (2.299)
1963:8- 0.580 -0.261 1.237
2001:12 (0.922) (-0.448) (2.236)
1963:8- 0.691 -0.391 1.534
2002:12 (1.278) (-0.707) (2.831)

 Panel D. Value-Weighted Average Volatility (NYSE/Amex/Nasdaq)

Period [lambda] [gamma]1 [gamma]2

1963:8- 0.018 -0.388 -0.494
1999:12 (0.042) (-2.347) (-0.255)
1963:8- -0.687 -0.361 -0.863
2001:12 (-1.892) (-2.168) (-0.447)
1963:8- -0.566 -0.200 -1.290
2002:12 (-1.516) (-1.269) (-0.655)

 Panel D. Value-Weighted Average Volatility (NYSE/Amex/Nasdaq)

Period [gamma]3 [gamma]4 [[delta].sub.1]

1963:8- 2.190 -0.565 -0.004
1999:12 (2.873) (-0.501) (-1.146)
1963:8- 2.357 -0.911 -0.004
2001:12 (3.055) (-0.815) (-1.015)
1963:8- 1.541 -0.788 0.000
2002:12 (2.080) (-0.702) (-0.148)

 Panel D. Value-Weighted Average Volatility (NYSE/Amex/Nasdaq)

Period [[delta].sub.2] [summation] [R.sup.2]
 [[beta].sub.t]

1963:8- -0.066 3.068 0.165
1999:12 (-7.378)
1963:8- -0.064 2.571 0.147
2001:12 (-7.216)
1963:8- -0.064 2.799 0.144
2002:12 (-7.282)

 Panel D. Value-Weighted Average Volatility (NYSE/Amex/Nasdaq)

Period [H.sub.1]: [H.sub.2]:
 [summation] [[beta].sub.t]=0
 [[beta].sub.t]=0 [for all]t>0

1963:8- 2.372 8.131
1999:12 (0.124) (0.229)
1963:8- 2.495 8.023
2001:12 (0.114) (0.236)
1963:8- 3.352 14.021
2002:12 (0.067) (0.029)

Table IV. Dynamic effects of innovations in Idiosyncratic volatility
on excess returns

This table contains estimation results of the regression:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [r.sup.e.sub.t] denotes the value-weighted portfolio excess
return, ut is the estimated innovation in volatility, and [IV.sub.t-7]
is the seven-period lagged volatility. We measure the volatility as
NYSE/Amex/Nasdaq value-weighted variance, which we calculate from the
difference between average volatility and market volatility by
using high frequency (daily) data, the market model residuals using
low frequency (monthly) data. We report the Newey-West adjusted
t-statistics in parentheses. We conduct the test of two hypotheses,
[H.sub.1] and [H.sub.2], using standard F-statistics and report
their p-values in parenthesis. [R.sup.2] is the adjusted [R.sup.2].

Panel A. Value-weighted Idiosyncratic Volatility (Constant Mean Model)

Period [[beta].sub.1] [[beta].sub.2] [[beta].sub.3]

1963:8- 1.063 -0.512 0.109
1999:12 (2.947) (-1.467) (0.200)
1963:8- 0.761 -0.471 0.139
2001:12 (1.632) (-1.196) (0.257)
1963:8- 0.794 -0.645 0.249
2002:12 (1.776) (-1.597) (0.413)

Panel A. Value-weighted Idiosyncratic Volatility (Constant Mean Model)

Period [[beta].sub.4] [[beta].sub.5] [[beta].sub.6]

1963:8- 0.157 0.654 0.820
1999:12 (0.273 (1.997) (1.775)
1963:8- 0.153 0.369 0.848
2001:12 (0.289) (1.100) (1.637)
1963:8- 0.264 0.377 0.914
2002:12 (0.482) (1.109) (1.736)

Panel A. Value-weighted Idiosyncratic Volatility (Constant Mean Model)

Period [lambda] [summation] [R.sup.2]
 [[beta].sub.t]

1963:8- 0.523 2.291 0.008
1999:12 (1.614)
1963:8- -0.200 1.798 0.004
2001:12 -(0.571)
1963:8- -0.243 1.953 0.009
2002:12 (-0.695)

Panel A. Value-weighted Idiosyncratic Volatility (Constant Mean Model)

Period [H.sub.1]:[summation] [H.sub.2]:[summation]
 [[beta].sub.t]=0 [[beta].sub.t]=0
 [for all]t>0

1963:8- 3.224 19.991
1999:12 (0.073) (0.003)
1963:8- 1.890 9.809
2001:12 (0.169) (0.133)
1963:8- 2.244 12.500
2002:12 (0.134) (0.052)

Panel B. Value-weighted Idiosvncratie Volatility (Market Model)

Period [[beta].sub.1] [[beta].sub.2] [[beta].sub.3]

1963:8- -0.034 -0.382 2.625
1999:12 (-0.030) (-0.322) (2.274)
1963:8- -0.046 -1.129 0.841
2001:12 (-0.066) (-1.572) (0.963)
1963:8- 0.117 -1.187 0.726
2002:12 (0.170) (-1.644) (0.877)

Panel B. Value-weighted Idiosvncratie Volatility (Market Model)

Period [[beta].sub.4] [[beta].sub.5] [[beta].sub.6]

1963:8- 0.667 1.993 0.386
1999:12 (0.645) (1.989) (0.339)
1963:8- 0.352 0.638 1.178
2001:12 (0.612) (0.972) (1.576)
1963:8- 0.522 0.657 1.159
2002:12 (0.902) (1.043) (1.541)

Panel B. Value-weighted Idiosvncratie Volatility (Market Model)

Period [lambda] [summation] [R.sup.2]
 [[beta].sub.t]

1963:8- 0.747 5.255 0.005
1999:12 (0.996)
1963:8- -0.634 1.834 0.011
2001:12 (-1.606)
1963:8- -0.698 1.995 0.012
2002:12 (-1.814)

Panel B. Value-weighted Idiosvncratie Volatility (Market Model)

Period [H.sub.1]:[summation] [H.sub.2]:[summation]
 [[beta].sub.t]=0 [[beta].sub.t]=0
 [for all]t>0

1963:8- 2.229 8.084
1999:12 (0.135) (0.232)
1963:8- 0.937 9.229
2001:12 (0.333) (0.161)
1963:8- 1.175 10.475
2002:12 (0.278) (0.106)

Panel C. Value-weighted Idiosyncratic Volatility (Market Model)

Period [[beta].sub.1] [[beta].sub.2] [[beta].sub.3]

1926:7- 3.001 0.420 -1.003
1999:12 (2.424) (0.575) (-1.464)
1926:7- 2.499 0.132 -0.888
2001:12 (2.233) (0.214) (-1.650)
1926:7- 2.532 0.113 -0.932
2002:12 (2.282) (0.183) (-1.739)

Panel C. Value-weighted Idiosyncratic Volatility (Market Model)

Period [[beta].sub.4] [[beta].sub.5] [[beta].sub.6]

1926:7- -0.512 0.194 -1.102
1999:12 (-0.725) (0.308) (-1.415)
1926:7- -0.189 0.049 -0.581
2001:12 (-0.322) (-0.096) (-0.882)
1926:7- -0.133 0.050 -0.592
2002:12 (-0.227) (0.098) (-0.895)

Panel C. Value-weighted Idiosyncratic Volatility (Market Model)

Period [lambda] [summation] [R.sup.2]
 [[beta].sub.t]

1926:7- -0.207 0.998 0.048
1999:12 (-0.311)
1926:7- -0.463 1.023 0.036
2001:12 (-0.986)
1926:7- -0.504 1.038 0.038
2002:12 (-1.088)

Panel C. Value-weighted Idiosyncratic Volatility (Market Model)

Period [H.sub.1]:[summation] [H.sub.2]:[summation]
 [[beta].sub.t]=0 [[beta].sub.t]=0
 [for all]t>0

1926:7- 0.098 14.409
1999:12 (0.754) (0.025)
1926:7- 0.173 11.149
2001:12 (0.678) (0.084)
1926:7- 0.180 11.881
2002:12 (0.672) (0.065)

Table V. Idiosyncratic volatility, fundamentals, and hedge component

This table contains estimates of the following regressions:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [IV.sub.t] denotes idiosyncratic volatility, [dy.sub.t],
[rtb.sub.t], [term.sub.t], [def.sub.t], and [cay.sub.t] denote
dividend yield, relative T-Bill, term spread, default spread, and
consumption-wealth ratio and time t. In Panels A, B, and C,
we report estimates and the Newey-West adjusted t-statistics are
reported in parenthesis. The p-value is the significant level of
F-test, in which the null hypothesis is that all coefficients of
fundamentals (state variables) are equal to zero. [R.sup.2] is the
adjusted [R.sup.2]. EWAV, VWAV, EWIV, VWIV, EWIVM, and VWIVM denote
monthly CRSP index equal- and value-weighted average stock volatility,
idiosyncratic volatility (constant mean return model) and
idiosyncratic volatility (market model), respectively. The sample
is from August 1963 to June 2001.

 Panel A. Average Volatility

 constant dy rtb term

EWAV 0.034 -3.020 -2.924 0.139
 (11.625) (-5.890) (-3.451) (2.274)
 -0.024 -3.044 -2.507 0.070
 (-1.041) (-5.972) (-2.922) (1.050)

VWAV 0.012 -1.122 0.188 -0.020
 (10.135) (-5.368) (0.543) (-0.785)
 0.044 -1.109 -0.042 0.019
 (4.757) (-5.370) (-0.122) (0.691)

 Panel A. Average Volatility

 def cay p-value [R.sup.2]

EWAV -0.461 0.000 0.088
 (-1.510)
 -0.281 0.161 0.000 0.099
 (-0.901) (2.553)

VWAV 0.243 0.000 0.057
 (1.957)
 0.144 -0.089 0.000 0.080
 (1.141) (-3.477)

Panel B. Idiosyncratic Volatility (Constant Mean Return Model)

 constant dy rtb term

EWIV 0.032 -2.962 -3.318 0.162
 (11.719) (-6.225) (-4.219) (2.855)
 -0.033 -2.989 -2.854 0.085
 (-1.538) (-6.338) (-3.596) (1.381)

VWIV 0.011 -1.010 -0.090 -0.011
 (9.227) (-5.047) (-0.271) (-0.471)
 0.036 -0.999 -0.271 0.019
 (4.021) (-5.034) (-0.813) (0.730)

Panel B. Idiosyncratic Volatility (Constant Mean Return Model)

 def cay p-value [R.sup.2]

EWIV -0.605 0.000 0.107
 (-2.135)
 -0.404 0.180 0.000 0.124
 (-1.403) (3.072)

VWIV 0.115 0.050
 (0.966)
 0.036 -0.070 0.000 0.065
 (0.301) (-2.858)

Panel C. Idiosyncratic Volatility (Market Model)

 constant dy rtb term

EWIVM 0.025 -2.830 -3.594 0.115
 (7.227) (-4.692) (-3.605) (1.598)
 -0.015 -2.847 -3.306 0.067
 (-0.557) (-4.725) (-3.260) (0.854)

VWIVM 0.007 -0.772 0.180 -0.004
 (9.075) (-5.549) (0.785) (-0.249)
 0.027 -0.763 0.037 0.020
 (4.390) (-5.548) (0.161) (1.092)

Panel C. Idiosyncratic Volatility (Market Model)

 def cay p-value [R.sup.2]

EWIVM -0.045 0.000 0.068
 (-0.125)
 0.079 0.112 0.000 0.071
 (0.216) (1.492)

VWIVM 0.157 0.060
 (1.901)
 0.096 -0.055 0.000 0.079
 (1.136) (-3.247)
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Author:Jiang, Xiaoquan; Lee, Bong-Soo
Publication:Financial Management
Geographic Code:1USA
Date:Jun 22, 2006
Words:13545
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