Country-Specific Idiosyncratic Risk and Global Equity Index Returns
C. James Hueng and Ruey Yau
Abstract:
The “idiosyncratic volatility puzzle” arises from the empirical evidence that stocks with
higher past idiosyncratic volatilities earn lower future returns. Studies have found that
this puzzle can be explained by certain time-series properties of the firm-specific
idiosyncratic shocks. In the country-level market index data, however, the puzzle does
not exist, which implies that the time-series properties of the country-specific
idiosyncratic shocks are different from those of the firm-specific idiosyncratic shocks
within a country.
We find that the differences are, first, that lagged idiosyncratic
volatility is a better proxy for expected idiosyncratic risk in the country-level data than in
the firm-level data. Second, unlike the firm-specific idiosyncratic skewness, the countryspecific idiosyncratic skewness is not significant enough to play a role in determining
index returns. Finally, return reversals documented in the firm-level data are not present
in the country-level index data. Instead, a momentum effect is found in the countryspecific index returns.
JEL Classification: G11, G12, G15
Keywords:
Idiosyncratic volatility puzzle; Idiosyncratic skewness; International
asset pricing
i
1. Introduction
The single-factor capital asset pricing model (CAPM) demonstrates that investors are
able to enjoy the benefit of reducing unsystematic risk from diversification while holding the
market portfolio in equilibrium (Sharpe 1964 and Lintner 1965). However, Merton (1987)
demonstrates that a sub-optimally diversified portfolio could be in equilibrium in a capital
market with incomplete information.
In reality, investors rarely hold well-diversified
portfolios. Campbell et al. (2001) suggest that an investor needs to hold at least 50 randomly
selected stocks to achieve complete portfolio diversification. Goetzmann and Kumar (2008),
however, examine more than 60,000 equity investment accounts from 1991 to 1996 and find
that less than ten percent of the investors hold more than ten stocks. The apparent lack of
diversification among most investors indicates that idiosyncratic risk should be priced because
under-diversified investors require compensation in the form of a higher return for bearing this
risk (Levy 1978; Merton, 1987).
Given the benchmark prediction of a positive relationship between idiosyncratic risk
and excess returns, the related empirical evidence is far from being conclusive. In particular,
the "idiosyncratic volatility puzzle" found by Ang et al. (2006) in the U.S. market has attracted
much research. This puzzle arises from the empirical evidence of cross-sectional analyses
showing that stocks with high idiosyncratic volatilities in the previous month have abysmally
low average monthly returns.
Ang et al. (2009) point out that the puzzle found by Ang et al. (2006) may be
dependent on the particular sample used. To explore the possibility of datasnooping, they
check whether the puzzle exists in other markets. They find the same puzzling evidence that
1
stocks with high idiosyncratic volatility tend to have low average returns in each of the G7
equity markets and in a larger sample of 23 developed markets.
Both studies (Ang et al., 2006, 2009) investigate the relationship between expected
returns and the associated risk within a country using cross-sectional firm-level data. The
discussion of the datasnooping problem of the puzzle has never been extended to another
investment avenue - the market for international equity indices. Country-level cross-sectional
analyses have used these global index data to discuss the international CAPM or international
market integration/segmentation (e.g., Li et al., 2003; Driessen and Laeven, 2007; You and
Daigler, 2010), but have never addressed the datasnooping issue of the puzzle in these data.
Studies suggesting international market segmentation such as Bali and Cakici (2010) find
evidence of a positive and significant relationship between country-specific idiosyncratic
volatilities in the previous month and future index returns. This absence of the abnormal
puzzle in the international equity index market simply confirms the normal positive risk-return
relationship and may not deserve much attention from the literature on the idiosyncratic
volatility puzzle.
However, the evidence that the puzzle does not exist in the international equity index
market provides a new path of research. Rather than discussing the datasnooping issue of the
idiosyncratic volatility puzzle, we use this evidence to investigate the differences in the timeseries properties between the firm-level and country-level data. Specifically, some studies
claim that the puzzle can be explained by certain time-series properties of the idiosyncratic
shocks. If these explanations are legitimate, the time series properties of the country-level data
must differ from those of the firm-level data since the puzzle does not exist in the international
2
index data. This would have important implications for different investment strategies in firmlevel stock portfolios and in international index portfolios.
For example, Huang et al. (2010) argue that the puzzle is caused by the omission of
lagged stock returns as a regressor in the cross-sectional regressions. They find that past
returns have a negative effect on current returns (return reversal). In addition, there is a
positive contemporaneous correlation between realized idiosyncratic volatility and stock
returns.
Therefore, if lagged returns are omitted from the regression, the effect of past
idiosyncratic volatility on expected returns is negatively biased. Once return reversals are
controlled for, they find a significantly positive relationship between the conditional
idiosyncratic volatility and expected returns.
Fu (2009), on the other hand, inspects the second moment of the idiosyncratic shocks.
He argues that since idiosyncratic volatilities are time-varying with a small average first-order
autocorrelation, lagged idiosyncratic volatility is not a good estimate of expected idiosyncratic
volatility. Therefore, the negative relationship between lagged idiosyncratic risk and excess
returns does not represent the expected risk-return relationship.
Instead, Fu models the
idiosyncratic volatility as an exponential GARCH (EGARCH) process, and uses the estimated
conditional idiosyncratic volatility to proxy for the expected idiosyncratic volatility. He finds
evidence of a positive relationship between the estimated conditional idiosyncratic volatilities
and stock returns.
Boyer et al. (2010) consider the investors’ preference for positive skewness in an
attempt to explain the puzzle. They provide evidence supporting the theory that expected
idiosyncratic skewness and expected returns are negatively correlated (Barberis and Huang,
3
2008; Mitton and Vorkink, 2007). In addition, they find that past idiosyncratic volatility is a
strong predictor of future idiosyncratic skewness and that the relationship between past
idiosyncratic volatility and expected idiosyncratic skewness is positive. Therefore, investors
may accept lower expected returns on stocks that have experienced high idiosyncratic volatility
because these stocks have higher expected idiosyncratic skewness. This provides a novel
explanation of the idiosyncratic volatility puzzle.
The empirical results from these studies suggest that the above time series properties
of the firm-level idiosyncratic shocks must differ from those of the country-specific
idiosyncratic shocks in Bali and Cakici (2010). Specifically, based on the arguments of Fu
(2009), the idiosyncratic volatilities of a specific country index return relative to the world
market must be highly autocorrelated, and therefore past idiosyncratic volatility is a good
predictor of expected idiosyncratic volatility. If not, then Bali and Cakici's (2010) test would
be invalid and a better measure of the expected idiosyncratic volatility needs to be used to test
the risk-return relationship in the country-level cross-sectional regressions.
Furthermore, the relationships among return, idiosyncratic volatility, and idiosyncratic
skewness in the firm-level data found by Boyer et al. (2010) may not hold in the international
index market. On the one hand, Bali and Cakici's (2010) results may simply imply that, unlike
the preference for lottery-like stocks in domestic asset portfolios, international investors do not
prefer positively skewed index securities in their under-diversified international portfolios.
However, on the other hand, if they do prefer positive skewness, then past idiosyncratic
volatility should not positively predict future idiosyncratic skewness in the international index
market; otherwise, the positive relationship between lagged idiosyncratic volatility and future
4
returns found by Bali and Cakici (2010) would imply that portfolios with higher idiosyncratic
skewness earn higher returns.
Using international equity index data, we find that, first, lagged idiosyncratic volatility
is not a bad predictor of future idiosyncratic volatility in the country-level index data, and
performs better than that in the firm-level data. It provides useful information on expected
idiosyncratic volatility just as the conditional idiosyncratic volatility does.
Second,
idiosyncratic skewness in the country-level index data is essentially zero. Therefore, it does
not play a role in determining the index returns and can be ignored in the pricing of
international portfolios. Finally, return reversals are not present in the country-level index data
and, instead, a momentum effect is found. This momentum effect, however, does not alter our
conclusions on the higher moments of the idiosyncratic shocks.
The next section discusses the time-series properties of the country-specific
idiosyncratic volatility and its relationship with the expected returns. Section 3 investigates the
role of idiosyncratic skewness in international asset pricing. Section 4 elaborates on the
economic meaning of our statistical findings and concludes the paper.
2. Returns and idiosyncratic volatilities
We start with a brief review of the idiosyncratic volatility puzzle. Let ri , d ,t denote the
daily idiosyncratic return of stock i on date d in month t. The idiosyncratic volatility in month
t is defined as the realized monthly standard deviation of the daily idiosyncratic shocks:
IVOLi ,t = Var ( ri , d ,t ) . Let Ri ,t denote the monthly return of stock i in month t. In deriving the
idiosyncratic shocks from the Fama-French (1993) three-factor model, Ang et al. (2006), using
5
U.S. firm-level data, and Ang et al. (2009), using firm-level data for several international
markets, run a cross-sectional regression for the following econometric specification in each
month t:
Ri ,t = γ 0,t + γ 1,t IVOLi ,t −1 + Γ t X i,t −1 + ε i ,t ,
(1)
where X i,t is a vector of other risk measures and firm characteristics. They find that the timeseries average of the estimated γ 1,t is negative and statistically significant, indicating that
investors accept a lower return on stocks with higher lagged idiosyncratic volatilities. This is
the empirical evidence known as "the idiosyncratic volatility puzzle" because, according to
Merton's (1987) theory, rational investors should require higher average returns to compensate
for their holding imperfectly diversified portfolios. They demand a premium for holding
stocks with high idiosyncratic volatilities, and therefore idiosyncratic risk should be priced to
compensate rational investors holding under-diversified portfolios (Malkiel and Xu 2006).
Fu (2009) argues that a valid test for the risk-return relationship should instead be
examined by the following regression:
Ri ,t = γ 0,t + γ 1,t Eˆ t −1[ IVOLi ,t ] + Γ t Eˆ t −1[ X i,t ] + ε i ,t .
(1')
That is, if idiosyncratic volatility is priced, we expect there to be a positive empirical
relationship between expected returns and expected idiosyncratic volatility.
Running (1)
instead of (1') would implicitly assume that IVOLi ,t −1 is a good predictor of Et −1[ IVOLi ,t ] . To
test whether this assumption is valid, Fu analyzes the persistence of the realized idiosyncratic
volatilities of U.S. stocks for the period 1963-2006. Using the Fama-French (1993) three6
factor model to obtain idiosyncratic returns, he shows that the first-order autocorrelation of
IVOLi ,t is small (the average across stocks is 0.330). In addition, he claims that if IVOLi ,t −1 is a
good predictor of Et −1[ IVOLi ,t ] , then IVOLi ,t should be modeled as a random walk process. He
uses the Augmented Dickey-Fuller t-test and shows that for almost 90% of the stocks in his
sample, the random walk hypothesis for the realized idiosyncratic volatility is rejected at the
1% significance level. Therefore, he claims that equation (1) run by Ang et al. (2006) is not a
valid test of the expected idiosyncratic risk-return relationship. Rather, a proper measure of
conditional volatility should be used instead of the lagged volatility.
To obtain an estimate of Et −1[ IVOLi ,t ] , Fu uses EGARCH models to obtain the monthly
conditional idiosyncratic variance (denoted as hi ,t ).
idiosyncratic volatility ( IVOLi ,t −1 ) in (1) with
By replacing the lagged realized
hi ,t , he finds that the time-series average of the
estimated γ 1,t is positive and significant. That is, there is a positive relationship between the
expected return and expected idiosyncratic volatility, and therefore, the idiosyncratic volatility
puzzle is explained.1
In a study of an international CAPM, Bali and Cakici (2010) test whether the countryspecific idiosyncratic risk is priced by using country-level aggregate market index data from
37 countries and a world market portfolio index. The daily idiosyncratic shocks in month t are
defined as the residuals from a regression of country i's daily market portfolio index returns
( Ri ,d ,t ) on the daily world market portfolio returns ( Rw ,d ,t ):
1
Spiegel and Wang (2005), who focus on the out-of-sample predictive power of idiosyncratic
volatility and liquidity using monthly data and EGARCH models, also find that stock returns
are increasing with the level of idiosyncratic volatility.
7
Ri , d ,t = µ i ,t + Betai ,t ⋅ Rw, d ,t + ri , d ,t , for d = 1, 2, . . ., Dt ,
(2)
where Dt is the number of trading days in month t and Betai ,t is the conditional world market
beta of country i in month t. They define the country-specific idiosyncratic volatility in month
t as the realized monthly standard deviation of the daily idiosyncratic shocks:
IVOLi ,t =
Dt
∑ (r
d =1
i , d ,t
2
− ri ,d ,t ) . They run a monthly cross-sectional regression of the returns to
2
country i’s market portfolio ( Ri ,t ) on lagged idiosyncratic volatility ( IVOLi ,t −1 ) and a vector of
other risk measures:
Ri ,t = γ 0,t + γ 1,t IVOLi ,t −1 + γ 2,t Betai ,t −1 + γ 3,t EPi ,t −1 + γ 4,t DYi ,t −1 + ε i ,t ,
(3)
where Betai ,t is the conditional world market beta from (2), EPi ,t is the natural logarithm of the
earnings-to-price ratio, and DYi ,t is the natural logarithm of the dividends-to-price ratio in
month t. They find that the time-series average of the estimated effect of IVOLi ,t −1 on Ri ,t (i.e.,
γˆ1,t ) is positive and statistically significant. This indicates that international investors hold
under-diversified international equity-index portfolios because the country-specific risk is
priced, and that the country-specific idiosyncratic volatility predicts a positive index return in
the next month. Therefore, the idiosyncratic volatility puzzle does not appear in the crosscountry market index data.
2
Since we use OLS regressions in (2) including a constant term, ri ,d ,t = 0 . Note that Bali and
Cakici (2010) ignore the scaling and do not divide the sum of squared residuals by the
number of days. We use this specification throughout the paper so that we can compare our
results with theirs.
8
According to Fu's argument, Bali and Cakici's (2010) test would be invalid unless the
lagged idiosyncratic volatility were a good predictor of the expected idiosyncratic volatility.
Therefore, our first step is to analyze the time-series property of the idiosyncratic volatility of
the country-level market index returns. We use the same data as those in Bali and Cakici
(2010), whose data end in September 2006, but update their data to November 2010. The data
are obtained from Datastream Global indices, and include U.S. dollar-denominated returns on
stock market indices for 37 countries plus the world market portfolio. There are 23 developed
markets and 14 developing or emerging markets.3
Table 1 shows the summary statistics of the monthly market index returns for each
country and the world market, including the means, standard deviations, and correlations with
the world market index returns. The next two columns show the time-series averages of EPi ,t
and DYi ,t for each country. We use all available data to calculate the summary statistics.4 The
starting month for each country is shown in the second column. The sample ends in November
2010 for all countries. Even with the updated data added, the summary statistics are in general
very similar to those in Bali and Cakici (2010): the emerging markets exhibit higher average
returns and higher standard deviations of returns, compared to the developed markets.
3
Based on Bali and Cakici's (2010) categorization, the 23 developed markets are Australia,
Austria, Belgium, Canada, Denmark, Finland, France, Germany, Greece, Hong Kong, Ireland,
Italy, Japan, the Netherlands, New Zealand, Norway, Portugal, Singapore, Spain, Sweden,
Switzerland, the United Kingdom, and United States. The 14 developing or emerging markets
are Argentina, Brazil, Chile, China, India, Korea, Malaysia, Mexico, the Philippines, Poland,
South Africa, Taiwan, Thailand, and Turkey.
4
Data for the earnings-to-price ratio ( EPi ,t ) and the dividends-to-price ratio ( DYi ,t ) are in
general shorter than the returns data.
9
To obtain the monthly data of the conditional world market beta ( Betai ,t ) and
idiosyncratic volatility ( IVOLi ,t ), we use daily data within each month to run time-series
regression (2). This generates the monthly Betai ,t and the daily idiosyncratic returns ( ri , d ,t ) .
The daily idiosyncratic returns are then used to calculate the realized monthly idiosyncratic
volatility ( IVOLi ,t ). The last two columns of Table 1 show the time series averages of Betai ,t
and IVOLi ,t for each country. Again, the results are in general consistent with those reported in
Bali and Cakici (2010): the country-specific idiosyncratic volatility is much higher in the
emerging markets than in the developed markets, while the cross-sectional differences in the
systematic risk for developed and emerging markets are not as significant as those in the
idiosyncratic risk.
To confirm Bali and Cakici's (2010) results on the idiosyncratic risk-return relationship
using the updated data, we run several versions of equation (3) like they do in their paper.
Table 2 reports the time-series averages of the estimated coefficients, their P-values based on
the Newey and West (1987) heteroscedasticity- and autocorrelation-adjusted t-statistics, and
the time-series averages of the R-squared.5 Consistent with the findings in Bali and Cakici
(2010), we find that, first of all, in all specifications, almost all the risk measures have a
positive effect on the returns. Secondly, the world systematic risk factor (i.e., the world market
beta) does not affect returns, indicating that country-specific factors provide more of an
5
The data availabilities are different either across variables within a country or across
countries. We require a minimum of ten observations in each cross-sectional regression.
10
explanation for the variation in index returns.6 Thirdly, the earnings-to-price ratio is the most
significant risk factor, and the effect of the dividends-to-price ratio disappears when the
earnings-to-price ratio is included in the regression. More importantly, the time-series average
of the estimated effect of IVOLi ,t −1 on Ri ,t , i.e., γˆ1,t , is positive and statistically significant at the
5% level in all cases. This confirms Bali and Cakici's (2010) finding that the idiosyncratic
volatility puzzle does not exist in the cross-country market index data.
However, is the above test valid? That is, according to Fu's argument, is IVOLi ,t −1 a
good estimate of expected idiosyncratic volatility Et −1[ IVOLi ,t ] ? The first column of Table 3
reports the first-order autocorrelation coefficients of IVOLi ,t for each country. The average of
the first-order autocorrelation coefficients of IVOLi ,t across these 37 countries is 0.552, which
is higher than that in Fu's U.S. firm-level data (0.330). That is, the idiosyncratic volatility is
more persistent in the country-level market index returns than in the U.S. firm-level data. The
second column of Table 3 reports the test statistics of the Augmented Dickey-Fuller unit-root ttest on IVOLi ,t . The results show that the test fails to reject the null hypothesis of a unit root in
13 of the 37 countries at the 1% significance level, compared to 90% rejection rate in Fu's
firm-level data. These two pieces of evidence in Table 3, of course, are not strong enough for
us to claim that the lagged idiosyncratic volatility ( IVOLi ,t −1 ) is a good proxy for the expected
idiosyncratic volatility ( Et −1 IVOLi ,t ) in the country-level data. However, it is a better proxy in
the country-level data than in the firm-level data. These observations may explain why the
6
Bekaert, Hodrick, and Zhang (2009) also show that local factors explain index returns
through a multiple-factor model.
11
idiosyncratic volatility puzzle in equation (3) exists in the firm-level data but not in the
country-level data.
Using the realized idiosyncratic volatility in month t-1 ( IVOLi ,t −1 ) as the forecast of the
idiosyncratic volatility in month t ( Et −1 IVOLi ,t ) implicitly assumes that the idiosyncratic
volatility follows a martingale process. The unit-root test results above show that this may not
be a proper assumption for all the countries in our sample. To relax this restrictive assumption,
we follow Huang et al. (2010) and use the best-fit autoregressive integrated moving average
(ARIMA) process to model the monthly conditional idiosyncratic volatility over a rolling
window. Specifically, we use the realized idiosyncratic volatility over the previous twentyfour months to find the best-fit ARIMA model using the Schwarz criterion (BIC). Then the
model is employed to predict the idiosyncratic volatility in the next month. We
replace
IVOLi ,t −1 in (3) with this estimated conditional idiosyncratic volatility ( Eˆ t −1 IVOLi ,t ) from the
best-fit ARIMA model and re-evaluate the relationship between the country-index returns and
the expected idiosyncratic volatilities:
Ri ,t = γ 0,t + γ 1,t Eˆt −1 IVOLi ,t + γ 2,t Betai ,t −1 + γ 3,t EPi ,t −1 + γ 4,t DYi ,t −1 + ε i ,t .
(3')
The results are reported in Table 4. The positive relationship between conditional expected
idiosyncratic volatilities and returns is again confirmed. Interestingly, the results are very
similar to those in Table 2: the effects of idiosyncratic volatilities on index returns are all
significantly positive and very similar in magnitudes; the world market beta is the least
significant risk factor; the earnings-to-price ratio is the most significant risk factor; and the
effect of the dividends-to-price ratio disappears when the earnings-to-price ratio is included in
12
the regression.
Judging from the average R-squared, using the conditional idiosyncratic
volatility estimated from the best-fit ARIMA model to replace the lagged idiosyncratic
volatility does not improve the in-sample forecast of the cross-country market index returns.
Bali and Cakici (2008), using firm-level data, show that, compared to the realized
monthly idiosyncratic volatility calculated from daily data, the conditional idiosyncratic
volatility estimated from a GARCH (1,1) model or an EGARCH(1,1) model using monthly
data is a more accurate proxy for expected future idiosyncratic volatility. 7 As mentioned
earlier, this is the strategy used by Fu (2009) to obtain the monthly conditional idiosyncratic
variance.
Therefore, to confirm the positive idiosyncratic risk-return relationship in the
country-level data, our next step is to follow their strategy and estimate the conditional
idiosyncratic volatility from an EGARCH (1,1) model by using monthly data. Specifically, we
first regress the monthly country index returns on the world returns, Ri ,t = µ i + ρ i Rw ,t + ri ,t , to
obtain the monthly country-specific idiosyncratic returns, and then model the idiosyncratic
returns as an AR-EGARCH(1,1) process:
ri ,t = α 0i + ∑ α ij ri ,t − j + ε i ,t ,
(4)
j
ln hi ,t = κ i + α i ⋅ ln hi ,t −1 + β i ⋅
7
ε i ,t −1
hi ,t −1
+γi ⋅
ε i ,t −1
hi ,t −1
⋅ I i+,t −1 ,
(5)
Foster and Nelson (1996) show that the conditional volatility estimated from the GARCH
models amounts to the conditional volatility estimated from a weighted rolling regression
using data from the preceding months.
13
where the autoregressive (AR) lag length
j is chosen by the Ljung-Box Q tests as the
minimum lag that renders the serially uncorrelated ε i ,t (at the 5% significance level) up to 24
lags from an OLS autoregression;
hi ,t is the conditional variance of ε i ,t , ε i ,t = hi ,t vi ,t ,
+
+
vi ,t ~ N (0,1) , and I i ,t = 1 if ε i ,t >0 and Ii ,t = 0 otherwise.
Table 5 reports the estimates of the EGARCH process, along with the means of the
estimated conditional idiosyncratic volatility hi ,t for each country. In general, the estimated
coefficients in the EGARCH process are statistically significant at the traditional significance
level. For most countries the conditional idiosyncratic volatility is highly persistent. The
relative conditional idiosyncratic volatilities ( hi ,t ) across countries are consistent with the
relative realized idiosyncratic volatilities ( IVOLi ,t ) across countries as reported in Table 1: the
country-specific idiosyncratic volatility is much higher in the emerging markets than in the
developed markets.
We replace IVOLi ,t −1 in (3) with
hi ,t and re-evaluate the relationship between the
country-index returns and expected idiosyncratic volatilities:
Ri ,t = γ 0,t + γ 1,t hi ,t + γ 2,t Betai ,t −1 + γ 3,t EPi ,t −1 + γ 4,t DYi ,t −1 + ε i ,t .
(3'')
The results are reported in Table 6. The positive relationship between conditional expected
idiosyncratic volatilities and index returns is again confirmed.8 By comparing the results in
8
Similar conclusions can also be found in Brockman et al. (2009), who apply Fu’s (2009)
EGARCH model to another set of international index data and find supporting evidence for a
positive and significant relationship between expected returns and idiosyncratic volatility.
14
Table 6 with those in Table 2 and Table 4, we can see that regressions (3), (3'), and (3'')
generate very similar results. Although (3'') yields a slightly higher R-squared than (3) and (3'),
the similarities in these results show that the lagged realized idiosyncratic volatility is a good
proxy for the conditional idiosyncratic volatility in the international market index data. It
provides as useful information as the conditional volatility does in forecasting international
index returns.
As argued by Huang et al. (2010), if return reversals exist and the lagged return is
omitted from the regressions, the effect of idiosyncratic volatility on expected returns would be
under-estimated. Even if this is the case, however, our conclusion above would not change
because we have found a positive and significant effect. To see the role of return reversals in
our analysis, we add the lagged returns to regressions (3), (3'), and (3'') and report the results in
Panels (A)-(C) of Table 7, respectively. Interestingly, return reversals do not exist in the
country-level index data.
Rather, we find a statistically significant momentum effect.
Therefore, omitting the lagged returns actually over-estimates the effect of conditional
idiosyncratic volatility on the expected returns. The magnitude of the momentum effect is,
however, relatively small. After controlling for the momentum effect, the size and the level of
significance of the effect of idiosyncratic volatility on the expected returns are slightly reduced.
However, the effect is still positive and marginally significant at the 10% level in almost all
cases.
In sum, our results are consistent with Fu's argument. As long as lagged realized
idiosyncratic volatilities proxy well for expected idiosyncratic volatilities, investors will expect
higher returns on portfolios that have experienced high idiosyncratic volatilities. The lagged
realized idiosyncratic volatility is as good a predictor of the expected idiosyncratic volatilities
15
as the conditional idiosyncratic volatility is for the international market index data. Our
analysis serves to validate Bali and Cakici's (2010) use of lagged idiosyncratic volatility to test
the risk-return relationship in the international market index data. These conclusions remain
robust after controlling for the momentum effect.
3. Returns, idiosyncratic volatilities, and idiosyncratic skewness
By incorporating the idea that investors prefer positive skewness, Boyer et al. (2010)
consider the role of idiosyncratic skewness in explaining the idiosyncratic volatility puzzle.
Using firm-level data in the U.S. stock market, they find that first, past idiosyncratic volatility
is, compared to the past idiosyncratic skewness, a strong predictor of future idiosyncratic
skewness; and second, expected idiosyncratic skewness and expected returns are negatively
correlated. Therefore, investors may accept lower expected returns on stocks that have high
lagged idiosyncratic volatilities because these stocks have higher expected idiosyncratic
skewness. The absence of the idiosyncratic volatility puzzle in the global equity index data
implies that Boyer et al.'s arguments may not hold in the international index market. If
international investors prefer positively skewed index securities in their under-diversified
international portfolios, then past idiosyncratic volatility should not positively predict future
idiosyncratic skewness in the international index market; otherwise, the positive relationship
between lagged idiosyncratic volatility and index returns found in Bali and Cakici (2010)
implies that portfolios with higher idiosyncratic skewness earn higher returns.
To verify Boyer et al.'s first argument (i.e., past idiosyncratic volatility is a strong
predictor of future idiosyncratic skewness) in the country-level index market, we follow their
steps and estimate the cross-sectional regression separately for each month t:
16
ISK i ,t = β 0,t + β1,t ISK i ,t −T + β 2,t IVOLi ,t −T + υ i ,t ,
(6)
where T is the forecast horizon. We use our country-level index data and equation (2) to
obtain the idiosyncratic shocks ( ri , d ,t ). By denoting S(t) as the set of trading days from the first
day of month t-T+1 through the end of month t, Boyer et al. define the historical estimate of
idiosyncratic volatility at time t as the standard deviation of the daily idiosyncratic shocks in
S(t): IVOLi ,t =
∑ (r
d ∈S ( t )
i , d ,t
9
− ri ,d ,t ) . Therefore, the definition of IVOLi ,t is different from that in
2
our earlier analyses unless the forecast horizon T = 1. The set of trading days S(t-T) would
cover the first day of month t-2T+1 through the end of month t-T. The historical estimate of

idiosyncratic skewness at time t is then defined as: ISK i ,t = 
∑ (r
 d ∈S (t )
i , d ,t
− ri , d ,t )
3

IVOL3i ,t  . Using

U.S. firm-level data, Boyer et al. find that if IVOLi ,t −T is included, the time series average of
βˆ1,t is insignificant and the time series average of βˆ2,t is significant. That is, past idiosyncratic
volatility is, compared to the past idiosyncratic skewness, a stronger predictor of future
idiosyncratic skewness.
Using the country-level index data, Table 8 reports, for horizons T = 1, 6, 12, 24, and
60, the time-series averages of the estimated coefficients, their P-values based on Newey and
West (1987) t-statistics, and the time-series averages of the R-squared. Apparently, all the
estimates are not only small in magnitudes but also highly insignificant. Neither lagged
9
Boyer et al. (2010) divide the sum of squared residuals by the number of trading days. We
do not do this scaling because the current expression is consistent with our analyses earlier in
the paper when the forecast horizon is one month (i.e. T=1). See Footnote 3.
17
idiosyncratic skewness nor lagged idiosyncratic volatility predicts future idiosyncratic
skewness. Therefore, Boyer et al.'s observation that past idiosyncratic volatility predicts future
idiosyncratic skewness is not present in the country-level data.
Next, using the country-level data, we verify the second argument by Boyer et al. that
the expected idiosyncratic skewness and expected returns are negatively correlated. Their
cross-sectional regression is specified as:
Ri ,t = γ 0,t + Λt −1Zi ,t −1 + γ t −1Eˆt −1[ ISKi ,t +T −1 ] + ε i ,t ,
(7)
where Zi,t is a vector of risk measures and Et [ ISKi ,t +T ] is the expected idiosyncratic skewness.10
Boyer et al. construct the measure of Et [ ISKi ,t +T ] from the conditional regression (6). However,
as shown in Table 8, the model fits badly, and one therefore cannot expect the conditional
idiosyncratic skewness from (6) to be a good measure of the expected idiosyncratic skewness.
To find a better measure of the expected idiosyncratic skewness, we adopt Fu's (2009) timeseries strategy and estimate the conditional idiosyncratic skewness from an EGARCH model.
For comparisons with our earlier results and for the sake of simplicity, we only focus on
estimating conditional idiosyncratic skewness with the horizon T = 1 in the following analyses.
Recall that in the previous section, we follow Fu and assume that vi ,t =
standard normal distribution.
ε i ,t
follows the
hi ,t
Although the EGARCH specification accommodates the
asymmetric property of volatility (whereby negative shocks increase volatility more than
10
Boyer et al. (2010) actually run this regression with portfolio returns and skewness formed
by sorting stock based on expected skewness. Here we instead use individual index returns
and skewness because we only have 37 indices.
18
positive shocks), that model is unable to explicitly estimate the skewness of the idiosyncratic
shocks. To model the skewness of the distribution, we relax the assumption of normality and
adopt a more flexible distribution, namely, the skewed student-t (ST) distribution proposed by
Hansen (1994). The ST distribution is a parsimonious two-parameter distribution, but also a
flexible one. It is able to model not only leptokurtosis but also asymmetry. The density
function of the ST distribution is:
η +1

2 − 2



1  σ ⋅ vi , t + µ 

 
σ ⋅ c ⋅ 1 +
 η − 2  1 + λ  



g ST (vi , t | η , λ ) = 
η +1

2 −


1  σ ⋅ vi ,t + µ   2

 
σ ⋅ c ⋅ 1 +
 η − 2  1 − λ  

for
vi , t ≥ −
µ
,
σ
for
vi ,t < −
µ
,
σ
η−2
where 2 < η < ∞ , − 1 < λ < 1 , µ = 4 ⋅ λ ⋅ c ⋅
, σ = 1 + 3 ⋅ λ2 − µ 2 , and c =
η −1
η +1
Γ

 2 
η 
π (η − 2)Γ  
2
.
 
The skewness sk =
M 3 − 3µσ 2 − µ 3
σ
3
, where the third raw moment M 3 =
16cλ (1 + λ 2 )(η − 2) 2
. The
(η − 1)(η − 3)
parameter η controls the tails and the peak of the density and λ controls the rate of descent of
the density around vi,t = 0. For details on the ST density, see the appendix in Hansen (1994).
To incorporate the conditional idiosyncratic skewness in our time-series model, we
specify the idiosyncratic shock in an autoregressive conditional density (ARCD) model
suggested by Hansen (1994). Hansen’s ARCD modeling strategy is to model the parameters in
the conditional density function as functions of the elements of the information set so that the
higher moments also depend on the conditioning information. He conjectures that since
19
GARCH models make the conditional second moment a function of the lagged errors, it is
reasonable to believe that this strategy could also work well for the other moments. Therefore,
we follow his suggestion and model the skewness parameter λ in the density function of the
ST distribution as:
λi ,t = −.99 +
.99 − ( −.99)
,
1 + exp( −ωi ,t )
ωi ,t = a + b ⋅ ε i ,t −1 + c ⋅ ε i2,t −1 ,
(8)
where the logistic transformation is used to set constraints that λi,t lies between .99 and -.99,
even though ωi ,t is allowed to vary over the entire real line. We call this extension of the
model in Section 2 the AR-EGARCH-ARCD model. The resulting conditional skewness is
denoted by ski ,t .11
The estimated conditional idiosyncratic skewness is then used as an additional
regressor in (3'') to test whether expected idiosyncratic skewness and expected returns are
negatively correlated in the country-level index market:
Ri ,t = γ 0,t + γ 1,t hi′,t + γ 2,t Betai ,t −1 + γ 3,t EPi ,t −1 + γ 4,t DYi ,t −1 + γ 5,t ski ,t −1 + ε i ,t ,
(3''')
where hi′,t is the conditional variance estimated from the new (AR-EGARCH-ARCD) model.
Panel (A) of Table 9 presents the results from (3'''). In Panel (B), we add the lagged returns as
an additional regressor to control for the momentum effect. The results in both panels show
that the relation between the returns and the expected idiosyncratic skewness is highly
insignificant. The other estimated coefficients are very similar to those reported in Table 6 and
11
The estimation results are not reported to save space but are available from the authors upon
request.
20
Panel (C) of Table 7. Therefore, idiosyncratic skewness does not play a significant role in the
country-level index market.
One possible explanation of the above results is that international investors do not
exhibit a pro-lottery preference in their under-diversified international portfolios. However, by
further investigating the idiosyncratic skewness, we find that the above results are more likely
to be due to the symmetry of the country-specific idiosyncratic shocks. In Table 10, we show
the means of the realized idiosyncratic skewness for T=1, 6, 12, 24, and 60. It can be seen that
the averages of ISK i ,t across these 37 countries are very small in magnitude. It is only 0.011
for T=1 in our country-level data, which is relatively small compared to that in the firm-level
data reported in Boyer et al., which is about 0.181.12 For the other horizons, the magnitudes
are even smaller.
In our estimations of the AR-EGARCH-ARCD model, although not reported here to
save space, many of the estimated coefficients ( â , b̂ , and ĉ ) in the conditional density
function (8) are statistically insignificant. This observation also casts doubt on the role of
idiosyncratic skewness in the international index pricing.
To see how appropriate this
modeling strategy is, we further experiment with two alternative specifications. The first one
removes the time-varying property of the skewness and imposes a constant skewness
parameter by setting b=c=0 in (8), which we denote as the AR-EGARCH-ST model.13 The
12
Boyer et al. (2010) reports a mean of 0.851 for the idiosyncratic skewness in the U.S. firm-
level data. However, their measure is ours multiplied by a scale of Dt . Assuming that there
are 22 trading days (Dt =22) yields a mean of 0.181.
13
We find that only 9 out of the 37 indices have a statistically significant (at the 5% level)
skewed distribution. The results are available from the authors upon request.
21
second alternative specification assumes a symmetric distribution by setting a=b=c=0 in (8),
denoted as the AR-EGARCH-t model, which is essentially an AR-EGARCH model with a
symmetric t-distribution. 14 Table 11 shows the estimation results from the cross-sectional
regression (3'') with the conditional idiosyncratic volatilities estimated from these three
alternative models (AR-EGARCH-ARCD, AR-EGARCH-ST, and AR-EGARCH-t).
The
results are very similar across these models. Therefore, whether we ignore the idiosyncratic
skewness or not, the cross-sectional evidence from Bali and Cakici's (2010) is not affected.
We conclude that the distribution of the idiosyncratic shocks in the international index market
is mostly symmetric and the idiosyncratic skewness is not significant enough to affect the
index returns.
4. Discussions and Conclusions
Theories show that under-diversified investors should be compensated for assuming
idiosyncratic risks.
Empirical studies testing this hypothesis often use a cross-sectional
regression of returns on realized idiosyncratic volatilities in the previous month and check for a
positive coefficient.
The empirical evidence that stocks with higher past idiosyncratic
volatilities earn lower future returns has evoked a series of studies trying to explain this
empirical puzzle. Studies researching the time series properties of the idiosyncratic returns for
answers claim that the puzzle can be explained by the fact that, firstly, past idiosyncratic
volatility is not a good predictor of expected idiosyncratic volatility; secondly, the estimate is
14
Not reported here but available upon request, the mean log-likelihoods from these three
specifications are so similar that the null of a symmetric distribution cannot be rejected by the
likelihood ratio test.
22
biased downward because of an omitted variable (past returns); and finally, investors' aversion
to idiosyncratic risk is outweighed by the preference for idiosyncratic skewness.
This paper finds that these time series properties of the idiosyncratic returns in the
firm-level data are different from those in the country-level index data, which explains why the
idiosyncratic volatility puzzle found in the former does not exist in the latter. First of all, in the
market for global equity indices, past idiosyncratic volatility is a good predictor of expected
idiosyncratic volatility. It provides just as useful information as the conditional idiosyncratic
volatility in predicting future returns. That is, country-specific volatilities are persistent. The
literature on international equity pricing suggests that the undiversifiable country-specific risk
may be due to factors such as purchasing power parity deviations, i.e., exchange rate and
inflation risk (Adler and Dumas, 1983), government restrictions on capital movements in
emerging markets (Henry, 2000), and asymmetric information across markets (Brennan and
Cao, 1997). Since these country-specific risk factors are most likely related to policies adopted
by individual countries, it is reasonable for uncertainty to be more persistent in these data,
especially in the emerging markets (Lewis, 2011). For example, before an emerging economy
announces that it is liberalizing its capital market, uncertainty regarding capital mobility and
exchange rate policies is persistently high. After the announcement, this type of uncertainty
should remain at a lower level. In addition, as information flows more slowly across borders
than across firms within a country, it is not surprising that the country-specific risk possesses a
higher degree of clustering, compared to the firm-specific risk relative to the individual
country's aggregate market risk. Therefore, past idiosyncratic volatility serves as a better
predictor of the expected idiosyncratic volatility in the global index market than in the stock
market within a country.
23
Secondly, in contrast to the finding of return reversals in the firm-level data, we report
that the international index returns exhibit return momentum. Positive autocorrelations of
index returns and short-term profits of momentum strategies in the international index market
have been well documented in the literature (e.g., Ahn et al., 2002; Bhojraj and Swaminathan,
2006). Antoniou et al. (2005) attribute this index return momentum to the introduction of
index futures, which has increased positive feedback trading in the spot markets. As positive
feedback traders respond to past index changes, positive autocorrelations of index returns are
shown over short horizons. This momentum effect indicates that the estimate of the expected
idiosyncratic risk-return relationship is actually biased upward in a standard international
CAPM model like equation (2) (Bali and Cakici, 2010) if the past return is omitted from the
regression. After adjusting the bias, however, our conclusions on the higher moments of the
country-specific idiosyncratic shocks remain robust.
Finally, we find that country-specific idiosyncratic skewness is not significant enough
to affect the country-level index returns. Unlike individual stock returns, the country-specific
idiosyncratic returns are mostly symmetrically distributed. Considering the globalization of
the financial markets, it is not reasonable to expect that an international equity index would
earn an abnormal extreme return. Therefore, under-diversified international portfolio investors
have no incentive to search for lottery-like equity indices. As a result, insignificant differences
in country-specific idiosyncratic skewness do not create significant differences in equity index
pricing. Index investors' preferences for idiosyncratic skewness do not outweigh their aversion
to idiosyncratic risk. Along with the findings of the momentum effects and more persistent
idiosyncratic volatilities, our results suggest that different investment strategies should be
adopted in forming global and domestic equity portfolios.
24
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28
Table 1: Summary Statistics of International Market Indices (Monthly Data)
The sample ends in November 2010 for all countries. The variable EP is the natural logarithm of the
earnings-to-price ratio, and DY is the natural logarithm of the dividends-to-price ratio. The
idiosyncratic volatility IVOLi ,t =
Dt
∑ (r
d =1
i , d ,t
− ri ,d ,t ) , where the idiosyncratic shocks are estimated by
2
equation (2): Ri , d ,t = µi ,t + Betai ,t ⋅ Rw, d ,t + ri , d ,t .
Country
Data
start
Argentina
Aug-93
Australia
Jan-73
Austria
Jan-73
Belgium
Jan-73
Brazil
Jul-94
Canada
Jan-73
Chile
Jul-89
China
Jul-93
Denmark
Jan-73
Finland
Mar-88
France
Jan-73
Germany
Jan-73
Greece
Jan-90
Hong Kong Jan-73
India
Jan-90
Ireland
Jan-73
Italy
Jan-73
Japan
Jan-73
Korea
Sep-87
Malaysia
Jan-86
Mexico
May-89
Netherlands Jan-73
New Zealand Jan-88
Norway
Jan-80
Philippines Nov-88
Poland
Mar-94
Portugal
Jan-90
Singapore
Jan-73
South Africa Jan-73
Spain
Mar-87
Sweden
Jan-82
Switzerland Jan-73
Taiwan
May-88
Thailand
Jan-87
Turkey
Jun-89
UK
Jan-73
US
Jan-73
WORLD
Jan-73
Market Index Returns
Correlation
Mean
Std
with WORLD
0.817
9.213
0.531
1.145
7.215
0.643
1.044
6.705
0.502
1.027
5.897
0.677
1.751 10.914
0.674
0.994
5.526
0.757
1.721
6.610
0.442
1.752 11.246
0.397
1.160
5.888
0.607
1.152
8.630
0.661
1.165
6.743
0.719
0.984
5.939
0.705
1.035 10.141
0.463
1.473
9.999
0.525
1.520 10.703
0.342
1.075
7.269
0.666
0.944
7.588
0.561
0.797
6.234
0.708
1.055 11.189
0.552
1.314
8.789
0.431
1.724
8.733
0.594
1.123
5.521
0.820
0.912
6.478
0.619
1.223
7.953
0.659
1.222
9.217
0.473
1.003 10.906
0.594
0.658
6.093
0.653
1.076
8.483
0.634
1.362
8.282
0.558
0.978
6.496
0.774
1.407
7.301
0.738
1.060
5.137
0.717
0.964 10.994
0.439
1.564 10.814
0.520
2.567 16.935
0.380
1.115
6.524
0.732
0.936
4.477
0.816
0.901
4.481
1.000
29
DY
EP
Beta
IVOL
0.872
1.371
0.612
1.264
0.948
1.052
1.191
1.022
0.615
0.882
1.262
0.909
0.946
1.247
0.345
1.251
0.968
0.120
0.588
0.967
0.615
1.383
1.545
0.888
0.251
0.463
1.056
0.912
1.306
1.107
0.907
0.720
0.545
1.012
1.069
1.414
1.069
---
-2.571
-2.804
-2.798
-2.539
-2.284
-2.657
-2.733
-3.269
-2.754
-2.775
-2.634
-2.600
-2.705
-2.764
-3.064
-2.753
-2.830
-3.480
-2.666
-2.902
-2.487
-2.441
-2.639
-2.365
-2.732
-3.085
-2.989
-2.754
-2.738
-2.643
-2.845
-2.532
-3.073
-2.518
-2.779
-3.169
-2.730
---
0.792
0.532
0.571
0.669
1.277
0.776
0.543
0.714
0.579
1.109
0.875
0.858
0.684
0.650
0.381
0.700
0.719
0.981
0.666
0.458
0.998
0.865
0.458
0.886
0.397
0.931
0.624
0.532
0.704
0.949
0.996
0.724
0.510
0.569
0.844
0.915
0.958
1.000
6.244
4.866
3.702
3.707
6.464
2.904
4.204
7.366
4.221
5.797
4.094
3.618
6.496
6.358
6.685
4.454
4.985
3.976
8.165
5.176
5.363
3.437
4.756
5.382
6.156
7.070
3.808
4.998
5.845
3.945
4.904
3.503
7.575
7.268
11.515
3.832
2.468
---
Table 2: Cross-Sectional Regressions
This table reports the cross-sectional regression results for (3):
Ri ,t = γ 0,t + γ 1,t IVOLi ,t −1 + γ 2,t Betai ,t −1 + γ 3,t EPi ,t −1 + γ 4,t DYi ,t −1 + ε i ,t .
The average intercepts, average slope coefficients, and average R2 are presented. The numbers
in parentheses are P-values calculated based on Newey and West (1987) t-statistics. A P-value
of 0.000 indicates that the P-value is nonzero, but smaller than 0.0005.
Constant
IVOLi ,t −1
0.761
0.106
(0.004)
(0.034)
Betai ,t −1
EPi ,t −1
DYi ,t −1
Avg. R2
0.113
0.696
0.109
0.087
(0.025)
(0.024)
(0.596)
0.188
2.440
0.125
0.645
(0.000)
(0.022)
(0.000)
0.194
0.336
0.098
0.404
(0.309)
(0.042)
(0.010)
2.511
0.144
0.092
0.708
(0.000)
(0.005)
(0.553)
(0.000)
0.282
0.109
0.091
(0.421)
(0.025)
(0.575)
2.596
0.143
0.099
0.723
-0.009
(0.001)
(0.006)
(0.527)
(0.001)
(0.967)
0.171
0.274
0.371
0.244
(0.027)
30
0.329
Table 3: First-Order Autocorrelation Coefficients and the Augmented Dickey-Fuller t-test
Statistics for Monthly Realized Idiosyncratic Volatility
ADF-t is the Augmented Dickey-Fuller t-statistic. The asterisk * indicates that the test fails to
reject the null hypothesis of a unit root at the 1% significance level.
Country
Argentina
Australia
Austria
Belgium
Brazil
Canada
Chile
China
Denmark
Finland
France
Germany
Greece
Hong Kong
India
Ireland
Italy
Japan
Korea
Malaysia
Mexico
Netherlands
New Zealand
Norway
Philippines
Poland
Portugal
Singapore
South Africa
Spain
Sweden
Switzerland
Taiwan
Thailand
Turkey
UK
US
1st-order autocorrelation
0.525
0.510
0.617
0.464
0.594
0.528
0.504
0.618
0.410
0.689
0.532
0.427
0.621
0.580
0.426
0.429
0.609
0.670
0.722
0.668
0.619
0.579
0.444
0.528
0.403
0.577
0.495
0.604
0.454
0.487
0.525
0.473
0.670
0.620
0.555
0.660
0.600
31
ADF-t
-6.293
-9.339
-4.380
-4.401
-4.305
-7.108
-6.429
-2.043*
-5.088
-3.159*
-5.652
-3.829*
-3.280*
-6.172
-6.360
-4.152
-4.525
-3.597*
-5.054
-2.973*
-3.679*
-4.478
-5.226
-6.237
-3.839*
-4.210
-3.247*
-4.413
-7.788
-5.130
-3.877*
-6.974
-3.171*
-3.195*
-3.580*
-4.964
-4.648
Table 4: Cross-Sectional Regressions
This table reports the cross-sectional regression results for (3'):
Ri ,t = γ 0,t + γ 1,t Eˆ t −1 IVOLi ,t + γ 2,t Betai ,t −1 + γ 3,t EPi ,t −1 + γ 4,t DYi ,t −1 + ε i ,t .
The average intercepts, average slope coefficients, and average R2 are presented. The numbers
in parentheses are P-values calculated based on Newey and West (1987) t-statistics. A P-value
of 0.000 indicates that the P-value is nonzero, but smaller than 0.0005.
Constant
Eˆ t −1 IVOLi ,t
0.685
0.098
(0.020)
(0.008)
Betai ,t −1
EPi ,t −1
DYi ,t −1
Avg. R2
0.109
0.552
0.095
0.189
(0.092)
(0.013)
(0.308)
0.186
2.167
0.081
0.518
(0.000)
(0.071)
(0.000)
0.194
0.217
0.095
0.430
(0.526)
(0.007)
(0.004)
1.866
0.099
0.184
0.489
(0.002)
(0.031)
(0.301)
(0.000)
0.271
0.066
0.099
0.219
0.408
(0.853)
(0.009)
(0.230)
(0.006)
1.628
0.092
0.235
0.449
0.142
(0.053)
(0.048)
(0.185)
(0.016)
(0.396)
32
0.166
0.240
0.326
Table 5: Estimation Results from the AR-EGARCH(1,1) Model
This table presents the key estimation results from the model: ri ,t = α 0i + ∑ α ij ri ,t − j + ε i ,t ,
j
ln hi ,t = κ i + α i ⋅ ln hi ,t −1 + β i ⋅
ε i ,t −1
hi ,t −1
+γi ⋅
ε i ,t −1
hi ,t −1
+
⋅ I i+,t −1 , ε i ,t = hi ,t vi ,t , vi ,t ~ N (0,1) , and I i ,t = 1 if ε i ,t
>0 and Ii+,t = 0 otherwise. The numbers in parentheses are P-values. A P-value of 0.000
indicates that the P-value is nonzero, but smaller than 0.0005.
Country
Argentina
Australia
Austria
Belgium
Brazil
Canada
Chile
China
Denmark
Finland
France
Germany
Greece
Hong Kong
India
Ireland
Italy
Japan
κ
β
α
γ
-0.001
0.954
-0.321
0.445
(0.994)
(0.000)
(0.001)
(0.023)
-0.089
0.995
-0.170
0.263
(0.007)
(0.000)
(0.000)
(0.001)
-0.045
0.932
-0.338
0.700
(0.639)
(0.000)
(0.000)
(0.000)
-0.079
0.950
-0.351
0.555
(0.276)
(0.000)
(0.000)
(0.000)
0.523
0.786
-0.606
0.731
(0.083)
(0.000)
(0.001)
(0.012)
1.756
0.257
-0.096
0.363
(0.026)
(0.398)
(0.395)
(0.054)
-0.032
0.961
-0.215
0.404
(0.715)
(0.000)
(0.021)
(0.015)
-0.094
0.948
-0.327
0.796
(0.548)
(0.000)
(0.008)
(0.001)
0.058
0.932
-0.213
0.358
(0.578)
(0.000)
(0.004)
(0.002)
-0.131
0.979
-0.271
0.521
(0.083)
(0.000)
(0.002)
(0.000)
-0.115
0.990
-0.195
0.363
(0.003)
(0.000)
(0.002)
(0.002)
-0.044
0.972
-0.131
0.317
(0.349)
(0.000)
(0.023)
(0.004)
-0.101
0.962
-0.348
0.616
(0.195)
(0.000)
(0.000)
(0.000)
-0.054
0.930
-0.467
0.819
(0.518)
(0.000)
(0.000)
(0.000)
0.102
0.947
-0.119
0.364
(0.574)
(0.000)
(0.156)
(0.023)
-0.105
0.976
-0.273
0.468
(0.024)
(0.000)
(0.000)
(0.000)
-0.116
0.988
-0.203
0.413
(0.004)
(0.000)
(0.000)
(0.000)
2.111
0.172
-0.425
0.823
(0.003)
(0.466)
(0.002)
(0.000)
33
Mean of
7.438
5.197
5.256
4.202
7.162
3.599
5.577
9.736
4.443
6.180
4.401
4.107
7.254
6.980
9.533
5.132
5.956
4.370
hi ,t
Table 5 (continued): Estimation Results for the AR-EGARCH (1,1) Model
Country
Korea
Malaysia
Mexico
Netherlands
New Zealand
Norway
Philippines
Poland
Portugal
Singapore
South Africa
Spain
Sweden
Switzerland
Taiwan
Thailand
Turkey
UK
US
κ
β
α
γ
-0.017
0.988
-0.243
0.158
(0.745)
(0.000)
(0.000)
(0.125)
-0.139
0.951
-0.420
0.778
(0.073)
(0.000)
(0.000)
(0.000)
-0.109
1.000
-0.095
0.253
(0.133)
(0.000)
(0.613)
(0.361)
0.025
0.925
-0.157
0.376
(0.795)
(0.000)
(0.016)
(0.003)
2.497
0.217
0.032
-0.098
(0.315)
(0.780)
(0.847)
(0.662)
-0.057
0.980
-0.132
0.311
(0.386)
(0.000)
(0.024)
(0.004)
-0.077
0.988
-0.213
0.312
(0.340)
(0.000)
(0.008)
(0.014)
5.513
-0.477
-0.546
1.035
(0.000)
(0.007)
(0.011)
(0.005)
0.010
0.960
-0.123
0.283
(0.922)
(0.000)
(0.126)
(0.038)
-0.125
0.967
-0.329
0.614
(0.026)
(0.000)
(0.000)
(0.000)
-0.037
0.982
-0.164
0.264
(0.585)
(0.000)
(0.006)
(0.014)
0.021
0.959
-0.228
0.238
(0.759)
(0.000)
(0.000)
(0.012)
0.078
0.919
-0.187
0.443
(0.745)
(0.000)
(0.181)
(0.086)
-0.077
0.950
-0.221
0.494
(0.246)
(0.000)
(0.002)
(0.000)
-0.115
1.004
-0.165
0.235
(0.007)
(0.000)
(0.021)
(0.013)
-0.045
0.986
-0.182
0.252
(0.429)
(0.000)
(0.000)
(0.001)
-0.098
1.008
-0.153
0.124
(0.005)
(0.000)
(0.002)
(0.008)
-0.114
0.995
-0.248
0.318
(0.001)
(0.000)
(0.000)
(0.003)
-0.138
0.977
-0.323
0.473
(0.000)
(0.000)
(0.000)
(0.000)
34
Mean of
hi ,t
7.811
6.612
6.318
3.143
4.819
5.501
7.700
7.602
4.554
5.886
6.738
3.929
4.774
3.516
8.385
8.747
14.001
3.814
2.485
Table 6: Cross-Sectional Regressions
This table reports the results for (3''):
Ri ,t = γ 0,t + γ 1,t hi ,t + γ 2,t Betai ,t −1 + γ 3,t EPi ,t −1 + γ 4,t DYi ,t −1 + ε i ,t .
The average intercepts, average slope coefficients, and average R2 are presented. The numbers
in parentheses are P-values calculated based on Newey and West (1987) t-statistics. A P-value
of 0.000 indicates that the P-value is nonzero, but smaller than 0.0005.
Constant
hi ,t
0.440
0.144
(0.134)
(0.004)
Betai ,t −1
EPi ,t −1
DYi ,t −1
Avg. R2
0.120
0.379
0.130
0.206
(0.240)
(0.011)
(0.188)
0.196
2.299
0.127
0.659
(0.000)
(0.016)
(0.000)
0.210
0.083
0.117
0.427
(0.802)
(0.015)
(0.012)
2.133
0.136
0.246
0.670
(0.000)
(0.010)
(0.107)
(0.000)
0.002
0.114
0.218
(0.995)
(0.027)
(0.166)
1.701
0.130
0.257
0.564
0.200
(0.015)
(0.013)
(0.108)
(0.005)
(0.215)
0.176
0.290
0.404
0.248
(0.018)
35
0.340
Table 7: Cross-Sectional Regressions
Panel (A) reports the cross-sectional regression results for (3) with the lagged return added; Panel (B)
reports the results for (3') with the lagged return added; and Panel (C) reports the results for (3'') with
the lagged return added.
The average intercepts, average slope coefficients, and average R2 are
presented. The numbers in parentheses are P-values calculated based on Newey and West (1987) tstatistics. A P-value of 0.000 indicates that the P-value is nonzero, but smaller than 0.0005.
Constant
Panel (A)
Panel (B)
Panel (C)
IVOLi ,t −1 Eˆ t −1 IVOLi ,t
hi ,t
Betai ,t −1
EPi ,t −1
DYi ,t −1
Ri ,t −1
Avg. R2
0.186
0.724
0.084
0.047
(0.005)
(0.082)
(0.011)
0.653
0.086
0.010
0.059
(0.025)
(0.074)
(0.950)
(0.001)
2.796
0.104
0.780
0.056
(0.000)
(0.026)
(0.000)
(0.002)
0.298
0.078
0.397
0.049
(0.305)
(0.101)
(0.001)
(0.010)
2.723
0.115
0.007
0.781
0.064
(0.000)
(0.017)
(0.970)
(0.000)
(0.000)
0.287
0.086
0.019
0.319
0.059
(0.378)
(0.081)
(0.903)
(0.027)
(0.001)
3.000
0.106
0.019
0.818
-0.135
0.059
(0.000)
(0.041)
(0.920)
(0.000)
(0.516)
(0.001)
0.611
0.094
0.043
(0.019)
(0.005)
(0.012)
0.475
0.096
0.086
0.053
(0.098)
(0.009)
(0.613)
(0.003)
2.079
0.076
0.514
0.050
(0.000)
(0.031)
(0.001)
(0.006)
0.129
0.093
0.440
0.047
(0.675)
(0.004)
(0.002)
(0.011)
1.743
0.113
0.017
0.465
0.056
(0.006)
(0.006)
(0.930)
(0.011)
(0.005)
-0.006
0.104
0.102
0.407
0.053
(0.985)
(0.005)
(0.536)
(0.005)
(0.004)
1.882
0.111
0.081
0.522
0.013
0.047
(0.020)
(0.010)
(0.693)
(0.014)
(0.943)
(0.023)
0.443
0.116
0.038
(0.100)
(0.016)
(0.033)
0.330
0.109
0.166
0.046
(0.268)
(0.036)
(0.295)
(0.007)
2.362
0.104
0.669
0.043
(0.000)
(0.037)
(0.000)
(0.020)
0.109
0.096
0.388
0.035
(0.734)
(0.052)
(0.012)
(0.055)
2.141
0.114
0.230
0.668
0.057
(0.000)
(0.030)
(0.101)
(0.000)
(0.001)
0.046
0.090
0.167
0.339
0.040
(0.897)
(0.104)
(0.253)
(0.035)
(0.019)
1.914
0.096
0.213
0.597
0.142
0.052
(0.006)
(0.068)
(0.137)
(0.006)
(0.277)
(0.004)
36
0.253
0.280
0.241
0.348
0.305
0.395
0.187
0.255
0.285
0.240
0.352
0.305
0.399
0.194
0.260
0.293
0.246
0.357
0.309
0.401
Table 8: Forecasting Idiosyncratic Skewness
This table reports, for horizons T = 1, 6, 12, 24, and 60, the time-series averages of the
estimated coefficients, their P-values (in parentheses) based on Newey and West (1987) tstatistics, and the time-series averages of R2 from cross-sectional regression (6):
ISK i ,t = β 0,t + β1,t ISK i ,t −T + β 2,t IVOLi ,t −T + υ i ,t .
Horizons
T=1
T=6
T = 12
T = 24
T = 60
Constant
ISK i ,t −T
IVOLi ,t −T
Avg. R2
0.117
0.003
0.013
0.001
(0.391)
(0.460)
(0.166)
-0.001
0.033
-3*10-5
(0.817)
(0.309)
(0.940)
-0.001
0.043
-7*10-5
(0.785)
(0.382)
(0.773)
0.004
0.161
-3*10-4
(0.386)
(0.130)
(0.171)
0.006
0.201
-2*10-4
(0.096)
(0.125)
(0.074)
37
0.156
0.145
0.145
0.204
Table 9: Cross-Sectional Regressions
This table reports the cross-sectional regression results for the following model:
Ri ,t = γ 0,t + γ 1,t hi′,t + γ 2,t Betai ,t −1 + γ 3,t EPi ,t −1 + γ 4,t DYi ,t −1 + γ 5,t ski ,t −1 + γ 6,t Ri ,t −1 + ε i ,t .
The average intercepts, average slope coefficients, and average R2 are presented. The numbers
in parentheses are P-values calculated based on Newey and West (1987) t-statistics. A P-value
of 0.000 indicates that the P-value is nonzero, but smaller than 0.0005.
Variables
Constant
Panel (A)
0.524
h′
0.128
(0.065)
(0.010)
0.431
0.119
0.213
-0.038
(0.156)
(0.025)
(0.185)
(0.825)
Panel (B)
Betai ,t −1
EPi ,t −1
DYi ,t −1
ski ,t −1
Ri ,t −1
0.008
Avg. R2
0.176
(0.963)
2.308
0.118
0.644
0.052
(0.000)
(0.028)
(0.002)
(0.818)
0.100
0.085
0.522
0.197
(0.767)
(0.089)
(0.003)
(0.275)
2.077
0.136
0.326
0.663
0.073
(0.000)
(0.013)
(0.049)
(0.000)
(0.747)
-0.004
0.083
0.216
0.504
0.179
(0.990)
(0.109)
(0.188)
(0.004)
(0.303)
1.327
0.126
0.391
0.487
0.298
0.128
(0.092)
(0.024)
(0.023)
(0.030)
(0.040)
(0.587)
0.247
0.274
0.227
0.348
0.295
0.396
0.381
0.102
0.168
-0.088
0.057
(0.179)
(0.049)
(0.259)
(0.581)
(0.001)
2.294
0.094
0.630
0.040
0.062
(0.000)
(0.038)
(0.002)
(0.840)
(0.001)
0.176
0.063
0.451
0.123
0.048
(0.580)
(0.225)
(0.009)
(0.439)
(0.007)
2.111
0.120
0.312
0.696
0.133
0.073
(0.000)
(0.011)
(0.018)
(0.000)
(0.535)
(0.000)
0.111
0.059
0.157
0.390
0.095
0.052
(0.757)
(0.317)
(0.287)
(0.027)
(0.534)
(0.004)
1.539
0.098
0.361
0.552
0.251
0.156
0.069
(0.041)
(0.056)
(0.015)
(0.014)
(0.058)
(0.467)
(0.000)
38
0.309
0.349
0.295
0.410
0.354
0.455
Table 10: Realized Idiosyncratic Skewness

The idiosyncratic skewness ISK i ,t = 
∑ (r
 d∈S ( t )
i , d ,t
− ri , d ,t )
3

IVOL3i ,t  , where S(t) is the set of trading

days from the first day of month t-T+1 through the end of month t and the idiosyncratic shocks
are estimated by equation (2): Ri , d ,t = µi ,t + Betai ,t ⋅ Rw, d ,t + ri , d ,t .
Horizon
Argentina
Australia
Austria
Belgium
Brazil
Canada
Chile
China
Denmark
Finland
France
Germany
Greece
Hong Kong
India
Ireland
Italy
Japan
Korea
Malaysia
Mexico
Netherlands
New Zealand
Norway
Philippines
Poland
Portugal
Singapore
South Africa
Spain
Sweden
Switzerland
Taiwan
Thailand
Turkey
UK
US
Average
T=1
T=6
T=12
T=24
T=60
0.017
-0.001
-0.003
0.012
0.013
0.000
0.020
0.036
0.007
-0.009
0.009
0.003
0.029
0.006
-0.009
0.007
0.016
0.019
0.018
0.030
-0.001
0.007
-0.002
0.016
0.017
0.011
0.021
0.028
0.007
-0.013
0.012
0.003
0.028
0.037
0.010
-0.004
0.010
0.011
0.001
-0.012
-0.008
0.003
0.012
-0.010
0.009
0.021
0.004
-0.018
0.000
-0.006
0.014
-0.016
0.000
-0.002
-0.002
0.016
0.007
0.006
0.012
0.000
-0.007
0.000
0.020
0.005
0.008
0.006
-0.011
-0.011
-0.001
0.003
0.008
0.011
0.008
-0.002
0.004
0.002
-0.005
-0.015
-0.008
0.002
0.007
-0.010
0.005
0.015
0.001
-0.016
-0.002
-0.009
0.008
-0.026
0.003
-0.002
-0.004
0.014
0.008
-0.001
0.009
-0.003
-0.009
-0.004
0.021
0.003
0.005
0.002
-0.013
-0.010
-0.003
0.000
0.005
0.004
0.008
-0.002
0.004
-0.0005
-0.013
-0.015
-0.006
0.000
0.004
-0.008
0.002
0.011
0.002
-0.014
-0.003
-0.007
0.005
-0.031
0.001
0.000
-0.004
0.014
0.010
-0.005
0.005
-0.003
-0.010
-0.005
0.025
0.002
0.002
0.000
-0.011
-0.007
-0.002
-0.001
0.003
0.000
0.006
0.000
0.005
-0.001
-0.020
-0.013
-0.002
0.001
0.005
-0.006
0.002
0.007
0.005
-0.009
-0.003
-0.005
0.003
-0.030
-0.003
-0.002
-0.003
0.012
0.013
-0.004
0.002
-0.001
-0.008
-0.004
0.025
0.001
0.000
-0.001
-0.008
-0.004
0.000
-0.001
0.002
0.004
0.004
-0.001
0.003
-0.001
39
Table 11: Cross-Sectional Regressions
This table reports the regression results of (3''):
Ri ,t = γ 0,t + γ 1,t hi ,t + γ 2,t Betai ,t −1 + γ 3,t EPi ,t −1 + γ 4,t DYi ,t −1 + ε i ,t ,
with hi ,t estimated from the following models:
(i)
AR-EGARCH-ARCD model - time-varying skewness parameter.
(ii)
AR-EGARCH-ST model - constant skewness parameter.
(iii) AR-EGARCH-t model - zero skewness, i.e., AR-EGARCH model with a t-distribution.
The average intercepts, average slope coefficients, and average R2 are presented. The numbers
in parentheses are P-values calculated based on Newey and West (1987) t-statistics.
Models
(i)
(ii)
(iii)
Constant
hi ,t
Betai ,t −1
EPi ,t −1
DYi ,t −1
Avg. R2
1.498
0.133
0.241
0.503
0.221
0.343
(0.029)
(0.006)
(0.147)
(0.010)
(0.203)
1.603
0.143
0.260
0.554
0.201
(0.023)
(0.005)
(0.125)
(0.006)
(0.213)
1.600
0.148
0.253
0.561
0.200
(0.023)
(0.004)
(0.146)
(0.005)
(0.235)
40
0.339
0.340