Is Idiosyncratic Volatility Related to Returns? Evidence from a Subset of Firms with Quality Idiosyncratic Volatility Estimates*

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1 Is Idiosyncratic Volatility Related to Returns? Evidence from a Subset of Firms with Quality Idiosyncratic Volatility Estimates* Mikael Bergbrant St. John s University Haimanot Kassa Miami University, and U.S. Securities and Exchange Commission This Version: March 12, 2018 Abstract Empirical studies investigating the relation between expected idiosyncratic volatility (IVOL) and returns find mixed results. Several papers report a negative relation using lagged realized IVOL to proxy for expectations, but Fu (2009) questions the validity of this proxy and proposes using forecasts from EGARCH models, resulting in a positive relation. However, recent studies show that this positive relation disappears when the forecasts are generated by out-of-sample models. We show that the proxies used in prior literature are noisy and propose using combinations of outof-sample IVOL forecasts that pass basic diagnostic tests. These high quality proxies uncover a significant positive relation. Key Words: Idiosyncratic Volatility, Priced Risk Factors, GARCH, EGARCH, Conditional Expected Volatility JEL Classification: G11, G12, G14 * We thank Doina Chichernea, Jared Delisle, Delroy Hunter, Patrick Kelly, Andre de Souza, Jidie Wintoki, Lei Zhou, and seminar participants at the University of South Florida. Mikael Bergbrant, bergbram@stjohns.edu, (813) , Department of Economics and Finance, Tobin College of Business, St. John s University, Queens, NY 11439, USA. Haimanot Kassa, kassah@miamioh.edu, (513) , Department of Finance, Farmer School of Business, Miami University, Oxford, OH 45056, USA; and U.S. Securities and Exchange Commission, 100 F Street N.E. Washington D.C , USA. The Securities and Exchange Commission disclaims responsibility for any private publication or statement of any SEC employee or Commissioner. This article expresses the author's views and does not necessarily reflect those of the Commission, the Commissioners, or other members of the staff.

2 1. Introduction Much research has been devoted to investigating the relation between expected idiosyncratic volatility (IVOL) and stock returns, but the evidence to date is mixed. In the first study to use firm level proxies for expected IVOL, Ang, Hodrick, Xing, and Zhang (2006, 2009) use lagged IVOL, calculated using daily returns in the previous month, to proxy for expectations (henceforth AHXZ_IVOL). The papers report a significant negative relation, giving rise to what has become known as the IVOL Puzzle. Since then, a multitude of studies have challenged the robustness of these findings (see survey in Hou and Loh, 2016). 1 For example, while Bali and Cakici (2008) finds that the negative relation is driven exclusively by small firms, other papers have found it to be related to microstructure noise, such as short term reversals (Huang, Liu, Rhee, and Zhang, 2009) and the maximum daily return in the prior month (Bali, Cakici, and Whitelaw, 2011). Fu (2009) criticizes AHXZ_IVOL from a theoretical perspective. It argues that lagged IVOL is a poor proxy for expectations because IVOL is time varying and the null hypothesis of a random walk is rejected for 90% of firms. Instead, the paper proposes using expected IVOL estimated from the GARCH family of models which account for the conditional time-variation of the underlying volatility process. In particular, the paper generates conditional volatility estimates (hereafter FU_IVOL) using exponential GARCH (or EGARCH) models that capture many of the dynamic properties of return volatility including asymmetry (i.e. leverage effect) and always generates a positive conditional volatility estimate. In addition to being theoretically sound, these models use monthly data (as opposed to daily data) and therefore alleviates several concerns with 1 Several studies attempt to explain the apparent puzzle using, for example, illiquidity (Bali and Cakici, 2008; Han and Lesmond, 2011), reversal (Huang, Liu, Rhee, and Zhang, 2009), earnings surprise (Jiang, Zu, and Yao, 2009), expected idiosyncratic skewness (Boyer, Mitton, and Vorkink, 2010), maximum daily return (Bali, Cakici, and Whitelaw, 2011), and arbitrage asymmetry (Stambaugh, Yu, and Yuan, 2015). 1

3 regards to microstructure noise that the AHXZ_IVOL proxy is subject to. Interestingly, and in stark contrast to prior literature, the expected IVOL proxy generated by Fu (2009) is strongly positively related to returns. Several empirical studies have since adopted the use of EGARCH models to generate proxies for expected IVOL (see, for example, Huang, Liu, Rhee, and Zhang, 2010; Spiegel and Wang, 2005; Chichernea, Ferguson, and Kassa, 2015). However, recent studies question the validity of the findings in Fu (2009) by showing that the positive relation between FU_IVOL and returns is a manifestation of a look-ahead bias (see, e.g., Fink, Fink, and He, 2012; Guo, Kassa, and Ferguson, 2014). In particular, these studies show that including the contemporaneous return observation when estimating the EGARCH parameters used to generate FU_IVOL has a profound impact on the positive relation between FU_IVOL and returns shown in Fu (2009). Using simulated data, Guo et al. (2014) illustrates the impact of the bias on the estimated relation between expected IVOL and returns and finds that the bias can cause the relation to appear positive even if it is non-existent in the data. Importantly, using actual stock return data, Fink et al. (2012) and Guo et al. (2014) find no significant relation between returns and out-of-sample conditional volatility forecasts (henceforth GKF_IVOL) that are free of a lookahead bias. This finding has been confirmed by additional studies (e.g., Bali, Scherbina, and Tang, 2016). Although a non-existing relation between expected IVOL and returns is consistent with efficient markets in which diversified investors seek compensation only for systematic risk, the fact that the conditional volatility estimates are sensitive to a look-ahead bias arising from the inclusion of one additional (i.e. the contemporaneous) return observation calls into question the quality/stability of the estimates. Perhaps, the poor quality of the estimates should not be surprising, given that prior literature (Fu, 2009; Fink et al., 2012; Guo et al., 2014) does not conduct 2

4 any diagnostic test on the models (with the exception of convergence) and therefore use volatility estimates from models that violate model assumptions and are not a good fit for the data. In this paper, we mitigate the noise inherent in prior expected IVOL proxies by creating proxies from a combination of out-of-sample conditional volatility forecasts from different EGARCH models that pass relevant diagnostic tests (henceforth BK_IVOL). Specifically, we use the mean conditional volatility forecast calculated from up to nine different out-of-sample EGARCH models that pass diagnostic tests and require at least five forecasts to be available in a given firm-month (the construction of our proxy is further explained in the next section). 2 We find that our quality proxy, BK_IVOL, is positively related to returns in the cross-section and the positive relation is both statistically significant and economically important. Firms with expected idiosyncratic volatility of one standard deviation above the mean have expected returns that are approximately 3.5% to 4.1% higher per year. Hence, our paper is the first to find a positive relation between unbiased proxies for expected IVOL and returns. By doing so, we qualitatively confirm the questioned findings in Fu (2009). Although inconsistent with traditional asset pricing theory, where agents only require compensation for systematic risk, the positive relation is consistent with theories that assume that investors are under-diversified and seek compensation for bearing idiosyncratic risk (e.g., Levy, 1978; Merton, 1987; and Malkiel and Xu, 2002). 3 Although our requirement for creating the quality expected IVOL proxy, BK_IVOL, result in a smaller sample of firms compared to prior studies, we find the sample to be representative. 2 The model diagnostic test we consider are normally distributed standardized residuals, covariance stationary parameters and the expected sign on the leverage effect, i.e. negative return shocks have a larger impact than positive return shocks on future volatility. We discuss this in more detail in the next section. 3 The idea of a positive relation is further supported by the undiversified holdings of individual investors. Looking at retail investors, Goetzmann and Kumar (2004) finds that more than 25% hold a single stock in their portfolios and fewer than 10% of the portfolios contain more than 10 stocks. This is in stark contrast to the approximately 50 stocks that Campbell, Lettau, Malkiel, and Zu (2001) argue that investors need in their portfolios to be diversified. 3

5 Summary statistics for our sample closely resembles that of prior studies, and all commonly used factors in our tests have the expected sign (and similar loadings) as found in prior studies. Importantly, AHXZ_IVOL is strongly negatively related to returns amongst the firms in our sample (with a similar coefficient as in the full CRSP universe), indicating that our positive results do not obtain because of the firms that are included in our sample (for which quality expected IVOL estimates can be estimated), but because of a superior measure of expected IVOL. The positive relation we document is in stark contrast to prior papers that also use unbiased forecasts from EGARCH models to proxy for expected IVOL but find no significant relation (e.g. Fink et al., 2012 and Guo et al. 2014). As such, we attempt to determine what drives this difference by relaxing our requirements for creating our quality expected IVOL proxy (BK_IVOL). We find that the difference arises mostly from the improved quality (i.e., less noise) of our proxy. As expected, we find that the results are sensitive to our diagnostic tests as, like Fink et al. (2012) and Guo et al. (2014), we find no significant relation between proxies for expected IVOL and returns when we exclude the requirement that models used to generate BK_IVOL pass diagnostic tests. Since models that do not pass basic diagnostic tests are more likely to generate noisy conditional volatility forecasts, this is the first evidence that our results are driven by reducing noise in our estimates. We also find that the results are sensitive to the number of models that pass the diagnostic tests for a given firm-month and, importantly, the economic (and statistical) magnitude of the results increases with the number of models that pass. Given that we expect more accurate estimates for firm-months in which several forecasts are available (to use when creating BK_IVOL), this also suggests that our results are driven by reducing noise in the estimates. Although the significant positive relation holds regardless of the way we select the quality estimate 4

6 (it holds for selections based on the Mean, Median, AIC, AICC, SBC, and HQC), the results are stronger when using combinations of forecasts (Mean or Median), again suggesting that higher quality (less noisy) estimates are the driver of our findings. Finally, we find that the results are robust to the number of observations required to estimate the model parameters (although the economic magnitude of the results increase with the number of observations used) and changing the maximum number of iterations for models to converge. We contribute to several strands of literature. First, we contribute to the literature on the impact of under-diversified investors on asset prices. Our finding that expected IVOL is positively related to returns imply that under-diversified investors influence asset prices, and suggests that there is more value to diversification than traditional asset pricing theory suggests. Second, we contribute to the literature that investigates the link between expected IVOL and returns at the firm level. While existing studies have mixed findings using a large cross-section of firms with noisy proxies (based on models that do not fit the data and/or allowing estimates from models that violate model assumptions) of expected IVOL, we uncover a strong positive relation for a subset of firms with quality expected IVOL estimates. Finally, we contribute to the narrower literature that investigates the relation between EGARCH proxies for expected IVOL and returns (Fu, 2009; Bergbrant, 2011; Fink et al., 2012; Guo et al., 2014). We confirm that the positive in-sample relation found by Fu (2009) also obtains for out-of-sample proxies after imposing requirement intended to ensure the quality of the models that the proxy is based on. The rest of the paper is organized as follows. The second section describes the data and construction of our expected IVOL proxy (BK_IVOL), while the third section presents the results. The fourth, and final, section concludes. 5

7 2. Data and variable descriptions We obtain holding period returns inclusive of dividends (RET), prices (P), shares outstanding (SHROUT), and volume (VOL) data for all firms traded on NYSE/AMEX, and NASDAQ from July of 1926 to December 2016 from the Center for Research in Security Prices (CRSP). Returns exceeding 300% are deleted to be consistent with prior literature (Fu, 2009; Bergbrant, 2011; Guo et al., 2014). 4 Accounting data, or more specifically book values, are obtained from Compustat s annual fundamentals file. Small-Minus-Big (SMB), High-Minus-Low (HML), the market premium (MKTRF), and the risk free rate (RF) are obtained from Ken French s website. 5 We follow the recent IVOL literature (Fu, 2009; Guo et al. 2014) in defining our control variables. Beta, size, and market-to-book ratios are defined as in Fama and French (1992) while momentum and turnover are constructed following Chordia, Subrahmanyam and Anshuman (2001). Momentum is calculated as the holding period return between month t 7 to t 2, where the last month s (t 1) return is excluded to avoid returns merely due to bid-ask bounce. Turnover is calculated as the average share turnover (monthly volume divided by the numbers of shares outstanding) in the past 36 months. 6 We also include the coefficient of variation of those turnovers over the estimation period. To avoid giving extreme observations heavy weight, all variables, except for returns and beta, are winsorized monthly at the.5% and 99.5%. Although we use all available data to create our variables, we have limited our empirical tests to the sample 4 This affects only 408 firm-month observations during our sample 5 We thank Ken French for making the data available. 6 We require at least 24 monthly returns to estimate the turnover variable. 6

8 period between July 1963 and December of 2016 due to limitations in accounting data prior to July Measuring Out-of-sample EGARCH Idiosyncratic Volatility Most papers investigating the relation between expected idiosyncratic volatility and return at the firm level use lagged realized IVOL (AHXZ_IVOL) to proxy for expectations, implicitly assuming that IVOL follows a random walk (Ang, Hodrick, Xing, and Zhang, 2006 and 2009). However, Fu (2009) points out that this is problematic as the null hypothesis of a random walk is rejected for 90% of the firms in its sample and argues that forecasting conditional volatility using models in the GARCH family is better. Particularly, Fu (2009) suggests that the EGARCH model proposed by Nelson (1991) provides a superior model of expected idiosyncratic volatility since it captures the asymmetric properties of volatility (e.g. leverage effects) and always generate positive volatility estimates. The EGARCH models are also more flexible than other ARCH and GARCH models and do not restrict the parameters to avoid negative values. 7 However, the estimates used in Fu (2009), FU_IVOL, have been shown to be biased as they use time series return observation up to and including time t in estimating the parameters of the model used for calculating the expected idiosyncratic volatility at time t when, in fact, the contemporaneous return should have been excluded to create out-of-sample forecasts (Fink, Fink, and He, 2012; Guo, Kassa, and Ferguson, 2014). We follow the methodology in Guo et al. (2014) and estimate unbiased (out-of-sample) forecasts of idiosyncratic volatility. Specifically, the EGARCH model that we fit is: R i,t RF t = a i + b i (R M,t R RF,t ) + s i SMB t + h i HML t + ε i,t 7 Pagan and Schwert (1990) compare different GARCH models and they find that the EGARCH model does the best in explaining monthly return volatility. 7

9 ln σ 2 2 i,t = a i + b i,l ln σ i,t l p l=1 ε i,t ~N(0, σ i,t 2 ) q + c i,k (θ ( ε i,t k σ i,t k ) + Υ ( ε i,t k σ i,t k ( 2 π ) 1/2 ) ) k=1 We estimate EGARCH (p,q), 1 p 3 and 1 q 3, for a total of nine models per firmmonth observation. We use data up to (and including) time t 1 to calculate the model parameters. We then use these model parameters to forecast the conditional volatility at time t. We do this iteratively using an expanding window, requiring at least 72 monthly observations to run the model. When estimating the model above, we allow for up to 1000 iterations for the model to converge Selecting High Quality EGARCH Idiosyncratic Volatility Estimates Although the use of lagged realized IVOL (AHXZ_IVOL) to proxy for expectations have been widely critiqued in the literature (see survey in Hou and Loh, 2016), and the novel EGARCH based proxy (FU_IVOL) used by Fu (2009) has been shown to be biased, there are several reasons to believe that the unbiased EGARCH based expected IVOL proxy used by prior literature (GKF_IVOL) is also a poor proxy for expected IVOL. 8 First, other than requiring model convergence, prior literature uses no other diagnostic tests to ensure the quality of the models from which the conditional volatility estimates are obtained. Convergence is a minimum/necessary condition for an acceptable model, but not a sufficient condition as convergence could occur at a local, rather than the global, maximum. Allowing estimates from models that converge at a local maxima would lead to noisy estimates, which would be less likely to pin down the true relation 8 It should be noted that although the FU_IVOL measure used by Fu (2009) is subject to the same critique regarding noise as the GKF_IVOL measure, the bias inherent in FU_IVOL could trump the noise and therefore lead to a positive relation between expected IVOL and return. 8

10 between expected IVOL and returns. We address this issue by applying diagnostic tests, i.e., only using conditional volatility forecasts from EGARCH models that produce standardized residuals that are normally distributed, parameters that are covariance stationary, and a leverage effect of the expected sign (negative return shocks have a larger impact than positive return shocks on future volatility). Second, previous literature estimates nine different EGARCH models ranging from EGARCH (1,1) to EGARCH (3,3). Although nine models are generated, these studies do not require any minimum number of forecasts to be available, potentially allowing volatility estimates for firm-months in which only one model converges. This issue is compounded by the fact that, as previously noted, the models are not subjected to any diagnostic tests (with the exception of convergence) leading to potentially poor (noisy) estimates of expected IVOL for many firms. If equity returns for US securities is (as assumed) fairly well-behaved on a monthly level and the EGARCH family of models is well suited to model the conditional volatility, then we would expect that many models will converge and pass relevant diagnostic tests. Requiring several models to converge would improve the quality of the volatility estimate. For instance, having more than one model that converged increases the likelihood that at least one model converged at the global maximum. We address this issue by requiring that a majority (five out of nine) of the EGARCH models pass the diagnostic tests in a given firm-month. 9 This approach gives us several models to use (or choose from) when creating (or selecting) our quality estimate. Finally, the previous literature selects the conditional volatility estimate from the nine EGARCH models using the lowest Akaike information criterion (AIC) score (Akaike, 1992). 9 If EGARCH models fit the data well, we expect several of the models to pass our diagnostic tests for a given firmmonth. 9

11 However, AIC may not always dominate other model selection criteria such as Bayesian information criterion (BIC) (Schwartz, 1978) and, in general, estimates based on model selection could be very unstable (Yang, 2003; Breiman, 1999). 10 We address this issue by using a combination of forecast approach, particularly a simple average (although the median works as well). This approach is similar to the combination of forecast approach heavily used in the timeseries forecasting literature (see, for example, Rapach, Strauss, and Zhou, 2010) and offers several advantages. For example, if several of the forecasts capture important non-overlapping information about expected IVOL, then a combination of forecast method will capture different components of expected IVOL. Alternatively, if a single IVOL estimate is very noisy and gives an implausible proxy for expected IVOL, combination of multiple forecasts can mitigates this noise - in short, there are gains from diversification. In addition, using the mean conditional volatility forecast should more closely relate to average investors expectations, as investors are likely to use a multitude of models to form expectations. 11 Overall, our proxy for expected IVOL is a simple average of at least five, and possibly up to nine, out-of-sample EGARCH idiosyncratic volatility estimates that converge and pass several diagnostic tests. Panel A of Table 1 reports the total number of models estimated as well as the final number of firm-months that have the required data to calculate our quality expected IVOL proxy (BK_IVOL). From July 1963 to December 2016, we estimate a total of 15.7 million EGARCH 10 Yang (2003) states that it is well known that when one of the models considered [among the choices] is a true model, with probability tending 1, BIC selects the true model and it performs asymptotically better than AIC; on the other hand, if none of the models being compared is the true model, AIC asymptotically outperforms BIC. For the reality of a finite sample, however, for either case, the answer to the question which criterion is better depends on how fast the approximation errors (bias) of the relevant models (depending on the sample size and the error variance) decreases. 11 The GARCH models had not been developed during the first half of the sample and would have been unlikely to be used in estimating IVOL. 10

12 models representing almost 1.75 million unique firm-month observations. Out of these, approximately 76% of the models (11.9 million) converge, but only about 20% of the converging models pass our diagnostic tests. The low proportion of models that pass the diagnostic tests highlights the importance of checking model diagnostics before using these models in forecasting and suggests that prior studies have overlooked this important fact, leading them to use noisy estimates. Further restricting the sample for which there are at least five models that pass these diagnostic tests result in data for 220,148 firm-month observations during our sample period. In our main tests, we use the average conditional volatility forecast for these firm-months, and our analysis is based on these 220,148 firm-month observations over the 642 month window. 12 In Panel B of Table 1, we report the percentage of converging models with different EGARCH(p,q) combinations. The table shows that EGARCH(1,1) models are the most likely to converge (93.6%), and EGARCH(3,3) is the least likely model to converge (48.6%). This pattern holds overall as we move from EGARCH(1,1) to EGARCH(3,3). In Panel C, we show the number of models that pass all diagnostic tests. Although a similar pattern is present in Panel C, the more advanced EGARCH models perform relatively better when judged on the percent of estimates that pass basic diagnostic tests. For example, we can see that 18.0% of estimates pass the diagnostic tests with the GARCH (3,1) whereas only 14.3% pass with EGARCH(1,1). Although our requirements of the models used to generate the conditional volatility forecasts results in fewer observations than the previous IVOL literature, the measure is available for a sufficient number of firm-month observations to conduct our tests. In Figure 1A, we show 12 We should note that we are not concerned with the final number of estimates that our diagnostic tests/filters produce, as long as the number is large enough to conduct our tests. This is because our goal is to test the relation between expected idiosyncratic volatility and returns for firms for which we have quality estimates of idiosyncratic volatility, and in order to do so we are willing to sacrifice the number of estimates. 11

13 the number of firms in each month for which BK_IVOL is available. Although relatively few firms have enough data to calculate the proxy in the first few years, we see a substantial jump in the number of firms included in our sample around 1979, when NASDAQ stocks are included. 13 Subsequent to the inclusion of NASDAQ firms, the number of firms for which BK_IVOL can be calculated in a given month generally exceeds Figure 1B shows the proportion of firms for which GKF_IVOL is estimated that has enough quality data so that BK_IVOL also can be estimated. Although the proportion varies slightly over time, there are no large sudden changes that would indicate that our requirements skews the data to certain time periods, but is generally representative of the population of listed firms. Although we have provided several theoretical arguments as to why our measure should be less noisy than prior proxies for expected IVOL, we have not yet shown any empirical evidence supporting this. In Figure 2, we plot the time series average of the standard deviation of both BK_IVOL and GKF_IVOL and see that the volatility of BK_IVOL is substantially smaller than GKF_IVOL. In unreported tests, we confirm that this also holds for individual firms. Focusing on firms for which the BK_IVOL measure is calculated during at least 10 months, the volatility of the estimates are on average 34% lower than the volatility of the GKF_IVOL estimates for the same firm. These results suggest that our measure BK_IVOL is less noisy GKF_IVOL. 3. Results Our objective is to examine the relation between high quality proxies for expected IVOL 13 NASDAQ stocks are included in CRSP in 1973, but cannot be included in our final sample until 72 months of data is available, as that is what we require to estimate the EGARCH models. 14 The number of firms decrease towards the end of the sample, similar to what we observe about the total number of firms reported in the CRSP database the number of firms decreased by about half from late 1990s (or early 2000s) to

14 and the cross section of stock returns. The existing literature reports mixed results: a negative result when using lagged realized IVOL (AHXZ_IVOL) to proxy for expectations (Ang et al., 2006, 2009), but a positive relation when using in-sample conditional volatility (FU_IVOL) estimated with EGARCH models (Fu, 2009). However, the positive relation disappears when using out-ofsample estimates of IVOL (GKF_IVOL) from EGARCH models without imposing requirement to ensure quality of the estimates (Guo et al., 2014). We use combinations of out-of-sample IVOL forecasts generated by different EGARCH models that pass basic diagnostic tests to help us pin down the real underlying relation, and find a positive empirical relation between expected IVOL and the cross section of stock returns Summary Statistics and Correlations We report summary statistics and correlations in Panels A and B of Tables 2 respectively. The mean raw return is 1.39% and the excess return is 0.98% during our sample period. The table also shows that the mean of our out-of-sample proxy for expected IVOL, BK_IVOL, is 10.14%, which is comparable to, albeit a bit smaller than, Fu s (2009) mean in-sample IVOL proxy of 12.67%. 15 Importantly, the standard deviation of BK_IVOL (5.98) is substantially lower than the reported by Fu (2009). Although Guo et al. (2014) does not report summary statistics, we replicate the proxy (GKF_IVOL) used in the paper and find that the volatility of GKF_IVOL is higher (8.34) than the volatility of our BK_IVOL measure (5.98). Since our sample of firms with quality estimates of expected IVOL (BK_IVOL) is relatively small in comparison to prior studies, we also present summary statistics for the complete CRSP/Compustat database during the sample 15 Fu s (2009) sample is from 1963 to 2006 whereas ours is from 1963 to Important to note is that the sample in Fu (2009) includes 2.8M observations of expected IVOL, while our quality sample consists of 220K observations. Additionally, as described before, the estimates used by Fu (2009) are in-sample whereas ours is out-of-sample. 13

15 period. The control variables are qualitatively similar to those of the complete CRSP/Compustat universe, although the average size of the firms that meet our diagnostic tests are larger (the mean log market value is 4.64 for all firm-months, and 5.55 for our final sample). In Panel B of Table 2, we report the correlation between BK_IVOL and returns and several firm characteristics for the pooled sample. We find that BK_IVOL is slightly positively correlated with returns, providing an initial confirmation of a positive, albeit weak, relation. Additionally, BK_IVOL is positively correlated with beta and illiquidity measures, and negatively correlated with firm-size, book-to-market and momentum, suggesting that (not surprisingly) risky, small, high growth, illiquid firms have high expected IVOL. Next, we use Fama-MacBeth (1973) regression to investigate the relationship between IVOL and returns, controlling for widely known firm characteristics that has been shown to explain the cross-section of stock returns Fama-MacBeth Regressions So far, we have established a positive correlation between quality out-of-sample proxies for expected IVOL (BK_IVOL) and returns (Panel B of Table 2). Next, we use Fama-MacBeth regression analysis to formally test whether BK_IVOL is positively related to future returns. Specifically, we employ the following Fama-MacBeth regression model: Ret it = α t + β t BK_IVOL it + θ t X it 1 + ε it where Ret it is stock i s excess return in month t, BK_IVOL it is the out-of-sample forecast of conditional volatility based on EGARCH models, and X it 1 is a set of lagged firm-level characteristic such as size, book-to-market, momentum, etc. that have been shown to explain the cross section of stock returns. We run this regression, cross-sectionally every month from July of 1963 to December of 2016 and report the average slopes in Table 3. Newey-West (1987) corrected 14

16 t-statistics with three lags are reported in the parenthesis In Model 1 of Table 3, we run a univariate model to investigate the economic magnitude of the correlation between BK_IVOL and returns. We find that BK_IVOL is significantly positively correlated with returns, confirming that firms with high expected IVOL have higher returns. Specifically, we find that the point estimate on BK_IVOL is basis points per unit of expected IVOL with a Newey-West adjusted t-statistic of Economically, this implies that a firm with one standard deviation (5.98) increase in idiosyncratic volatility increases the expected return by 34.1 basis points per month (5.98*0.057) which equals 4.09% per year. It is worth pointing out that although this coefficient is large in magnitude, it is substantially lower than the point estimates from using in-sample proxies for expected IVOL, as reported in Fu (2009) and Guo et al. (2014) 16. However, the magnitude and the statistical significance of our result is substantially higher than the out-of-sample (insignificant) point estimate reported in Guo et al. Table 5, p. 288 (1.4 basis points, with a t-statistic of ) In Model 2 of Table 3, we add market beta to the baseline model and continue to find a positive coefficient on BK_IVOL confirming our initial result that stocks with higher expected idiosyncratic volatility have higher returns even after controlling for systematic risk. As in prior literature, we observe that the relation between beta and returns appears to be flat. In Model 3, we add log market capitalization LN(ME) and log book-to-market LN(BE/ME) in addition to beta as control variables and we continue to find a positive coefficient on BK_IVOL. Further, size and book-to-market are significant with the expected signs, in that small and value firms earn higher returns on average. In Model 4, we add momentum RET(-2, -7), turnover LN(TURN) and the 16 Fu (2009) reports a point estimate of 11 basis points (Table 5, p. 31), whereas Guo et al. (2014) report a point estimate of 13.8 basis points (Table 4, p. 286). 15

17 coefficient of variation of turnover LN(CVTURN) as additional control variables and again we continue to find a positive coefficient on BK_IVOL. The coefficient on BK_IVOL in the full specification model (Model 4) is not substantially different in magnitude from the univariate model (Model 1). With a coefficient of 0.049, it implies that a firm with one standard deviation (5.98) increase in idiosyncratic volatility increases the expected return by 29.3 basis points per month (5.98*0.049) which equals 3.52% per year. The results are unchanged when we remove the marketbeta from the specification (Models 5 and 6.) 17 All control variables are significant with the expected signs, suggesting that our sample of firms is representative of the population. Using in-sample expected IVOL proxies (FU_IVOL), some studies report that small stocks and high beta stocks have lower returns i.e., some studies report a positive size effect and/or a negative sign on market beta (e.g. Fu, 2009; Fink, et al. 2012; Chichernea et al. 2015). However, using our out-of-sample measure, BK_IVOL, we find that the coefficients on market beta and firmsize in Table 3 are similar to the mainstream literature the relation between market beta and returns is flat but the relation between firm size and returns is negative. These results are consistent with the suggestion in Guo et al. (2014) that the change in sign for size and beta is likely a result of the look-ahead bias in prior studies. Overall, the results in Table 3 show that there is a positive, significant relation between our quality out-of-sample conditional volatility proxies for expected IVOL (BK_IVOL) and stock returns. Importantly, the magnitude of the coefficient in BK_IVOL does not change much when we add additional controls, suggesting that the positive relation between expected IVOL and returns is robust, and clearly distinct from that of the controls. The results indicate that the data 17 Fu (2009) does not include the market beta in the models that also include expected idiosyncratic volatility, so models 5 and 6 are primarily produced for comparison purposes. 16

18 corroborates the positive trade-off between idiosyncratic risk and return theorized by Merton (1978) and Malkiel and Xu (2002). Next, we investigate the driver of our findings What drives our novel findings? We have shown that firms with high expected IVOL have higher returns. Our measure of expected IVOL (BK_IVOL), as previously noted, differs from prior proxies of expected IVOL in several meaningful ways that are designed to increase the quality of the proxy. First, we have ensured the quality of the models from which the forecasts are derived by requiring that the models pass basic diagnostic tests. Second, we require several forecasts (five out of nine) to be available to calculate BK_IVOL. Finally, we calculate our proxy using the mean of the available forecasts. Given the stark contrast between our results (positive relation) and those of prior studies that use forecasts from out-of-sample EGARCH models to proxy for expected IVOL (no relation), we attempt to relax our requirements for calculating our quality proxy (BK_IVOL) to determine what drives the difference. First, we start by estimating a proxy for expected IVOL without requiring that models pass diagnostic tests, similar to Guo et al. (2014). Secondly, we change the requirement of the minimum number of available forecasts to range from one to six and investigate the relation between the resulting proxy for expected IVOL and returns. Finally, we change the way in which we combine (or choose) the forecast(s) to use in calculating BK_IVOL. Given the difference between our results and past papers, we expect that relaxing at least one of these criteria will have a very large impact on the relation between expected IVOL and returns. We start with Model 1 of Table 4 where we do not impose the diagnostic test requirement. As expected, we find that the results are sensitive to our diagnostic tests, as, like Fink et a. (2012) and Guo et al. (2014), we find no significant relation between the proxy for expected IVOL and returns when we exclude the diagnostic tests. Since models that do not pass basic diagnostic tests 17

19 are likely to generate noisy conditional volatility forecasts, this is the first evidence that our results are driven by reducing the noise in our proxy. In the following Models of Table 4, we perturb the minimum number of forecasts that are required to generate BK_IVOL, to range from one to six. Although averaging over several models could be argued to create better estimates, one possible drawback is that requiring several models to pass diagnostic tests could bias the sample towards firms with simple EGARCH structures (such as GARCH 1,1). 18 In addition, requiring several forecasts to pass the diagnostic tests also significantly reduces the sample size, and therefore possibly the power of the tests. We find that the results are quantitatively sensitive to the number of forecast that are available for a given firmmonth, as the economic magnitude of the results increases monotonically with the number of forecasts required to calculate BK_IVOL. Requiring six forecasts to calculate BK_IVOL (Model 6) results in only 119,739 observations but produces the largest coefficient. Interestingly, the positive relation is significant for any requirement of available forecasts above one (the relation is insignificant in Model 2). Given that we expect less noisy estimates for firm-months in which several forecasts can be used (chosen from) when calculating BK_IVOL, this also suggests that our results are driven by reducing noise in the BK_IVOL measure. Note that we reported the results from requiring at least five models in Table 3, so we do not replicate those findings in this table. Overall, this table shows that the relation between expected IVOL and returns is sensitive to the diagnostic tests we apply (Model 1) as well as the minimum number of forecasts required to create BK_IVOL (Models 2-6). Next, we investigate whether the main result we report in Table 3 is sensitive to the way 18 If the true model is a GRACH (1,1) several of the models could pass diagnostic tests, even though higher level EGARCH terms might be insignificant. However, if the EGARCH (3,3) is the true model, most other models might fail the diagnostic tests, and therefore the IVOL estimate from those models would not make the final sample. 18

20 we create/select BK_IVOL out of the available forecasts (where forecasts are based on models that pass basic diagnostic tests). Up to this point, we have used the average conditional volatility forecast from all models that pass the diagnostic tests (as long as at least 5 models pass) unless we specified otherwise (Table 4). Using the average IVOL should closely relate to average investors expectations, as investors are likely to use a multitude of models to form expectations. In addition, using combination of multiple forecasts can mitigates noise from poor forecasts and could capture different aspects of conditional volatility. In Model 1 of Table 5, we instead use the median conditional volatility forecast and find that the results are very similar to when using the average of the forecasts (Model 4 of Table 3). Specifically, we find that the median BK_IVOL is positively correlated with returns and the magnitude and significance is similar to when using the mean (coefficient of with a t-statistic of 3.01). In the remaining models, we use different selection criteria (AIC, AICC, SBC, and HQC) to find the best model. Overall, the positive relation we uncovered is generally robust to the model selection criteria, as we continue to find a significant, positive relation regardless of how we choose the model. However, it is important to note that the magnitude of the coefficient is much stronger when using the mean or median estimate as compared to selecting one particular model Is our sample representative? In this section, we investigate whether our sample is representative of the population of publicly traded firms, as well as the firms used in prior studies. Although we have not purposefully selected particular firms into our sample, it is possible that the time frame of our study, and/or the requirement we impose in order to calculate BK_IVOL biases our sample. Given that our sample size is relatively small compared to prior literature, this could be inadvertently driving the results. This concern could be especially troublesome, as, for example, Stambaugh et al. (2015) argue that 19

21 the negative relation between lagged realized IVOL (AHXZ_IVOL) and returns found in prior studies could be driven by overpriced firms, while underpriced firms might exhibit a positive relation. To the extent that our sample could be biased towards underpriced firms, the positive relation we uncover might be sample specific. Other papers have similarly argued that the negative relation between AHXZ_IVOL and returns is driven by a small subset of firms, and that the relation could even be positive for specific subsets of firms.to the extent that our sample inadvertently biases the sample toward one of these groups, our results might not be generalizable to the population of listed firms. To address this, we start by creating the lagged realized IVOL proxy used in Ang et al. (2006) and several subsequent papers that find a negative relation between lagged realized IVOL (AHXZ_IVOL) and returns. In Model 1 of Table 6, we confirm the negative relation between AHXZ_IVOL and returns during our sample period. Additionally, we replicate the Guo et al. measure (GKF_IVOL) and confirm their no relation result (Model 2 of Table 6). Perhaps this result should not be surprising given that relaxing the requirement of models to pass diagnostic tests (Model 1 of Table 4) and eliminating the minimum number of forecasts required (Model 2 of Table 4) is sufficient to diminish the positive relation between expected IVOL and returns. However, the result is reassuring, as it indicates that the time period used in our paper is not the driver of our findings. More importantly, we limit the sample to firm-month observations included in our sample (for which the BK_IVOL measure is available) and study the effect of AHXZ_IVOL for those firm-months. 19 Model 3 of Table 6 shows that the coefficient on AHXZ_IVOL remains negative 19 We note that we cannot estimate the GKF_IVOL measure for the observations where BK_IVOL are available, as the GKF_IVOL measure for those firm-months would be identical to Model 2 in Table 5. 20

22 and significant in our sample of firms, and of similar magnitude as in Model 1.. Taken together, these tests present strong evidence that the composition/characteristics of the firms in our sample is not driving our result, but it is driven by the quality of our BK_IVOL estimates. In Model 4, we include both the AHXZ_IVOL and our BK_IVOL measures, and find that including AHXZ_IVOL has little impact on the relation between our high quality BK_IVOL measure and returns. The relation remains positive and significant and the magnitude is similar to before. Interestingly, the AHXZ_IVOL measure remains negative and significant, suggesting that AHXZ_IVOL and BK_IVOL could be proxies for different things. This is consistent with papers that argue that the negative relation between AHXZ_IVOL and returns is driven by microstructure noise Robustness In this section, we provide additional robustness tests by analyzing the impact of the number of observations required to estimate the EGARCH models, as well as the number of iterations allowed for the models to converge. As indicated earlier, the main result we report in Table 3 requires at least six years of data (i.e., 72 time-series monthly return observations) to estimate the EGARCH parameters used to forecast conditional volatility. However, it could be argued that a substantially larger number of observations is needed to generate quality parameter estimates in the EGARCH models. 20 Since increasing the number of observations substantially reduces the number of estimates that are generated, it is a difficult tradeoff between the number of observations required and the final number of observations. However, if the number of observations is important to the quality of the 20 The literature emphasizes the importance of a long time series to obtain quality estimates of the GARCH parameters (see, e.g., Scruggs, 1998; Lundblad, 2007) 21

23 estimates, we should expect that changing the number of observations required would drastically impact the findings. In order to analyze whether the number of observations used in the estimation is important for our findings, we now perturb this assumption to study the effect of this requirement on the relationship between BK_IVOL and returns. As before, we do not alter the other requirements, including the minimum number of models that pass diagnostic tests (five out of nine) and how we create BK_IVOL from the converging models (we use the average). We provide the results in Table 7. In Model 1 of Table 7, we require only 36 monthly return observations to estimate the parameters of the EGARCH models used to forecast conditional volatility (and ultimately our proxy for expected IVOL, BK_IVOL). We still find that firms with high expected IVOL have high returns. The results are similar when we require 48 months in Model 2, 60 months in Model 3, 84 months in Model 4, and 96 months in Model 5, but the magnitude (point estimate) almost doubles from to as we increase the minimum number of monthly time series observations from 36 to 96 months. 21 This is especially noteworthy, as the number of estimates that are included in the models decrease from 200,552 (Model 1) to 130,144 (Model 5). Taken together, the results from Table 5 suggest that the statistical significance between BK_IVOL and returns is robust to the minimum number of time series monthly observations used to estimate BK_IVOL. However, to the extent that more observations results in better models, this suggest that the higher quality expected IVOL estimates have a stronger positive relation to returns. In Table 8 we allow for a different maximum number of iterations compared to the main tests, which were set to 1000 iteration. This is important, as prior papers have pointed out that the 21 The result for 72 months is omitted here as it is our main result reported in Model 4 of Table 3. 22

24 results could be sensitive to the number of iterations allowed (for example, Guo et al., 2014). We find that as the maximum number of iterations allowed for convergence increases, the positive relation between BK_IVOL and returns generally strengthens. Although this could be due to better estimates from models that require more iteration to converge, it could also be that these findings are mechanical. Increasing the maximum number of iterations increases the number of firm-month observations and therefore could increase the power of the tests. 4. Summary and Conclusion Prior studies have found conflicting evidence on the relation between expected idiosyncratic volatility (IVOL) and returns. Most studies use lagged realized IVOL to proxy for expected IVOL and find a puzzling negative relation with returns. Fu (2009) attempts to overcome this surprising result by conjecturing that the negative relation might arise from the use of a poor proxy for expected IVOL, and instead uses EGARCH models to forecast expected IVOL. Doing so, Fu (2009) finds a strong positive relation. However, subsequent studies show that the forecast used by Fu (2009) are biased (in-sample) and they find no significant relation when using unbiased forecasts. We argue that the differences are due to the noise inherent in IVOL estimates based on the EGARCH model, and apply several criteria (including diagnostic tests) to ensure the quality of the models that are used to generate the forecasts. We revisit the relation between expected IVOL and returns using our quality proxy for expectations (BK_IVOL), and uncover a significant positive relation. The relation we find is robust, in that we can relax several of our criteria without qualitatively changing the results. However, the drawback of our methodology is that we only investigate the relation between expected IVOL and returns for the subset of firm-months for which we can get quality estimates. This is not an issue for the current paper, but it does present problems for papers who want to 23