Our data include daily and monthly returns of NYSE, AMEX, and NASDAQ common stocks with share codes 10 or 11 from July 1926 to December 2014. We obtain daily and monthly returns, daily trading volume, and share code data from the CRSP and the book value of individual companies from Compustat. We follow the procedure adjusting for delisting firms used by Shumway [30] in daily and monthly returns. We use the one-month Treasury bill rate as the risk-free rate. Moreover, we take into account the Size, B/M, OP, AGR, and Zero Volume in the explaining of the IV puzzle. Size is the natural logarithm of market capitalization and B/M is the natural logarithm of book-to-market in month t. OP is the book-equity-deflated operating profitability and AGR is the total asset growth rate. To consider the impact of infrequent trading, we also define Zero_Volume as the stocks with zero trading volume.

We measure realized idiosyncratic volatility following the approach in Ang et al. [6] [7] and Bali and Cakici [12]. Specifically, for one month, we perform the following Fama and French [28] three-factor regression for firms that have more than 15 daily return observations in that month:

r d , m i − r f = α m i + β MKT i , m ( MKT d , m − r f ) + β SMB i , m SMB d , m + β HML i , m HML d , m + ε d , m i (1)

IV m i _FF3FM = Var ( ε d , m i ) (2)

where, for day d in month m, r d , m i is stock i’s market return, MKT d , m − r f is the market excess return, SMB d , m and HML d , m are the returns on portfolios formed to capture size and book-to-market effects, respectively³. Specifically, SMB d , m is the return on a diversified portfolio of small stocks minus the return on a diversified portfolio of big stocks, HML d , m is the difference between the returns on diversified portfolios of high and low stocks. ε d , m i denotes the resulting residual. We use the standard deviation of daily residuals in month m to measure the individual stock’s idiosyncratic volatility IV m i _FF3FM for this month.

A five-factor model that adds profitability (RMW) and investment (CMA) factors performs better than the three-factor model [29] [31]. Therefore, we also consider a idiosyncratic volatility proxy based on the residuals of following Fama and French [31] five-factor model:

r d , m i − r f = α m i + β MKT i , m ( MKT d , m − r f ) + β SMB i , m SMB d , m + β HML i , m HML d , m     + β RMW i , m RMW d , m + β CMA i , m CMA d , m + ε d , m i (3)

IV m i _FF5FM = Var ( ε d , m i ) (4)

where RMW d , m is the difference between the returns on diversified portfolios of stocks with robust and weak profitability, and CMA d , m is the difference between the returns on diversified portfolios of stocks of low (conservative) and high (aggressive) investment firms. IV m i _FF5FM isthe individual stock’s idiosyncratic volatility for m month derived from the Equation (4).

To see whether the IV anomaly exists only for a specific period of time. In other words, whether the IV puzzle can be explained by data snooping bias. We divide the whole sample period (07/1926-12/2014) into two subsample periods (07/1926-06/1963 and 07/1963-12/2014). Table 1 presents the descriptive statistics and Spearman rank correlation for the main variables involved in this paper. Panel A reports the number and the proportion of Zero_Volumestocks during 07/1926-12/2014, 07/1926-06/1963 and 07/1963-12/2014. We notice that the number of Zero_Volume stocks is 17359 and account for 71.99% of the total common stocks. In the whole sample (07/1926-12/2014) and two subsamples (07/1926-06/1963 and 07/1963-12/2014), the proportion of Zero_Volume stocks is 12.95%, 25.77% and 11.07%, respectively. This implies that the number of

stocks with zero trading volume is quite substantial and Zero_Volume stocks shouldn’t be ignored for the research on empirical asset pricing especially anomalies in stock market, which is in accordance with the argument of Liu [22] that understanding the role of stocks with zero daily trading volumes is an important key to explain anomalies.

Panel B and C report the number, means, standard deviations, minimum and maximum, and pairwise correlations for main variables during 07/1926-12/2014, respectively. IV_FF3FM is the idiosyncratic volatility defined as the standard deviation of residuals from the regression of daily return with respect to local Fama-French three-factor model in the previous month. The daily return for IV_FF3FM is based on CRSP original daily return, while is rearranged CRSP daily return after considering the influence of infrequent trading for IV_FF3FM_Infre. Size is the natural logarithm of market capitalization. B/M is the natural logarithm of book-to-market in month t. OP is the book-equity-deflated operating profitability and AGR is the total asset growth rate. We see that the value of IV_FF3FM_Infre is higher than IV_FF3FM in termsof mean, standard deviations, minimum and maximum. As expected, the IV_FF3FM has a downward bias compared with new idiosyncratic volatility measure considering the influence of infrequent trading, which provides strong evidence that the calibration for the daily return of Zero_Volume stocks is feasible. Panel C shows that IV_FF3FM_Infre is highly correlated with IV_FF3FM and the coefficient of association is 0.9897, which indicates that IV_FF3FM_Infre could be a reasonable proxy for idiosyncratic risk. Moreover, the correlation between IV_FF3FM and Size means small firms have high idiosyncratic volatility. The IV_FF3FM is also negatively correlated with OP, indicating that firms with low profitability have higher volatility. All the representation of Spearman rank correlation of IV_FF3FM is tenable for the IV_FF3FM_Infre.

We duplicate the research method of Ang et al. [6] [7] and examine the IV puzzle during 07/1926-06/1963 and 07/1963-12/2014. In this process, idiosyncratic volatility is measured by IV_FF3FM, which is the standard deviation of residuals from Fama-French three-factor model. At the end of each month, stocks are allocated to ten groups (Low to High) according to IV_FF3FM and using CRSP breakpoints. We compute the average returns and intercepts from the Fama-French three factor regressions for decile portfolios in Table 2. Panel A of Table 2 shows that the difference of return (−0.28) between the highest decile IV portfolio and the lowest decile IV portfolio is inapparent for portfolios formed by value-weight during 07/1926-06/1963. Interestingly, the difference of return (0.18) between highest and lowest portfolio presents a positive trend when equal-weighted portfolios are formed, although it is significant at close to the 10% level. In stark contrast to Panel A, the H-L returns are −1.12 (t = −3.20) and −0.03 (t = −0.07) for value-weighted and equal-weighted decile portfolios during 07/1963-12/2014 in Panel B, implying that portfolios with high idiosyncratic volatility have lower return than portfolios with low idiosyncratic volatility. The Alpha has similar performance for the two samples. The results show that the idiosyncratic volatility puzzle is not robust during 07/1926-06/1963 and only exists during 07/1963-12/2014. Therefore, we infer that the idiosyncratic volatility puzzle is a result of data snooping bias.

To further examine the time-specific performance of the IV puzzle, we divide 07/1963-12/2014 into two subsamples in average, the first subsample period is 07/1963-12/1989 and the second subsample period is 01/1990-12/2014. Table 3 presents that the average returns and intercepts from the Fama-French three

factor regressions for decile portfolios sorted by idiosyncratic volatility during 07/1963-12/1989 and 01/1990-12/2014. We see that H-L value-weighted portfolio return is significantly negative (−1.26) during 07/1963-12/1989, however, is indistinctive (−0.97) during 01/1990-12/2014. When the portfolios are formed

by equal-weight, the difference of return between highest and lowest idiosyncratic volatility portfolio is negative (−0.28) for the first subsample but is positive (0.23) for the second subsample, although neither is significant. Similarity, the H-L Alpha is significant negative (−1.88 and −0.97) for value-weighted and equal-weighted portfolio during 07/1963-12/1989. The result suggests that the idiosyncratic volatility puzzle only exists in the period of July 1963 to December 1989. That is to say, even for the second subsample, the negative relation between idiosyncratic volatility and expected returns is not robust, we can draw the conclusion that the anomaly really is a time-specific phenomenon.

In addition, to form decile portfolios with a relatively more balanced average market share, we also use the NYSE breakpoints which start with Fama and French [33] ⁴ to examine the idiosyncratic volatility puzzle. Table 4 reports the average return and Alpha of portfolios sorted by IV_FF3FM and based on NYSE breakpoints during 07/1963-12/1989 and 01/1990-12/2014. For value-weighted portfolios, the H-L return is −0.56% (t = −1.63) during 07/1963-12/1989 and is −0.54% (t = −1.09) during 01/1990-12/2014. For equal-weighted portfolios, the H-L return is −0.08% (t = −0.25) and 0.14% (t = 0.18), respectively. In total, for NYSE breakpoints, there is no negative and significant relation between idiosyncratic volatility and expected returns. However, in the first subsample period (07/1963-12/1989), the t-statistics is very close to the 10% significant level, so we still believe that the idiosyncratic volatility puzzle is significant during this period.

A five-factor model that adds profitability and investment factors to the three-factor model of Fama and French [28] largely absorbs the patterns in average returns [31] [34]. Especially in the anomalies of dissecting, the five-factor model shrinks the troublesome intercepts of the three-factor model [29]. Therefore, we will try again to examine the idiosyncratic volatility puzzle based on the five-factor model in this section. Different from the above measure of idiosyncratic volatility, we will use IV_FF5FM which is the standard deviation of residuals from Fama-French five-factor model to measure idiosyncratic volatility. To be more practical, we apply the NYSE breakpoints to allocate stocks. Table 5 reports the average return and Alpha of portfolios sorted by IV_FF5FM and based on NYSE breakpoints during 07/1963-12/1989 and 01/1990-12/2014. The difference of return between highest and lowest idiosyncratic volatility portfolio is significantly negative (−0.58, t = −1.76) during 07/1963-12/1989 for value-weighted portfolio, however, is inapparent (−0.47, t = −0.97) during 01/1990-12/2014. Similarity, the H-L Alpha is also significantly negative (−0.92, t = −6.43) during 07/1963-12/1989, however, is not significant (−0.20, t = −0.84) during 01/1990-12/2014. The results show that the well-known idiosyncratic volatility puzzle (i.e., a negative relation between the monthly realized idiosyncratic volatility in the previous month and the value-weighted portfolio return in the subsequent month) exists during 07/1963-12/1989. The results of equal-weighted portfolio are also support our conjecture that the idiosyncratic volatility puzzle is a time-specific anomaly.

In addition, compared with Table 4, the intercept term in Table 5 has a massive improvement. Specifically, five of the ten value-weighted portfolios are significant (0.24, 0.26, 0.24, −0.27, −0.91) during 01/1990-12/2014 in Table 4, however, all are not significant in Table 5. The Alpha of highest value-weighted portfolio is −1.09 (t = −9.40) during 07/1963-12/1989 in Table 4, however,

shrinks to −0.97 (t = −8.82) in Table 5. A similar pattern holds when portfolios are formed by equal-weight. All of these imply that the five-factor model has a good power in explaining anomalies. Therefore, we will emphatically focus on the five-factor model in the remainder.

In calculating a daily return, we need the closing price for previous day and current day. However, the information of closing prices can be missing due to no trading. In view of this case, CRSP generally uses the average of bid and ask prices to calculate the daily return. For zero trading volume stocks with high expected returns, if we use the daily returns of CRSP to estimate idiosyncratic volatility, the series of daily returns will tend to be smooth and the measure of idiosyncratic volatility will have a lower estimation which has been confirmed in the Section 2.3. Ultimately, it will have a real impact on the relationship between idiosyncratic volatility and returns. Therefore, we will try to use the rearranged CRSP daily return after considering the influence of infrequent trading and a five-factor model to estimate the idiosyncratic volatility (Here, we call it as IV_FF5FM_Infre) and test the idiosyncratic volatility puzzle during 07/1963-12/1989 again. Table 6 reports the average return and Alpha of portfolios sorted by IV_FF5FM_Infre during 07/1963-12/1989. Whether using CRSP breakpoints or NYSE breakpoints, the return of H-L portfolio is significantly negative (−1.25, −0.58) for value-weight portfolios. Similarly, the difference of intercept between the highest and lowest portfolio is also significantly negative (−1.67, −0.92). The result suggests that the negative relationship between idiosyncratic volatility and returns is not alleviated when infrequent trading is considered.

The omission of the previous month’s stock returns can lead to a negatively biased estimate of the relation [11]. As a consequence, we construct a model following the approach in Huang et al. [11], which is similar to Fama and Macbeth [35] and Fama and French [31], with the exceptions that we include realized idiosyncratic volatility and the prior month’s individual stock returns. We test whether the data mining issue of idiosyncratic volatility puzzle can be explained by short-term return reversals. We run the following cross-section regression:

R i , t + 1 − R f , t + 1 = α + β 1 IV FF5FM + β 2 R i , t + β 3 Size + β 4 B / M     + β 5 OP + β 6 AGR + ε i , t + 1 (5)

where R i , t + 1 is stock i’s return in month t + 1 , and R f , t + 1 is the one-month Treasury bill rate in month t + 1 . The first element is a constant in the right-hand side of equation. IV_FF5FM is realized idiosyncratic volatility, given as the standard deviation of the residual from a five-factor model. We include other variables known to explain the cross-sectional variation in stock returns, such as the previous month’s return R i , t , Size, B/M, OP and AGR. Size is the natural logarithm of market capitalization and B/M is the natural logarithm of book-to-market in month t. OP is the book-equity-deflated operating profitability and AGR is the total asset growth rate. ε i , t + 1 is the model residual.

We conduct cross-sectional regressions with Equation (5) each month and then report the time-series averages of estimated coefficients in Panel A of Table 7. To make a better comparison for different sample periods, we repeat the

cross-sectional regression with Equation (5) in Panel B of Table 7 for the period 01/1990-12/2014, which has been proved to be free of idiosyncratic volatility anomaly in Section 3, so we needn’t to worry about the effect of prior monthly returns. Moreover, we also use IV_FF3FM, Size and B/M as regressor during 07/1926-06/1963 in Panel C. Consistent with the results of the foregoing portfolio analysis, idiosyncratic volatility anomaly only exists in the period 07/1963-12/1989. Specifically, the coefficient on idiosyncratic volatility is significant and negative at the 5% level (−0.1729, t = −2.29) when only IV_FF5FM is included in the model during 07/1963-12/1989. However, the coefficient on idiosyncratic volatility is insignificant (−0.0293, −0.0279) during the period

01/1990-12/2014 and 07/1926-06/1963. This again supports our conjecture that the idiosyncratic volatility puzzle is a result of data snooping bias. The third row in Panel A of Table 7 indicates that the coefficient on idiosyncratic volatility is not significant (−0.0940, t = −1.07) once we control for return reversals. We also control for several explanatory variables that might be related to past returns or idiosyncratic volatility in our cross-sectional regression, such as Size, B/M, OP and AGR. Interestingly, the coefficient on idiosyncratic volatility is still significant negative (−0.2344, t = −4.22) when ignore the prior monthly returns. The coefficient becomes insignificant (−0.1212, t = −0.86) once we include stock returns in the previous month. The result confirms the critical role of short-term return reversals in explaining idiosyncratic volatility puzzle.

To mitigate the effect of short-term return reversals, we leave a one-month gap between the portfolio formation date and holding period. Then we form ten value-weighted and equal-weighted portfolios every month by sorting all stocks with IV_FF5FM, which is a proxy for idiosyncratic risk given as the standard deviation of the residual from a five-factor model. Table 8 reports the average return and Alpha of portfolios sorted by IV_FF5FM and based on NYSE breakpoints during 07/1963-12/1989 and 01/1990-12/2014. We find that idiosyncraticrisk is no longer negatively related to expected returns (i.e. the idiosyncratic volatility anomaly disappears) when the short-term return reversals are considered during 07/1963-12/1989 in Panel A. Specifically, the H-L portfolio return is negative but insignificant (−0.28, t = −0.83) for value-weighted portfolios. The same is true for equal-weighted portfolios (−0.22, t = −0.61). Although the difference of Alpha between the highest and lowest portfolio is still significantly negative (−0.51, t = −3.45), the significance compared with Table 5 (−0.92, t = −6.43) is greatly diminished. As previously found, the negative correlation between idiosyncratic volatility and expected stock returns is not significant during 01/1990-12/2014 in Panel B. All the results suggest that the idiosyncratic volatility puzzle is driven by the force of short-term return reversals and is a data snooping bias.