We report the descriptive statistics for three volatility series in Table 1. MVOL is monthly market volatility (MVOL). The MVOL for a given month m is the standard deviation of the daily value-weighted market returns for the past 22 trading days ending on the last trading day of month m. IVOL^EW and IVOL^VW are respectively the equal-weighted (EW) and value-weighted (VW) average idiosyncratic volatility across all firms, where IVOL is the standard deviation of residuals from (1). Table 1 shows that both IVOL^EW and IVOL^VW are 0.0188 and 0.0127 respectively, which is only about half compared to the IVOL in China, the biggest emerging stock market in the world [31] . The results support the view that stock markets in developed countries might be more stable than those in emerging stock markets. Moreover, the results also indicate that small firms seem to have higher idiosyncratic volatility than big firms as suggested by the higher mean of IVOL^EW compared with IVOL^VW. This is consistent with results in other markets particularly the U.S. However, IVOL^EW is less variable than IVOL^VW as indicated by its lower coefficient of variation (CV). MVOL on the other hand is more variable than IVOL^EW having a higher CV, but it has a similar CV as IVOL^VW.

As expected, our volatility measures are highly correlated as shown in Panel B, with correlation coefficients ranging from 0.849 to 0.913.

Panel C displays the autocorrelation structure of the three volatility series. As serial correlation is fairly high in all three series, we test for the presence of unit roots using the augmented Dickey and Fuller [32] test. Panel D shows that we can reject the presence of unit roots for all three series, whether or not a trend is included. Hence our analysis of the volatility series will be in levels instead of first differences.

Figure 1 plots IVOL^EW, IVOL^VW, and MVOL. As indicated in Figure 1, there does not appear to be any long-term trend in any of these volatility series.

Instead, the plots suggest episodic behavior in these series. There was an upward trend in all three volatility series from 1991 to 2001, a downward trend until 2006, a spike in 2009 followed by a decreasing trend thereafter. The pattern of our volatility measures in Figure 1 are very similar to the volatility figures for the French stock market reported by Duncan and Kabundi [33] .

As a formal test for the presence of trends, we begin by estimating the following OLS model:

V O L t = b 0 + b 1 t + b 2 V O L t − 1 + ε t (3)

where VOL represents IVOL^EW, IVOL^VW, and MVOL, and t is time. The estimated time trend b₁ parameter and its t-statistic are reported in Table 2. Over the full sample period from 1991: 08 to 2012: 06, the standard t-test shows a statistically significant positive trend in IVOL^EW but no trend for both IVOL^VW and MVOL. However, Vogelsang [34] points out that when errors in the trend regression are persistent, the student t-value often rejects the null hypothesis of no trend. As a consequence Bunzel and Vogelsang [25] developed the t-dan test which we employ in this study.³ This test is valid whether or not a unit root exists in the error terms. The t-dan test also has better power than the standard t-test while retaining its good size properties. The t-dan test statistics reported in Table 2 confirm the absence of a trend in either IVOL^VW or MVOL but a positive trend in IVOL^EW which indicates that the IVOL of small stocks have been trending upwards. However, this apparent trend in IVOL^EW could be simply due to the spike in volatility towards the end of 2008. Nonetheless, the fact that MVOL and IVOL^VW are relatively flat over the study period, implies that there is also no trend in correlations among stocks and that the benefits from diversification would have likely remained the same on average over the study period, which also means that the number of stocks needed to attain a certain level of diversification would also have remained the same.

In this section, we test for regime-switching behavior in idiosyncratic volatility. We follow Bekaert et al.’s [22] method to further test whether or not our volatility series in the French stock market is characterized by a stationary process that occasionally switches between high- and low-volatility regimes. Bekaert et al. [22] argue that the upward trends of idiosyncratic volatility in the U.S. and 22 other developed markets were driven by the chosen starting- and ending time points. For example, if the starting point is in a low volatility period, while the end point is in a high volatility period, then the trend test would easily show a positive trend. Bekaert et al. [22] thus suggest that idiosyncratic volatility in the U.S. and 22 other developed markets are best characterized by a stationary process that occasionally switches between high- and low-volatility regimes. A regime-switching behavior in idiosyncratic volatility also appears evident in emerging markets with Nartea, et al. [35] documenting evidence of such behavior in the Chinese stock market, the world’s largest emerging market.

To test for regime switching behavior in idiosyncratic volatility in the French stock market, we let volatility, y_t, follow an AR(1) model where all parameters can take on one of two values depending on the realization of a discrete regime variable, s_t. The regime variable follows a Markov chain with constant transition probabilities. Indexing the current regime by i the model is

y t = ( 1 − b i ) μ i + b i y t − 1 + σ i e t , i ∈ { 0 , 1 } (4)

with e_t ~ N (0,1). In the model, we force regime 0 (regime 1) to be the low (high) volatility regime and the mean levels (μ_i) of the volatility series of both regimes to be nonnegative (i.e. μ₁ > μ₀ > 0).

The transition probability matrix, Φ, is 2 × 2, where each probability represents P [ s t = i | s t − 1 = j ] , with i , j ∈ { 1 , 2 } :

Φ = ( p 11 1 − p 11 1 − p 22 p 22 )

The model has only 8 parameters, { μ 0 , μ 1 , b 0 , b 1 , σ 0 , σ 1 , p 11 , p 22 } .

The estimation results for each volatility series (y_t = IVOL^EW, IVOL^VW, and MVOL) are reported in Table 3. We use the robust White (1980) covariance

to compute the standard errors. Table 3 shows that levels for both EW and VW IVOL in the low volatility regime (μ₀) at 0.0258 (IVOL^EW) and 0.0105 (IVOL^VW) respectively, which are both statistically significant. The corresponding levels in the high volatility regime (μ₁) at 0.0163 and 0.0181 are also both statistically significant. The differences between the levels of the two regimes for both volatility series are also statistically significant as indicated by the results of the Wald test. Our results also show higher volatility in regime 1(σ₁) compared with regime 0 (σ₀) for both EW and VW IVOL. The EW (VW) volatility in regime 1 is 0.0040 (0.0035) compared with 0.0013 (0.0013) for regime 0. Thus we find that idiosyncratic volatility in the French stock market conforms with a stationary autoregressive process that occasionally switches between high and low-variance regimes. This is consistent with the behavior of idiosyncratic volatility in developed stock markets [22] and in the world’s largest emerging market [35] .

Figure 2 shows the smoothed probabilities of being in regime 0 for our three volatility series. Unlike the evidence reported by Bekaert et al. [22] in the U.S. stock market, we find that both high- and low-idiosyncratic volatility regimes in the French stock market have the propensity to stay for a period before switching to another. We observe this phenomenon several times over the study period. For example, Panel a of Figure 2 shows that IVOL^EW was in a high

volatility regime from 1992 to mid-2000, consistent with Morel’s [36] findings in the French stock market. And then, it shifts to a low volatility regime until the early of 2006. Furthermore, the IVOL^EW remained in a high volatility regime during the 2008 financial crisis, until the beginning of 2009 when it switched again to a low volatility regime. Panel b of Figure 2 shows that IVOL^VW switched between high- and low volatility regimes more frequently than IVOL^EW. IVOL^VW stayed in the low volatility regime for most of the 20 year testing period, for example from 1992 to March 1998, January 2003 to January 2008, and September 2009 to the end of the study period. If we define y_t to be in regime 0 if the probability of being in regime 0 is higher than 0.5, and vice versa for regime 1, then there are 4 regime switches in IVOL^VW and 3 regime switches in IVOL^EW over the 20-year study period. Our results indicate that big firms have become more volatile after the integration of the European stock market, but the volatily of small firms seem to becoming more stable. The results are consistent with Bekaert and Harvey’s (2000) suggestion that market liberalization could not only reduce the volatility in emerging markets, but also in developed stock markets. Panel c of Figure 2 shows that MVOL was switching from low volatility regime to high volatility regime unpredictably over the study period. It is hard to observe a clear pattern for the MVOL, but the regime of the MVOL partly follows the regime of IVOL^VW reported in the panel b.

Figure 2 confirms an insignificant time series trend in IVOL in the French stock market consistent with results reported in Table 2. For example, the absence of a trend in both IVOL^VW and MVOL reported in Table 2 is consistent with both panels b and c in Figure 2, where both series start and end in the low volatility regimes over our study period. Panel a of Figure 2 shows that our testing period starts from a high-level of IVOL and ends in a low-level of IVOL, which indicates that the significant positive trend in IVOL^EW reported in Table 2 is not due to the choice of sample period. However, results reported in Table 3 still suggest a significant regime-switching behavior for IVOL^EW.

We also find it interesting that IVOL^VW and IVOL^EW exhibit a divergence in the period from 1992 to 1999, with IVOL^EW being on a high-volatility regime while IVOL^VW was on a low-volatility regime. We suggest that this could be due to the boom in high-tech stocks over this period. As high-tech stocks are normally smaller in size and more volatile than traditional listed firms, we expect IVOL^EW to be more volatile than IVOL^VW before the high-tech bubble burst around year 2000. We also find that both IVOL^VW and IVOL^EW were on a high volatility regime during the recent 2008 financial crisis. This is consistent with previous findings in the literature wherein stock markets are more volatile during the financial crisis period than other periods [35] . Finally, we find that both IVOL^VW and IVOL^EW show a convergent behavior after 2002 in the French stock market.

In sum, the results from Table 3 and Figure 2 indicate evidence of episodic behavior in all three volatility series, consistent with occasional regime shifts throughout the study period.

In this section we examine the presence of an IVOL effect in the French stock market. Table 4 shows the average monthly returns and FF-3 alpha of EW and VW portfolios sorted according to idiosyncratic volatility. Though both the EW and VW return spreads between high and low IVOL portfolios are consistently negative at −0.91% and −0.71% per month respectively, they are not statistically significant. We only report the FF-3 alpha of each portfolio in the third column. Stotz et al. [36] report that there are not big differences between Fama and French’s alpha and Jensen’s alpha. The EW alpha spread is likewise negative at −0.34% per month but also statistically insignificant. The exception is the statistically significant VW alpha spread at −0.58% per month. This appears to be consistent with the anomalous and puzzling evidence documented by Ang et al. [1] [4] and Brockman and Yan [37] for the U.S. market. However it is not as high as the −1.31% per month reported by Ang et al. [1] for the U.S.

Before we test the robustness of this apparent negative IVOL effect in the French stock market, we report the average of the monthly averages of various

characteristics of the IVOL-sorted portfolios in Table 5. We report values for IVOL, size (SIZE), BM (Value), momentum (MOM), and short-term reversal (REV). These variables are as defined previously. The high IVOL portfolio has three times as much IVOL as the low IVOL portfolio and the difference is highly statistically significant as expected. High (low) IVOL stocks also tend to be small (big) stocks. These results are consistent with Drew et al.’s [38] findings in the German and UK stock markets, where they find that small firms have higher IVOLs than big firms. High (low) IVOL stocks also tend to be previous losers (winners) in the past 11 months. However, there is no significant difference in the value and short-term reversal variables between high and low IVOL portfolios. We formally control these variables using firm-level cross-sectional regressions in the next section.

We begin with univariate regressions on IVOL and our control variables. Table 6 reports the time-series averages of the slope coefficients over the 251 months from 1991:08 to 2012:06 with Newey-West [39] t-statistics in parenthesis. The univariate regression shows a statistically significant negative relation between IVOL and the cross-section of one-month ahead stock returns. The results also

show significant momentum and BM effects with previous winners and stocks with high BM exhibiting higher returns. However we find no size and short-term reversal effects. The absence of a size effect is consistent with Morel [40] who reports a significant size effect from March 1996 to July 1996 which disappears thereafter.

Next we control the size, BM, reversal, and momentum effects individually with bi-variate Fama-MacBeth cross-sectional regressions and then simultaneously in a multivariate regression. We report the results in Table 7. The bivariate regressions show that none of our control variables can individually explain the negative IVOL effect. However if we control for all four variables simultaneously, the IVOL coefficient becomes insignificant! Therefore, firm-level cross-sectional regression results indicate that the apparent negative IVOL effect in the French stock market is not robust and can be explained by our control variables.