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

Transcription

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: December 19, 2017 Abstract Recent literature finds mixed results on the relation between proxies for expected idiosyncratic volatility and stock returns. Depending on the proxy of choice, studies have shown both a negative or positive relation, but the results have been questioned as they are not robust and appear driven by microstructure noise or estimation bias. We develop quality estimates of expected idiosyncratic volatility that overcome the problems associated with prior studies, and find a strong positive relation between these proxies and returns. Key Words: Idiosyncratic Volatility, Priced Risk Factors, GARCH, EGARCH, Conditional Expected Volatility JEL Classification: G11, G12, G14 * We thank Jared Delisle, Delroy Hunter, 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, as a matter of policy, disclaims responsibility for any private publication or statement by any of its employees. The views expressed herein are those of the author and do not necessarily reflect the views of the Commission or of the author s colleagues on the staff of the Commission.

2 1. Introduction Much research has been devoted to investigating the relation between expected idiosyncratic volatility (IV) and stock returns, but the evidence to date is mixed. In the first study to use firm level proxies for expected IV, Ang, Hodrick, Xing, and Zhang (2006, 2009) use lagged IV (calculated using daily returns in the previous month) to proxy for expectations. The paper reports a significant negative relation, giving rise to what has become known as the IV Puzzle. Since then, a multitude of studies have challenged the robustness of these finding, often attributing them to microstructure noise (see survey in Hou and Loh, 2016). 1 In stark contrast to the negative relation found in prior firm level studies, Fu (2009) finds a strong positive relation between expected IV and returns. 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). The different (opposing) findings appear due to the novel proxy that Fu (2009) uses for expected IV. Fu (2009) argues that lagged IV (as used in prior studies) is a poor proxy for expectations, because IV is time varying, and instead uses exponential GARCH (or EGARCH) models to estimate conditional IV which is then used to proxy for expectations (henceforth referred to as EIVOL). In addition to being theoretically sound, the EGARCH models fit by Fu (2009) use monthly data (as opposed to daily data) which alleviates several concerns with regards to 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), arbitrage asymmetry (Stambaugh, Yu, and Yuan, 2015). 1

3 microstructure noise. Several empirical studies have since adopted the use of EGARCH models to generate proxies for expected IV (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 results in Fu (2009) and show that the positive relation between EIVOL 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 the inclusion of the contemporaneous return observation when estimating the EGARCH parameters used to forecast EIVOL drives the positive relation in Fu (2009). Importantly, these studies find no significant relation between returns and out-of-sample EIVOL (that are free of a look-ahead bias). This finding has been confirmed by additional studies (e.g., Bali, Scherbina, and Tang, 2016). Although a non-existing relation between expected IV 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 and motivates our research question: if we can generate quality estimates of expected IV, what is the relation between those estimates and returns? We show that high quality proxies for expected IV derived from out-of-sample EGARCH models are positively related with returns. There are several reasons to believe that the EIVOL estimates produced by the previous literature are poor proxies for expected idiosyncratic volatility. First, other than requiring model convergence, prior literature uses no other diagnostic tests to ensure the quality of the models that 2

4 the estimates are based on. Second, previous literature estimates 9 different EGARCH models ranging from EGARCH (1,1) to EGARCH (3,3), and selects the conditional volatility estimate from the model with the lowest Akaike information criterion (AIC) score. Although 9 models are generated, the selection criteria does not require any minimum number of converging models, 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 idiosyncratic volatility for many firms. If equity data for US securities is (as assumed) fairly well-behaved on a monthly level and the EGARCH family of models is well suited to estimate 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 as, regardless of selection criteria, several models would compete to generate the best estimate. Finally, Fu (2009) requires a minimum of 30 monthly observations in order to estimate the EGARCH parameters. This is problematic, given that the literature emphasizes the importance of a very long time series to obtain quality estimates of the GARCH parameters (see, e.g., Scruggs, 1998; Lundblad, 2007). Guo et al. (2014) attempts to mitigate this effect by requiring a minimum of 60 observations, but even this is arguably too small to produce reliable estimates. 2 We compute high quality out-of-sample conditional volatility estimates by first applying basic diagnostic tests to the individual models: in addition to requiring model convergence, we require the EGARCH models to produce standardized residuals that are normally distributed, 2 If the goal of a study is to test the relation between idiosyncratic volatility and returns for the largest cross-section available (and potentially use the measure as a control in future studies) the tradeoff between quality estimates and the quantity of available estimates is important and could justify the selections used in prior studies. 3

5 parameters that are covariance stationary, and a leverage effect of the expected sign, i.e., negative return shocks have a larger impact than positive return shocks on future volatility. We then take the estimates that pass the diagnostic tests and apply several filters (to the firm-month): we require that at least half (5 out of 9) of the EGARCH models pass the diagnostic tests for a given firm/month and we require a minimum of 72 observations of time series data to calculate the EGARCH parameters. We then take the mean of the estimates that meet these requirements. Using the mean idiosyncratic volatility should more closely relate to average investors expectations, as investors are likely to use a multitude of models to form expectations. 3 It should be noted 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. Our final sample consists of approximately 204,000 firm month observations over a 642 month window. We use Fama McBeth (1973) regression to test the relationship between EIVOL and returns, and find that the EIVOL estimates that our method produces are positively related to returns. Hence, our paper is the first to confirm the main findings in Fu (2009) using bias-free, outof-sample estimates of EIVOL. Although the economic magnitude of our result is smaller than Fu (2009), the results we report are economically important. Firms with expected idiosyncratic volatility of one standard deviation above the mean have expected returns that are approximately 4% higher per year. Further, our findings are consistent with theory (Merton, 1987; Malkiel and 3 The GARCH models had not been developed during the first half of the sample, and would have been unlikely to be used in estimating IV 4

6 Xu, 2002) that argues that expected idiosyncratic volatility should be related to returns if a substantial number of investors are under-diversified, something that has been confirmed in the literature (Goetzmann and Kumar, 2008). 4 We note that the diagnostic tests and selection techniques should bias us against finding a significant relation between idiosyncratic volatility and returns. This is because the filters we apply biases our sample to larger firms, and the pricing of many factors, including measures of idiosyncratic volatility, have been shown to be particularly (if not only) driven by small firms (see, for example, Bali and Cakici, 2009; Bergbrant, 2011). With the exception of firm size, the firm characteristics in our sample do not differ considerably from prior studies, suggesting that our sample is representative. We attempt to relax our filters and diagnostic tests to determine what drives the difference in the findings compared to prior studies. Regarding the filters, the results are sensitive to the requirement that several models converge for a given firm-month. Importantly, the economic (and statistical) magnitude of the results increases with the number of models that converge. Interestingly, the results are not sensitive to the number of observations required to estimate the EGARCH parameters (although the economic magnitude of the results increase with the observations included) or the particular choice of EIVOL estimate (it holds for selections based on the Mean, Median, AICC and SBC). We also find that the results are robust to changing the number of iterations for models to converge. As expected, we find that the results are sensitive to our main diagnostic tests, as our results are similar to those presented by Guo et al. (2014) when we exclude 4 Goetzmann and Kumar (2008) find that less than 10% of retail investors portfolios contain more than 10 stocks. 5

7 the diagnostic tests. We contribute to several strands of literature. First, we contribute to the literature on the impact of under-diversified investors. Our finding that idiosyncratic volatility is positively related to returns imply that under-diversified investors are the marginal traders that influence asset prices. Second, we contribute to literature that investigates the link between idiosyncratic volatility and returns at the firm level. While existing studies have mixed findings using a large cross-section of firms with dubious quality of the expected IV estimates, we find that limiting the sample to firms for which quality estimates are available, uncovers a strong positive relation. Finally, we contribute to the narrower literature that investigates the relation between EGARCH proxies for expected IV and returns (Fu, 2009; Bergbrant, 2011; Fink, Fink, and He, 2012; Guo, Kassa, and Ferguson, 2014). We confirm that a positive relation obtains after applying several diagnostic tests and filters to ensure the quality of out-of-sample idiosyncratic volatility estimates. The rest of the paper is organized as follows. The second section describes our data and methodology. The third section presents the results and the fourth section concludes. 2. Data and Methodology To conduct the various tests, three sets of data are required. The first set contains the test asset, i.e., stock returns. The second set contains our measure of expected idiosyncratic volatility, and the third set contains control variables that have been shown to explain the cross-section of returns. The data are obtained from three sources. Daily and monthly data for all firms traded on NYSE/AMEX, and NASDAQ from July of 1926 to December 2016 is obtained through the Center for Research in Security Prices (CRSP). Accounting data is obtained from Compustat. We obtain 6

8 data on holding period returns (RET), prices (P), shares outstanding (SHROUT), and volume (VOL). Returns exceeding 300% are deleted to be consistent with prior literature. 5 Holding period returns include capital gains as well as dividend yields. Accounting data, or more specifically book values, are obtained from Compustats annual fundamentals file. The Fama and French (1993) factors, Small-Minus-Big (SMB) and High-Minus-Low (HML), as well as the market premium and the proxy for the risk free rate are obtained from Professor Kenneth French s website. 6 Due to limitations in accounting data prior to July 1963, we have limited our empirical tests to the sample period between July 1963 and December of Measuring High Quality EGARCH Idiosyncratic Volatility Most papers investigating the relation between expected idiosyncratic volatility and return use lagged idiosyncratic volatility to proxy for expectations, implicitly assuming that idiosyncratic volatility follows a random walk (Ang, Hodrick, Xing, and Zhang, 2006 and 2009). However, Fu (2009) points out that this is problematic. Fu (2009) rejects the null hypothesis of a random walk for 90% of the firms in his sample and argues that forecasting idiosyncratic volatility using models in the GARCH family is better. Particularly, Fu (2009) suggests that the EGARCH model proposed by Nelson (1991) would provide a superior model of expected idiosyncratic volatility since it captures the asymmetric properties of volatility (e.g. leverage effects). 7 The EGARCH models are also more flexible than other ARCH and GARCH models and do not restrict the parameters to 5 This only affects 408 monthly observations during out sample 6 We thank Ken French for making the data available. 7 It is important to note that estimates of idiosyncratic volatility should not be judged based on their correlation with realized idiosyncratic volatility as realizations can be a poor proxy for expectations (see e.g. Elton, 1999). This is likely to pose a severe problem for idiosyncratic volatility due to microstructure effects. 7

9 avoid negative values. 8 However, the estimates used in Fu (2009) have been shown to be biased as they use time series return observation up to and including 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 (Guo, Kassa, and Ferguson, 2014). We follow the methodology in Guo, Kassa, and Ferguson (2014) and estimate unbiased 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 (1) ε i,t ~N(0, σ i,t 2 ) p ln σ 2 2 i,t = a i + b i,l ln σ l=1 i,t l + c i,k (θ ( ε i,t k ) + Υ ( ε i,t k σ i,t k σ i,t k ( 2 π ) 1/2 ) ) (2) q k=1 We estimate EGARCH (p,q), 1 p 3 and 1 q 3, for a total of 9 models per firm-month observation. We use data up to (and including) time t 1 to calculate the model parameters. We then use those model parameters to forecast the conditional (expected) idiosyncratic volatility at time t. We do this iteratively using expanding window. In order to generate high quality expected IV (EIVOL) estimates, several diagnostic tests are applied to individual models, and filters are applied to the firm-months. Out of the population of IV estimates (9 for each firm-month) generated by the EGARCH models, we only keep estimates from models that pass basic diagnostic tests. First, following prior 8 Pagan and Schwert (1990) compare different GARCH models and they find that the EGARCH model does the best in explaining monthly return volatility. 8

10 literature, we require that the model converges. Second, we require that the standardized residuals of the EGARCH model are normally distributed. Third, we require that the models are covariance stationary. Finally, the EGARCH model was developed in order to account for the leverage effect the empirical observation that negative return shocks have a larger impact on future volatility than positive shocks. This suggests that the coefficient theta in equation (2) above should be negative, so we make this a formal requirement (we do not require that the coefficient on theta is significant). For firm-months with models that pass the diagnostic tests, we then apply two filters. First, if EGARCH models fit the data well, we expect several of the models to pass our diagnostic tests for a given firm-month. Requiring several of the models to pass the diagnostic tests increases the likelihood that the EGARCH models fit the data, and gives us several models to use (chose from) in generating (or selecting) our EIVOL estimate. We therefore require that a majority of the models meet the criteria above (a minimum of 5 out of 9 models). Second, we require at least 72 observations of time-series return data to estimate EGARCH idiosyncratic volatility. Note that, although our empirical tests are conducted on the July 1963 to December 2016 sample due to limitations imposed by accounting variables, we use all the available time-series return data to estimate the conditional volatility, starting from July In our main tests, EIVOL is the mean of the IV estimates from models that pass the diagnostic tests for a firm-month that passes the filters. Panel A of Table 1 reports the resulting firm-month observations when we follow the outlined sample selection criteria. From July 1963 to December 2016, we estimate a total of 25.9 million EGARCH idiosyncratic volatility models representing 2.89 million unique firm-month 9

11 observations. Out of these, approximately 60% of the models (15.7 million) converge, but only about 28% of the converging models pass our diagnostic tests. When we further require at least 72 months of time-series return observation to estimate idiosyncratic volatility, we end up with 2.3 million estimates which accounts for 0.63 million unique firm-month observations. Further restricting the sample for which there are at least five models that pass these filters result in data for 204,297 firm-year observations during our sample period. We use the average EGARCH idiosyncratic volatility for these firm-months, and our analysis is based on these 204,297 firmmonth observations. 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 (72.5%), and EGARCH(3,3) is the least likely model to converge (41.6%). This pattern holds overall as we move from EGARCH(1,1) to EGARCH(3,3). In Panel C, we show the number of estimates that pass all diagnostic tests. Although the same pattern is present as in Panel B, the more advanced EGARCH models perform relatively better when judged on quality estimates. As shown, between 13.7% and 20.9% of estimates pass the diagnostic tests. Figure 1A shows the number of firms that are included in our basic model in each month. As can be seen, the number of firms included is small in the first years (before 1979), and then increases substantially as enough time has passed since the inclusion of NASDAQ stocks, for them to appear in the sample (as noted previously, we require 72 months of data to estimate EIVOL). Figure 1B shows the proportion of firms with at least one converging EIVOL estimate that pass the additional diagnostic tests and filters. Although the proportion varies slightly over time, there 10

12 are no large sudden changes that would indicate that our diagnostic tests/filters skews the data to certain time periods, but is generally representative of the sample of traded firms Control Variables We estimate beta, size, and market-to-book ratio following Fama and French (1992). Momentum and turnover are measured following Chordia, Subrahmanyam and Anshuman (2001), among others. Momentum is calculated as the holding period return between month t 7 to t 2. We exclude the last month s (t 1) return to avoid returns merely due to bid-ask bounce. Consistent with prior literature, we also include Turnover, calculated as the average share turnover (monthly volume divided by the numbers of shares outstanding) in the past 36 months 9. and 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%. 3. Results Our objective is to examine the relation between high quality idiosyncratic volatility estimates and the cross section of stock returns. The existing literature reports mixed results: a negative result when using lagged IV to proxy for expectations (Ang, Hodrick, Xing, and Zhang, 2006, 2009), but a positive relation when using in-sample idiosyncratic volatility estimated with EGARCH models (Fu, 2009). However, the positive relation disappears when using out-of-sample estimates of idiosyncratic volatility from EGARCH models without imposing requirement to ensure quality of 9 We require a minimum of 24 months of data to calculate turnover 11

13 the estimates (Guo et al., 2014). We use quality out-of-sample estimates from a subset of EGARCH models that pass relevant diagnostic tests and filters to help us pin down the real underlying relation. We find a positive empirical relation between out-of-sample idiosyncratic volatility estimates and the cross section of stock returns Summary Statistics and Correlations We report summary statistics and correlation matrix in Panels A and B of Tables 2 respectively. The table shows that the mean (median) return is 1.40% (0.65%) suggesting that the cross-section of stock returns are positively skewed. The mean (median) out-of-sample idiosyncratic volatility (EIVOL) is 10.13% (8.61). In comparison, Fu (2009) reports a mean (median) in-sample EIVOL of 12.67% (10.29%). The control variables are qualitatively similar to those reported in prior literature with the exception that the size of the firms that meet our diagnostic tests and pass our filters are somewhat larger (the mean log of MV is 5.06 for all firm/months, and 5.56 for our final sample). This is not surprising given that the return series of large firms are likely less noisy and hence can be more accurately modelled, and therefore more likely to produce quality estimates of EIVOL. The larger size of firms in our sample could bias against finding a significant relation between idiosyncratic volatility and returns since the pricing of many factors, including measures of idiosyncratic volatility, have been shown to be particularly (if not only) driven by small firms (see, for example, Bali and Cakici, 2009; Bergbrant, 2011). In unreported tests, we find that all industries (based on Fama and French 17 industry classifications) are reasonably represented in our final sample. The industry with the highest relative representation (17.15% of the firm/months make the cut) is Utilities, followed by Financials (15.1%), while the lowest representation is from Steel (7.3%) and Food (7.8%). 12

14 In Panel B of Table 2, we report the correlation between EIVOL and returns and other firm characteristics for the pooled sample. EIVOL and returns are positively correlated, providing the initial confirmation of a positive albeit weak relation between them. Additionally, EIVOL is positively correlated with Beta and liquidity measures, and negatively correlated with firm-size, book-to-market and momentum, suggesting that (not surprisingly) risky, small, growth, illiquid firms have high idiosyncratic volatility. Next, we investigate the relationship between idiosyncratic volatility and returns using Fama-MacBeth regression controlling for widely known firm characteristics that explain the cross-section of stock returns Fama-MacBeth Regressions So far, we have established a positive correlation between out of sample estimates of expected EGARCH idiosyncratic volatility (EIVOL) and returns (Panel B of Table 2). Next, we use Fama- MacBeth regression analysis to formally test whether EIVOL is positively related to future returns. Specifically, we employ the following Fama-MaceBeth regression model: Ret it = α t + β t EIVOL it + θ t X it 1 + ε it where Ret it is stock i s excess return in month t, EIVOL it is the out-of-sample estimate of expected EGARCH idiosyncratic volatility, 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 on approximately 200,000 firm-month observations from 1963 to 2016 and report the average slopes and associated t-statistics in Table 3. We follow the model specifications in Fu (2009). In column (1) of Table 3, we run a 13

15 baseline univariate model to investigate the relationship between EIVOL and returns by suppressing other firm characteristics. We find that EIVOL is positively correlated with returns, confirming that firms with high expected idiosyncratic volatility have higher returns. Specifically, we find that the point estimate on EIVOL is basis points per unit of IV with a Newey-West adjusted t-statistic of Economically, this implies that a firm with one standard deviation (6.03) increase in idiosyncratic volatility increases the expected return by 36.8 basis points per month (6.03*0.061) which equals 4.41% per year. Although large in magnitude, the coefficient is substantially lower than point estimates from using in sample idiosyncratic volatility as reported in Fu (2009) and Guo et al. (2014). Fu reports a point estimate of 11 basis points (Table 5, p. 31), whereas Guo et al. report a point estimate of 13.8 basis points (Table 4, p. 286). However, the magnitude and the statistical significance of our result is substantially higher than the out of sample point estimate reported in Guo et al. Table 5, p. 288 (1.4 basis points, with a t-statistic and adjusted R-square of 1.4%.) In column (2) of Table 3, we add market beta to the baseline model and we continue to find a positive coefficient on EIVOL confirming our initial result that stocks with higher 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 column (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 EIVOL. Further, size and book-tomarket are significant in the model with the expected signs. In column (4), we add momentum RET(-2, -7), turnover LN(TURN) and coefficient of variation of turnover LN(CVTURN) as additional control variables and again we continue to find a positive coefficient on EIVOL. The results are unchanged when we remove the market-beta from the specification (columns (5) and 14

16 (6).) 10 Importantly, all control variables are significant with the expected signs, suggesting that our sample of firms is representative of the population. We should note that when using in sample idiosyncratic volatility estimates, some studies report that small stocks and high beta stocks have lower returns a positive size effect and a negative sign on market beta (e.g. Fu,2009; Chichernea et al. 2015). However, using out-of-sample measure of EIVOL, we find that the coefficients on market beta and firm-size in Table 3 are similar to the mainstream literature, in that 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 an artifact of the look-ahead bias in prior studies. Overall, the results in Table 3 show that there is a positive, significant relation between out-of-sample EGARCH idiosyncratic volatility and stock returns. The data collaborates the positive idiosyncratic risk and return trade-off theorized by Merton (1978), and more recently by Malkiel and Xu (2002). Next, we will investigate the robustness of these results Is One Filter Particularly Important to Our Finding? Thus far, we have shown that high idiosyncratic risk firms have higher returns, i.e. a positive relation between out-of-sample idiosyncratic volatility and returns. Our results thus far require a minimum of five (out of nine) models to pass diagnostic tests for a given firm-month and a minimum of 72 monthly return observations to estimate the models. In addition, we did not engage in model selection but used the average conditional volatility from the models as the proxy for 10 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. 15

17 expected idiosyncratic volatility. A natural next step is to investigate the relationship between EIVOL and returns by varying these filters one at a time, leaving the other filters unchanged. First, we change the requirement of the minimum number of models that must pass the diagnostic tests to range from one to six and investigate the relation between EIVOL and returns. Then, we perturb the time-series monthly observation requirement to allow for models ranging from 36 to 96 observations and study the relation between EIVOL and returns. Finally, we change the criteria for how we use the quality estimates of different models to estimate EIVOL (median, AIC, AICC, SBC, and HQC). Given that our findings of a positive relation is in stark contrast to prior studies using out-of-sample EIVOL estimates, we expect that relaxing at least one filter would have a large impact Varying the Number Models that Pass Diagnostic Tests In Table 4, we investigate the sensitivity of our result the finding that high EIVOL stocks earn high returns by relaxing the requirement that at least five, out of possible nine, models pass the diagnostic tests when we estimate EIVOL. If the EGARCH family of models represents the best models for estimating time-varying volatility for equity securities, we would expect that different EGARCH specifications would generate similar quality estimates. If only a few of the nine models pass diagnostic tests, we would be cautious to consider the output from those models. In column (1) of Table 4, we don t make a restriction on the number of models, i.e. we use the mean of all the models that pass the diagnostic tests without imposing a minimum requirement of models to pass. Consistent with prior literature (Guo et al., 2014), we find an insignificant relation between these idiosyncratic volatility estimates and returns with a point estimate on EIVOL of and a t-statistic of In column (2), we impose the restriction that at least two 16

18 models out of a possible nine pass the diagnostics test. We find that EIVOL is marginally positively related to returns with a point estimate of and a t-statistic of Further, columns 3-5 show that the coefficient on EIVOL increases monotonically as we requiring more models to meet the criteria. For any requirement above 3, the findings show a positive relation that is economically large and statistically significant. Note that, we reported the results from requiring at least five models in our main test in Table 3, so we do not replicate those findings in Table 6. These results provide several important insights. First, the requirement that several different EGARCH models pass the diagnostic tests and filters is important to find a positive relation between EIVOL and returns. Although this is not surprising, it is a testament to the sometimes noisy estimates that can be generated by GARCH models, and the importance to carefully analyze the output of those models as well as the sensitivity of the models with respect to the ARCH and GARCH terms included. Second, the point estimates are monotonically increasing in the number of estimates that pass the diagnostic tests. To the extent that more models passing the diagnostic tests and filters is indicative of quality IV estimates, higher quality estimates show a stronger positive relation between expected IV and returns. Hence, these results illustrate the importance of separating quality estimates from noisy estimates Varying the Minimum Number of Return Observations to Estimate EIVOL As we indicated earlier, the main result we report in Table 3 requires at least 72 time-series monthly return observations to estimate EGARCH idiosyncratic volatility. Although 72 observations would be considered by many researchers as a small number for estimation of EGARCH models,we now perturb this assumption to study the effect of this requirement on the relationship between EIVOL and returns. As before, we don t alter the other requirements, including the minimum number of 17

19 models that pass diagnostic tests (five out of nine) and how we choose the EIVOL estimate from those models (we use the average EIVOL). We provide the results are in Table 5. In column (1) of Table 5, we require only 36 monthly return observations to estimate EIVOL, and we then use these estimates to study the relationship between EIVOL and the returns using Fama-MacBeth regression. With only 36 monthly observations, we still find that high idiosyncratic volatility stocks have high returns. The results are similar when we require 48 months in column 2, 60 months in column 3, 84 months in column 4, and 96 months in column 5. The result for 72 months is omitted here as it is reported in our main result in column 4 of Table 3. Taken together, the results from Table 5 suggest that the statistical significance between EIVOL and returns does not seem to be highly sensitive to the minimum number of time series monthly observations used to estimate EIVOL, but the magnitude (point estimate) doubles from to as we increase the minimum number of monthly time series observations from 36 to 96 months. This is especially noteworthy, as the number of estimates that are included in the models decrease from 184,335 (column 1) to 121,643 (column 5) Varying the Criteria for Choosing the EIVOL Estimate Next, we investigate whether the main result we report in Table 3 is sensitive to the way we select the EIVOL estimate out of the converged models. Up to this point, we used the average estimate of EIVOL from all models that pass the diagnostic tests (as long as more than 5 models pass) unless we specified otherwise. Using the mean idiosyncratic volatility should closely relate to average investors expectations, as investors are likely to use a multitude of models to form expectations. In column 1 of Table 6, we instead use the median estimated EIVOL from these models. The results are very similar to when using the mean of the estimates (Table 3, column 4). 18

20 Specifically, we find that the median EIVOL is positively correlated with returns (coefficient of with a t-statistic of 2.56). In the remaining models, we use different selection criteria (AIC, AICC, SBC, and HQC) to find the best model, and then use the EIVOL estimate from that model in the Fama MacBeth test. With the exception of using the AIC for model selection, we continue to find a positive relation between EIVOL and returns. Overall, the positive relation uncovered in this paper is generally robust to the choice of the estimates. Important to note is that the magnitude of the coefficient is much stronger when using the mean or median estimate as compared to selecting one particular model (based on AIC, AICC, SBC, or HQC). This could be due to each information criteria generally penalizing more complex models (and the more complex models generate better estimates) or that investors use different models to form their expectations and therefore the combination of estimates (mean or median) reflect investors beliefs more accurately than a single model Robustness In Table 7 we allow for a different number of iterations compared to the main tests (200). This is important, as prior papers have pointed out that the results could be sensitive to the number of iterations allowed. We find that as the number of iterations allowed for convergence increases, the positive relation between EIVOL and returns monotonically increases. Although this could be due to better estimates from models that require more iteration to converge, it might be more likely that these findings are mechanical. Increasing the maximum number of iterations increases the number of observations (firm-months) with sufficient data (5/9 models that meet diagnostic tests) and therefore increase the power of the tests. 19

21 Recently, Bali, Cakici, and Whitelaw (2011) find that including the maximum daily return in the prior month (MAX) as a control variable significantly impacts the relation between lagged idiosyncratic volatility and returns. Importantly, the paper reports that the relation between lagged idiosyncratic volatility and returns turns positive after controlling for MAX. Although the change in sign when including MAX suggests that microstructure noise could have impacted the relation between lagged idiosyncratic volatility and returns (which is not likely to be an issue when using EGARCH models on monthly data), we nonetheless include MAX as a control variable and report the results in column 1 of Table 8. We first confirm the findings in Bali, Cakici, and Whitelaw (2011) that MAX is important determinant of the cross-section of returns. Importantly, we still find a positive relation between EIVOL and returns without any measurable difference in the magnitude of the point estimate of EIVOL, suggesting that MAX and EIVOL capture distinctly different risks. In columns 2 and 3, we closely replicate prior papers by excluding the requirement that models pass diagnostic tests. Our results are very close to the results presented by Guo, Kassa, and Ferguson (2014) and help emphasize that the difference is not driven by a different time period, but by our quest to find quality estimates of expected idiosyncratic volatility. 4. Summary and Conclusion Prior studies have found conflicting evidence on the relation between idiosyncratic volatility and returns. Many studies use lagged idiosyncratic volatility to proxy for expected idiosyncratic volatility and find a puzzling negative relation with returns. Fu (2009) attempts to overcome this anomalous result by conjecturing that the negative relation might arise from the use of a poor proxy 20

22 for idiosyncratic volatility, and instead uses EGARCH models to estimate idiosyncratic volatility. Doing so, Fu (2009) finds a positive relation. However, subsequent studies show that the results in Fu (2009) are based on biased EGARCH idiosyncratic volatility estimates and they find no significant relation when using unbiased estimates. We conjecture that the differences are due to EGARCH idiosyncratic volatility estimates beeing noisy and apply several diagnostic tests and filters to ensure the quality of the estimates. We revisit the relation between EGARCH idiosyncratic volatility and returns using only observations that pass our diagnostic tests and filters, and uncover a significant positive relation. The relation we find is robust, in that we can relax several of our filters without qualitatively changing the results. However, the drawback of our methodology is that we only investigate the relation between idiosyncratic volatility and returns for the subset of firm-months for which we can get quality estimates. This is not an issue for the current paper since our goal is to investigate the relation between idiosyncratic volatility and returns, but it does present problems for papers who want to adequately control for the impact of idiosyncratic volatility on returns in future crosssectional studies. We hope that future research will provide guidance as to how quality idiosyncratic volatility estimate can be generated for a larger sample of firms (perhaps by using different models for different firms). We also leave it to future research to analyze if the relation we uncover is driven by a particular subset of firms. It should also be noted that a relation between idiosyncratic volatility and returns might not be evidence of idiosyncratic volatility being priced, but simply that we have not adequately identified all priced factors. This is something we wish to explore in future research. 21

23 Reference List: Amihud, Y., & Mendelson, H. (1986). Liquidity and stock returns. Financial Analysts Journal, Amihud, Y. (2002). Illiquidity and stock returns: cross-section and time-series effects. Journal of financial markets, 5(1), Ang, A., Hodrick, R. J., Xing, Y., & Zhang, X. (2006). The cross section of volatility and expected returns. The Journal of Finance, 61(1), Ang, A., Hodrick, R. J., Xing, Y., & Zhang, X. (2009). High idiosyncratic volatility and low returns: International and further US evidence. Journal of Financial Economics, 91(1), Bali, T. G., & Cakici, N. (2008). Idiosyncratic Volatility and the Cross Section of Expected Returns., Journal of Financial and Quantitative Analysis, 43(1), Bali, T. G., Scherbina, A., & Tang, Y. (2009). Unusual news events and the cross-section of stock returns., Management Science., Forthcoming Bali, Turan G., Nusret Cakici, and Robert F. Whitelaw. "Maxing out: Stocks as lotteries and the cross-section of expected returns." Journal of Financial Economics 99.2 (2011): Bergbrant, M. C. (2011). Is Idiosyncratic Volatility Really Priced?. Working Paper, St. Johns University Boyer, B.; T. Mitton; and K. Vorkink. (2010) Expected Idiosyncratic Skewness. Review of Financial Studies, 23, Brennan, M. J., & Subrahmanyam, A. (1996). Market microstructure and asset pricing: On the compensation for illiquidity in stock returns. Journal of financial economics, 41(3), Chichernea, D. C., Ferguson, M. F., & Kassa, H. (2015). Idiosyncratic risk, investor base, and returns. Financial Management, 44(2), Chordia, T., Subrahmanyam, A., & Anshuman, V. R. (2001). Trading activity and expected stock returns. Journal of Financial Economics, 59(1), Datar, V. T., Naik, N. Y., & Radcliffe, R. (1998). Liquidity and stock returns: An alternative test. Journal of Financial Markets, 1(2), Dimson, E. (1979). Risk measurement when shares are subject to infrequent trading. Journal of Financial Economics, 7(2),

24 Fama, E. F., & French, K. R. (1992). The cross section of expected stock returns. the Journal of Finance, 47(2), Fama, E. F., & MacBeth, J. D. (1973). Risk, return, and equilibrium: Empirical tests. Journal of political economy, 81(3), Fink, J. D., Fink, K. E., & He, H. (2012). Expected idiosyncratic volatility measures and expected returns. Financial Management, 41(3), Fu, F. (2009). Idiosyncratic risk and the cross-section of expected stock returns. Journal of Financial Economics, 91(1), Goetzmann, W. N., & Kumar, A. (2008). Equity Portfolio Diversification. Review of Finance, Goyal, A., & Santa-Clara, P. (2003). Idiosyncratic risk matters!. The Journal of Finance, 58(3), Guo, H., Kassa, H., & Ferguson, M. F. (2014). On the relation between EGARCH idiosyncratic volatility and expected stock returns. Journal of Financial and Quantitative Analysis, 49(01), Han, Y., & Lesmond, D. (2011). Liquidity biases and the pricing of cross-sectional idiosyncratic volatility. Review of Financial Studies, 24(5), Hou, K., & Loh, R. K. (2016). Have we solved the idiosyncratic volatility puzzle?. Journal of Financial Economics, 121(1), Huang, W., Liu, Q., Rhee, S. G., & Zhang, L. (2009). Return reversals, idiosyncratic risk, and expected returns. The Review of Financial Studies, 23(1), Jegadeesh, N., & Titman, S. (1993). Returns to buying winners and selling losers: Implications for stock market efficiency. The Journal of finance, 48(1), Jegadeesh, N., & Titman, S. (2001). Profitability of momentum strategies: An evaluation of alternative explanations. The Journal of finance, 56(2), Jiang, G.; D. Xu; and T. Yao. (2009) The Information Content of Idiosyncratic Volatility. Journal of Financial and Quantitative Analysis, 44, Lundblad, C. (2007). The risk return tradeoff in the long run: Journal of Financial Economics, 85(1), Malkiel, B. G., & Xu, Y. (2002). Idiosyncratic risk and security returns. University of Texas at Dallas (November 2002). 23

25 Merton, R. C. (1987). A simple model of capital market equilibrium with incomplete information. The Journal of Finance, 42(3), Nelson, D. B. (1991). Conditional heteroskedasticity in asset returns: A new approach. Econometrica: Journal of the Econometric Society, Pástor, Ľ., & Stambaugh, R. F. (2003). Liquidity risk and expected stock returns. Journal of Political economy, 111(3), Pagan, A. R., & Schwert, G. W. (1990). Alternative models for conditional stock volatility. Journal of econometrics, 45(1), Scruggs, J. T. (1998). Resolving the puzzling intertemporal relation between the market risk premium and conditional market variance: A two-factor approach. The Journal of Finance, 53(2), Spiegel, M. I., and Wang, X. (2005) "Cross-sectional variation in stock returns: Liquidity and idiosyncratic risk." Yale School of Management. Stambaugh, R. F., Yu, J., & Yuan, Y. (2015). Arbitrage asymmetry and the idiosyncratic volatility puzzle. The Journal of Finance, 70(5), Wei, S. X., & Zhang, C. (2005). Idiosyncratic risk does not matter: A re-examination of the relationship between average returns and average volatilities. Journal of Banking & Finance, 29(3),

26 Table 1: Summary of Estimates Panel A shows the total number of EGARCH models run (column 3), and resulting firm/months (column 4) that meet the criteria given in column 2. Panel B shows the percentage of EGARCH models that converge while Panel C shows the percentage of EGACH models that pass the diagnostic tests. Panel A: Diagnostic Tests Filters p p Total Estimates Firm/Months Total EIVOL Estimates (1963: :12) 25,887,236 2,891,173 Converging Estimates 15,666,941 2,700,992 Estimates that pass diagnostic tests 4,478,055 1,324,550 Estimates with at least 72 observations 2,316, ,436 Firms with at least 5 estimates for a given month 204,297 Panel B (converging estimates): q % 72.1% 68.3% % 58.3% 51.3% % 43.4% 41.6% Panel C (estimates that pass all diagnostic tests): q % 19.4% 18.7% % 16.8% 15.1% % 13.7% 13.7% 25

27 Table 2: Summary Statistics and Correlation Matrix This table reports the time series average of cross-sectional summary statistics. RET is stock return. RETRF is stock return in excess of the risk free rate. BETA is the loading on the market risk. LN(ME) is the log market cap. LN(BE/ME) is the log book-to-market ratio. RET(-7,-2) is the average of the compound return over the previous 2 nd to 7 th months. LN(TURN) is the log of turnover. LN(CVTURN) is the log of the coefficient of variation of turnover. EIVOL is the average conditional expected volatility from nine different EGARCH models that pass diagnostic tests (requiring that at least five models pass the diagnostic tests for a given firm/month). EIVOL (of 1) is the same as EIVOL, except that it only requires that one model pass the diagnostic tests.the data is from 1963:07 to 2016:12. Panel A: Summary Statistics Mean Median Std P25 P75 N RET ,297 RETRF ,297 BETA ,389 LN(ME) ,960 LN(BE/ME) ,338 RET(-7,-2) ,528 LN(TURN) ,712 LN(CVTURN) ,451 EIVOL ,297 EIVOL (of 1) ,304 Panel B: Correlation Matrix EIVOL RET BETA LN(ME) LN(BE/ME) RET(-7,-2) LN(TURN) RET 0.02 BETA LN(ME) LN(BE/ME) RET(-7,-2) LN(TURN) LN(CVTURN)