Skewness of Firm Fundamentals, Firm Growth, and

Cross-Sectional Stock Returns∗

Yuecheng Jia Shu Yan†

December 2015

Abstract

We present a novel interpretation of the conditional sample skewness of firm fundamentals

as a proxy of the expected growth rate of firm cash flows and therefore being positively

related to the expected stock return. Empirically, we document significant evidence that the

skewness of firm fundamentals positively predicts cross-sectional stock returns and future

firm growth. Our findings cannot be explained by existing predictors and risk factors. The

evidence further indicates that the alternative skewness measures of firm fundamentals are

proxies of different factors driving firm cash flows.

∗We would like to thank Yuzhao Zhang for helpful comments. All errors are our own.†Both Jia and Yan are at Department of Finance, Spears School of Business, Oklahoma State University,

Stillwater, OK 74078, USA. Phone: 405-744-5089(Yan). Fax: 405-744-5180. Please address correspondenceto yuecheng.jia@okstate.edu or yanshu@okstate.edu.

Skewness of Firm Fundamentals, Firm Growth, and

Cross-Sectional Stock Returns

Abstract

We present a novel interpretation of the conditional sample skewness of firm fundamentals

as a proxy of the expected growth rate of firm cash flows and therefore being positively

related to the expected stock return. Empirically, we document significant evidence that the

skewness of firm fundamentals positively predicts cross-sectional stock returns and future

firm growth. Our findings cannot be explained by existing predictors and risk factors. The

evidence further indicates that the alternative skewness measures of firm fundamentals are

proxies of different factors driving firm cash flows.

JEL Classification: G12

Keyword : Skewness, fundamental, growth, stock return, earnings, profitability

1 Introduction

Fama and French (2006, 2008) point out that most stock return anomalies, no matter whether

they are rational or irrational, are consistent with the basic stock valuation equation, which

is a mathematical identity that relates firm cash flows and stock returns (e.g., Campbell and

Shiller (1988) and Vuolteenaho (2002)). According to the equation, higher expected growth

rate of future cash flows implies higher expected stock return if the book-to-market ratio is

fixed.1 Fama and French argue that the anomaly variables such as book-to-market ratio, net

stock issuance, accruals, lagged stock return, asset growth, and profitability are proxies of

expected cash flows.

One notable feature of the aforementioned return predictors is that they are the first

moments of firm fundamentals or stock returns. In this paper, we propose the conditional

sample skewness of firm fundamentals as a new proxy of expected growth rate of cash flows.

We demonstrate analytically and numerically that, for very general data-generating spec-

ifications, the conditional sample skewness of firm cash flows is positively correlated with

the expected growth rate of the underlying process.2 Based on the basic stock valuation

equation, this interpretation of skewness of firm fundamentals implies our main testable hy-

pothesis: The conditional sample skewness of firm fundamentals positively predicts future

stock returns.

Beyond the first moments, researchers have examined whether the second moments of

firm fundamentals can predict stock returns and firm performance (e.g., Diether, Malloy, and

Scherbina (2002), Johnson (2004), Dichev and Tang (2009), and Gow and Taylor (2009)).3

To our knowledge, our paper is the first to examine the return predictability of the skewness

1This conclusion can also be drawn from the q-theory of investment (e.g., Hou, Xue, and Zhang (2014)).The valuation equation is a tautology and can be interpreted in alternative ways.

2In addition, we show that a high value of the conditional sample skewness implies high acceleration rateand high level of the underlying process. Both effects strengthen the positive relation between the conditionalsample skewness of firm fundamentals and future stock returns.

3There is also a literature on return predictability by stock volatility, the second moment of stock returns.See, for example, Goyal and Santa-Clara (2003), Ang, Hodrick, Xing, and Zhang (2006), and Jiang, Xu, andYao (2009).

1

of firm fundamentals. Using the conditional sample skewness of firm fundamentals to infer

the expected growth rate of firm cash flows has a couple of advantages over other econo-

metric approaches. First it is very easy to calculate. Second, the approach does not make

any assumptions about the underlying data-generating mechanism and is robust to many

alternative model specifications.

Our paper also differs from the previous studies on the return predictability of stock re-

turn skewness in several ways.4 First, our analysis is “preference-free” as we do not require

investors to prefer positive skewness in stock returns. Second, in contrast to the negative

relation between the return skewness and future stock returns documented in the litera-

ture, our model dictates a positive relation between the skewness of firm fundamentals and

future stock returns. Third, as we will see later, incorporating the return skewness as a

control variable in the empirical analysis does not change our results for the skewness of firm

fundamentals.

To test our hypothesis, we use three measures of cash flows: gross profitability of Novy-

Marx (2013) (GP), earnings per share (EPS), and analyst earnings forecasts (AF ).5 We

denote the skewness of these three measures by SKGP , SKEPS, and SKAF . Unlike the first

two skewness measures which are time-series estimates, SKAF is the cross-sectional skewness

of analysts’ forecasts, and does not conform to our argument that the conditional sample

skewness is a proxy of expected growth rate of the underlying cash flow process. To see that

the return predictability of SKAF is consistent with that of SKGP and SKEPS, we draw

support from the literature of stock analysts.6 Both theory and empirical evidence in the

4The literature on stock return (co)skewness dates back to the seminal work of Kraus and Litzenberger(1976). Recent studies include Harvey and Siddique (2000), Dittmar (2002), Barone-Adesi, Gagliardini, andUrga (2004), Chung, Johnson, and Schill (2006), Mitton and Vorkink (2007), Boyer, Mitton, and Vorkink(2010), Engle (2011), Chang, Christoffersen, and Jacobs (2013), Conrad, Dittmar, and Ghysels (2013), andChabi-Yo, Leisen, and Renault (2014).

5We have also considered alternative cash flow measures including ROE and earnings surprises. Theresults for the alternative measures are very similar and available upon request.

6The important and relevant articles in this literature include Scharfstein and Stein (1990), Avery andChevalier (1999), Hong, Kubik, and Solomon (2000), Clement and Tse (2005), Clarke and Subramanian(2006), and Evgeniou, Fang, Hogarth, and Karelaia (2013).

2

literature indicate that less skilled analysts tend to herd while more skilled analysts tend

to be bolder and are likely to deviate from the consensus. Because skewness is driven by

outliers, SKAF is likely to pick up the forecasts of more skilled analysts, which are supposed

to be more accurate. Consequently, SKAF should be positively correlated with expected

future earnings and predict future stock returns.

Strongly supporting our hypothesis, all three skewness measures are positively significant

in predicting cross-sectional stock returns. For example, when stocks are sorted on SKGP

into decile portfolios, the equal-weighted average next-quarter return increases from decile 1

to decile 10. The H-L spread between deciles 10 and 1 is 1.55% per quarter and statistically

significant. Value-weighting stock returns and adjusting average returns by the conventional

risk factors do not change the results. The predictability evidence is corroborated by the

Fama-MacBeth regressions, even in the presence of other return predictors including the

level of GP .

To strengthen our argument, we further test whether the skewness measures positively

predict future firm cash flows, which are measured by ROE and GP . The evidence over-

whelmingly supports our argument. All three skewness measures can significantly forecast

future ROE and GP for up to 4 quarters. If we sort stocks on SKGP into decile portfolios,

the next-quarter H-L spread in ROE is 1.20% and statistically significant. The next-quarter

H-L spread in GP is much higher at 3.76%. We find similar evidence for SKEPS but weaker

evidence for SKAF . These results suggest that the alternative skewness measures are related

to different aspects of firm cash flows.

We further examine the differences across the alternative skewness measures by comparing

their return predictability. Only for the sample of stocks with valid observations of SKAF ,

the predictability of SKEPS is subsumed by that of SKAF . We don’t find SKGP and SKAF

to dominate each other. For the larger sample of stocks with valid observations of SKGP

and SKEPS, neither skewness measure dominates the other. These findings further support

the view that the alternative skewness measures are proxies of different factors of firm cash

3

flows.

An alternative motivation of our main hypothesis is the positive impact of cash flow

skewness on firm growth option. In the literature, a firm’s growth options are treated as

options written on the firm’s cash flow process (e.g., Berk, Green, and Naik (1999) and

Carlson, Fisher, and Giammarino (2004)). A standard option pricing argument illustrates

that everything else fixed, higher skewness of firm cash flows leads to higher value of firm

growth option. Following the approach of Bernardo, Chowdhry, and Goyal (2007), this

implies higher stock risk and therefore higher stock return. We also test this argument with

two proxies of growth option: market asset-to-book asset ratio and Tobin’s q. We find strong

evidence supporting this argument as well.

Studies of higher moments of firm fundamentals or macroeconomic variables are scarce.

Colacito, Ghysels, and Meng (2013) show that the skewness of forecasts on the GDP growth

rate made by professional forecasters can predict stock market return. In a separate study, we

consider the skewness of aggregate stock market and find that it can predict stock market

return. The current paper differs from these two studies in a couple of ways. First, we

consider the skewness of firm fundamentals and individual stock returns. Second but more

importantly, we do not make specific assumptions on the data-generating processes and

investors’ utility functions as our approach is based on the basic stock valuation equation.

Also related to our paper, Scherbina (2008) constructs a non-parametric skewness mea-

sure of analysts’ earnings forecasts as the difference of median and mean forecasts and finds

that it negatively predicts future stock returns, opposite to the positive relation we find for

SKAF .7 In addition to the methods of measuring skewness of analyst forecasts, there are

a couple of important differences between our papers. First, Scherbina’s evidence is weak

for large firms while our evidence is very strong for large firms. Second, she argues that her

findings are caused by negative information withheld by analysts being incorporated in stock

7The non-parametric skewness measure is not an accurate proxy of population skewness for many proba-bility distributions. One such example is a bi-modal distribution where the non-parametric skewness measurecan have the wrong sign.

4

prices with a significant delay while we argue that more skilled analysts are more likely to

issue bolder (but more accurate) forecasts.

The rest of the paper is organized as following. In Section 2, we show why the conditional

sample skewness of firm fundamentals contains information on the firm cash flows and as a

result positively predicts stock returns. We describe the data and econometric methodology

in Section 3. Section 4 discusses the empirical evidence. Section 5 concludes.

2 Information Content of Conditional Sample Skew-

ness of Firm Fundamentals

In this section, we first demonstrate a novel approach to infer sampling properties of general

time-series processes from the conditional sample skewness. Then we apply the results to

firm fundamentals, and derive the relation between the conditional sample skewness of firm

fundamentals and expected stock return.

2.1 Conditional Skewness of Small Samples

To estimate the conditional skewness of a time-series process at time t using the past sample

of size n, {xt−n+1, ..., xt}, the standard formula is

b̂ =m3

s3=

1n

∑ni=1(xt−n+i − x̄)3[

1n−1

∑ni=1(xt−n+i − x̄)2

]3/2 , (1)

where x̄ is the sample mean, s is the sample standard deviation, and m3 is the sample third

central moment. Alternative formulas can be used but they do not affect our results.

We show next that the estimated b̂ is informative about the order of the sample obser-

vations of the change of x. Define the change of x as ∆xt = xt − xt−1. For the ease of

presentation, assume xt−n = 0. Using xt =∑n

i=1∆xt−n+i, we can express the first three

5

sample moments as

x̄ =1

n

n∑i=1

i∑j=1

∆xt−n+j

=1

n

n∑i=1

(n− i+ 1)∆xt−n+i, (2)

s2 =1

n− 1

n∑i=1

(i∑

j=1

∆xt−n+j − x̄

)2

=1

n− 1

n∑i=1

(i∑

j=1

j − 1

n∆xt−n+j −

n∑j=i+1

(1− j − 1

n

)∆xt−n+j

)2

, (3)

m3 =1

n

n∑i=1

(i∑

j=1

∆xt−n+j − x̄

)3

=1

n

n∑i=1

(i∑

j=1

j − 1

n∆xt−n+j −

n∑j=i+1

(1− j − 1

n

)∆xt−n+j

)3

. (4)

In the sample mean x̄, earlier observations of ∆xt−n+i are clearly over-weighed than later

observations. To see how the location of an observation affects its weight in s2 and m3, we

consider two examples. For n = 3, simple calculations show

s2 =2

3

(∆x2

2 +∆x2∆x3 +∆x23

),

m3 =1

9(∆x3 −∆x2)

(2∆x2

2 + 3∆x2∆x3 + 2∆x23

).

In this case, s2 is symmetric with respect to ∆x2 and ∆x3 while m3 is monotonically in-

creasing in ∆x3 −∆x2. For n = 4, we can write

s2 =1

4

(3∆x2

2 +∆x23 + 3∆x2

4 + 4∆x2∆x3 + 2∆x2∆x4 + 4∆x3∆x4

),

m3 =3

8(∆x4 −∆x2)

(∆x2

2 +∆x24 + 2∆x2∆x3 + 2∆x2∆x4 + 2∆x3∆x4

).

In this case, s2 is symmetric with respect to ∆x2 and ∆x4. m3 is monotonically increasing in

∆x4 −∆x2 if the second part of m3 is positive, which is the case when ∆x3 = 0. These two

6

examples suggest that the sign and magnitude of m3 depend on the order of observations

{∆xi} while s2 is not. This seems intuitive from the construction of s2 and m3. The second

moment is insensitive to whether an observation occurs early or late in the sample but the

third moment may over-weighs/under-weighs an observation depending on its location in

the sample. Taken together, a high value of b̂ seems to suggest relatively high (low) values

for more recent (earlier) observations of ∆xt.

It is messy to extend the above examples to general settings without specifying the

underlying data-generating process. In the following, we consider the class of AR(1) processes

xt = ρxt−1 + ut, (5)

where ρ ≤ 1 is a constant and ut is an iid standard white noise process. Note that xt is a

random walk when ρ = 1. The initial value x0 is set to be zero for simplicity. There is no

constant term on the right-hand side although including one does not change the results.

Instead of providing analytical proofs, we conduct the following numerical analysis. To be

consistent with our later empirical work, we consider n = 8, 12, 16, and 20 and ρ = 0.9, 0.95,

and 1.8 To take into account of sampling errors, we use the Monte Carlo simulation method

to examine the correlations between the conditional sample skewness and cross-sections of

sample observations of ∆xt. The steps are detailed as following.

• Step 1: For fixed n and ρ, independently generate N = 1, 000, 000 paths of xt according

to equation (5). Denote the observations of the ith path by {xit}nt=0.

• Step 2: For the ith path, compute the sample skewness b̂i.

8The small sample sizes are appropriate when we consider low-frequency financial accounting data suchas the quarterly earnings. Using larger sample sizes to estimate the conditional skewness is problematic ifthe underlying data-generating mechanism is time-varying and non-stable. The near-unit-root or unit-rootspecification for x is also reasonable as most financial accounting variables are highly persistent. Negativevalues may arise in the AR(1) process, which are undesirable since most accounting variables are non-negative. This problem can be solved by taking logarithm. In most practical cases, the results are notaffected by the log transformation.

7

• Step 3: For each value of t = 2, ..., n, compute the correlation of b̂i and ∆xit across the

N sample paths and denote it by c(t).

Figure 1 shows the plots of c(t) as a function of t for different values of n and ρ. Several

patterns are important to note. First, for every (n, ρ) pair, the value of c(t) is negative

during the first half of the sample but positive during the second half of the sample. Second,

c(t) is monotonically increasing in t for the cases of n = 8, from less than −0.2 to over 0.2

when ρ = 1. For the cases of n = 12, 16, 20, c(t) is monotonically increasing except for the

two ends of the sample. The minimum and maximum of c(t) still occur near the beginning

and ending of the sample, respectively. Third, when n is fixed, the increasing pattern of

c(t) becomes more significant as ρ increases to 1. Fourth, when ρ is fixed, the shape of

c(t) becomes flatter as n increases. The minimum and maximum of c(t) are located further

away from the first and last observations. These correlation patterns of c(t) are not sensitive

to the iid assumption for ut as we have checked various heteroskedastic specifications for

ut. We have also considered numerous alternative ARMA(p,q) specifications for xt and find

qualitatively similar results.

The numerical results confirm our conjecture that the conditional sample skewness b̂ is

informative about the order of observations of ∆xt at least for small sample size up to 20. A

high value of b̂ suggests that the recent growth rates are likely high while the earlier growth

rates are likely low. A low value of b̂ suggests the opposite. Moreover, b̂ is not only an

indicator of the current growth rate of xt but also positively associated with the acceleration

rate (change of growth rate) of xt.

In addition to the growth rate of xt, it is also interesting to examine the correlations of b̂

with the levels of xt. To this end, we use the same Step 1 an Step 2 of the previous analysis.

But in the new Step 3, we calculate the cross-sectional correlations of b̂i with xit (denoted

by C(t)) instead of ∆xit. Figure 2 shows the plots of C(t). We make several comments.

First, C(t) is mostly negative except when t is close to the sample end n. Second, C(t) is

initially decreasing but becomes increasing as t increases, with the maximum attained at

8

t = n. The near-convexity of C(t) is consistent with the near-monotonicity of c(t) shown in

Figure 1. Third, with fixed n, C(t) becomes smaller when ρ increases. Fourth, with fixed ρ,

C(t) becomes larger when n increases. Overall, the most important finding is that a higher

value of b̂ is more likely associated with a higher value of the last sample observation than

past observations and the effect is stronger for lower ρ and higher n.

Having seen how b̂ is related to the past growth rates of x, an important question is:

What does b̂ tell us about the expected future growth rate of x. If ∆xt is iid over time,

the above results are not useful for prediction purpose because knowing b̂ and therefore the

order of the past observations of ∆xt does not provide additional information about future

∆xt. In real financial data, however, ∆xt is often non-iid, and b̂ can be informative about

the expected growth of x. As an example, consider the following process

∆xt = ut + εt (6)

ut = µ+ θut−1 + et (7)

where µ and 0 < θ < 1 are constants, and εt and et are iid standard white noise processes.

In this model, ut is the expected growth rate of x and follows an AR(1) process which is

unobserved. A high value of ∆xt implies a high value of ut and consequently higher future

growths of x due to the persistence of the growth rate process.

Estimates and predictions of this type of models can be obtained using methods such as

the Kalman filters. But there are some serious issues with the parametric approach. First,

accurate estimates of such models require long time-series data, which are not available.

Second, the models are not stable over time. This can happen, for example, when there

are structural breaks in the underlying data-generating process. Third, the models are

likely misspecified. Alternative ARMA specifications or regime-switching models can provide

similar fit of the same data.

Using the conditional sample skewness b̂ to imply the expected growth rate of x circum-

9

vents these problems. It doesn’t need long time series to estimate. More importantly, it

doesn’t rely on any parametric models. It allows many different types of model specifica-

tions. As long as the growth rate of x is positively autocorrelated to a certain extent, a high

value of b̂ implies that the future growth rate of x is also likely high. The same argument

also implies potentially high acceleration and level of x when b̂ is high.

2.2 Skewness of Firm Fundamentals and Stock Returns

We now apply the previous results for general time series to firm fundamentals. Let At

denote a measure of the fundamental cash flows of a firm at time t. We do not impose any

parametric specifications on At other than that it satisfies certain time-series properties so

that, as shown in the last section, the conditional sample skewness is positively related to

the growth rate, acceleration, and level of At.

Suppose that we have calculated b̂ using the historical data of size n ≤ 20, {At−n+1, ..., At}.

As we will argue next, there are three channels through which b̂ positively predicts future

stock returns. First, we have demonstrated that a high value of b̂ implies that the recent

growth rates of A are likely high. If the growth rate of A is persistent, then the expected

future growth rate of A will also be high. According to the basic stock valuation equation,

higher expected growth rate of firm cash flows implies higher stock returns. Consequently, b̂

positively predicts future stock returns.

Second, we have also shown previously that a high value of b̂ implies positive acceleration

of A, in other words, increasing growth rate of A during the recent sample period. If the

positive acceleration persists into the future, then the expected future growth rate of A will

be even higher than that implied by the previous point. This second-order effect strengthens

the positive return predictability of b̂.

Lastly, recall that a high value of b̂ also implies that the current level of A is potentially

high. To see how this fact leads to higher future stock returns, we resort to the literature

of firm growth options. In this literature, the firm value is decomposed into two parts: firm

10

fundamental cash flows and growth option. We follow the approach of Bernardo, Chowdhry,

and Goyal (2007) by treating the growth option as a call option on A. It is well-known that

the call option value is an increasing function of the underlying process. Hence a high value

of b̂ likely causes the value of firm growth option to increase, which leads to more weight

of the growth option in the firm total value. As demonstrated in Bernardo, Chowdhry, and

Goyal (2007), the risk and therefore return of the growth option are higher than those of

the firm cash flows. Combining the above results leads to the third channel that b̂ positively

predicts future stock returns.

The effects of all three channels are in the same direction regardless which of them is

at work. We therefore conclude that the conditional sample skewness of firm fundamentals

positively predicts future stock returns.

3 Data and Methodology

In this section, we first show the definitions of skewness measures of firm fundamentals. We

then describe the data. Finally, we discuss our econometric methods.

3.1 Definition of Skewness Measures

In quarter t, we follow Gu and Wu (2003) to define skewness of GP and EPS as the standard

skewness coefficient of lagged observations during the rolling window of quarters t − n to

t− 1:

SKGP,t =n

(n− 1)(n− 2)

t−1∑τ=t−n

(GPτ − µGP

sGP

)3

, (8)

SKEPS,t =n

(n− 1)(n− 2)

t−1∑τ=t−n

(EPSτ − µEPS

sEPS

)3

, (9)

where µGP (µEPS) and sGP (sEPS) are, respectively, the sample average and standard devia-

tion of GP (EPS) for the rolling window. In the benchmark case reported in the paper, we

11

fix n = 8. The results for n up to 20 are very similar and available upon request.

Note that we don’t use the quarter t GP and EPS in the definitions because they not

reported until quarter t + 1. When examining whether the skewness of earnings skewness

up to quarter t predicts the stock returns in quarter t + 1, using future information that is

available in quarter t+1 but not in quarter t biases the statistical inference. We in fact have

conducted (unreported) empirical analysis without skipping quarter t and have found even

stronger (but biased) results.

To define the skewness of analysts’ forecasts, we follow Diether, Malloy, and Scherbina

(2002) to use the most recent valid analysts’ annual earning forecasts by the end of quarter

t.9 We restrict our sample to stocks with at least three analyst forecasts in a quarter so that

skewness is well defined. The skewness across analysts is then

SKAF,t =N

(N − 1)(N − 2)

N∑j=1

(epsj − µeps

seps

)3

, (10)

where epsj is the earnings forecast of the jth analyst, µeps and seps are, respectively, the

sample average and standard deviation of all forecasts, and N is the number of analysts.

Note that we don’t skip quarter t as for the first two skewness measures of firm fundamentals

because analyst forecasts are known before quarter t+ 1.

3.2 Data

Stock return and accounting data are obtained from CRSP and COMPUSTAT. The I/B/E/S

provides analysts’ earnings forecasts. We consider all NYSE, AMEX and NASDAQ firms

in the CRSP monthly stock return files up to December, 2013 except financial stocks (four

digit SIC codes between 6000 and 6999) and stocks with share price less than $5. To be

9Our measure is updated every quarter while Diether, Malloy, and Scherbina (2002) construct theirdispersion in analysts’ forecasts at monthly frequency. We have also considered dropping the forecasts madein the last five days of quarter t to make sure all forecasts are observed. The results are almost identical tothose reported in paper.

12

included in the sample of SKGP or SKEPS, we require a firm to have at least 16 quarters of

gross profitability or earnings data during 1971–2013. The construction of each observation

of skewness measure needs observations of 8 consecutive quarters. We begin portfolio sorting

and regression analysis for SKGP and SKEPS from Q1, 1973, and for SKAF from Q1, 1985.

For each quarter, the accounting variables are defined as follows.

• GP : Following Novy-Marx (2013), gross profitability is quarterly revenues minus quar-

terly cost of goods sold scaled by quarterly asset total.

• EPS: Quarterly earnings per share before extraordinary items.

• MC: Market capitalization is the quarter-end shares outstanding multiplied by the

stock price.

• BM : Book-to-market ratio is the ratio of quarterly book equity to quarter-end market

capitalization. Quarterly book equity is constructed by following Hou, Xue, and Zhang

(2014) (footnote 9), which is basically a quarterly version of book equity of Davis, Fama,

and French (2000).

• ROE: Return on equity is defined as income before extraordinary items (IBQ) divided

by 1-quarter-lagged book equity.

• MABA: Market asset-to-book asset ratio is defined as [Total Asset−Total Book Com-

mon Equity+Market Equity]/Total Assets.

• Tobin’s q: It is defined as [Market Equity+Preferred Stock+Current Liabilities−Current

Assets Total+Long−Term Debt]/Total Assets.

• Disp: Dispersion of analysts’ forecasts is defined in almost the same way as Diether,

Malloy, and Scherbina (2002) using the detail history file of I/B/E/S. The only dif-

ference is that we use quarterly frequency instead of monthly frequency. If an analyst

makes more than one forecast in a given quarter, only the most recent forecast is used

in the calculation.

13

Firm size and book-to-market ratio are standard control variables in asset pricing studies.

ROE is a popular measure of firm cash flows and has been shown to predict stock returns

(e.g., Hou, Xue, and Zhang (2014)). We follow Cao, Simin, and Zhao (2008) to use MABA

and Tobin’s q as proxies of firm growth options. We have also considered the dispersion

in analysts’ forecasts defined in Diether, Malloy, and Scherbina (2002) as a control variable

when we examine the return predictability of SKAF . But controlling the dispersion does

not change the results. So we do not include it in the paper. The variables related to stock

returns are defined in the following.

• MOM : Momentum for month t is defined as the cumulative return between months

t− 6 and t− 1. We follow the convention in the literature by skipping month t when

MOM is used to predict returns in month t + 1. We have also used the cumulative

return between months t− 11 and t− 1 and obtained similar results.

• Idvol: Idiosyncratic volatility is, following Jiang, Xu, and Yao (2009), the standard

deviation of the residuals of the Fama and French (1993) 3-factor model using daily

returns in the quarter.

• Idskew: Following Harvey and Siddique (2000) and Bali, Cakici, and Whitelaw (2011),

it is defined as the skewness of the regression residuals of the market model augmented

by the squared market excess return. We use daily returns in the quarter to estimate

the regression.

• Prskew: It is predicted idiosyncratic skewness defined in Boyer, Mitton, and Vorkink

(2010). We obtain the Prskew data from Brian Boyer’s website.

• MAX: Following Bali, Cakici, and Whitelaw (2011), it is the average of the two highest

daily returns in quarter t. Note that we use quarterly frequency instead of monthly

frequency.

14

We use Iddvol as a control because a number of studies have documented that it predicts

returns. The skewness measures of stock returns, Idskew, Prskew, and MAX are good

controls to evaluate additional return explanatory power of skewness of firm fundamentals.

We have also considered total return skewness of daily stock returns in the quarter and

obtained similar results. We winsorize all the variables at 1% and 99% levels although the

results do not change significantly without winsorizing or winsorizing at 0.5% and 99.5%

levels.

3.3 Econometric Methods

We rely on the portfolio sorts and cross-sectional regressions of Fama and MacBeth (1973)

for our empirical investigation. For single portfolio sorts, we rank stocks on a skewness

measure of firm fundamentals into decile portfolios and then consider both equally-weighted

and value-weighted portfolio returns. If the skewness is positively related to stock returns,

we expect an increasing pattern of portfolio returns from decile 1 to decile 10. For double

portfolio sorts, we first rank stocks into quintiles by a control variable such as MC and

then further sort stocks within each portfolio into quintiles by the skewness measure. If the

control variable can explain the predictability of skewness, we expect the increasing pattern

of returns in skewness to be much less significant in each quintile of the control variable. To

compute t-statistics of average portfolio returns, we use the Newey-West adjusted standard

errors because of the persistence in the portfolios.

For the Fama-MacBeth regressions, we expect the estimated average coefficient of the

skewness measure to be positive and significant. The cross-sectional regressions allow us to

estimate the marginal effect of the skewness measure when controlling for other variables

known to predict stock returns. In the most general specification, we include all the control

variables in the regression. If the skewness measure captures information about expected

stock returns beyond that in other variables, the coefficient of the skewness measure should

be significant even in the presence of all the control variables.

15

We also use the Fama-MacBeth regression approach to compare the explanatory power

of different skewness measures. To do so, we include two or three skewness measures in one

regression. If the coefficient of one skewness measure is no longer significant in the presence

of another skewness measure, it indicates that the later skewness measure dominates the

first measure in the sense that it subsumes all the explanatory power of the first measure.

4 Empirical Evidence

4.1 Data Descriptions

The data of skewness measures are unbalanced. There are 350,050, 384,402, and 162,782

firm-quarter observations for SKGP , SKEPS, and SKAF , respectively. The sample size

for SKAF is less than half of those for the other two measures because not only the sample

period for the I/B/E/S data is shorter but also many small firms do not have enough analyst

coverage to compute skewness.

Table 1 reports the average contemporaneous cross-sectional correlations of quarterly

skewness measures and some control variables. Amongst the three skewness measures, SKGP

and SKEPS are mildly correlated with correlation coefficient of 0.31. The forward-looking

measure, SKAF , is essentially uncorrelated with the other two time-series measures. The

results indicate that different skewness measures have different information contents about

firm cash flows.

SKGP is mildly correlated with MOM and level of GP but uncorrelated with other

controls. SKEPS seems to be slightly correlated with the control variables but all correlation

coefficients are below 0.17. In contrast, the forward-looking skewness, SKAF , is uncorrelated

with all the control variables.

16

4.2 Single Portfolio Sorts

Table 2 reports the average time-series returns and characteristics of the decile portfolios

formed by sorting stocks on the three skewness measures. When sorted on SKGP as in panel

A, the average equal-weighted quarterly return increases from decile 1 (2.99%) to decile 10

(4.54%). The average H-L spread between is 1.55% per quarter (or 6.20% per year) and

highly significant (t = 5.67). To make sure that the significant H-L spread is not driven by

higher stock risks, we estimate the risk-adjusted α using the four-factor model of Carhart

(1997). The risk-adjusted H-L spread is even higher at 1.69% and significant. The results for

value-weighted returns are very similar to those for equal-weighted returns, implying that

the findings are not dominated by small stocks.

Looking at the characteristics of the decile portfolios, low-SKGP stocks have low past

return, GP , and ROE but slightly high book-to-market ratio and idiosyncratic volatility.

One reason of these patterns in control variables is that low-SKGP stocks are past under-

performers in terms of profitability. To make sure that the return predictability of SKGP is

not a result of the firm characteristics, we will use double portfolio sorts and Fama-MacBeth

regressions.

The results of portfolios sorts on SKEPS in panel B are very close to those for SKGP .

The unadjusted and adjusted H-L spreads for SKEPS are actually slightly higher than those

for SKGP . The firm characteristics of the decile portfolios also exhibit similar patterns as

those in panel A.

Panel C shows the results for SKAF . The portfolio returns are generally increasing from

decile 1 to decile 10. All average equal-weighted/value-weighted unadjusted/adjusted H-L

spreads are higher than 1.26% per quarter and statistically significant. The numbers for

SKAF are smaller and less significant than those for SKGP and SKEPS. Interestingly, we do

not observe any significant patterns in firm characteristics. There are several reasons why the

results for SKAF are different from those for SKGP and SKEPS. First, the sample period for

SKAF is shorter. Second, the stocks for SKAF tend to be larger and better followed firms.

17

Third, as we see in Table 1, SKAF are uncorrelated with firm characteristics.

Overall, in spite of the obvious differences across the three skewness measures, we find

consistent positive predictive relation between skewness of firm fundamentals and future

stock returns, confirming our hypothesis. The results are robust regardless whether the

returns are equal-weighted or value-weighted, and unadjusted or risk-adjusted.

4.3 Double Portfolio Sorts

We now investigate whether the predictability of the skewness measures are caused by firm

characteristics. We use the double portfolio sort approach by first sorting stocks on firm

characteristics and then sorting on the skewness measures. Table 3 reports the average equal-

weighted returns of double-sorted portfolios for the six characteristics reported in Table 2.

The results for value-weighted returns are very similar and not shown for brevity. We have

also considered a number of other control variables and their results are available upon

requests.

First, we consider the results for SKGP in panel A. When stocks are initially ranked by

MC, the H-L spreads of skewness quintiles show a decreasing pattern from MC quintile 1

(2.51%) to MC quintile 5 (0.58%), suggesting that the predictability of SKGP is stronger

for small stocks. In contrast, the predictability of SKGP is stronger for high momentum,

GP , and idiosyncratic volatility stocks but there is no clear pattern for BM and ROE. No

matter which firm characteristic is considered, all H-L spreads remain positive and most of

them are statistically significant. The evidence indicates that the return predictive power of

SKGP can not be explained the firm characteristics.

The results for SKEPS in panel B are similar to those for SKGP with a few differences.

The predictability of SKEPS is strong for low BM and high ROE stocks. The H-L spreads

for GP quintiles exhibit a U-shape pattern. If any firm characteristic can explain SKEPS,

it is ROE because only one of the five H-L spreads is significant. Some loss of statistical

significance can be attributed to the higher standard errors due to smaller sample sizes. Close

18

inspection of the ROE quintiles reveals non-linear interactions among stock return, SKEPS,

and ROE. We will get a clearer picture when we estimate Fama-MacBeth regressions with

other control variables.

Finally, we consider the results for SKAF in panel C. In all cases but one, the H-L spreads

are positive, many of which are significant. The relations between H-L spreads and the firm

characteristics are not generally linear. For example, the H-L spreads for BM quintiles are

hump-shaped. In sum, no characteristic seems to be able to explain the predictive power of

SKAF .

4.4 Fama-MacBeth Regressions

We further examine the return predictability of the skewness measures with the Fama-

MacBeth regressions, which allow us to control for other return predictors. The results are

reported in Table 4. For each skewness measure, we estimate eight regressions. The first

model uses a skewness as the only explanatory variable. Models (2)-(7) consider the six

control variables, one at a time. Because of different sample sizes for different measures,

we reestimate these models for each skewness measure. Model (8) includes the skewness

measure and all six control variables.

First, let us look at the results for SKGP in panel A. The average coefficient of SKGP

in model (1) is positive and significant at the 1% level (0.24 and t = 6.18). Every control

variables but MC is significant when it is used alone to forecast returns. In model (8) where

all controls are incorporated, the average coefficient of SKGP is smaller in magnitude than

that in model (1) but still significant at the 1% level (0.11 and t = 3.87). Interestingly, the

average coefficient for MC is now significant at the 10% level and has the same negative sign

as that documented in the literature.

Next, as shown in panel B, the estimation results for SKEPS are very similar to those

for SKGP . By itself, SKEPS positively predicts stock returns in model (1). Even when all

controls are included in model (8), the average coefficient of SKEPS remains positive and

19

significant at the 5% level (0.09 and t = 2.37).

Finally, panel C shows the results for SKAF . The average coefficient of SKAF in model

(1) is 0.21 and significant at the 5% level (t = 2.60). Among the control variables, MOM ,

GP , and ROE are significant while MC, BM , and Idvol are insignificant in predicting stock

returns by themselves. In model (8), the average coefficient of SKAF is even larger and more

significant (0.28 and t = 3.31) than that in model (1).

Among the control variables, GP and ROE are consistently significant return predictors

in all three samples, confirming the evidence documented in Novy-Marx (2013) and Hou,

Xue, and Zhang (2014). In sum, the results of Fama-MacBeth regressions are consistent with

those of portfolio sorts. All three skewness measures of firm fundamentals positively predict

stock returns. While in the presence of control variables some evidence is not as significant

as in the portfolios sorts, the overall return predictability by the skewness measures cannot

be explained by other predictors.

4.5 Skewness and Future Firm Profitability

We now test the implication of our model that the skewness of firm fundamentals is positively

related to future profitability or growth of firm cash flows. We proxy growth rate by ROE

and GP . We look forward for four quarters. We use both portfolio sorts and Fama-MacBeth

regressions to examine the issue.

Table 5 reports the average equal-weighted future ROE and GP of decile portfolios

formed by sorting stocks on the skewness measures. Value-weighted results are very similar

and not reported for brevity. The results of panel A for SKGP indicate that high-skewness

stocks have higher profitability in future four quarters. The H-L spreads of both ROE and

GP are positive and significant at the 1% level for all four quarters. The H-L spreads decline

gradually as horizon increases, suggesting mean reversion in firm performance.

Panel B shows the results for SKEPS. The results are similar to those for SKGP as

stocks with high values of SKEPS have higher profitability in terms of ROE and GP in the

20

futures. There are a couple of small differences. First, the H-L spreads in ROE in panel

B are larger than those in panel A. Second, the H-L spreads in GP in panel B are smaller

than those in panel A. These results are not surprising as the skewness of earnings should

be more significant in predicting ROE than the skewness of GP while the opposite is true

for the skewness of GP .

The results for SKAF shown in panel C are in start contrast to those for SKGP and

SKEPS. Although the H-L spreads in ROE and GP are all positive and significant at the

10% level, they are much smaller than those in panels A and B. Looking closely, the positive

H-L spread are mostly driven by the high profitability of decile 10. There is no increasing

pattern in ROE or GP from decile 1 to decile 10. The results of the portfolio sorts strongly

support our argument for SKGP and SKEPS but the evidence for SKAF is weak.

We reexamine the above evidence using the Fama-MacBeth regressions and report the

regression estimates in Table 6. We only show the results where the dependent variable is

the next-quarter ROE and GP as the estimates for long horizons up to four quarter are

very similar. The regression results confirm what we have found with the portfolio sorts: the

skewness measures positively predicts future profitability. The results are robust to inclusion

of control variables. Interestingly, SKAF becomes more significant in predicting ROE and

GP when the control variables are included. Overall the evidence in this section confirms

our hypothesis that the skewness measures forecast future profitability.

4.6 Skewness and Future Growth Option

We have argued for a second channel through which the skewness of firm fundamentals is

positively related to expected stock return: skewness is positively related to the growth

option. We test this argument using two measures of growth option, MABA and Tobin’s q.

Again we conduct both portfolio sorts and Fama-MacBeth regressions.

Table 7 reports the average equal-weighted future MABA and Tobin’s q up to four

quarters of decile portfolios formed by sorting stocks on the skewness measures. The results

21

of portfolio sorts generally support our argument that higher skewness implies higher growth

option. The evidence is very strong for SKGP and SKEPS as the H-L spreads are all positive

and significant at the 1% level for all four future quarters and both growth option proxies.

The evidence for SKAF is weaker but is still consistent with that for SKGP and SKEPS.

The H-L spreads are much smaller than those for SKGP and SKEPS albeit still positive and

marginally significant in most cases, particularly in the short run.

In Table 8, we present the estimates of Fama-MacBeth regressions where the dependent

variable is the next-quarter MABA or Tobin’s q. The results for the two proxies of growth

options are very similar. When a skewness measure is the only predictor, its estimated

coefficient is positive. The estimates for SKGP and SKEPS are significant at the 1% level

while the estimate for SKAF is significant at the 10% level. When all control variables

including the lagged value of the growth option proxy are incorporated, the coefficients on

the skewness measures are much smaller and less significant. The estimates for SKGP and

SKEPS are still significant at the 10% level but the estimate for SKAF becomes insignificant.

Taken together with the evidence of portfolio sorts, these results are consistent with our

hypothesis that high skewness of firm fundamentals implies high growth option. But this

effect seems less significant than that on the growth rate of cash flows shown earlier.

4.7 Controlling for Skewness of Stock Returns

One concern about our main findings so far is whether the return predictability of the

skewness of firm fundamentals is related to the return predictability of the skewness of stock

returns. We address this concern by incorporating three popular return skewness measures

(Max, Idskew, and Prskew) in the Fama-MacBeth regressions of the fundamental skewness

measures. Table 9 reports the estimation results of the Fama-MacBeth regressions.

In models (1)–(3), we only use one of the three return skewness measures. MAX and

Prskew are significant but Idskew is insignificant in predicting returns. However, the sign

of average coefficient for MAX changes signs for different samples. Model (4) use all three

22

measures, Max, Idskew, and Prskew. MAX and Idskew are significant in the sample of

SKGP while Prskew is significant in the samples of SKEPS and SKAF . For our samples,

the return skewness measures do not appear to consistently predict stock returns.

We next combine the skewness of fundamentals with the return skewness measures. In

model (4), we only include one skewness measure of fundamentals. The average coefficients

of all three skewness measures of fundamentals are significant at the 1% level but among the

skewness measures of returns only MAX is significant at the 1% level for the SKGP sample

and Prskew is significant at the 10% level in the SKEPS and SKAF samples. Model (5)

augments model (4) by including all the control variables that we used in Table 3. MAX

and Prskew are still significant in some cases. Most importantly, the estimates for all three

skewness measures of fundamentals are significant at the 1% level even when all the control

variables are incorporated. The evidence of this section indicates that our findings for SKGP ,

SKEPS, and SKAF can not be explained by the skewness measures of stock returns.

4.8 Comparison of Alternative Skewness Measures

Given the different constructions of the skewness measures, it is interesting to investigate

their relative return predictive power. To do this, we estimate Fama-MacBeth regressions

with multiple skewness measures as explanatory variables. The estimation results are re-

ported in Table 10.

In models (1)–(3), we compare one skewness measure against another. Model (4) includes

all three measures. There is no control variables in models (1)–(4). We include all control

variables in model (5) together with all three skewness measures. The estimated coefficients

of the control variables are not shown for brevity.

In model (1), the average coefficients of SKGP and SKEPS are both positive (0.19 and

0.13) and significant at the 5% level, indicating that the two time-series skewness measures

do not dominate each other. In model (2), the average coefficient of SKAF is significant

but the average coefficient of SKEPS is insignificant. This suggests that for the sample of

23

stocks with valid SKAF , the return predictability of SKEPS is subsumed by that of SKAF .

In contrast, the estimates of model (3) show that both SKGP and SKAF are significant in

positively predicting returns when they are considered together. The estimates of model (4)

are consistent with those of the first three models as SKGP and SKAF are significant but

SKEPS is no longer significant. Finally, when all the control variables are incorporated in

model (5), the average coefficients of SKGP and SKAF remain significant at the 1% level

and the average coefficient of SKEPS is insignificant.

Despite the loss of significance in the joint regressions above, we should not dismiss the

return predictability of SKEPS. One reason is that the sample of stocks with valid SKAF

is much smaller than that for SKEPS. Furthermore, as seen in the estimates of model (1),

for the larger sample of stocks with valid SKGP and SKEPS, the predictability of SKEPS

is not subsumed by that of SKGP . In summary, the evidence suggests that the alternative

skewness measures capture different factors driving the firm cash flows.

5 Conclusions

We present a novel interpretation of sample skewness of a general time series as a proxy

of the growth rate of the underlying process. Applying this interpretation to the firm cash

flows together with the basic stock valuation equation, we posit a positive relation between

the skewness of firm fundamentals and expected stock return.

Using three skewness measures of firm fundamentals based on gross profitability, earnings,

and analysts’ earnings forecasts, we find strong evidence supporting our hypothesis. The

skewness measures positively predict not only cross-sectional stock returns but also future

firm performance in terms of growth rate and growth option. The evidence is robust to

control variables including the skewness of stock returns. The evidence also suggests that

the alternative skewness measures capture different factors of the underlying firm cash flow

process.

24

Because our findings are rooted in the basic stock valuation equation, we are, in the spirit

of Fama and French (2006, 2008), agnostic about whether the return predictability of the

skewness measures is rational or irrational. Given the strong evidence of skewness in firm

cash flows, our results highlight the importance of incorporating the skewness measures of

firm fundamentals in asset pricing research.

25

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Figure 1: Correlations of Sample Skewness and Changes of Sample ObservationsThis figure presents the plots of the correlations of estimated sample skewness and changesof sample observations. The data-generating process is xt = ρxt−1 + ut, where ρ ≤ 1 is aconstant and ut is an iid standard white noise process. The initial value x0 is set to be zero.In step 1, we independently generate N = 1, 000, 000 paths of xt. Denote the observations ofthe ith path by {xit}nt=0. In step 2, we compute, for the ith path, the sample skewness b̂i forthe ith path. In the last step 3, for each value of t = 2, ..., n, we compute the cross-sectionalcorrelation of b̂i and ∆xit and denote it by c(t). The four rows of the panels correspond ton = 8, 12, 16, and 20, respectively while the three columns of the panels correspond to ρ =0.9, 0.95, and 1, respectively.

2 4 6 8

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30

Figure 2: Correlations of Sample Skewness and Levels of Sample ObservationsThis figure presents the plots of the correlations of estimated sample skewness and levelsof sample observations. The data-generating process is xt = ρxt−1 + ut, where ρ ≤ 1 is aconstant and ut is an iid standard white noise process. The initial value x0 is set to be zero.In step 1, we independently generate N = 1, 000, 000 paths of xt. Denote the observations ofthe ith path by {xit}nt=0. In step 2, we compute, for the ith path, the sample skewness b̂i forthe ith path. In the last step 3, for each value of t = 2, ..., n, we compute the cross-sectionalcorrelation of b̂i and xit and denote it by C(t). The four rows of the panels correspond ton = 8, 12, 16, and 20, respectively while the three columns of the panels correspond to ρ =0.9, 0.95, and 1, respectively.

2 4 6 8−0.4

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t

C(t

)

n =8, ρ =0.95

2 4 6 8−0.4

−0.2

0

0.2

t

C(t

)

n =8, ρ =1

2 4 6 8 10 12−0.4

−0.2

0

0.2

t

C(t

)

n =12, ρ =0.9

2 4 6 8 10 12−0.4

−0.2

0

0.2

t

C(t

)

n =12, ρ =0.95

2 4 6 8 10 12−0.4

−0.2

0

0.2

t

C(t

)

n =12, ρ =1

2 4 6 8 10 12 14 16−0.4

−0.2

0

0.2

t

C(t

)

n =16, ρ =0.9

2 4 6 8 10 12 14 16−0.4

−0.2

0

0.2

t

C(t

)

n =16, ρ =0.95

2 4 6 8 10 12 14 16−0.4

−0.2

0

0.2

t

C(t

)

n =16, ρ =1

2 4 6 8 10 12 14 16 18 20−0.4

−0.2

0

0.2

t

C(t

)

n =20, ρ =0.9

2 4 6 8 10 12 14 16 18 20−0.4

−0.2

0

0.2

t

C(t

)

n =20, ρ =0.95

2 4 6 8 10 12 14 16 18 20−0.4

−0.2

0

0.2

t

C(t

)

n =20, ρ =1

31

Table 1: Correlation Matrix of Skewness Measures and Firm CharacteristicsThis table reports the average contemporaneous cross-section correlations of quarterly skew-ness measures and control variables. SKGP is the skewness of gross profitability, SKEPS

is the skewness of earnings per share, and SKAF is the skewness of analyst annual earningforecasts. MC is the market capitalization, BM is the book-to-market ratio, MOM is thecumulative return of month t−6 to t−1, GP is the gross profitability, ROE is the return onequity, Idvol is the idiosyncratic volatility, Rskew is the skewness of stock returns, and Dispis the dispersion of analysts’ earnings forecasts. The detailed definitions of the variables areshown in Section 3. The sample period is Q1, 1973 – Q4, 2013 for all variables except SKAF

and Disp for which the sample period is Q1, 1985 – Q4, 2013.

SKGP SKEPS SKAF MC BM MOM GP ROE IdvolSKGP 1SKEPS 0.31 1SKAF -0.01 -0.01 1MC 0.04 0.08 0.01 1BM -0.07 -0.13 0.00 -0.16 1

MOM 0.15 0.17 -0.01 0.10 -0.13 1GP 0.21 0.15 -0.01 -0.01 -0.11 0.12 1ROE 0.05 0.13 0.00 0.05 -0.04 0.13 0.15 1Idvol -0.02 -0.09 -0.01 -0.36 -0.04 -0.05 -0.04 -0.05 1

32

Table 2: Returns and Characteristics of Decile Portfolios Sorted on Skewness MeasuresThis table reports the average returns and characteristics of decile portfolios formed by sort-ing stocks on the skewness measures. Panels A–C are for SKGP , SKEPS, and SKAF , respec-tively. The decile portfolios are formed every quarter fromQ1, 1973 toQ4, 2013 for SKGP andSKEPS, and from Q1, 1985 to Q4, 2013 for SKAF . Portfolio 1(10) is the portfolio of stockswith the lowest(highest) skewness. EW(VW) Ret is the equal-weighted(value-weighted) av-erage quarterly portfolio return. EW(VW) α is the equal-weighted(value-weighted) averageabnormal return with respect to the Carhart (1997) 4-factor model. The row H-L reportsthe differences of returns and αs between decile 10 and decile 1. The corresponding Newey-West t-statistics are shown in the last row. Returns, αs, MOM , and Idvol are shown inpercentage. MC is in $ billion.

Panel A: SKGP

EW EW VW VWDecile Ret α Ret α SKGP MC BM MOM GP ROE Idvol

1 2.99 0.16 3.11 0.31 -3.63 5.60 0.91 9.39 0.07 0.01 11.062 3.10 0.61 3.14 0.67 -2.20 5.58 0.89 12.23 0.09 0.02 10.643 3.03 0.39 3.01 0.41 -1.39 5.51 0.94 13.08 0.09 0.02 10.604 3.40 0.43 3.43 0.51 -0.76 5.53 0.87 15.13 0.09 0.02 10.525 3.41 0.81 3.44 0.87 -0.20 5.54 0.86 15.95 0.10 0.03 10.486 3.84 0.94 3.82 0.97 0.29 5.54 0.89 19.03 0.10 0.03 10.487 4.38 1.33 4.27 1.30 0.79 5.56 0.88 20.03 0.10 0.03 10.438 4.11 1.48 4.01 1.45 1.38 5.60 0.87 23.57 0.11 0.03 10.409 4.17 1.67 4.07 1.64 2.15 5.59 0.78 26.67 0.11 0.03 10.5710 4.54 1.85 4.41 1.76 3.67 5.64 0.72 30.85 0.12 0.04 10.88H-L 1.55 1.69 1.30 1.45t-stat. 5.67 5.14 4.85 4.17

Panel B: SKEPS

EW EW VW VWDecile Ret α Ret α SKEPS MC BM MOM GP ROE Idvol

1 2.43 -0.27 2.62 -0.14 -3.45 5.19 0.97 6.14 0.06 -0.02 11.942 3.09 0.52 3.09 0.51 -1.93 5.34 0.95 10.02 0.06 0.01 11.073 3.01 0.51 3.00 0.54 -1.13 5.34 0.93 11.33 0.07 0.01 10.964 3.18 0.70 3.24 0.81 -0.51 5.37 0.91 12.87 0.07 0.02 10.685 3.22 0.79 3.23 0.86 0.01 5.39 0.89 14.56 0.08 0.02 10.596 3.37 0.98 3.39 1.01 0.47 5.39 0.87 17.09 0.08 0.03 10.437 3.58 1.26 3.51 1.26 0.95 5.49 0.81 19.10 0.08 0.04 10.388 3.64 1.16 3.57 1.19 1.53 5.56 0.78 21.49 0.09 0.04 10.289 3.96 1.56 3.81 1.49 2.30 5.60 0.73 23.88 0.10 0.05 10.3410 4.09 1.63 3.98 1.62 3.85 5.64 0.66 30.14 0.11 0.06 10.63H-L 1.66 1.89 1.36 1.75t-stat. 4.20 3.97 3.44 3.61

33

Table 2 – Continued

Panel C: SKAF

EW EW VW VWDecile Ret α Ret α SKAF MC BM MOM GP ROE Idvol

1 3.37 0.57 3.38 0.66 -2.23 6.83 0.59 19.02 0.10 0.03 10.062 3.32 0.49 3.30 0.55 -1.30 6.73 0.61 18.74 0.10 0.03 10.103 3.44 0.59 3.46 0.71 -0.76 6.82 0.61 18.98 0.10 0.03 9.984 3.58 0.80 3.63 0.93 -0.37 6.79 0.67 18.05 0.10 0.03 9.885 3.28 0.47 3.26 0.53 -0.03 6.78 0.62 16.41 0.10 0.03 9.886 3.01 0.25 3.00 0.33 0.28 6.80 0.62 16.26 0.10 0.02 9.887 3.58 0.87 3.56 0.93 0.61 6.85 0.61 15.34 0.10 0.02 9.898 3.66 0.95 3.63 0.99 1.00 6.81 0.62 14.92 0.10 0.02 9.949 3.88 1.23 3.87 1.29 1.50 6.71 0.61 15.40 0.10 0.02 10.0210 4.66 1.90 4.63 1.95 2.37 6.90 0.55 18.35 0.11 0.03 9.93H-L 1.30 1.33 1.26 1.28t-stat. 2.15 2.83 2.10 2.73

34

Table 3: Double Portfolio Sorts of Stocks on Skewness Measures and CharacteristicsThis table reports the equal-weighted average returns of portfolios formed by double sortingstocks on the skewness measures and characteristics. Panels A–C are for SKGP , SKEPS,and SKAF , respectively. For each firm characteristic, we first stocks into quintiles usingthe characteristic, and then within each quintile, we further sort stocks into quintiles basedon the skewness measure of interest. The row H-L shows the differences of average returnsbetween quintile 5 and quintile 1. The corresponding Newey-West t-statistics are shown inthe last row.

Panel A: SKGP

SKGP MC Quintile BM QuintileQuintile Low 2 3 4 High Low 2 3 4 HighLow 2.40 3.07 3.31 3.29 3.16 2.00 2.54 2.77 3.66 4.232 2.67 3.04 3.24 3.45 3.16 2.17 2.83 3.21 3.75 4.173 3.45 3.58 4.14 3.75 3.53 2.35 3.43 4.21 4.02 3.924 4.19 4.29 4.06 3.84 3.54 2.95 3.51 4.15 4.60 4.94

High 4.91 4.64 4.63 4.44 3.74 3.42 3.91 4.34 5.05 5.27H-L 2.51 1.56 1.32 1.14 0.58 1.42 1.37 1.58 1.39 1.04t-stat. 3.04 4.41 3.40 3.07 1.96 4.01 5.21 4.17 3.07 2.71SKGP MOM Quintile GP QuintileQuintile Low 2 3 4 High Low 2 3 4 HighLow 2.23 2.74 3.61 3.83 4.21 1.96 3.07 3.46 3.80 4.172 1.52 3.26 3.39 3.62 4.35 2.09 2.85 3.15 3.68 4.393 1.96 2.99 3.83 4.07 4.76 2.52 3.20 3.41 4.42 4.464 2.43 3.91 4.13 4.38 5.14 2.56 3.59 4.61 4.08 5.05

High 2.73 3.49 3.87 4.62 5.21 2.67 3.58 4.01 4.45 5.58H-L 0.49 0.74 0.26 0.80 1.00 0.71 0.51 0.56 0.65 1.42t-stat. 1.08 2.76 1.32 2.72 3.36 2.02 1.37 1.56 1.96 4.55SKGP ROE Quintile Idvol QuintileQuintile Low 2 3 4 High Low 2 3 4 HighLow 1.58 3.24 3.66 3.72 3.65 3.29 3.22 3.67 3.19 1.502 1.37 3.26 3.17 4.12 3.67 3.44 3.64 3.27 2.96 1.343 1.81 3.52 3.94 4.23 4.46 3.67 3.83 4.26 3.47 2.544 1.61 3.72 4.56 4.55 4.93 3.99 4.17 4.43 3.89 2.19

High 3.16 3.89 4.64 4.60 4.65 4.16 4.32 5.01 4.88 3.58H-L 1.58 0.65 0.98 0.88 1.00 0.87 1.09 1.34 1.69 2.07t-stat. 1.78 2.27 3.19 3.65 3.30 3.34 4.76 3.41 3.39 2.72

35

Table 3 – Continued

Panel B: SKEPS

SKEPS MC Quintile BM QuintileQuintile Low 2 3 4 High Low 2 3 4 HighLow 1.83 2.54 3.11 3.20 2.76 1.02 2.39 2.73 3.27 3.902 2.66 2.83 3.26 3.40 2.98 1.64 2.65 3.14 3.81 3.923 2.85 3.29 3.52 3.50 3.15 2.35 2.79 3.63 3.84 4.054 3.52 3.57 3.87 3.40 3.37 2.38 3.32 3.65 4.14 4.33

High 4.31 4.37 4.29 4.14 3.27 3.22 3.42 4.19 4.64 4.99H-L 2.49 1.82 1.18 0.94 0.51 2.20 1.03 1.46 1.36 1.09t-stat. 5.50 3.67 2.77 2.23 1.70 5.53 2.73 3.47 3.36 2.52SKEPS MOM Quintile GP QuintileQuintile Low 2 3 4 High Low 2 3 4 HighLow 1.72 3.09 3.17 3.40 4.28 1.31 2.77 3.02 4.04 4.002 1.81 2.91 3.33 3.66 4.51 1.79 2.64 3.37 3.65 4.333 2.01 3.05 3.29 3.81 4.33 2.15 2.77 3.28 3.31 4.404 1.78 3.13 3.82 4.22 4.31 2.46 2.90 3.29 3.80 4.61

High 1.84 3.12 3.66 4.05 5.06 2.70 3.02 3.39 3.98 4.95H-L 0.13 0.03 0.49 0.65 0.78 1.38 0.25 0.37 -0.06 0.96t-stat. 0.27 0.08 1.44 2.24 2.38 2.48 0.56 1.07 -0.18 2.64SKEPS ROE Quintile Idvol QuintileQuintile Low 2 3 4 High Low 2 3 4 HighLow 1.41 3.34 3.92 4.03 4.00 3.07 3.21 3.48 2.77 0.912 1.83 3.67 3.75 3.96 4.19 3.46 3.24 3.30 3.02 1.503 1.45 3.46 3.92 4.03 4.59 3.44 3.66 3.92 3.27 1.794 1.96 3.45 3.91 4.41 4.10 3.48 3.60 4.11 3.85 1.92

High 1.76 3.44 4.30 4.02 4.94 3.66 4.38 4.38 4.14 2.09H-L 0.35 0.11 0.39 -0.01 0.95 0.60 1.17 0.90 1.38 1.18t-stat. 0.78 0.02 1.31 -0.05 2.82 2.48 3.86 2.34 2.70 2.02

36

Table 3 – Continued

Panel C: SKAF

SKAF MC Quintile BM QuintileQuintile Low 2 3 4 High Low 2 3 4 HighLow 3.17 3.93 3.15 3.17 3.07 3.04 3.05 3.00 3.54 3.762 2.66 3.76 3.44 3.60 3.54 2.64 2.73 3.89 3.91 4.613 2.90 3.50 3.41 3.23 3.26 2.94 2.73 2.84 3.67 3.554 3.44 4.00 3.46 3.68 3.14 3.52 3.22 3.30 3.93 4.19

High 4.05 4.27 3.67 4.57 4.14 3.63 4.08 4.32 4.08 4.31H-L 0.88 0.34 0.51 1.41 1.08 0.59 1.03 1.32 0.53 0.55t-stat. 1.56 0.72 0.97 2.38 2.28 1.09 1.96 2.70 1.03 1.10SKAF MOM Quintile GP QuintileQuintile Low 2 3 4 High Low 2 3 4 HighLow 2.31 2.77 4.01 3.32 4.16 2.43 2.47 3.37 3.37 4.662 2.63 3.74 3.31 3.59 4.34 1.94 3.44 3.55 3.72 4.483 1.96 3.35 3.23 3.52 4.01 1.94 2.89 3.26 3.86 4.354 2.50 3.84 3.48 3.81 4.89 2.86 3.09 3.44 3.65 4.81

High 3.09 4.34 3.66 4.05 5.41 2.83 3.98 4.18 4.12 5.30H-L 0.77 1.57 -0.34 0.73 1.24 0.41 1.51 0.81 0.76 0.65t-stat. 1.42 3.93 -0.94 1.54 2.21 0.62 3.05 1.49 1.72 1.22SKAF ROE Quintile Idol QuintileQuintile Low 2 3 4 High Low 2 3 4 HighLow 0.28 2.71 3.61 4.21 4.99 3.57 3.36 3.51 2.68 2.342 0.18 3.40 4.05 4.58 5.00 3.48 3.83 3.57 3.79 2.373 -0.20 2.64 4.25 4.29 4.82 3.32 3.40 3.38 3.01 1.234 0.67 3.11 4.00 4.82 5.14 3.86 3.32 3.71 3.75 2.34

High 0.62 3.48 4.50 5.31 5.87 4.03 4.27 4.51 4.79 3.29H-L 0.34 0.77 0.89 1.10 0.88 0.45 0.90 0.99 2.11 0.95t-stat. 0.69 1.90 1.71 2.95 1.65 1.36 2.12 1.72 2.96 1.09

37

Table 4: Fama-MacBeth RegressionsThis table reports the average estimated coefficients and corresponding t-statistics of Fama-MacBeth regressions for the three skewness measures of firm fundamentals. Panels A–C arefor SKGP , SKEPS, and SKAF , respectively. *, **, and *** indicate statistical significance at10%, 5%, and 1% levels, respectively. The dependent variable of the regressions is the nextquarter stock return. For each of models (1)–(7), there is one independent variable. Model(8) includes all variables.

(1) (2) (3) (4) (5) (6) (7) (8)Panel A: SKGP

SKGP 0.24*** 0.11***(6.18) (3.87)

MC 0.23 -0.26*(0.90) (-1.76)

BM 0.93*** 1.42***(2.85) (4.32)

MOM 1.98*** 0.10***(3.01) (3.36)

GP 9.05*** 8.34***(4.01) (4.07)

ROE 5.80** 3.64**(2.36) (2.24)

Idvol -0.16*** -0.19***(-2.72) (-3.22)

Panel B: SKEPS

SKEPS 0.25*** 0.09**(4.73) (2.37)

MC 0.03 -0.26**(0.33) (-2.41)

BM 0.88*** 1.02***(2.74) (3.63)

MOM 2.03*** 0.92**(3.15) (2.45)

GP 10.24*** 8.57***(5.27) (5.75)

ROE 3.75*** 5.80***(3.36) (3.96)

Idvol -0.16*** -0.18***(-3.13) (-3.71)

38

Table 4 – Continued

(1) (2) (3) (4) (5) (6) (7) (8)Panel C: SKAF

SKAF 0.21** 0.28***(2.36) (3.31)

MC 0.05 -0.28(0.27) (-1.57)

BM 0.12 0.94**(0.23) (2.11)

MOM 1.47** 1.18*(1.98) (1.73)

GP 10.10*** 5.51**(4.24) (2.57)

ROE 7.44*** 8.86***(4.70) (4.78)

Idvol -0.15 -0.11(-1.62) (-1.36)

39

Table 5: Future Profitability of Decile Portfolios Sorted on Skewness MeasuresThis table reports the average equal-weighted future profitability, measured by ROE andGP , of decile portfolios formed by sorting stocks on the skewness measures. Panels A–C arefor SKGP , SKEPS, and SKAF , respectively. We consider four future quarters (t+1, ..., t+4).All numbers are reported in percentage. The decile portfolios are formed every quarter fromQ1, 1973 for SKGP and SKEPS, and from Q1, 1985 for SKAF . Portfolio 1(10) is the portfolioof stocks with the lowest(highest) earnings skewness. The row H-L reports the differences ofprofitability between decile 10 and decile 1. The corresponding Newey-West t-statistics areshown in the last row.

ROE GPDecile t+1 t+2 t+3 t+4 t+1 t+2 t+3 t+4

Panel A: SKGP

1 -0.06 -0.14 -0.18 -0.24 7.59 7.66 7.66 7.832 0.49 0.43 0.37 0.32 8.66 8.70 8.67 8.733 0.57 0.53 0.46 0.36 9.04 9.02 8.98 9.024 0.70 0.59 0.52 0.43 9.30 9.24 9.17 9.205 0.73 0.63 0.55 0.45 9.64 9.53 9.51 9.506 0.76 0.69 0.60 0.51 9.92 9.82 9.72 9.677 0.84 0.74 0.67 0.54 10.25 10.17 10.06 9.988 0.97 0.87 0.79 0.66 10.83 10.71 10.61 10.469 1.04 0.93 0.84 0.69 10.94 10.79 10.66 10.5710 1.14 1.03 0.92 0.80 11.34 11.12 10.91 10.75H-L 1.20 1.16 1.11 1.03 3.76 3.46 3.25 2.92t-stat. 11.54 10.34 9.08 8.26 18.87 17.42 18.48 14.94

Panel B: SKEPS

1 -0.70 -0.73 -0.74 -0.70 6.77 6.81 6.88 6.932 0.15 0.12 0.08 0.06 7.55 7.47 7.48 7.543 0.36 0.29 0.24 0.22 7.84 7.83 7.71 7.814 0.56 0.49 0.43 0.41 8.16 8.00 7.92 7.935 0.66 0.54 0.45 0.43 8.39 8.23 8.19 8.116 0.80 0.69 0.60 0.55 8.66 8.43 8.29 8.267 1.02 0.92 0.80 0.71 9.04 9.09 8.84 8.678 1.24 1.12 1.05 0.95 9.44 9.34 9.22 9.089 1.43 1.32 1.21 1.11 9.51 9.28 9.20 9.1110 1.85 1.72 1.57 1.43 9.73 9.47 9.26 9.16H-L 2.55 2.45 2.31 2.13 2.96 2.65 2.38 2.23t-stat. 11.75 11.44 10.68 9.71 4.42 3.73 3.55 3.14

40

Table 5 – Continued

ROE GPDecile t+1 t+2 t+3 t+4 t+1 t+2 t+3 t+4

Panel C: SKAF

1 0.83 0.73 0.62 0.73 9.99 9.82 9.79 9.732 0.70 0.62 0.60 0.59 9.78 9.70 9.68 9.643 0.78 0.69 0.64 0.60 9.72 9.59 9.56 9.524 0.72 0.76 0.65 0.56 9.54 9.49 9.48 9.455 0.70 0.61 0.64 0.62 9.49 9.47 9.40 9.346 0.70 0.65 0.58 0.52 9.46 9.45 9.35 9.297 0.71 0.72 0.62 0.64 9.46 9.31 9.39 9.378 0.68 0.62 0.57 0.60 9.31 9.34 9.34 9.279 0.71 0.66 0.65 0.56 9.58 9.61 9.51 9.5510 1.02 0.93 0.87 0.91 10.24 10.23 10.16 10.18H-L 0.19 0.20 0.25 0.18 0.25 0.41 0.36 0.45t-stat. 2.34 2.24 2.88 2.62 1.75 2.42 2.21 2.51

41

Table 6: Fama-MacBeth Regressions of Future ProfitabilityThis table reports the average estimated coefficients and corresponding t-statistics of Fama-MacBeth regressions of future profitability on skewness measures of firm fundamentals. Pan-els A and B consider ROE and GP , respectively. For each skewness measure, the first regres-sion only uses the skewness measure while the second regression includes all control variables.*, **, and *** indicate statistical significance at 10%, 5%, and 1% levels, respectively.

Panel A: ROESKGP MC BM MOM GP ROE Idvol

0.004***(15.17)0.002*** 0.002*** 0.061*** 0.009*** 0.123*** 0.061*** -0.001***(8.42) (8.97) (3.05) (14.70) (10.50) (10.48) (7.34)SKEPS

0.004***(17.86)0.002*** 0.001*** 0.001** 0.009*** 0.106*** 0.058*** -0.001***(14.31) (10.35) (2.20) (13.67) (9.70) (9.46) (-10.69)SKAF

0.001*(1.83)

0.001*** 0.001*** -0.002** 0.008*** 0.098*** 0.156*** -0.001***(3.32) (2.69) (-2.11) (9.29) (9.60) (9.40) (-10.43)

Panel B: GPSKGP MC BM MOM GP ROE Idvol

0.006***(21.78)0.001*** -0.001 -0.003*** 0.010*** 0.707*** 0.039*** -0.001***(6.63) (-0.91) (-8.03) (8.55) (28.03) (3.58) (-7.16)SKEPS

0.005***(3.14)0.001 -0.001*** -0.003*** 0.008*** 0.684*** 0.054*** -0.001***(0.82) (-4.86) (-9.84) (4.61) (19.56) (3.22) (-4.07)SKAF

0.001(0.37)0.001** -0.001*** -0.004*** 0.009*** 0.749*** 0.082*** -0.001(2.55) (-3.22) (-7.22) (9.52) (55.89) (5.54) (-0.91)

42

Table 7: Future Growth Option of Decile Portfolios Sorted on Skewness MeasuresThis table reports the average equal-weighted future growth option, measured byMABA andTobin’s q, of decile portfolios formed by sorting stocks on the skewness measures. PanelsA–C are for SKGP , SKEPS, and SKAF , respectively. We consider four future quarters(t + 1, ..., t + 4). All numbers are reported in percentage. The decile portfolios are formedevery quarter from Q1, 1973 for SKGP and SKEPS, and from Q1, 1985 for SKAF . Portfolio1(10) is the portfolio of stocks with the lowest(highest) earnings skewness. The row H-Lreports the differences of profitability between decile 10 and decile 1. The correspondingNewey-West t-statistics are shown in the last row.

MABA Tobin’s qDecile t+1 t+2 t+3 t+4 t+1 t+2 t+3 t+4

Panel A: SKGP

1 1.89 1.87 1.85 1.83 1.31 1.29 1.27 1.252 1.84 1.83 1.80 1.78 1.26 1.25 1.22 1.203 1.84 1.83 1.80 1.78 1.26 1.25 1.22 1.204 1.88 1.85 1.82 1.80 1.29 1.27 1.24 1.215 1.90 1.89 1.86 1.83 1.32 1.30 1.27 1.246 1.93 1.92 1.88 1.84 1.34 1.32 1.29 1.257 1.92 1.93 1.88 1.85 1.32 1.93 1.28 1.258 1.95 1.94 1.97 1.95 1.36 1.35 1.95 1.949 2.06 2.03 1.99 1.96 1.46 1.43 1.39 1.3610 2.30 2.27 2.22 2.17 1.72 1.69 1.63 1.59H-L 0.41 0.40 0.36 0.34 0.41 0.40 0.36 0.34t-stat. 8.51 8.35 8.07 7.66 7.88 7.74 7.50 7.15

Panel B: SKEPS

1 1.80 1.79 1.77 1.75 1.22 1.20 1.18 1.162 1.78 1.78 1.75 1.73 1.20 1.20 1.17 1.143 1.86 1.83 1.81 1.78 1.28 1.25 1.22 1.194 1.85 1.83 1.79 1.77 1.27 1.24 1.21 1.185 1.89 1.86 1.83 1.80 1.30 1.27 1.25 1.216 1.90 1.88 1.85 1.83 1.31 1.29 1.27 1.247 1.99 1.95 1.91 1.88 1.40 1.36 1.32 1.298 2.05 2.01 1.97 1.92 1.45 1.42 1.37 1.339 2.15 2.11 2.06 2.02 1.55 1.52 1.46 1.4310 2.41 2.34 2.27 2.19 1.82 1.76 1.68 1.61H-L 0.61 0.56 0.50 0.44 0.60 0.56 0.50 0.45t-stat. 10.22 9.57 9.33 9.21 9.72 9.69 9.25 8.79

43

Table 7 – Continued

MABA Tobin’s qDecile t+1 t+2 t+3 t+4 t+1 t+2 t+3 t+4

Panel C: SKAF

1 2.17 2.17 2.14 2.16 1.68 1.63 1.59 1.592 2.22 2.17 2.12 2.09 1.62 1.57 1.52 1.533 2.19 2.15 2.10 2.07 1.60 1.55 1.50 1.514 2.18 2.11 2.08 2.06 1.59 1.53 1.50 1.515 2.15 2.11 2.07 2.03 1.58 1.54 1.49 1.496 2.13 2.08 2.04 2.01 1.54 1.49 1.46 1.467 2.16 2.11 2.09 2.07 1.57 1.53 1.51 1.528 3.58 2.12 3.53 2.03 1.57 1.53 2.99 1.489 2.17 3.56 2.07 2.04 1.58 1.53 1.49 1.4910 2.31 2.23 2.19 2.17 1.76 1.69 1.64 1.60H-L 0.14 0.06 0.04 0.01 0.08 0.06 0.06 0.01t-stat. 2.35 1.94 1.79 0.36 2.76 2.31 2.02 0.95

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Table 8: Fama-MacBeth Regressions of Future Growth OptionThis table reports the average estimated coefficients and corresponding t-statistics of Fama-MacBeth regressions of future growth option on skewness measures of firm fundamentals.Panels A and B consider MABA and Tobin’s q, respectively. For each skewness measure,the first regression only uses the skewness measure while the second regression contains allcontrol variables, including the lagged value of the growth option proxy. *, **, and ***indicate statistical significance at 10%, 5%, and 1% levels, respectively.

Panel A: MABASKGP MC BM MOM GP ROE Idvol MABA

0.046***(11.17)0.009* 0.040*** -0.097*** 0.479*** 0.118 -0.170*** 0.002 0.804***(1.89) (3.03) (-4.36) (14.76) (1.32) (-3.80) (0.46) (31.78)SKEPS

0.078***(10.05)0.007* 0.032*** -0.059*** 0.474*** 0.082 -0.171*** 0.001 0.812***(1.73) (4.06) (-4.70) (13.23) (1.00) (-3.76) (0.33) (27.99)SKAF

0.039*(1.69)0.002 0.044*** -0.043*** 0.595*** 0.553*** -0.058 -0.002 0.801***(0.57) (2.77) (-4.53) (16.65) (3.12) (-1.65) (-0.69) (30.19)

Panel B: Tobin’s qSKGP MC BM MOM GP ROE Idvol Tobin’s q

0.044***(9.51)0.008* 0.042*** -0.092*** 0.474*** 0.129 -0.168*** 0.001 0.800***(1.88) (3.96) (-4.35) (14.35) (1.42) (-3.64) (0.30) (31.39)

SKEPS

0.076***(9.88)0.006* 0.039*** -0.057*** 0.469*** 0.003 -0.163*** 0.001 0.817***(1.72) (4.19) (-4.78) (12.72) (0.04) (-3.31) (0.32) (30.55)SKAF

0.040*(1.78)0.001 0.049*** -0.038*** 0.589*** 0.439*** -0.055 -0.002 0.796***(0.42) (2.95) (-3.39) (16.23) (2.73) (-1.48) (-0.96) (30.16)

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Table 9: Comparison of Skewness of Fundamentals and Skewness of Stock ReturnsThis table reports the average estimated coefficients and corresponding t-statistics of Fama-MacBeth regressions of future returns on skewness measures of firm fundamentals and stockreturns. The dependent variable of the regressions is the next quarter stock return. Thethree skewness measures of stock returns are Max, Idskew, and Prskew. Panels A–C arefor SKGP , SKEPS, and SKAF , respectively. Models (1)–(5) do not use any control variableswhile model (6) include all control variables in Table 3. The estimates for the controlvariables are not reported for brevity. *, **, and *** indicate statistical significance at 10%,5%, and 1% levels, respectively.

(1) (2) (3) (4) (5) (6)Panel A: SKGP

MAX -0.23*** -0.28*** -0.26*** 0.05(-6.09) (-6.23) (-5.64) (0.02)

Idskew 0.03 0.15** -0.11 0.04(0.26) (2.07) (-0.40) (0.68)

Prskew -0.81* 0.11 -0.54 -0.42(-1.69) (0.19) (-0.69) (-0.84)

SKGP 0.24*** 0.12***(6.05) (3.12)

Controls No No No No No YesPanel B: SKEPS

MAX 0.06*** 0.04 0.03 0.05*(2.61) (1.38) (1.26) (1.74)

Idskew 0.02 0.02 0.03 0.09(0.21) (0.20) (0.25) (1.28)

Prskew -0.73* -0.75* -0.65* -0.81*(-1.83) (-1.88) (-1.71) (-1.77)

SKEPS 0.25*** 0.09***(4.12) (2.88)

Controls No No No No No YesPanel C: SKAF

MAX 0.08** 0.03 0.02 0.06(2.19) (1.11) (0.93) (1.14)

Idskew 0.12 0.06 0.07 0.10(0.82) (0.42) (0.56) (0.88)

Prskew -1.68* -1.78* -1.70* -1.79**(-1.89) (-1.76) (-1.84) (-2.02)

SKAF 0.22*** 0.23***(3.02) (3.35)

Controls No No No No No Yes

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Table 10: Comparison of Return Predictability of Alternative Skewness MeasuresThis table reports the average estimated coefficients and corresponding t-statistics of Fama-MacBeth regressions with multiple skewness measures. The dependent variable of the re-gressions is the next quarter stock return. Models (1)–(4) do not use any control variableswhile Model (5) includes all control variables used in Table 3. The estimates for the controlvariables are not reported for brevity. *, **, and *** indicate statistical significance at 10%,5%, and 1% levels, respectively.

(1) (2) (3) (4) (5)SKGP 0.19*** 0.17*** 0.16*** 0.08***

(5.67) (3.10) (3.74) (2.42)SKEPS 0.13** 0.01 0.04 -0.04

(2.03) (0.13) (0.57) (-0.82)SKAF 0.22*** 0.16** 0.18** 0.25***

(2.86) (2.45) (2.46) (2.77)Controls No No No No Yes

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