# Expected idiosyncratic volatility measures and expected returns.

We find that idiosyncratic volatility forecasts using information available to traders at the time of the forecast are not related to expected returns. The positive relation documented in a number of other papers only exists when forward-looking information is incorporated into the volatility estimate. That positive relation is driven by the realized idiosyncratic volatility component that cannot be forecasted by investors. Our findings are robust m several different empirical tests, volatility forecasting models and time periods.**********

Depending upon the study, idiosyncratic volatility is positively related, negatively related, or unrelated to expected returns. For example, Fu (2009) finds that expected returns are positively related to idiosyncratic volatility forecasts in the cross-section. (1) In contrast, Ang et al. (2006, 2009) find a negative relation between lagged idiosyncratic volatility and subsequent returns. Resolving this apparent inconsistency is important for two reasons. First, theoretical asset pricing papers suggest a positive idiosyncratic volatility-return relationship (see Merton (1987) and Malkiel and Xu (2002), for example). Second, any predictable relation may give rise to exploitable trading opportunities. We demonstrate that the measurement of idiosyncratic volatility is crucial to inferences concerning its relationship with expected returns. In particular, that look-ahead bias can induce a spurious positive relation. The ramifications of the look-ahead bias are evident for a broad range of potential idiosyncratic volatility definitions.

Fu (2009) defines expected idiosyncratic volatility as the best conditional volatility forecast from a set of models given monthly return information available to traders at the time of the forecast. However, the datasets used to form the volatility forecasts in Fu (2009) include forward-looking information. Specifically, one forward-looking return observation is included in the estimation. We find that when idiosyncratic volatility forecasts are properly determined from information available to traders, there is no relationship between these forecasts and expected returns. Bali and Cakici (2008) contest the robustness of the Ang et al. (2006) relationship, suggesting the result to be largely driven by small firms. Huang et al. (2010) and Hart and Lesmond (2011) demonstrate that the negative association between expected returns and lagged realized idiosyncratic volatility of Ang et al. (2006) are driven by return reversals and liquidity effects, respectively. Thus, we see that there does not exist a relationship between expected return and idiosyncratic volatility formed with information sets available to traders, whether we use the estimation methods as in Fu (2009) or as in Ang et al. (2006). When we incorporate forward-looking information, however, we find a strong positive relationship in the cross-section. This result holds whether we form idiosyncratic volatility using the estimation methods in either Fu (2009) or Ang et al. (2006). Use of the full information set may be appropriate when testing theoretical models such as Merton (1987) and Malkiel and Xu (2002), as their models assume that all parameters in the model are known. Our findings therefore confirm the positive association between expected idiosyncratic volatility and expected return put forth in those theories, given the strong informational assumptions about the agents.

The pricing implications of idiosyncratic volatility have long been considered. The capital asset pricing model (CAPM) model of Sharpe (1964) and Lintner (1965) and extended by Black (1972) provides a theoretical framework in which idiosyncratic volatility is irrelevant for asset pricing, and this is generally supported in early empirical work (e.g., see Fama and MacBeth, 1973). Subsequent research presents more sophisticated factor models, but these alternative characterizations of the portfolio problem reinforce the asset pricing insignificance of idiosyncratic risk. In contrast, Fu (2009) provides an argument for a positive relationship between expected idiosyncratic volatility and asset returns, save for the problem of forward-looking information.

We estimate the relation between idiosyncratic volatility and returns as follows. First, we estimate abnormal returns to positions formed from portfolios sorted by forecasted idiosyncratic volatility, where forecasted idiosyncratic volatility is variously defined using both backward and forward-looking information to form the forecasts. Secondly, we estimate Fama-MacBeth regressions to test the relationship of the variously defined idiosyncratic volatility forecasts with cross-sectional returns. Further, we repeat these tests for several different time periods. Our results robustly demonstrate that even one additional forward-looking observation significantly affects inference concerning the expected idiosyncratic volatility to return relationship. When expected idiosyncratic volatility is formed from information available to a trader, no positive relationship with expected returns is present in the cross-section. However, consistent with Fu (2009), we find that the inclusion of forward-looking information into any of a broad variety of idiosyncratic volatility estimates does indicate a significant positive relationship.

The rest of the paper proceeds as follows. In Section I, we introduce several measures of idiosyncratic volatility common in the literature. In Section II, we present our data, including the idiosyncratic volatility measures that we test, and discuss their appropriate use. In Section III, we examine the cross-sectional relationships between idiosyncratic volatility measures, control variables, and expected returns. We provide a series of robustness checks with an emphasis on the stability of the idiosyncratic volatility to expected return relationship through time in Section IV. In Section V, we present our conclusion.

I. Measuring Idiosyncratic Volatility

A straightforward way to estimate firm-level monthly idiosyncratic volatility is to use a GARCH-type model over a rolling or expanding time window using monthly data. While the standard GARCH model assumes a symmetric response of volatility to returns, the EGARCH model allows for an asymmetry in that relationship. (2) Consider an EGARCH (p,q) model of the following form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1)

where [R.sub.it] is the return to asset i in month t [member of]{1, ..., T}, [r.sub.ft] is the risk free rate in month t. [[beta].sub.ik] are the sensitivities of firm i to the K monthly factors indexed by k. For a particular choice of p and q, we may estimate a sequence of in-sample monthly idiosyncratic volatility forecasts, or use the estimated parameters to forecast out-of-sample.

Use of the EGARCH model seems prudent; there exists significant evidence of an asymmetric effect of asset returns on volatility, though these studies tend to focus on in-sample performance. (3) Further, the use of monthly data mitigates the microstructure concerns of Bali and Cakici (2008). However, there are shortcomings to this approach as well. Out-of-sample performance of this family of models has occasionally been called into question, though Andersen and Bollerslev (1998) provide a convincing defense of the forecasting ability of GARCH models.

Expected idiosyncratic volatility may be estimated by a forecast from the EGARCH (p,q) model in Equation (1) using:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2)

where [PHI] is the information set. When we use a generic information set, we will not include a time subscript on [PHI], as in Equation (2). A subscript indicates the (inclusive) most recent data available in the information set. For example, [[PHI].sub.t] indicates that the most recent data available in the information set includes time t variables, while [[PHI].sub.T] indicates all data is included in the information set. Spiegel and Wang (2005), Fu (2009), Huang et al. (2010), Brockman, Schutte, and Yu (2010), Eiling (2011), and Peterson and Smedema (2011) produce expected volatility forecasts using Equation (2), with varying information sets. (4) Each of these papers uses the Fama and French (1992, 1993) factors--the excess market return (MKTRF), the returns to small firms less the return to large firms (small minus big, SMB), and the returns to high book-to-market firms less the returns to low book-to-market firms (high minus low, HML).

Fu (2009) forms expected idiosyncratic volatilities using the functional form in Equation (2). Since these forecasts include one forward-looking return observation, Fu (2009) estimates Equation (2) as [[??].sup.2.sub.i,t]|[[PHI].sub.t]. (5) Brockman et al. (2010), Peterson and Smedema (2011), and Eiling (2011) use [[??].sub.i,t]|[[PHI].sub.t] for all t [member of] {1, ..., T) for their idiosyncratic volatility estimates. All three papers find idiosyncratic volatility to be priced when using information from the full dataset, consistent with both the findings of Fu (2009) and our results in Section III. Huang et al. (2010) use a rolling 30-month window to estimate EGARCH(p,q) models (for the variable denoted IV4 in their paper), as opposed to an expanding window of observations. However, they choose their optimal p and q for the model using the full dataset, also inducing a look-ahead bias. (6) Therefore, Huang et al. (2010) use [[??].sub.i,t]|[[PHI].sub.t] as well.

An alternative to using Equation (1) is to estimate firm-level monthly idiosyncratic volatility directly from the daily return observations in that month. Typically, a linear factor model of the form:

[R.sub.i[tau]] - [r.sub.f[tau]] = [[alpha].sub.i] + [M.summation over (j=1)] [[beta].sub.ij] [F.sub.j[tau]] + [[epsilon].sub.i[tau]] [[epsilon].sub.i[tau]] ~ iid(0, [[sigma].sup.2.sub.it]), (3)

For asset i is estimated for month t. Trading days in month t are indexed by [tau], [[beta].sub.ij] are the sensitivities of firm i to the M factors indexed by j. [R.sub.i[tau]] is the return to asset i during day [tau]. [r.sub.f[tau]] is the daily risk free rate. The idiosyncratic volatility for month t is then defined as [[??].sub.i,t] the sample standard deviation of the residuals in the month. This estimate is often adjusted to provide a value on a monthly scale.

Idiosyncratic volatility forecasts may be made using the information embedded in Equation (3). The simplest way to do this is to assume (counterfactually) that [[??].sub.i,t] follows a random walk. Then,

[[??].sub.i,t]|[[PHI].sub.t] = [[??].sub.i,t]. (4)

Ang et al. (2006) define idiosyncratic volatility according to Equations (3) and (4). Using previous month idiosyncratic volatilities as determined by the Fama-French three factor model, they form quintile portfolios. They find that the highest idiosyncratic volatility quintile underperforms the lowest quintile by 1.06% per month. Ang et al. (2009) find a similar relationship exists in developed international markets.

Estimation of monthly idiosyncratic volatilities from daily data as in Equation (3) is a simple and robust method for the estimation of these quantities. In particular, such estimates have the appeal of a stable, simple model and that the estimate [[??].sub.i,t] has known statistical properties. There are some drawbacks to this estimation method, however. First, there is no formal link between volatility estimates in adjacent months. Since there exists strong evidence that volatility exhibits some form of autocorrelation, inefficient estimates result. Further, Bali and Cakici (2008) uncover what appears to be significant microstructural noise, most pronounced in small firms, spuriously driving the idiosyncratic volatility estimates obtained from daily data in Equation (3).

Alternatively, if we assume that the volatility series does not follow a random walk, we may model [[??].sub.i,t] using an autoregressive process, and create out-of-sample forecasts from the resulting parameter estimates. For example, if we construct a sample of estimates of [[??].sub.i,t] (formed using Equation (3)), then we may fit an ARMA model of the form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5)

where [[epsilon].sub.it] is the time series error for asset i in month t. We may form a time t forecast of expected idiosyncratic volatility for one month ahead as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6)

Chua, Goh, and Zhang (2010) examine expected idiosyncratic volatility for a special case of this specification, for p = 2 and q = 0 and find that expected idiosyncratic volatility is not priced, except jointly with unexpected idiosyncratic volatility.

Bali and Cakici (2008) also use a monthly return-derived estimate of idiosyncratic volatility similar to Equation (3). Specifically, they estimate

[R.sub.it] - [r.sub.ft] = [[alpha].sub.i] + [M.summation over (j=1)] [[beta].sub.ij] [F.sub.jt] + [[epsilon].sub.it] [[epsilon].sub.it] ~ iid (0, [[sigma].sup.2.sub.it]), (7)

for asset i,. t indexes months, and [R.sub.it] is the return to asset i during month t. [r.sub.ft] is the monthly risk free rate. [[beta].sub.ij] are the sensitivities of firm i to the M factors indexed by j. The factors are taken to be Fama-French monthly factors, and [[??].sub.i,t] | [[PHI].sub.t-1] is [[??].sub.i,t] estimated from the preceding 60 months of returns.

Given the different sources of error of the various idiosyncratic volatility estimates, examination of the cross-sectional implications of a range of volatility forecasts may yield stronger conclusions than examination of any model individually. As we see in Section III, our finding that idiosyncratic volatility forecasts conditional on time t-1 information are not priced is robust to the forecast model chosen. Further, the finding that expected idiosyncratic volatility and expected return have a positive relationship using forward-looking information is similarly robust to the estimation method.

II. Data and Variables

A. Volatility Variables

Our dataset includes monthly and daily data on stocks traded on the NYSE, Amex, and NASDAQ during the period of July 1963 to December 2008, for a total of 546 months. The return data are from Center for Research in Security Prices (CRSP), and firm accounting variables are from Compustat. Consistent with previous studies, monthly observations with returns greater than 300% are deleted from the sample.

We estimate our first set of idiosyncratic volatilities using EGARCH(p,q) models consistent with Equation (2). First, using the entire return series we select the best fitting EGARCH (p,q) model for each stock using the AIC criterion. We evaluate nine combinations of p,q [member of] {12,3}. We estimate the in-sample conditional variance and take the square root. The result is [[??].sub.i,t] | [[PHI].sub.T] We denote this variable of expected idiosyncratic volatility formed from monthly returns using information set [[PHI].sub.T] as M_EGARCH(T). For all volatility estimates, we specify our factor model as in Fama and French (1992). (7)

We use a second EGARCH(p,q) implementation to capture, [[??].sub.i,t] | [[PHI].sub.t] For this variable, we follow the methodology outlined in Fu (2009) and fit nine different EGARCH([p.bar],q) models for each p,q [member of] {1,2,3} combination. However, for each stock at each time period-we use information available up to time t to forecast time t volatility. This in-sample estimate corresponds with the implementation of Fu (2009), as described in Fu (2010). Following Fu (2009) we only include firms in the sample once they have 30 observations, therefore our first observations in the analysis are in January of 1966. We use the AIC criterion to select the best model for each firm each month, and compute the expected idiosyncratic volatility from that model. We denote [[??].sub.i,t] | [[PHI].sub.t] computed this way M_EGARCH(t).

Our third EGARCH implementation proceeds similarly to our second, but uses one less return observation. We then forecast expected idiosyncratic volatility one month ahead. We fit nine EGARCH models as in M_EGARCH(t) and use the AIC criterion to determine the optimal forecast, thus estimating [[??].sub.i,t] | [[PHI].sub.t-1]. This is a forecast made using only information available to traders at the time of the investment decision, and so we denote our volatility estimate computed this way by M_EGARCH(t-1).

Implementation of EGARCH idiosyncratic volatility forecasting naturally gives rise to methodological concerns. For example, the EGARCH family of models may require a significant number of observations to be reliable, as noted for example in Lundblad (2007). The median number of observations for an EGARCH forecast in our dataset is 106.

Our dataset consists of about 2.5 million firm-month observations, from which we estimate almost 22 million EGARCH forecasts (so as to produce forecasts for all nine p, q combinations). While we find that 27% of our EGARCH models estimated in our formation of M_EGARCH(t-1) do not converge, in only 0.78% of cases do all nine EGARCH models fail to converge, resulting in a missing M_EGARCH(t-1). In 75% of our cases, six or more EGARCH models successfully converge. Table I reports the frequency with which each EGARCH(p,q) model is chosen for both M_EGARCH(t-1) and M_EGARCH(t) using the AIC criterion. As expected, the convergence characteristics for both variables are quite similar. Notably, and consistent with Fu (2009), the three EGARCH (3,q) models are a significant plurality of the models chosen, comprising about 40% of the total. In unreported results, we exclude all firm-month observations in which four or more EGARCH models fail to converge. Our results are not changed in any meaningful way when we use this alternative dataset.

Guo, Kassa, Ferguson (2010) also note the look-ahead bias of Fu (2009). However, Fu (2010) provides a refutation of the methodology of Guo et al. (2010), asserting that their results are fatally affected by two computational issues. Specifically, Fu (2010) claims that Guo et al. (2010) a) include observations in which the EGARCH models have not converged properly, and b) do not allow for a sufficiently high maximum number of iterations for the EGARCH estimations. For our dataset, we only accept EGARCH observations for which convergence has been properly achieved. (8) Further, while Fu (2010) finds that using a maximum number of iterations of 500 is sufficient, we allow for a maximum of 2,000 iterations in our computations. (9) We include the average number of iterations necessary to achieve convergence for each model in Table I and the results there provide strong support for the claim in Fu (2010) that the 100 maximum iterations used in Guo et al. (2010) is insufficient to achieve proper convergence.

Our second set of idiosyncratic volatilities we estimate using models consistent with Equation (3). We estimate monthly realized idiosyncratic volatility for individual stocks from daily returns. Specifically, we regress daily excess returns of stock i for a given month on the Fama-French three factor model, recording the resulting residuals and computing their standard deviation. We then multiply the standard deviation by the square root of the number of trading days in the month. We denote [[??].sub.i,t] | [[PHI].sub.t] constructed this way using daily data by D_SQRET(t). We also examine [[??].sub.i,t] | [[PHI].sub.t-1], which is just the previous month's estimate of idiosyncratic volatility, as our corresponding variable when we test portfolio management implications. This is the variable of interest in Ang et al. (2006). We term this lagged estimate D_SQRET(t-1).

We also report forecasts of future idiosyncratic volatilities formed using the autoregressive model presented in Equation (6). In this case, we fit an ARMA (p,q) model to an expanding window of observations. For each firm i and each month t, we use all previous observations to estimate the parameters of all ARMA (p,q) models for p,q [member of] {0,1,2,3}. We then use the AIC criterion to choose the best model and forecast the next month's idiosyncratic volatility. We do not make forecasts for the first 30 months of an asset's trading history, to avoid poor small sample forecasts that may contaminate our results. For these ARMA(p,q) forecasts, the expected value [[??].sub.i,t] | [[PHI].sub.t-1] is defined D_SQRT_ARMA (t-1).

Consistent with Equation (7), we also estimate [[??].sub.i,t] | [[PHI].sub.t-1] using the preceding 60 monthly returns. Naturally, contemporaneous returns are not used in this estimate, which is denoted M_SQRET(t-1).

We also compute an unexpected idiosyncratic volatility series for each firm. We define "unexpected" relative to the information set of a trader at time t-1. To form our unexpected volatility series for each firm i, we estimate the regression

[M_EGARCH(T).sub.i,t] = [[alpha].sub.i] + [[beta].sub.i][(M_EGARCH(t - 1).sub.i,1]) + [e.sub.i,t]. (8)

Estimates of [e.sub.i,t] give the estimated unexpected idiosyncratic volatility at time t for firm i. (10) We denote these estimates UNEXPECTED. (11)

Panel A of Table II provides the descriptive statistics for our idiosyncratic volatility variables. To reduce the effect of extreme observations, we winsorize all volatility variables monthly at the 0.5% and 99.5% level. As expected, M_EGARCH(t-1) is considerably more volatile than either M_EGARCH(t) or M_EGARCH(T). It is also worth noting that the mean and median of M_EGARCH(t-1) is lower than the mean of M_EGARCH(T).

Since volatility is a latent variable, even volatility estimates that use the full sample period are still only estimates. Of course, M_EGARCH(T), M_EGARCH(t), M_EGARCH(t-1), D_SQRET_ARMA(t-1), and D_SQRET(t) are all estimates of the same quantity--firm level idiosyncratic volatility. Importantly, however, estimates using forward-looking information may be much more precise than estimates that would be available to investors at the time of the trade decision. Therefore, while the full information set estimates are appropriate for asset pricing model evaluation, they are not appropriate for evaluation of trading strategies. Even a single additional forward-looking observation in the forecast can provide substantial increases in the precision of the volatility estimate beyond what is available to investors. Volatility estimates other than those that include time T information calculate rolling parameters, and estimation errors in parameters likely drive a lot of the variation in limited information settings. (12) Small additions to the information set (such as one additional observation at t) can significantly improve the estimate.

For example, consider a firm such as Microsoft (CRSP PERMNO 10107). During our sample period, Microsoft transforms from a small computer company to one of the largest companies in the world. Panel A of Figure 1 illustrates M_EGARCH(t) and M_EGARCH(t-1) for this firm. The standard deviation of M_EGARCH(t-1) for Microsoft is 5.90. This is 231% larger than the corresponding standard deviation of M_EGARCH(t) for Microsoft, which is 2.55. The inclusion of even one additional observation in the estimation yields significantly more precise estimates of idiosyncratic volatility. The relative stability of M_EGARCH(t) is not limited to early in Microsoft's corporate life, though it is pronounced in that region (also recall, we exclude the first 30 observations of volatility forecasts to increase the stability of the results). Limiting our attention to the January 2002-December 2008 period, the standard deviation of M_EGARCH(t) is 1.40. The standard deviation of M_EGARCH(t-1) is 1.75, 25% greater. By this point in the sample Microsoft is a mature company with stable cash flows, yet still, one additional return observation provides notably greater precision in the volatility estimates. M_EGARCH(T), shown in Panel B for comparison, exhibits substantially less variation. The standard deviation over the entire time period for M_EGARCH(T) is 0.82. The differences between M_EGARCH(t-1) and M_EGARCH(T) that are apparent in Figure 1 underscore the importance of choosing the information set appropriate to the financial question at hand.

[FIGURE 1 OMITTED]

The pattern of increased precision of M_EGARCH(t) relative to M_EGARCH(t-1) is present throughout our sample. When we take the standard deviation of M_EGARCH(t) and M_EGARCH(t-1) for each firm, the averages of these standard deviations are 7.15 and 9.91, respectively, making the average standard deviation of M_EGARCH(t-1) 38% greater. This effect is not driven by young firms. If we confine ourselves only to firms that have at least ten years of monthly observations in our sample, the effect is still present. While the forecasts are naturally more precise in this case, the average of firm-level standard deviations of M_EGARCH(t) and M_EGARCH(t-1) are 5.57 and 7.72, respectively. M_EGARCH(t-1) variability remains about 38% greater. (13) We see similar results if we consider the medians rather than averages.

Table III provides a correlation matrix for the idiosyncratic volatility variables. M_EGARCH(t-1) and M_EGARCH(t) have correlations of only about 0.52, which provides us a preliminary indication that the one additional return observation used in M_EGARCH(t) may have a greater effect on the forecast than intuition might suggest. The correlation between M_EGARCH(t-1) and D_SQRET(t) is only about 25%. All correlations are significant at the 1% level.

B. Control Variables

In our portfolio sorts and cross-sectional regressions in Section III we incorporate several control variables that have been demonstrated to correlate with expected returns in previous literature. We include the systematic risk betas from the Fama-French model and denote the market beta, small-minus-big beta and high-minus-low beta as MKTBETA, SMBBETA, and HMLBETA, respectively. We measure a firm's size by the market value of equity (ME) at the end of the fiscal year as in Fama and French (1992), which equals monthly price per share at closing multiplied by the shares outstanding in June. We construct Book to market ratio (BEME) as the fiscal year-end book value of common equity over the calendar year-end market equity (December). Book equity equals the total common equity plus investment tax credit and deferred taxes, minus preferred stock liquidation value (or redemption value).

We also include control variables to catch the momentum effects, liquidity effects and asset growth effects. Momentum factor CRETURN is represented by the cumulative returns from month (t-7) to (t-2). So that we may replicate experiments in Fu (2009), we include a pair of turnover variables, as in Chordia, Subrahmanyam, and Anshuman (2001). These include the share turnover, measured by the number of shares traded divided by the number of shares outstanding in the second to last month (TURN), and the coefficient of variation of turnover, calculated over the past 36 months beginning in the second to last month (CVTURN). The liquidity measure we use more broadly in the paper, PS_LIQ, is the historical liquidity beta due to Pastor and Stambaugh (2003). Asset growth effect (AG) is captured by the percentage change of total assets as defined by Cooper, Gulen, and Schill (2008). We take natural logs for ME, BEME, TURN, CVTURN, and AG, and following Fu (2009), winsorize them at .5% and 99.5% levels. We denote these log transformations as LNME, LNBEME, LNTURN, LNCVTURN, and LNAG. Panel B of Table II provides descriptive statistics for the percentage return and control variables for our data period of July 1963 to December 2008.

III. Expected Idiosyncratic Volatility in the Cross Section

A. Portfolio Analysis

If contemporaneous idiosyncratic volatility is priced, then if high quality volatility forecasts are available it may be that these forecasts are positively related to expected returns as well. Indeed, if perfect forecasts were available, they would necessarily be priced. In this section, we first focus our attention on the portfolio manager's trading problem, and test simple relationships of idiosyncratic volatility forecasts using information set [[PHI].sub.t-1] with abnormal returns (relative to the Fama-French three factor model) via portfolio sorts of the volatility variables. To demonstrate the importance of the limited information assumption, we compare these variables to their counterparts formed using information sets [PHI] and [PHI]T. Then, in Section III.B, we take a more comprehensive view, by conditioning on all relevant control variables simultaneously, and estimating Fama-MacBeth regressions on our full range of expected idiosyncratic volatility variables.

Portfolio sorts provide a simple and powerful mechanism by which we may examine the relationships between abnormal returns and our various volatility variables. While they have the drawback of limiting the number of variables on which we may simultaneously condition, they have the benefit of making no functional form assumptions about the relationships between the variables.

In Table IV we report Fama-French three-factor alphas for value-weighted portfolios sorted on idiosyncratic volatility by quintile. We report alphas for portfolios sorting on the volatility variable only in the first row, labeled "Exchange Traded Stocks" (our entire dataset is limited to exchange traded stocks). The remaining rows report the results of double sorts--we first sort portfolios on a control variable. Then, within each control variable quintile, we form quintiles by sorting on the relevant idiosyncratic volatility variable. For this procedure we follow Ang et al. (2006).

The top section of Panel A in Table IV provides the results of the portfolio sorts for M_EGARCH(t), which corresponds to the primary variable of interest in Fu (2009). Consistent with that paper, the 5-1 portfolio over all stocks is positive and both statistically and economically significant. We use a lag of four months in the construction of Newey-West t-statistics for all reported variables for all of our portfolio sorts. (14) It is instructive to note the similarity of this quintile sort over all stocks to the corresponding decile sort in the last row of table VI on p. 33 of Fu (2009). For example, the last two deciles reported in that row (the row labeled "FF alphas") are 0.13 and 1.45. Our excess return for our largest quintile is 0.60. (15)

The lower section of Panel A in Table IV provides the portfolio sorts using the corresponding volatility variable that may be estimated with information available to a trader, M_EGARCH(t-1). The differences from the M_EGARCH(t) table are striking. Not only is the 5 1 sort over all stocks not positive, there is even some weak evidence of a negative relationship. (16) To address the origins of this weak negative relationship, we first perform a double sort of size and idiosyncratic volatility. Consistent with Ang et al. (2006), we determine size using one-month lagged market capitalization. This is the most intuitive variable to try, but our weak negative result appears to be little affected by a sort on size. Motivated by Huang et al. (2010), we next perform a double sort on LAGRET and idiosyncratic volatility. We find that this eliminates the small residual pricing power the variable had retained. For comparison, we next perform a double sort on liquidity and idiosyncratic volatility, consistent with Han and Lesmond (2011). We find that this sort also eliminates the small residual pricing power the variable had retained. These results provide quite a counterpoint to M_EGARCH(t). From the standpoint of the portfolio manager's trading problem, forecasted idiosyncratic volatility appears to have no relationship to abnormal profits, save for its correlation with either lagged returns or liquidity. (17)

For completeness, Panel B provides similar results for the remaining idiosyncratic volatility variables, though only a subset of these variables are reasonable proxies for volatility forecasts of market participants. For example, while Ang et al. (2006) show that sorts D SQRET(t-1) can generate abnormal returns, D_SQRET(t-1) is not a reasonable proxy for market participants' best volatility forecast, since volatility does not follow a random walk. (18) The first two tables in Panel B provide results for the remaining volatility variables that use forward-looking information. While the 5-1 portfolio in the M_EGARCH(T) case only exhibits a weak positive relationship at the 10% significance level, strong confidence is achieved (Newey-West t-statistic of about 8.9) when we control for firm size. It is interesting to note the high level of significance of the size control variable, relative to the other controls, for volatility variables that contain forward-looking information. A detailed look at the size-volatility sorts (unreported) indicates that these high levels of significance are largely driven by firms that jointly fall in the smallest size quintile and largest volatility quintile. The findings for D_SQRET(t) are very similar, though there is a surprising lack of significance when we control for lagged returns. In line with the results in Panel A, the inclusion of forward-looking information generally yields a positive relationship between idiosyncratic volatility and abnormal returns.

The remaining sections of Panel B provide results for portfolio sorts using time t-1 information. Consistent with the existing literature, the 5-1 portfolios of D_SQRET(t-1), D_SQRET_ARMA(t-1), and M_SQRET(t-1) all yield strong negative returns. The results for D_SQRET(t-1) closely correspond to table VII on p. 286 of Ang et al. (2006). Reassuringly, our double sorts provide similar results to those reported there, despite our addition of several years of data relative to their dataset. Because idiosyncratic volatility does not follow a random walk, only D_SQRET_ARMA(t-1) of the t-1 information set group can reasonably be considered a forecast of idiosyncratic volatility, albeit an inefficient one relative to M_EGARCH(t-1). Given its 85% correlation with D_SQRET(t-1), the negativity is not surprising. As we see in Section III.B, when a host of control variables are considered jointly, none of these volatility variables retain statistical or economic significance.

Any double-sort analysis of this nature has the significant weakness that it can only condition on expected idiosyncratic volatility and one other variable. Many control variables are relevant to the analysis, and they are jointly associated with abnormal returns. To reach firm conclusions, more sophisticated conditioning analysis is required than simple portfolio sorts, and so in the second part of this section we estimate Fama-MacBeth regressions. At the cost of an assumed functional form, this method will allow us to control for many factors that contribute to cross-sectional return variation simultaneously, and help us to examine more completely the true effect of expected idiosyncratic volatility on expected returns.

B. Fama-MacBeth Regressions

Like the portfolio analysis in Section III.A, our analysis here takes place at a monthly frequency. Following Fama and MacBeth (1973), we estimate the following cross-sectional regressions for each month t = 1, ..., T:

[R.sub.it] = [[lambda].sub.0t] + [j.summation over (j-1)][[lambda].sub.j], [X.sub.jit] + [e.sub.it]. (9)

i = 1, ... [N.sub.t] indexes firms in the sample at time t. As our dataset consists of data from January 1966 to December 2008, we have 516 total months in our sample. The set of [X.sub.jit] variables will include our volatility variables and control variables introduced in Section II. Additionally, we will add to the regressions each firm's lagged return, denoted LAGRET, as Huang et al. (2010) find that return reversals explain the negative relationship between D_SQRET(t-1) and expected return. We estimate the effect of the independent variables by averaging them over T months, and construct Newey-West t-statistics to establish confidence limits. As in section III.A, we use a lag of four months in the construction of our Newey-West t-statistics.

For our first set of cross-sectional regressions, we equally weight all firms. We present these results in Table V. In this table, we use a limited set of control variables, to replicate as closely as possible the equal-weighted Fama-MacBeth regressions of Fu (2009). Notably, we exclude the lagged return variables of Huang et al. (2010) as well as MKTBETA, SMBBETA, and HMLBETA. For completeness, we present cross-sectional regressions for all of our volatility variables. The positive coefficients on the volatility variables that include forward-looking information provide significant contrast to those forecasts formed using the time t-1 information set. Model 2 in Table V corresponds to Model 5 in table V of Fu (2009). With a coefficient of 0.32, we find an even greater positive coefficient on this variable than Fu (2009) does (Fu estimates 0.15), though with quite similar results. Our M_EGARCH(t) coefficient also compares closely to the coefficient on the EGARCH forecast in Models 8 and 9 in table I of Huang et al. (2010)--monthly derived EGARCH forecasts that have a look-ahead bias. The coefficients on the EGARCH forecasts in those models in Huang et al. (2010) are 0.25 and 0.26, respectively; the small differences may be explained by our more comprehensive set of control variables and slightly longer time period. This result is also similar to the estimated Fama-MacBeth coefficients on expected idiosyncratic volatility presented in table XII of Spiegel and Wang (2005). They find a coefficient as large as 0.27, depending on the model specification. (19) Of the variables formed using the time t-1 information set, only D_SQRET(t-1) is significantly negative, consistent with the results of Ang et al. (2006). The broad insignificance of the coefficients on idiosyncratic volatility variables that are limited to time t-1 information is in direct contrast to Fu (2009), but is consistent with Brennan and Wang (2010), who find that lagged idiosyncratic volatility estimated (assuming constant volatility) from the preceding 60 monthly returns are not priced in the cross section. (20)

Table VI presents similar equal-weighted cross-sectional regressions, but with our full set of control variables. This complete set of control variables removes measures of turnover as proxies for liquidity, and instead uses PS_LIQ beta as the appropriate measure. (21) MKTBETA, SMBBETA, and HMLBETA are included as measures of systematic risk, LNAG is included as a measure of asset growth, and LAGRET is included due to its established importance. (22)

Despite the more comprehensive set of control variables, the results in Table VI broadly align with those in Table V. All variables that contain forward-looking information are significantly positive. With the addition of the comprehensive set of control variables, none of the volatility variables formed using time t-1 information are significantly negative in these equal-weighted regressions. The only variable that was significantly negative in Table V, D_SQRET(t-1), is now insignificant at any conventional level.

We include one additional volatility variable in Table VI, UNEXPECTED. The coefficient on UNEXPECTED in Model 8 is higher and more significant than any of the other variables. In unreported results we include both UNEXPECTED and M_EGARCH(T) as independent variables in the Fama-MacBeth regression, and find that the significance of M_EGARCH(T) disappears while the significance of UNEXPECTED remains. (23)

In Table VII, we present the same set of equal-weighted regressions as in Table VI, but exclude LAGRET. (24) In Table VII, Model 5 exhibits weak significance where none was seen in Table VI. Even more notable, Models 6 and 7 exhibit significance at the 1% and 5% levels respectively, though they were insignificant in Table VI. In equal-weighted regressions, controlling for lagged return appears to eliminate the significance of volatility variables formed using [[PHI].sub.t-1], consistent with Huang et al. (2010).

In Table VIII, we present value-weighted Fama-MacBeth regressions for our idiosyncratic volatility variables with the comprehensive set of control variables, including LAGRET. Value weights for cross-sectional regressions including out-of-sample volatility forecasts are determined by lagged market capitalization. For regressions that include forward-looking volatility forecasts, contemporaneous market capitalization is used to determine the weights, for consistency. The results broadly align with the results in Table VI, though there are some interesting exceptions. In all cases, idiosyncratic volatility variables that include forward-looking information are positive and statistically significant. However, D_SQRET(t-1) and D_SQRET_ARMA(t-1) retain their negative significance. This is consistent with Han and Lesmond (2011). Huang et al. (2010) find that lagged returns explain the negative relation between lagged idiosyncratic volatility and expected returns, while Han and Lesmond (2011) do not. It appears that lagged returns explain the negative relation between lagged idiosyncratic volatility and expected returns, but only in an equal-weighted cross section. When larger firms drive the results in a value-weighted cross section, lagged returns are insufficient to explain away the negative pricing power of D_SQRET(t-1). It is in this case that controlling for liquidity as in Han and Lesmond (2011) becomes important. It is interesting to note, however, that the PS_LIQ variable (which we include as a control) is insufficient to negate the pricing power of D_SQRET(t-1). As we show in Section IV, computing D_SQRET(t-1) from the midpoints of the bid-ask spread does render that variable insignificant in value-weighted cross-sectional Fama-MacBeth regressions. Robust to the choice of value- or equal-weighted regressions is the finding that in-sample idiosyncratic volatility estimates have a positive relation with expected returns, while out-of-sample estimates do not.

Some interesting patterns are evident among the control variable coefficient estimates when the forward-looking idiosyncratic volatility measures are included in the regressions--when compared to those estimates where the volatility variables included are limited to time t-1 information. In the equal-weighted regressions in Table VI, MKTBETA and SMBBETA are significantly negative for the in-sample models (Models 1-3), and insignificant for the remaining models. Also in the equal-weighted regressions, HMLBETA is significantly positive for the in-sample models and insignificant for the others. This pattern is seen elsewhere in the literature; see for example the BETA variable on p. 2554, table IV, Panel A of Peterson and Smedema (2011) in their comparison of D SQRET(t-1) (they label this variable IV1) and M_EGARCH(T) (they label this variable IV4). We also see a switch in the sign on the coefficient for LNME--negative for the forecast variables, positive for the in-sample variables. This may also be seen in Peterson and Smedema (2011), table IV, as well as table V on p. 31 of Fu (2009). Fu's (2009) Models 5 and 7 in that table contain forward-looking information and positive LNME coefficients. Fu's (2009) Model 6 contains lagged information, and the coefficient is negative. The LNME coefficient difference may also be seen in table I, on p. 156 of Huang et al. (2010).

The control variable coefficient pattern in Models 1-3 is remarkably consistent with Merton (1987), which assumes that the idiosyncratic volatility parameters are known. If the volatility estimates that use t-1 information are irrelevant for explaining cross-sectional returns, but in-sample volatility estimates are important, then Models 4-7 of Table VI have an omitted variable bias. In Models 1-3, we see that MKTBETA is significantly negative and LNME significantly positive, consistent with equation (29) and equation (31) of Merton (1987). While Merton (1987) did not control for other systematic factors, it is reasonable that the omitted variable bias would affect those variables as well. These changes are less severe in the value-weighted regressions of Table VIII, but still consistent with Merton (1987). MKTBETA is negative and generally weakly significant when the omitted variable bias is rectified through the inclusion of the in-sample volatility estimates, while LNME is insignificant across most of the models.

IV. Robustness

To examine whether our results are time period specific, Tables IX and X repeat our Fama-MacBeth analysis for two time periods, chosen to align with Han and Lesmond (2011). (25) For the sake of brevity, we confine our analysis to the M_EGARCH and D_SQRT variables. Table IX provides equal-weighted results, while Table X provides value-weighted results. We find that volatilities formed using forward-looking information are uniformly positive and statistically significant, with roughly similar coefficient magnitudes in all regressions. In the equal-weighted regressions the volatility variables formed from the t-1 information are not significant at even the 10% level in either time period, and their coefficients are near zero.

In the value-weighted regressions, we find M_EGARCH(t-1) fails to provide significance even at the 10% level in either time period, in direct contrast to M_EGARCH(t). Consistent with Han and Lesmond (2011), we find that D_SQRET(t-1) is insignificant in the 1992-2008 period, but negative and significant in the 1966-1991 period. Han and Lesmond (2011) find that when they compute D SQRET(t-1) using the bid-ask midpoint returns, that the significance of the negative coefficient disappears. The necessary bid-ask information is only available in CRSP since 1993, which is a subset of our second period. In Model 9, we compute D_SQRET(t-1) this way, and find that the (insignificant) coefficient on this variable is only 41% as large as the coefficient on D_SQRET(t-1). We are unable to produce a similar variable for the 1966-1991 time period, but it seems reasonable that doing so would similarly reduce the magnitude of the D_SQRET(t-1) variable. (26)

In Table XI we provide value-weighted abnormal return of portfolio sorts for M EGARCH(t-1) and D_SQRET(t-1), similar to those presented in Section III.A, for the 1966-1991 and 1992-2008 time periods. Absent the bid-ask midpoint return construction, the consistent negativity of the abnormal returns to the 5-1 portfolios sorted on D_SQRET(t-1) for both time periods is in line with both Ang et al. (2006) and our findings in Section III. M_EGARCH(t-1) from 1992 to 2008 is statistically insignificant, in line with our findings in Section III. However, the 1966-1991 period exhibits significant negative abnormal returns to the 5-1 portfolio, even when controlling for size, lagged returns, or liquidity.

It appears that this negativity is driven by an unusual period in the 1980s. To see this issue explicitly, we re-estimate the portfolio sort in Table XI in the following way. First, we sort all firms each month into quintiles based on M_EGARCH(t-1), as we do in Tables IV and XI. For each quintile, we construct a value-weighted portfolio return. We then form a 5-1 long/short portfolio from the quintiles. Finally, we estimate the following model:

[r.sub.it] = [2008.summation over ([[alpha].sub.[tau]])] + [s.sub.i][SMB.sub.t] + [h.sub.i][HML.sub.t] + [[beta].sub.i]MKTRF (10)

where SMB, HML, and MKTRF are the Fama-French factors--small minus big, high minus low, and the excess return of the market, respectively, [r.sub.it] is the raw 5-1 portfolio return. The [[alpha].sub.[tau]] term is an estimate of the abnormal return to the 5-1 portfolio for year [tau]. Figure 2 illustrates the [[alpha].sub.[tau]] terms for the single sort 5-1 portfolio, where the sorting variable is M_EGARCH(t-1). The decade of the 1980s stands out as an unusual extended period of negative abnormal returns to the 5-1 portfolio, and is the period driving the negativity seen in the M_EGARCH(t-1) results in Table XI over the period of 1966-1991. Indeed, if we split this period further into two subperiods, the 5-1 (single sort) hedge portfolio exhibits an abnormal return from 1966 to 1979 of -0.003 with a t-statistic of -0.02, in line with the complete dataset. However, from 1980 to 1991 the 5-1 hedge portfolio exhibits an abnormal return of -0.79 with a t-statistic of -4.55. The reasons for the distinctly different pricing behavior of idiosyncratic volatility over this period are unclear, but give rise to interesting questions as to the origins of the time variation in the pricing of idiosyncratic volatility. The large year-to-year variations in the abnormal returns illustrated in Figure 2 further underscore the uncertainty in the pricing of out-of-sample forecasts of idiosyncratic volatility.

For completeness, we also provide in Table XI abnormal returns of portfolios sorted on D_SQRET(t-1) using quote midpoint returns from 1993 to 2008. We find significant negative abnormal returns to the 5-1 portfolio, though this negativity becomes insignificant when we control for liquidity using the Pastor-Stambaugh liquidity measure.

Our discovery of an insignificant relationship between out-of-sample forecasted idiosyncratic volatility and expected returns--while finding a positive relationship when the full sample is employed in volatility estimation--may stem from two possible sources. First, there may be no relationship between forecasted idiosyncratic volatility and expected returns. Alternatively, we may have poor proxies for market participant forecasts of idiosyncratic volatility. We proxy for the degree of imperfection of market participant forecasts through the use of the EGARCH forecasts (formed from monthly data) as well as ARMA forecasts (formed from daily data), and in neither case find a significant relationship between volatility forecasts and expected returns. (27)

[FIGURE 2 OMITTED]

It may be that the greater noise in volatility forecasts of firms with relatively few observations is contaminating our result. Certainly it is more difficult to forecast the idiosyncratic volatility of young firms than mature ones. As a robustness check, in unreported results, we re-estimate our equal-weighted Fama-MacBeth regressions, excluding the first five years (60 monthly observations) of volatility estimates for all firms. This eliminates all firms less than five years from consideration, and allows our volatility forecasts double the previous minimum number of observations to form an estimate of expected idiosyncratic volatility. Even when restricting our analysis to mature firms, we find no relationship between out-of-sample idiosyncratic volatility forecasts and expected return. Further, all of the in-sample forecasts retain similar relationships to those estimated using the full dataset.

V. Conclusion

There are two broad questions concerning the relationship between expected idiosyncratic volatility and expected returns. First, does the data support theoretical asset pricing models such as Merton (1987) and Malkiel and Xu (2002), where investors are assumed to know all parameters of the model, including the idiosyncratic volatility of assets? Second, if there is a positive relationship between expected idiosyncratic volatility and expected returns, as suggested in the literature, is it possible to form portfolios using idiosyncratic volatility forecasts to earn abnormal returns? We provide a comprehensive set of results that clarify the role of idiosyncratic volatility in asset pricing by answering these two questions.

The first of these questions has been answered well in earlier papers, and we confirm their results. We confirm earlier papers such as Malkiel and Xu (1997) and Malkiel and Xu (2002) in determining that when we get precise estimates of idiosyncratic volatility and make strong assumptions about agents' knowledge of parameters in the model, expected idiosyncratic volatility and expected returns appear to be positively related. This relationship appears to be strong, and robust to idiosyncratic volatility measurement.

Our answer to the second of these questions clarifies previous findings. Though expected idiosyncratic volatility does affect expected returns positively in the cross-section, it is the portion of this volatility that is unexpected from the vantage point of a trader making a decision that is driving the result. The positive association of expected returns with expected idiosyncratic volatility that Fu (2009) finds is explained by contamination of the volatility forecast with this unexpected idiosyncratic volatility--the byproduct of a look-ahead bias. Consistently, we find a similar positive relationship when we use contemporaneous realized idiosyncratic volatility (as opposed to Ang et al. (2006) who use lagged realized idiosyncratic volatility). We provide the first comprehensive examination of the implications of this look-ahead bias.

Our results complement well the findings of both Huang et al. (2010) and Han and Lesmond (2011). Huang et al. (2010) find that when return reversals are taken into account, lagged idiosyncratic volatility estimated from daily data has no association with expected returns. Han and Lesmond (2011) find that lagged idiosyncratic volatility estimated from daily data has a liquidity-induced bias that overstates the relationship with future returns. We buttress these findings by concluding that EGARCH forecasts of expected idiosyncratic volatility formed from lagged monthly data--where bid-ask bounce and return reversal concerns are considerably less important--are also unrelated to expected returns. By examining a broad spectrum of idiosyncratic volatility measures, we augment the literature by presenting a comprehensive set of cross-sectional results relating expected returns to idiosyncratic volatility measures. We see that the inclusion of forward-looking information induces positivity into the relationship between expected returns and idiosyncratic volatility. This induced positive relationship is present in Fu (2009) and is present when we use contemporaneous data in the idiosyncratic volatility measure of Ang et al. (2006).

Our results rejecting the relation between out-of-sample expected idiosyncratic volatility and expected returns appear to be robust to several commonly employed methods of forecasting volatility, though there does appear to be some unpredictable variation through time. This robustness extends to mature firms and is not dependent upon the asset pricing model.

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(1) Also in the cross-section, Malkiel and Xu (1997) find a positive relationship between idiosyncratic-volatility sorted portfolios and holding period returns, though they do not report on the statistical significance of these findings.

(2) There is a substantial literature describing the nature of this asymmetry. For example, see Black (1976) and Christie (1982), or more recently, Bekaert and Wu (2000) and Daouk and Ng (2011).

(3) See for example, Nelson (1991) and Hentschel (1995).

(4) This is one of several specifications considered by Huang et al. (2010). Their primary findings are that return reversals affect estimates of the relationship between idiosyncratic volatility and expected returns when ivol isestimated as in Ang et al. (2006). They determine, however, that estimates of idiosyncratic volatility using monthly return data are unaffected by such reversals.

(5) We thank Fangjian Fu for helpful correspondence that allowed us to better understand the estimation issues in Fu (2009).

(6) We thank Qianqiu Liu for insightful correspondence concerning this possible look-ahead bias.

(7) Fama-French factors are taken from the WRDS data service.

(8) Consistent with Fu (2010), we employ the SAS "autoreg" procedure for our estimates, and only use estimates for which a status of "0 Converged" has been reported.

(9) We are indebted to Maria Schutte, who has generated an independent set of out-of-sample EGARCH forecasts of the sort we use in this paper. Her willingness to provide cross-verification of the properties of the forecasts was instrumental in the development of this paper.

(10) It is tempting to simply define unexpected idiosyncratic volatility as M_EGARCH(T) - M_EGARCH(t-1). However, defining unexpected volatility this way assumes [[alpha].sub.i] = 0 and [[beta].sub.i] = 1 for all i in Equation (8). Regressions estimated with our data set indicate that this is not a good approximation. If we do impose these constraints, the resulting unexpected volatility series is highly predictable and has a strong negative association with M_EGARCH(t - 1). By defining unexpected idiosyncratic volatility as in Equation (8), we ensure orthogonality of expected and unexpected idiosyncratic volatility.

(11) It is worth noting the performance of the regressions in Equation (8) for our data set. These regressions may be interpreted as forecast evaluation regressions. The mean [R.sup.2] of the 21,036 firm level regressions we estimated were roughly 13% each for the regressions.

(12) We thank an anonymous referee for this observation.

(13) Naturally, M_EGARCH(T) is still the most precise of the estimates, in this case the average standard deviation among all firms is 4.29.

(14) Davidson and Mackinnon (1993, p. 607-614) suggest a Newey-West lag parameter of [n.sup..25]. Computing this value with our 516 observations and truncating gives us a lag length of four, which also corresponds to the lag length chosen in Ang et al. (2009).

(15) In unreported results, when we repeat our analysis using deciles rather than quintiles, our decile results from M_EGARCH(t) have a 98% correlation with Fu's (2009) FF-alpha decile results reported in Table VI on p. 33 of Fu (2009).

(16) Similarly, Gulen, Xing, and Zhang (2011) find a high degree of in-sample predictability of the value premium, but no significant out-of-sample predictability.

(17) Vozlyublennaia (2012) find, using a GARCH-in-Mean framework for individual securities that for only a small proportion of market securities is there a significant risk-return relationship.

(18) The negativity of D SQRET(t-1) may be surprising given the weak positive correlation between raw returns and D_SQRET(t-1) we present in Table III. There are several reasons why we would not necessarily expect these relationships to exhibit the same sign. For example, once the systematic factors have been removed, the result is an unpredictable sign change in the raw versus systematic returns. Further, the sorts are delineated by quintiles, and the values that separate the quintiles fluctuate through time. There is no such relative ranking in the determination of the unconditional correlation.

(19) Spiegel and Wang (2005) use a shorter data set and a more limited set of control variables than we employ here. Xiaotong Wang thanks Fangjian Fu for sharing his SAS code in the acknowledgements of their paper.

(20) See model 4, Table VII, p. 3463 of Brennan and Wang (2010).

(21) We thank an anonymous referee for this suggestion.

(22) See Lipson, Mortal, and Schill (2010) for a detailed discussion of the relationship between asset growth and idiosyncratic volatility.

(23) In earlier versions of the paper, we repeat the experiment in Table V using the Chen, Roll, and Ross (1986) factor model. We use the factors published by Liu and Zhang (2008). All results are in line with the results provided in Tables V and VI.

(24) We thank the Editor for this suggestion.

(25) We thank an anonymous referee for suggesting appropriate time periods for a dynamic robustness check.

(26) Han and Lesmond (2011) augment the CRSP data, and demonstrate D_SQRET(t-1) using the bid-ask midpoint returns is not significantly related to expected returns in the 1984-1991 period.

(27) Consistent with Heston and Sadka (2008), idiosyncratic volatility estimates appear to exhibit some seasonality, with January and February idiosyncratic volatility estimates typically higher than other months. We therefore also replicate the Fama-Macbeth regressions of M_EGARCH(t-1) excluding data from the first quarter of the year. M_EGARCH (t-1) remains insignificant in the cross-sectional regressions, and all in-sample estimates remain significant with similar magnitudes. However, the estimates derived from daily return observations exhibit some negative significance. This latter result is likely explained by liquidity (which is also seasonal), as in Han and Lesmond (2011).

The authors would like to thank Geert Bekaert, Pamela Drake, and Qianqiu Liu for helpful discussions and/or correspondence, and John Battipaglia for excellent research assistance. We especially thank Maria Schutte and James Weston for important conversations that significantly strengthened this paper. The suggestions of two anonymous referees and Marc Lipson (Editor) led to significant improvements in this paper All volatility data is provided at http://people.jmu.edu/hehx/index.html.

* Jason D. Fink is the Chandler/Universal Professor of Banking at James Madison University in Harrisonburg, VA. Kristin E. Fink is a Professor at James Madison University in Harrisonburg. VA. Hui He is an Assistant Professor at James Madison University in Harrisonburg, VA.

Table I. Convergence Table for EGARCH Volatility Estimates This table presents the frequencies of the EGARCH (p, q) models used to create M_EGARCH(t-1) and M_EGARCH(t). Among the EGARCH (p, q) models that successfully converge, we use the AIC criterion to choose best fitting model. The models are indexed in a pq format. For example 12 indicates the frequency of the p = 1, q = 2 model. We also report the average iterations used for the EGARCH models to converge. EGARCH Frequency Percent (%) Average (p,q) Iterations Panel A: M_EGARCH (t-1) 11 215,695 9.16 112.03 12 175,254 7.44 128.07 13 195,439 8.30 150.48 21 304,303 12.92 131.83 22 225,410 9.57 247.42 23 226,681 9.62 241.26 31 387,905 16.47 198.47 32 300,246 12.75 294.42 33 324,837 13.79 265.34 Panel B. M_EGARCH (t) 11 212,305 9.09 110.90 12 172,510 7.39 126.54 13 192,926 8.26 148.89 21 301,712 12.92 130.93 22 223,508 9.57 246.87 23 225,213 9.65 240.69 31 385,196 16.50 197.91 32 298,085 12.77 294.12 33 322,950 13.83 264.93 Table II. Summary Statistics Panel A reports descriptive statistics for our various idiosyncratic volatility estimates, while Panel B reports these statistics for our other variables. The whole sample is from July 1963 to December 2008 for firms traded on the NYSE, Amex and NASDAQ exchanges, when available. Market data is collected from CRSP, whereas accounting variables are collected from Compustat. M_EGARCH(T) are EGARCH(p,q) generated estimates of monthly idiosyncratic volatility using the AIC criterion to choose the number of tags. The entire dataset is used to find these estimates. D_SQRET(t) are monthly estimated idiosyn cratic risks estimated from daily returns. M_EGARCH(t) are EGARCH(p,q) expected idiosyncratic volatili ties formed using information through time t to forecast time t values. M_EGARCH(t-1) are EGARCH(p,q) expected idiosyncratic volatilities formed using information through time t-1 to forecast time t values. The AIC criterion is used to choose the optimal number of lags for each stock at each time period. D_SQRET(t-1) is the lagged idiosyncratic volatility estimate using daily data as in Ang et al. (2006). D_SQRET ARMA(t-1) are expected idiosyncratic volatilities formed using D_SQRET(t) information through time t-1 to fore cast time t values. The forecasts are made using ARMA (p,q) model where the AIC criterion is used to choose the optimal number of lags for each stock at each time period. UNEXPECTED is time t un expected volatility estimated by the residuals of a regression of M_EGARCH(T) on M_EGARCH(t-1). RET is the percentage return to the firm over the one-month holding period. MKTBETA, SMLBETA, HML BETA, and SP_LIQ are beta estimates from time series regressions with Fama-French three factors and the liquidity series from Pastor and Stambaugh (2003) using the previous 60 month data on a rolling basis. LNME is the natural log of a firm's market equity at the end of the fiscal year. LNBEME is the log of the fiscal year-end book value of common equity over the calendar year-end market equity. CRETURN is represented by the cumulative returns from month (t-7) to (t-2) and is expressed as a percentage return. LNAG is the log of the percentage change of total assets of the firm. Variable Mean Median Std Dev Panel A. Volatility measures (%) Idiosyncratic Volatility Measures: M_EGARCH(T) 12.4202 10.0620 9.1393 M_EGARCH(t) 11.3914 9.0128 9.1277 D_SQRET(t) 13.1582 9.6954 13.1706 M_EGARCH(t-1) 11.8861 8.9806 13.9432 M_SQRET(t-1) 11.6313 9.7189 7.6343 D_SQRET(t-1) 13.1582 9.6954 13.1706 D_SQRET_ARMA(t-1) 12.6496 10.0967 10.4883 M_SQRET_ARMA(t-l) 11.0447 9.2561 7.1923 UNEXPECTED -0.0037 -0.4613 5.7547 Panel B. Return and Control Variables RET 1.0533 0.0000 17.6566 MKTBETA 0.9389 0.8945 1.3241 SMBBETA 0.8281 0.6335 2.3912 HMLBETA 0.2805 0.3159 1.6391 PS_LIQ -0.0033 -0.0021 0.6513 LNME 4.5617 4.4049 2.1536 LNBEME -0.3735 -0.3563 1.0632 CRETURN 7.9195 6.0858 39.1656 LNAG 0.1024 0.0743 0.2916 Variable Lower Upper N Quartile Quartile Panel A. Volatility measures (%) Idiosyncratic Volatility Measures: M_EGARCH(T) 6.7048 15.2397 2,996,213 M_EGARCH(t) 5.7865 14.0599 2,355,988 D_SQRET(t) 5.8524 16.1413 3,067,364 M_EGARCH(t-1) 5.7038 14.1894 2,355,988 M_SQRET(t-1) 6.7003 14.4735 1,838,453 D_SQRET(t-1) 5.8524 16.1413 3,067,364 D_SQRET-ARMA(t-1) 6.6614 15.5168 2,323,498 M_SQRET ARMA(t-l) 6.4970 13.6177 1,457,410 UNEXPECTED -2.2012 1.4043 2,351,503 Panel B. Return and Control Variables RET -6.4516 6.7423 3,107,879 MKTBETA 0.5089 1.3089 1,838,508 SMBBETA 0.1420 1.3241 1,838,508 HMLBETA -0.2201 0.8002 1,838,508 PS_LIQ -0.1949 0.1984 1,838,508 LNME 2.9588 6.0573 1,781,488 LNBEME -0.9737 0.1952 1,688,241 CRETURN -12.2420 25.0512 1,781,488 LNAG -0.0191 0.1901 1,738,229 Table III. Correlation Matrix of Idiosyncratic Volatility Variables Volatility estimates are from January 1966 to December 2008 for firms traded on the NYSE, Amex and NASDAQ exchanges, when available. Return data is collected from CRSP M_EGARCH(T) are EGARCH(p,q) generated estimates of monthly idiosyncratic volatility using the AIC criterion to choose the number of lags. The entire dataset is used to find these estimates. D_SQRET(t) are monthly estimated idiosyncratic risks estimated from daily returns. M_EGARCH(t) are EGARCH(p,q) expected idiosyncratic volatilities formed using information through time t to forecast time t values. M_EGARCH(t-1) are EGARCH(p,q) expected idiosyncratic volatilities formed using information through time t-1 to forecast time t values. The AIC criterion is used to choose the optimal number of lags for each stock at each time period. D_SQRET(t-1) is the lagged idiosyncratic volatility estimate using daily data as in Ang et al. (2006). D_SQRET_ARMA(t-1) are expected idiosyncratic volatilities formed using D_SQRET(t) information through time t-1 to forecast time t values. The forecasts are made using ARMA (p,q) model where the AIC criterion is used to choose the optimal number of lags for each stock at each time period. UNEXPECTED is time t unexpected volatility estimated by the residuals of a regression of M_EGARCH(T) on M_EGARCH(t-1). All correlations are significant at the 1% level. Idiosyncratic Volatility Measures M-EGARCH M_EGARCH D_SQRET M_EGARCH (T) (t) (t) (t-1) M_EGARCH(T) 1 M_EGARCH(t) 0.6041 1 D_SQRET(t) 0.5057 0.3515 1 M_EGARCH(t-1) 0.4108 0.5199 0.2522 1 AI1_SQRET(t-1) 0.6475 0.3191 0.4533 0.5905 D_SQRET(t-1) 0.4765 0.3130 0.6499 0.2753 D_SQRET_ARMA(t-1) 0.5421 0.3543 0.6907 0.3192 M_SQRET_ARMA(t-1) 0.6589 0.5025 0.4618 0.6148 UNEXPECTED 0.6516 0.2118 0.1876 -0.0001 RET(t) 0.1335 0.1394 0.1283 0.0089 M_SQRET D-SQRET D_SQRET M_SQRET (t-1) (t-1) ARMA ARMA M_EGARCH(T) M_EGARCH(t) D_SQRET(t) M_EGARCH(t-1) M_SQRET(t-1) 1 D_SQRET(t-1) 0.4754 1 D_SQRET_ARMA(t-1) 0.5800 0.8527 1 M_SQRET_ARMA(t-1) 0.9988 0.4850 0.5918 1 UNEXPECTED 0.0332 0.1345 0.1150 0.0373 RET(t) 0.0099 0.0156 0.0232 0.0098 UNEXPECTED Ret (t) M_EGARCH(T) M_EGARCH(t) D_SQRET(t) M_EGARCH(t-1) M_SQRET(t-1) D_SQRET(t-1) D_SQRET_ARMA(t-1) M_SQRET_ARMA(t-1) UNEXPECTED 1 RET(t) 0.1874 1 Table IV. Abnormal Returns of Portfolios Sorted on Idiosyncratic Volatility The table reports Fama-French (1993) three-factor alphas for value-weighted portfolios sorted on idiosyn cratic volatilities from January 1966 to December 2008. We first report alphas for portfolios sorting on the idiosyncratic volatility measures only on the row labeled "Exchange Traded Stocks." The column "5-1" refers to the difference in the Fama-French alphas between portfolios 5 and 1. In the rows controlling for size, book-to-market, liquidity, asset growth and lagged returns, we perform a double sort. We first sort stocks based on the first characteristic into quintiles each month, and then within each quintile, we sort stocks based on the specific idiosyncratic volatility measure. We average the five idiosyncratic volatility portfolios over the each of the five characteristic portfolios as in Ang et.al (2006). We follow Ang et al. (2006) and use the historical liquidity beta from Pdstor and Stambaugh (2003) as our liquidity measure. We report in the Panel A, the abnormal returns of portfolios sorted on the idiosyncratic volatility estimate using the in-sample EGARCH estimates, M_EGARCH(t), the out-of-sample EGARCH forecast, M_EGARCH(t-1); Panel B represents the full-sample EGARCH, M_EGARCH(T), the lagged idiosyncratic volatility estimate using daily data as in Ang et al. (2006), D_SQRET(t-1), the contemporaneous idiosyncratic volatility esti mates using daily data, D_SQRET(t), the ARMA forecast of D_SQRET(t), D_SQRET ARMA(t-1), and the volatility estimates using monthly data as in Bali and Cakici (2008), M_SQRET(t-1). Newey-West (1987) adjusted t-statistics with a lag of four months are in parentheses. Panel A Variable Ranking on M_EGARCH(t) 1 Low 2 3 Exchange Traded Stocks 0.040 0.020 -0.053 (0.959) (0.456) (-0.899) Controlling for Size -0.671 *** -0.708 *** -0.660 *** (-11.456) (-12.434) (-10.872) Controlling for Lag Return 0.012 -0.022 -0.087 (0.314) (-0.510) (-1.558) Controlling for Liquidity 0.038 -0.019 -0.107 * (0.903) (-0.453) (-1.939) 4 5 High 5-1 Exchange Traded Stocks 0.012 0.603 *** 0.563 ** (0.131) (2.865) (2.359) Controlling for Size -0.308 *** 2.354 *** 3.025 *** (-4.252) (11.170) (12.883) Controlling for Lag Return 0.048 0.903 *** 0.891 *** (0.531) (4.846) (4.307) Controlling for Liquidity 0.017 0.766 *** 0.728 *** (0.172) (4.449) (3.677) Variable Ranking on M_EGARCH (t-1) 1 Low 2 3 Exchange Traded Stocks 0.033 0.015 0.0829 (0.707) (0.345) (1.5350) Controlling for Size -0.007 0.023 -0.0188 (-0.102) (0.384) (-0.3045) Controlling for Lag Return 0.034 0.049 0.0545 (0.724) (1.163) (1.1392) Controlling for Liquidity 0.071 0.033 0.0217 (1.501) (0.758) (0.4242) 4 5 High 5-1 Exchange Traded Stocks -0.010 -0.226 * -0.260 * (-0.113) (-1.924) (-1.773) Controlling for Size -0.023 -0.224 ** -0.218 * (-0.321) (-2.098) (-1.711) Controlling for Lag Return -0.031 -0.158 -0.192 (-0.415) (-1.615) (-1.589) Controlling for Liquidity -0.007 -0.120 -0.192 (-0.106) (-1.109) (-1.368) Panel B Variable Ranking on M_EGARCH(T) 1 Low 2 3 Exchange Traded Stocks 0.007 -0.002 0.032 (0.156) (-0.045) (0.536) Controlling for Size -0.481 *** -0.479 *** -0.475 *** (-8.145) (-7.536) (-7.60) Controlling for Lag Return -0.058 0.007 0.036 (-1.431) (0.176) (0.570) Controlling for Liquidity 0.054 -0.044 0.036 (1.198) (-0.998) (0.510) Ranking on D-SQRET(t) Exchange Traded Stocks 0.014 0.064 0.147 ** (-0.244) (1.492) (1.991) Controlling for Size -0.696 *** -0.997 *** -0.757 *** (-10.137) (-12.461) (-10.434) Controlling for Lag Return -0.025 0.065 0.004 (-0.419) (1.340) (0.063) Controlling for Liquidity -0.012 0.066 0.032 (-0.204) (1.245) (0.376) Ranking on D-SQRET(t-1) Exchange Traded Stocks 0.084 * 0.046 0.032 (1.844) (1.170) (0.492) Controlling for Size 0.081 0.165 ** 0.087 (1.012) (2.241) (1.217) Controlling for Lag Return 0.101 ** 0.075 * -0.094 (2.433) (1.930) (-1.548) Controlling for Liquidity 0.102 ** 0.083 * -0.033 (1.938) (1.899) (-0.526) Ranking on D_SQRET_ARMA(t-1) Exchange Traded Stocks 0.080 * 0.032 0.014 (1.757) (0.582) (0.176) Controlling for Size 0.132 * 0.134 * 0.0726 (1.923) (1.889) (1.111) Controlling for Lag Return 0.112 0.050 -0.055 (2.869) (1.020) (-0.763) Controlling for Liquidity 0.089 * 0.024 0.038 (1.790) (0.506) (0.542) Ranking on M SQRET(t-1) Exchange Traded Stocks 0.053 0.029 0.024 (0.930) (0.514) (0.316) Controlling for Size 0.148 ** 0.077 0.021 (2.139) (1.007) (0.275) Controlling for Lag Return 0.056 0.043 0.017 (1.305) (0.874) (0.231) Controlling for Liquidity 0.054 0.043 0.080 (1.020) (0.945) (1.134) Variable Ranking on M_EGARCH(T) 4 5 High 5-1 Exchange Traded Stocks 0.000 0.477 ** 0.470 * (-0.005) (2.262) (1.923) Controlling for Size -0.227 *** 1.466 *** 1.946 *** (-3.143) (7.550) (8.854) Controlling for Lag Return -0.101 0.639 *** 0.698 (-1.188) (3.561) (3.403) Controlling for Liquidity 0.010 0.409 ** 0.355 * (0.110) (2.250) (1.684) Ranking on D-SQRET(t) Exchange Traded Stocks -0.085 0.559 0.573 (-0.593) (1.397) (1.298) Controlling for Size -0.077 2.268 *** 2.964 (-0.860) (8.643) (9.780) Controlling for Lag Return 0.068 0.438 0.462 (0.541) (1.150) (1.083) Controlling for Liquidity -0.016 1.090 *** 1.101 *** (-0.104) (3.021) (2.766) Ranking on D-SQRET(t-1) Exchange Traded Stocks -0.336 *** -1.166 -1.250 (-3.271) (-6.955) (-6.319) Controlling for Size -0.151 * -0.924 *** -1.005 *** (-1.880) (-7.132) (-5.994) Controlling for Lag Return -0.239 *** -1.125 *** -1.226 *** (-3.085) (-8.582) (-8.253) Controlling for Liquidity -0.204 ** -0.857 *** -0.959 *** (-2.226) (-6.494) (-5.957) Ranking on D_SQRET_ARMA(t-1) Exchange Traded Stocks -0.380 ** -0.993 *** -1.073 (-2.810) (-5.409) (-5.163) Controlling for Size -0.071 -0.621 *** -0.753 *** (-0.857) (-4.133) (-4.074) Controlling for Lag Return -0.302 *** -0.925 *** -1.037 *** (-2.832) (-5.704) (-5.638) Controlling for Liquidity -0.245 ** -0.873 *** -0.962 *** (-2.470) (-4.882) (-4.652) Ranking on M SQRET(t-1) Exchange Traded Stocks 0.043 -0.483 *** -0.537 *** (0.351) (-3.049) (-2.724) Controlling for Size -0.114 -0.297 ** -0.445 *** (-1.558) (-2.287) (-2.587) Controlling for Lag Return 0.048 -0.425 *** -0.481 (0.460) (-3.353) (-3.158) Controlling for Liquidity -0.129 -0.345 ** -0.399 ** (-1.246) (-2.375) (-2.180) *** Significant at the 0.01 level. ** Significant at the 0.05 level. * Significant at the 0.10 level. Table V. Equal-Weighted Fama-MacBeth Regressions with Fu (2009)'s Specification This table reports the results of equal-weighted Fama- MacBeth cross-section regressions for our vari ous idiosyncratic volatility estimates, utilizing a limited set of control variables, to align closely with the regressions in Fu (2009) Table 5. Volatility and other variable data are from January 1966 to December 2008 for firms traded on the NYSE, Amex and NASDAQ exchanges, when available. Market data is collected from CRSP and accounting variables are collected from Compustat. All idiosyncratic volatility estimates in this table are derived from the Fama-French factor model. M_EGARCH(T) are EGARCH(p,q) generated estimates of monthly idiosyncratic volatility using the AIC criterion to choose the number of lags. The entire dataset is used to find these estimates. D_SQRET(t) are monthly estimated idiosyncratic risks estimated from daily returns. M_EGARCH(t) are EGARCH(p,q) expected idiosyncratic volatilities formed using information through time t to forecast time t values. M_EGARCH(t-1) are EGARCH(p,q) expected idiosyncratic volatilities formed using information through time t-1 to forecast time t values. The AIC criterion is used to choose the optimal number of lags for each stock at each time period. M_SQRET(t-1) are idiosyncratic volatilities estimated as the standard deviation of residuals from Fama-French time series regressions using the previous 60 month data on a rolling basis. D_SQRET(t-1) is the lagged idiosyncratic volatility estimate using daily data as in Ang et al. (2006). D_SQRET ARMA(t-1) are expected idiosyncratic volatilities formed using D_SQRET(t) information through time t-1 to forecast time t values. The forecasts are made using ARMA (p,q) model where the AIC criterion is used to choose the optimal number of lags for each stock at each time period. RET is the percentage return to the firm over the one-month holding period. LNME is the natural log of a firm's market equity at the end of the fiscal year. LNBEME is the log of the fiscal year-end book value of common equity over the calendar year- end market equity. CRETURN is represented by the cumulative returns from month (t-7) to (t-2) and is expressed as a percentage return. The log of the number of shares traded divided by the number of shares outstanding in the second to last month is LNTURN, and the log of the coefficient of variation of turnover calculated over the past 36 months beginning in the second to last month is LNCVTURN. Newey- West (1987) adjusted t-statistics with a lag of four months are in parentheses. (1) (2) M_EGARCH(T) (x 10) 2.402 *** (14.049) M_EGARCH(t) (x 10) 3.181 *** (18.936) D_SQRET(t) (x 10) M_EGARCH(t-1) (x 10) M_SQRET(t-1) (x 10) D_SQRET(t-1) (x 10) D_SQRET_ARMA(t-1) (x 10) LNME 0.223 *** 0.310 *** (5.251) (6.592) LNBEME 0.462 *** 0.565 *** (8.277) (9.920) CRETURN 0.007 *** 0.006 *** (4.410) (3.466) LNTURN -0.581 *** -0.707 *** (-9.017) (-10.547) LNCVTURN -0.773 *** -0.859 *** (-8.101) (-8.862) [R.sup.2] 0.071 0.087 (3) (4) M_EGARCH(T) (x 10) M_EGARCH(t) (x 10) D_SQRET(t) (x 10) 3.193 *** (10.041) M_EGARCH(t-1) (x 10) 0.009 (0.230) M_SQRET(t-1) (x 10) D_SQRET(t-1) (x 10) D_SQRET_ARMA(t-1) (x 10) LNME 0.407 *** -0.161 *** (10.630) (-3.616) LNBEME 0.387 *** 0.183 *** (6.886) (3.288) CRETURN 0.013 *** 0.006 *** (7.140) (3.263) LNTURN -0.552 *** -0.172 ** (-8.344) (-2.284) LNCVTURN -0.791 *** -0.421 *** (-8.537) (-5.551) [R.sup.2] 0.102 0.052 (5) (6) M_EGARCH(T) (x 10) M_EGARCH(t) (x 10) D_SQRET(t) (x 10) M_EGARCH(t-1) (x 10) M_SQRET(t-1) (x 10) -0.153 (-1.196) D_SQRET(t-1) (x 10) -0.254 *** (-3.654) D_SQRET_ARMA(t-1) (x 10) LNME -0.148 *** -0.179 *** (-4.125) (-4.239) LNBEME 0.176 *** 0.170 *** (3.349) (3.069) CRETURN 0.006 *** 0.006 *** (3.249) (3.538) LNTURN -0.162 ** -0.137 * (-2.518) (-1.861) LNCVTURN -0.402 *** -0.364 *** (-6.007) (-4.929) [R.sup.2] 0.051 0.056 (7) M_EGARCH(T) (x 10) M_EGARCH(t) (x 10) D_SQRET(t) (x 10) M_EGARCH(t-1) (x 10) M_SQRET(t-1) (x 10) D_SQRET(t-1) (x 10) D_SQRET_ARMA(t-1) (x 10) -0.159 (-1.195) LNME -0.174 *** (-4.795) LNBEME 0.164 *** (3.090) CRETURN 0.006 (3.733) LNTURN -0.141 ** (-2.0526) LNCVTURN -0.391 *** (-5.686) [R.sup.2] 0.058 *** Significant at the 0.01 level. ** Significant at the 0.05 level. * Significant at the 0.10 level. Table VI. Equal-Weighted Farna-MacBeth Regressions with all Control Variables This table reports the results of equal-weighted Fama- MacBeth cross-section regressions for our various idiosyncratic volatility estimates formed from the Fama- French model. Volatility and other variable data are from January 1966 to December 2008 for firms traded on the NYSE, Amex and NASDAQ exchanges, when available. Market data is collected from C RSP and accounting variables are collected from Compustat. All id iosyncratic volatility estimates in this table are derived from the Fama-French factor model. M_EGARCH(T) are EGARCH(p,q) generated estimates of monthly idiosyncratic volatility using the AIC criterion to choose the number of lags. The entire dataset is used to find these estimates. D_SQRET(t) are monthly estimated idiosyncratic risks estimated from daily returns. M_EGARCH(t) are EGARCH(p,q) expected idiosyn cratic volatilities formed using information through time t to forecast time t values. M_EGARCH(t-1) are EGARCH(p,q) expected idiosyncratic volatilities formed using information through time t-1 to forecast time t values. The AIC criterion is used to choose the optimal number of lags for each stock at each time period. M_SQRET(t-1) are idiosyncratic volatilities estimated as the standard deviation of residuals from Fama- French time series regressions using the previous 60 month data on a rolling basis. D_SQRET(t-1) is the lagged idiosyncratic volatility estimate using daily data as in Ang et al. (2006). D_SQRET ARMA(t-1) are expected idiosyncratic volatilities formed using D_SQRET(t) information through time t-1 to forecast time t values. The forecasts are made using ARMA (p,q) model where the AIC criterion is used to choose the optimal number of lags for each stock at each time period. UNEXPECTED is time t unexpected volatility estimated by the residuals of a regression of M_EGARCH(T) on M_EGARCH(t-l). RET is the percentage return to the firm over the one-month holding period. LAGRET is the percentage return to the firm over the preceding one month holding period. MKTBETA, SMLBETA, HMLBET4, and PS LIQ are beta estimates from time series regressions with Fama-French three factors and the liquidity series from pastor and Stam baugh (2003) using the previous 60 month data on a rolling basis. LNME is the natural log of a firm's market equity at the end of the fiscal year. LNBEME is the log of tire fiscal year-end book value of common equity over the calendar year- end market equity. CRETURN is represented by the cumulative returns from month (t-7) to (t-2) and is expressed as a percentage return. LNAG is the log of the percentage change of total assets of the firm. Newey-West (1987) adjusted t-statistics with a lag of four months are in parentheses. (1) (2) M_EGARCH(T) (x 10) 2.329 *** (14.117) M_EGARCH(t) (x 10) 3.210 *** (19.813) D_SQRET(t) (x 10) M_EGARCH(t-1) (x 10) M_SQRET(t-1) (x 10) D_SQRET(t-1) (x 10) D_SQRET_ARAMA(t-1) (x 10) UNEXPECTED (x 10) MKTBET4 -0.473 *** -0.660 *** (-4.351) (-6.051) SMBBETA -0.355 *** -0.480 *** (-5.257) (-6.989) HMLBETA 0.270 *** 0.338 *** (3.777) (4.539) LNME 0.269 *** 0.366 *** (9.272) (11.670) LNBEME 0.431 *** 0.544 *** (9.571) (11.898) CRETURN 0.004 ** 0.003 * (2.521) (1.776) PS_LIQ -0.259 ** -0.382 *** (-2.372) (-3.435) LNAG -0.947 *** -1.059 *** (-7.548) (-8.274) LAGRET -0.068 *** -0.070 *** (-14.991) (-15.25) [R.sup.2] 0.071 0.096 (3) (4) M_EGARCH(T) (x 10) M_EGARCH(t) (x 10) D_SQRET(t) (x 10) 3.069 *** (11.851) M_EGARCH(t-1) (x 10) -0.019 (-0.452) M_SQRET(t-1) (x 10) D_SQRET(t-1) (x 10) D_SQRET_ARAMA(t-1) (x 10) UNEXPECTED (x 10) MKTBET4 -0.577 *** 0.017 (-5.282) (0.140) SMBBETA -0.513 *** -0.050 (-7.860) (-0.641) HMLBETA 0.240 *** 0.075 (3.353) (1.010) LNME 0.373 *** -0.076 ** (12.487) (-2.242) LNBEME 0.361 *** 0.184 *** (7.541) (3.855) CRETURN 0.011 *** 0.003 * (6.052) (1.765) PS_LIQ -0.280 ** -0.013 (-2.478) (-0.112) LNAG -0.431 *** -0.929 *** (-3.458) (-7.764) LAGRET -0.065 *** -0.070 *** (-14.837) (-14.623) [R.sup.2] 0.112 0.064 (5) (6) M_EGARCH(T) (x 10) M_EGARCH(t) (x 10) D_SQRET(t) (x 10) M_EGARCH(t-1) (x 10) M_SQRET(t-1) (x 10) -0.123 (-1.025) D_SQRET(t-1)(x 10) -0.058 (-0.804) D_SQRET_ARAMA(t-1) (x 10) UNEXPECTED (x 10) MKTBET4 0.039 0.018 (0.324) (0.146) SMBBETA -0.035 -0.042 (-0.491) (-0.573) HMLBETA 0.079 0.071 (1.110) (0.980) LNME -0.091 *** -0.061 ** (-2.937) (-1.9671) LNBEME 0.165 *** 0.183 *** (3.483) (3.770) CRETURN 0.003 * 0.004 ** (1.853) (2.201) PS_LIQ -0.002 0.003 (-0.013) (0.027) LNAG -0.916 *** -0.899 *** (-7.637) (-7.930 LAGRET -0.070 *** -0.070 *** (-14.524) (-14.113) [R.sup.2] 0.066 0.067 (7) (8) M_EGARCH(T) (x 10) M_EGARCH(t) (x 10) D_SQRET(t) (x 10) M_EGARCH(t-1) (x 10) M_SQRET(t-1) (x 10) D_SQRET(t-1) (x 10) D_SQRET_ARAMA(t-1) (x 10) -0.165 (-1.373) UNEXPECTED (x 10) 3.778 *** (23.036) MKTBET4 0.041 -0.015 (0.359) (-0.122) SMBBETA -0.026 -0.066 (-0.375) (-0.840) HMLBETA 0.068 0.079 (0.960) (1.061) LNME -0.075 ** -0.061 * (-2.559) (-1.720 LNBEME 0.177 *** 0.165 *** (3.681) (3.449) CRETURN 0.004 ** 0.005 *** (2.194) (2.999) PS_LIQ 0.011 -0.011 (0.102) (-0.100) LNAG -0.903 *** -0.759 *** (-7.940) (-6.545) LAGRET -0.070 *** -0.067 *** (-14.48) (-14.673) [R.sup.2] 0.069 0.084 *** Significant at the 0.01 level. ** Significant at the 0.05 level. * Significant at the 0.10 level. Table VII. Equal-Weighted Fama-MacBeth Regressions with all Control Variables Except Lagged Returns This table reports the results of equal-weighted Fama-MacBeth cross- section regressions with all control variables except lagged returns for our various idiosyncratic volatility estimates formed from the Fama French model. Volatility and other variable data are from January 1966 to December 2008 for firms traded on the NYSE, Amex and NASDAQ exchanges, when available. Market data is collected from CRSP and accounting variables are collected from Compustat. All idiosyncratic volatility estimates in this table are derived from the Fama-French factor model. M_EGARCH(T) are EGARCH(p,q) generated estimates of monthly idiosyncratic volatility using the AIC criterion to choose the number of lags. The entire dataset is used to find these estimates. D_SQRET(t) are monthly estimated idiosyncratic risks estimated from daily returns. M_EGARCH(t) are EGARCH(p,q) expected idiosyncratic volatilities formed using informa tion through time t to forecast time t values. M_EGARCH(t-1) are EGARCH(p,q) expected idiosyncratic volatilities formed using information through time t-1 to forecast time t values. The AIC criterion is used to choose the optimal number of lags for each stock at each time period. M_SQRET(t-1) are idiosyncratic volatilities estimated as the standard deviation of residuals from Fama-French time series regressions using the previous 60 month data on a rolling basis. D_SQRET(t-1) is the lagged idiosyncratic volatility estimate using daily data as in Ang et al. (2006). D_SQRET ARMA(t-1) are expected idiosyncratic volatilities formed using D_SQRET(t) information through time t-1 to forecast time t values. The forecasts are made using ARMA (p,q) model where the AIC criterion is used to choose the optimal number of lags for each stock at each time period. UNEXPECTED is time t unexpected volatility estimated by the residuals of a regression of M_EGARCH(T) on M_EGARCH(t-1). RET is the percentage return to the firm over the one- month holding period. MKTBETA, SMLBETA, HMLBETA and PS_LIQ are beta estimates from time series regressions with Fama-French three factors and the liquidity series from Pastor and Stambaugh (2003) using the previous 60 month data on a rolling basis. LNME is the natural log of a firm's market equity at the end of the fiscal year. LNBEME is the log of the fiscal year-end book value of common equity over the calendar year end market equity. CRETURN is represented by the cumulative returns from month (t-7) to (t-2) and is expressed as a percentage return. LNAG is the log of the percentage change of total assets of the firm. Newey-West (1987) adjusted t-statistics with a bandwidth of four months are in parentheses. (1) (2) (3) (4) M_EGARCH 2.372 *** (T)(x 10) (14.553) M_EGARCH 3.212 *** (1)(x 10) (19.829) D_SQRET 0.312 *** (t)(x 10) (12.276) M_EGARCH -0.029 (t-1)(x 10) (-0.698) M_SQRET (t-1)(x 10) D_SQRET (t-1)(x 10) D_SQRET_ARMA (t-I) (x l0) UNEXPECTED (x 10) MKTBETA -0.501 *** -0.684 *** -0.600 *** 0.001 (-5.204) (-7.026) (-6.012) (0.012) SMBBETA -0.359 *** -0.479 *** -0.528 *** -0.038 (-5.660) (-7.462) (-8.285) (-0.518) HMLBETA 0.265 *** 0.332 *** 0.234 *** 0.064 (4.117) (4.922) (3.611) (0.967) LNME 0.269 *** 0.362 *** 0.380 *** -0.087 *** (9.533) (11.838) (12.953) (-2.610) LNBEME 0.420 *** 0.529 *** 0.352 *** 0.167 *** (9.511) (11.791) (7.469) (3.565) CRETURN 0.006 *** 0.004 *** 0.012 *** 0.004 ** (3.376) (2.5932) (7.154) (2.440) PS LIQ -0.208 * -0.327 *** -0.223 * 0.051 (-1.832) (-2.818) (-1.888) (0.436) LNAG -0.915 *** -1.024 *** -0.386 *** -0.902 *** (-7.388) (-8.137) (-3.117) (-7.610) [R.sup.2] 0.074 0.088 0.105 0.057 (5) (6) (7) (8) M_EGARCH (T)(x 10) M_EGARCH (1)(x 10) D_SQRET (t)(x 10) M_EGARCH (t-1)(x 10) M_SQRET -0.193 * (t-1)(x 10) (-1.6912) D_SQRET -0.200 *** (t-1)(x 10) (-3.0832) D_SQRET_ARMA -0.225 ** (t-I) (x l0) (-1.989) UNEXPECTED 3.833 *** (x 10) (23.374) MKTBETA 0.037 0.026 0.038 -0.042 (0.352) (0.245) (0.376) (-0.377) SMBBETA -0.015 -0.018 -0.009 -0.065 (-0.221) (-0.260) (-0.141) (-0.880) HMLBETA 0.062 0.058 0.056 0.072 (1.004) (0.915) (0.915) (1.076) LNME -0.109 *** -0.091 *** -0.093 *** -0.068 ** (-3.680) (-2.929) (-3.264) (-1.961) LNBEME 0.142 *** 0.161 *** 0.157 *** 0.150 *** (3.055) (3.3673) (3.324) (3.193) CRETURN 0.005 *** 0.005 *** 0.005 *** 0.006 (2.611) (2.779) (2.946) (3.895) PS LIQ 0.070 0.090 0.086 0.042 (0.617) (0.771) (0.746) (0.366) LNAG -0.887 *** -0.902 *** -0.886 *** -0.724 (-7.456) (-8.038) (-7.880) (-6.354) [R.sup.2] 0.059 0.060 0.061 0.077 *** Significant at the 0.01 level. ** Significant at the 0.05 level. * Significant at the 0.10 level. Table VIII. Value-weighted Farna-MacBeth Regressions with all Control Variables This table reports the results of value-weighted Fama-MacBeth cross- section regressions for our various idiosyncratic volatility estimates formed from the Fama-French model. Volatility and other variable data are from January 1966 to December 2008 for firms traded on the NYSE, Amex and NASDAQ exchanges, when available. Market data is collected from CRSP and accounting variables are collected from Com pustat. All idiosyncratic volatility estimates in this table are derived from the Fama-French factor model. M_EGARCH(T) are EGARCH(p,g) generated estimates of monthly idiosyncratic volatility using the AIC criterion to choose the number of lags. The entire dataset is used to find these estimates. D_SQRET(t) are monthly estimated idiosyncratic risks estimated from daily returns as in Ang et. al (2003) M_EGARCH(t) are EGARCH(p,g) expected idiosyncratic volatilities formed using information through time t to forecast time t values. AI EGARCH(t-1) are EGARCH(p,y) expected idiosyncratic volatilities formed using information through time t-1 to forecast time t values. The AIC criterion is used to choose the optimal number of lags for each stock at each time period. M_SQRET(t-1) are idiosyncratic volatilities estimated as the standard deviation of residuals from Fama-French time series regressions using the previous 60 month data on a rolling basis. D_SQRET(t-1) is the lagged idiosyncratic volatility estimate using daily data as in Ang et al. (2006). D_SQRET-ARMA(t-1) are expected idiosyncratic volatilities formed using D_SQRET(t) information through time t-1 to forecast time t values. The forecasts are made using ARMA (p,q) model where the AIC criterion is used to choose the optimal number of lags for each stock at each time period. UNEXPECTED is time t unexpected volatility estimated by the residuals of a regression of M_EGARCH(T) on M_EGARCH(t-1). RET is the percentage return to the firm over the one-month holding period. LAGRET is the percentage return to the firm over the preceding one month holding period. MKTBETA, SMLBETA, HMLBETA and PS LIQ are beta estimates from time series regressions with Fama-French three factors and the liquidity series from Pdstor and Stambaugh (2003) using the previous 60 month data on a rolling basis. LNME is the natural log of a firm's market equity at the end of the fiscal year. LNBEME is the log of the fiscal year-end book value of common equity over the calendar year end market equity. CRETURN is represented by the cumulative returns from month (t-7) to (t-2) and is expressed as a percentage return. LNAG is the log of the percentage change of total assets of the firm. Newey-West (1987) adjusted t-statistics with a lag of four months are in parentheses. (1) (2) (3) (4) M_EGARCH 1.947 *** (T)(x10) (10.410) M_EGARCH 2.146 *** (t)(x10) (13.085) D_SQRET 2.602 *** (t)(x10) (7.5193) M_EGARCH -0.093 (t-1)(x10) (-1.216) M_SQRET (t-1)(x10) D_SQRET (t-1) (x10) D_SQRET_ARMA (t-1)(x10) UNEXPECTED (x10) MKTBETA -0.205 -0.271 * -0.261 * -0.049 (-1.295) (-1.669) (-1.717) (-0.302) SMBBET4 -0.114 -0.155 -0.094 -0.03 (-1.184) (-1.604) (-1.077) (-0.306) HMLBETA 0.157 0.181 * 0.122 0.102 -1.448 -1.682 -1.217 -0.914 LNME -0.048 -0.036 -0.012 -0.052 (-1.310) (-0.971) (-0.271) (-1.483) LNBEME 0.107 0.121 * 0.091 0.086 -1.5 -1.73 -1.339 -1.213 CRETURN 0.004 0.003 0.004 0.006 * -1.322 -1.08 -1.445 -1.947 PS_LIQ -0.083 -0.143 -0.22 -0.006 (-0.460) (-0.772) (-1.224) (-0.033) LNAG -0.672 *** -0.709 *** -0.562 *** -0.531 *** (-4.241) (-4.574) (-3.619) (-3.435) LAGRET -0.044 *** -0.044 *** -0.048 *** -0.041 *** (-8.405) (-8.277) (-8.842) (-7.7533) [R.sup.2] 0.169 0.171 0.195 0.161 (5) (6) (7) (8) M_EGARCH (T)(x10) M_EGARCH (t)(x10) D_SQRET (t)(x10) M_EGARCH (t-1)(x10) M_SQRET 0.001 (t-1)(x10) (0.053) D_SQRET -0.036 *** (t-1) (x10) (-3.128) D_SQRET_ARMA -0.045 ** (t-1)(x10) (-2.278) UNEXPECTED 0.261 *** (x10) -13.58 MKTBETA -0.068 -0.008 0.01 0.149 (-0.423) (-0.048) -0.063 -0.8805 SMBBET4 -52 -0.008 0.003 0.146 (-0.522) (-0.074) -0.035 -1.422 HMLBETA 0.109 0.091 0.079 0.005 -0.979 -0.804 -0.719 -0.047 LNME -0.045 -0.073** -0.079 ** -0.190 *** (-1.207) (-2.106) (-2.156) (-5.274) LNBEME 0.086 0.077 0.073 0.047 -1.229 -1.085 -1.023 -0.651 CRETURN 0.006 ** 0.006 * 0.006 ** 0.005 * -2.045 -1.916 -2.057 -1.795 PS_LIQ -0.011 0.024 0.045 0.044 (-0.062) -0.132 -0.243 -0.239 LNAG -0.555 *** -0.516 *** -0.512 *** -0.415 ** (-3.599) (-3.309) (-3.342) (-2.568) LAGRET -0.041 *** -0.040 *** -0.041 *** -0.042 *** (-7.802) (-7.291) (-7.676) (-7.974) [R.sup.2] 0.164 0.163 0.165 0.166 *** Significant at the 0.01 level. ** Significant at the 0.05 level. * Significant at the 0.10 level. Table IX. Equal-weighted Fama-MacBeth Regressions with all Control Variables (1966-1991 and 1992-2008) This table reports the results of equal-weighted Fama-MacBeth cross- section regressions for our various idiosyncratic volatility estimates for two subperiods from January 1966 to December 1991 and from January 1992 to December 2008. Volatility and other variable data are for firms traded on the NYSE, Amex and NASDAQ exchanges, when available. Market data is collected from CRSP and accounting variables are collected from Compustat. All idiosyncratic volatility estimates in this table are derived from the Fama-French factor model. M_EGARCH(t) are EGARCH(p,q) expected idiosyncratic volatilities formed using information through time t to forecast time t values. M_EGARCH(t-1) are EGARCH(p,q) expected idiosyncratic volatilities formed using information through time t-1 to forecast time t values. The AIC criterion is used to choose the optimal number of lags for each stock at each time period. D_SQRET(t) are monthly estimated idiosyncratic risks estimated from daily returns. D_SQRET(t-1) is the lagged idiosyncratic volatility estimate using daily data as in Ang et al. (2006). RET is the percentage return to the firm over the one- month holding period. LAGRET is the percentage return to the firm over the preceding one month holding period. MKTBETA, SMLBETA, HMLBETA and PS LIQ are beta estimates from time series regressions with Fama- French three factors and the liquidity series from Pastor and Stambaugh (2003) using the previous 60 month data on a rolling basis. LNME is the natural log of a firm's market equity at the end of the fiscal year. LNBEME is the log of the fiscal year-end book value of common equity over the calendar year end market equity. CRETURN is represented by the cumulative returns from month (t-7) to (t-2) and is expressed as a percentage return. LNAG is the log of the percentage change of total assets of the firm. Newey-West (1987) adjusted t- statistics with a lag of four months are in parentheses. Note: Model 9 is estimated using the time period 1993-2008 because bid-ask spread data in CRSP is limited to this time period. 1966-1991 (1) (2) (3) M_EGARCH 3.369 *** (t)(x10) (16.435) D_SQRET 3.701 *** (t)(x10) (10.224) M_EGARCH -0.080 (t-1)(x10) (-1.534) D_SQRET (t-1)(x10) D_SQRET (t-1) Using quote mid point returns (x10) MKTBETA -0.725 *** -0.783 *** -0.004 (-4.767) (-5.247) (-0.025) SMBBETA -0.458 *** -0.659 *** -0.019 (-4.643) (-7.391) (-0.170) HMLBETA 0.281 *** 0.195 ** 0.089 (3.224) (2.209) (1.057) LNME 0.363 *** 0.355 *** -0.044 (8.185) (10.799) (-0.997) LNBEME 0.596 *** 0.394 ** 0.248 *** (9.060) (5.620) (3.649) CRETURN 0.002 0.012 *** 0.003 (0.943) (4.694) (1.138) PS_LIQ -0.306 ** -0.278 * 0.077 (-2.030) (-1.744) (0.486) LNAG -0.971 *** -0.276 -0.924 *** (-5.268) (-1.488) (-5.475) LAGRET -0.086 *** -0.079 *** -0.083 *** (-13.790) (-13.283) (-13.019) [R.sup.2] 0.104 0.1269 0.073 1966-1991 1992-2008 (4) (5) (6) M_EGARCH 2.988 *** (t)(x10) (11.581) D_SQRET 2.192 *** (t)(x10) (6.892) M_EGARCH (t-1)(x10) D_SQRET -0.147 (t-1)(x10) (-1.479) D_SQRET (t-1) Using quote mid point returns (x10) MKTBETA -0.010 -0.569 *** -0.291 ** (-0.063) (-3.739) (-1.978) SMBBETA -0.009 -0.510 *** -0.309 *** (-0.083) (-5.668) (-3.652) HMLBETA 0.089 0.418 *** 0.304 ** (1.073) (3.229) (2.559) LNME -0.040 0.372 *** 0.399 *** (-0.940) (8.450) (7.258) LNBEME 0.243 *** 0.471 *** 0.316 *** (3.471) (7.9817) (5.289) CRETURN 0.003 0.004 * 0.009 *** (1.302) (1.849) (3.846) PS_LIQ 0.089 -0.488 *** -0.282 * (0.571) (-3.008) (-1.831) LNAG -0.906 ** -1.182 *** -0.645 *** (-5.630) (-7.035) (-4.464) LAGRET -0.084 *** -0.048 *** -0.047 *** (-12.357) (-9.546) (-8.732) [R.sup.2] 0.075 0.083 0.092 1992-2008 (7) (8) (9) * M_EGARCH (t)(x10) D_SQRET (t)(x10) M_EGARCH 0.067 (t-1)(x10) (1.015) D_SQRET 0.007 (t-1)(x10) (0.650) D_SQRET -0.017 * (t-1) (-1.687) Using quote mid point returns (x10) MKTBETA 0.047 0.057 0.091 (0.2621) (0.318) (0.524) SMBBETA -0.092 -0.089 -0.079 (-0.8922) (-0.875) (-0.775) HMLBETA 0.057 0.046 -0.005 (0.424) (0.358) (-0.041) LNME -0.121 ** -0.091 ** -0.106 ** (-2.265) (-1.984) (-2.049) LNBEME 0.094 0.101 0.193 ** (1.539) (1.634) (2.045) CRETURN 0.004 0.005 ** 0.004 (1.484) (2.018) (1.618) PS_LIQ -0.137 -0.116 -0.153 (-0.837) (-0.716) (-0.925) LNAG -0.935 *** -0.889 *** -0.802 *** (-5.544) (-5.690) (-4.201) LAGRET -0.050 *** -0.051 *** -0.048 *** (-8.555) (-8.337) (-8.100) [R.sup.2] 0.053 0.056 0.055 *** Significant at the 0.01 level. ** Significant at the 0.05 level. * Significant at the 0.10 level. Table X. Value-weighted Fama-MacBeth Regressions with all Control Variables (1966-1991 and 1992-2008) This table reports the results of value-weighted Fama-MacBeth cross- section regressions for our various idiosyncratic volatility estimates for two subperiods from January 1966 to December 1991 and from January 1992 to December 2008. Volatility and other variable data are for firms traded on the NYSE, Amex and NASDAQ exchanges, when available. Market data is collected from CRSP and accounting variables are collected from Compustat. All idiosyncratic volatility estimates in this table are derived from the Fama-French factor model. M_EGARCH(t) are EGARCH(p,q) expected idiosyncratic volatilities formed using information through time t to forecast time t values. M_EGARCH(t-1) are EGARCH(p,q) expected idiosyncratic volatilities formed using information through time t-1 to forecast time t values. The AIC criterion is used to choose the optimal number of lags for each stock at each time period. D_SQRET(t) are monthly estimated idiosyncratic risks estimated from daily returns. D_SQRET(t-1) is the lagged idiosyncratic volatility estimate using daily data as in Ang et al. (2006). RET is the percentage return to the firm over the one-month holding period. LAGRET is the percentage return to the firm over the preceding one month holding period. MKTBETA, SMLBETA, HMLBETA and PS LIQ are beta estimates from time series regressions with Fama-French three factors and the liquidity series from Pastor and Stambaugh (2003) using the previous 60 month data on a rolling basis. LNME is the natural log of a firm's market equity at the end of the fiscal year. LNBEME is the log of the fiscal year-end book value of common equity over the calendar year-end market equity. CRETURN is represented by the cumulative returns from month (t-7) to (t-2) and is expressed as a percentage return. LNAG is the log of the percentage change of total assets of the firm. Newey-West (1987) adjusted t- statistics with a lag of four months are in parentheses. Note: Model 9 is estimated using the time period 1993-2008 because bid-ask spread data in CRSP is limited to this time period. 1966-1991 (1) (2) (3) M_EGARCH 2.397 *** (t)(x10) (10.748) D_SQRET 3.463 *** (t)(x10) (7.496) M_EGARCH -0.161 * (t-1)(x10) (-1.592) D_SQRET (t-1)(x10) D_SQRET (t-1) Using quote mid-point returns (x10) MKTBETA -0.557 *** -0.664 *** -0.230 (-2.683) (-3.472) (-1.112) SMBBETA -0.207 -0.241 ** -0.042 (-1.593) (-2.028) (-0.315) HMLBETA 0.250 ** 0.257 ** 0.158 (2.184) (2.325) (1.397) LNME -0.005 0.071 -0.060 (-0.109) (1.269) (-1.321) LNBEME 0.126 0.089 0.054 (1.153) (0.838) (0.486) CRETURN 0.004 0.007 * 0.007 * (1.001) (1.665) (1.681) PS_LIQ -0.327 -0.430 ** -0.101 (-1.379) (-1.910) (-0.422) LNAG -0.976 *** -0.807 *** -0.792 *** (-4.424) (-3.658) (-3.609) LAGRET -0.051 *** -0.052 *** -0.049 *** (-6.989) (-7.269) (-6.619) [R.sup.2] 0.177 0.205 0.165 1966-1991 1992-2008 (4) (5) (6) M_EGARCH 1.797 *** (t)(x10) (7.849) D_SQRET 1.405 *** (t)(x10) (3.028) M_EGARCH (t-1)(x10) D_SQRET -0.429 *** (t-1)(x10) (-2.697) D_SQRET (t-1) Using quote mid-point returns (x10) MKTBETA -0.195 0.127 0.299 (-0.947) (0.520) (1.370) SMBBETA -0.028 -0.082 0.111 (-0.206) (-0.573) (0.929) HMLBETA 0.163 0.084 -0.067 (1.432) (0.418) (-0.373) LNME -0.080 * -0.078 -0.128 * (-1.801) (-1.348) (-1.780) LNBEME 0.046 0.114 0.095 (0.420) (1.580) (1.309) CRETURN 0.007 * 0.002 0.001 (1.704) (0.452) (0.250) PS_LIQ -0.076 0.1141 0.0720 (-0.318) (0.397) (0.250) LNAG -0.779 *** -0.339 * -0.221 (-3.535) (-1.750) (-1.120) LAGRET -0.047 *** -0.034 *** -0.041 *** (-6.022) (-4.584) (-5.075) [R.sup.2] 0.168 0.177 0.205 1992-2008 (7) (8) (9) * M_EGARCH (t)(x10) D_SQRET (t)(x10) M_EGARCH 0.002 (t-1)(x10) (0.0l5) D_SQRET -0.271 (t-1)(x10) (-1.605) D_SQRET -0.113 (t-1) (-0.880) Using quote mid-point returns (x10) MKTBETA 0.203 0.252 0.240 (0.796) (1.001) (0.908) SMBBETA -0.014 0.020 -0.016 (-0.095) (0.134) (-0.099) HMLBETA 0.024 -0.008 -0.047 (0.112) (-0.038) (-0.197) LNME -0.040 -0.063 -0.060 (-0.738) (-1.151) (-0.994) LNBEME 0.130 * 0.119 0.080 (1.830) (1.638) (0.884) CRETURN 0.004 0.004 0.003 (0.976) (0.901) (0.630) PS_LIQ 0.1260 0.1633 0.1760 (0.426) (0.560) (0.555) LNAG -0.168 -0.150 -0.247 (-0.865) (-0.756) (-1.079) LAGRET -0.029 *** -0.030 *** -0.029 (-4.224) (-4.220) (-4.052) [R.sup.2] 0.165 0.168 0.159 *** Significant at the 0.01 level. ** Significant at the 0.05 level. * Significant at the 0.10 level. Table XI. Abnormal Returns of Portfolios Sorted on Idiosyncratic Volatility (1966-1991 and 1992-2008) The table reports Fama-French (1993) three-factor alphas for value- weighted portfolios sorted on idiosyncratic volatilities for two subperiods from January 1966 to December 1991 and from January 1992 to December 2008. We first report alphas for portfolios sorting on the idiosyncratic volatility measures only on the row labeled "Exchange Traded Stocks." The column "5-1" refers to the difference in the Fama- French alphas between portfolios 5 and 1. In the rows controlling for size, book-to-market, liquidity and lagged returns (Lagret), we perform a double sort. We first sort stocks based on the first characteristic (size, liquidity, and lagret) into quintiles each month, and then within each quintile, we sort stocks based on the specific idiosyncratic volatility measure. We average the five idiosyncratic volatility portfolios over the each of the five characteristic portfolios as in Ang et al. (2006). We follow Ang et al. (2006) and use the historical liquidity beta from Pastor and Stambaugh (2003) as our liquidity measure. We report the abnormal returns of portfolios sorted on the idiosyncratic volatility estimate the out-of-sample EGARCH forecast, M_EGARCH(t-1) in Panel A, and in Panel B, the lagged idiosyncratic volatility estimate using daily data as in Ang et al. (2006), D_SQRET(t-1). Newey-West (1987) adjusted t- statistics with a bandwidth of four months are in parentheses. Panel A Variable Ranking on M EGARCH(t-1): 1966-1991 1 Low 2 3 Exchange Traded Stocks 0.063 0.049 0.053 (1.453) (1.215) (0.785) Controlling for Size -0.031 0.033 -0.050 (-0.543) (0.677) (-0.850) Controlling for Lagret 0.105 ** 0.074 0.062 (2.154) (1.577) (1.108) Controlling for Liquidity 0.113 *** 0.056 0.041 (2.779) (1.091) (0.690) 4 5 High 5-1 Exchange Traded Stocks -0.114 -0.287 *** -0.350 *** (-1.225) (-2.608) (-2.594) Controlling for Size -0.078 -0.347 *** -0.316 *** (-1.157) (-3.770) (-3.057) Controlling for Lagret -0.128 -0.201 ** -0.306 *** (-1.619) (-1.883) (-2.346) Controlling for Liquidity -0.065 -0.238 ** -0.350 ** (-0.815) (-2.049) (-2.543) Variable Ranking on M EGARCH(t-1): 1992-2008 1 Low 2 3 Exchange Traded Stocks 0.012 -0.010 0.133 (0.140) (-0.153) (1.406) Controlling for Size 0.104 0.071 0.072 (0.944) (0.731) (0.649) Controlling for Lagret -0.065 0.033 0.044 (-0.749) (0.433) (0.529) Controlling for 0.050 0.044 0.033 Liquidity (0.683) (0.801) (0.344) 4 5 High 5-1 Exchange Traded Stocks 0.138 -0.198 -0.210 (0.953) (-1.012) (-0.898) Controlling for Size 0.075 -0.071 -0.176 (0.526) (-0.357) (-0.883) Controlling for Lagret 0.117 -0.132 -0.067 (0.857) (-0.829) (-0.330) Controlling for 0.074 0.008 -0.042 Liquidity (0.609) (0.044) (-0.199) Panel B Variable Ranking on D_SQRET(t-1): 1966-1991 1 Low 2 3 Exchange Traded Stocks 0.042 0.113 *** 0.072 (0.995) (2.587) (1.042) Controlling for Size 0.086 0.225 *** 0.115 * (1.006) (3.318) (1.653) Controlling for Lagret 0.122 *** 0.099 ** -0.026 (2.580) (2.209) (-0.383) Controlling for 0.103** 0.131 ** 0.025 Liquidity (2.348) (2.408) (0.368) 4 5 High 5-1 Exchange Traded Stocks -0.321 *** -1.328 *** -1.370 *** (-3.524) (-9.507) (-8.446) Controlling for Size -0.150 ** -1.069 *** -1.155 *** (-2.037) (-8.701) (-7.784) Controlling for Lagret -0.277 *** -1.220 *** -1.342 *** (-3.698) (-10.428) (-10.358) Controlling for -0.212 ** -1.106 *** -1.209 *** Liquidity (-1.993) (-8.134) (-7.931) Variable Ranking on D_SQRET(t-1) 1992-2008 1 Low 2 3 Exchange Traded Stocks 0.177 ** -0.037 -0.036 (2.180) (-0.630) (-0.313) Controlling for Size 0.141 0.135 0.092 (1.450) (1.229) (0.748) Controlling for Lagret 0.079 0.043 -0.208 * (1.103) (0.577) (-1.928) Controlling for Liquidity 0.142 ** 0.057 -0.085 (1.962) (0.801) (-0.798) 4 5 High 5-1 Exchange Traded Stocks -0.409 ** -0.992 *** -1.169 *** (-2.356) (-3.428) (-3.492) Controlling for Size -0.144 -0.744 *** -0.885 *** (-0.856) (-3.224) (-3.444) Controlling for Lagret -0.204 -1.022 *** -1.100 (-1.442) (-4.397) (-4.063) Controlling for Liquidity -0.182 -0.559 ** -0.701 *** (-1.338) (-2.442) (-2.703) Variable Ranking on D_SQRET(t-1) using quote mid-point returns 1993-2008 1 Low 2 3 Exchange Traded Stocks 0.205 * -0.055 -0.128 (1.888) (-0.754) (-1.243) Controlling for Size 0.239 ** 0.097 0.025 (2.025) (0.774) (0.175) Controlling for Lagret 0.055 0.019 -0.275 *** (0.663) (0.243) (-2.729) Controlling for 0.193 ** 0.052 -0.132 Liquidity (2.252) (0.511) (-1.087) 4 5 High 5-1 Exchange Traded Stocks -0.375 ** -0.389 * -0.604 ** (-2.002) (-1.841) (-2.225) Controlling for Size -0.158 -0.771 *** -1.028 *** (-0.871) (-3.238) (-3.779) Controlling for Lagret -0.163 -0.554 *** -0.613 *** (-1.082) (-2.786) (-2.523) Controlling for -0.105 -0.219 -0.425 * Liquidity (-0.716) (-1.059) (-1.715) *** Significant at the 0.01 level. ** Significant at the 0.05 level. * Significant at the 0.10 level.

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Author: | Fink, Jason D.; Fink, Kristin E.; He, Hui |
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Publication: | Financial Management |

Article Type: | Report |

Geographic Code: | 1USA |

Date: | Sep 22, 2012 |

Words: | 19385 |

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