Stock Crash and R2 around a Catastrophic Event:
Evidence from the Great East Japan Earthquake

Abstract: We investigate the eﬀects of opacity on stock price synchronicity, and frequency andseverity of a stock crash. We develop a simple analytical model for valuing stocks in the presenceof management earnings forecasts and market-wide information. Stock price synchronicity is
predicted to increase with opacity. Our model also reveals that stock crashes are more frequentand more severe for opaque ﬁrms. These predictions are conﬁrmed empirically. Further, ourmodel predicts that after a catastrophic event, stock price becomes synchronous with the market
and the frequency and severity of a stock crash increase because of severely limited investor
attention. We use the Great East Japan Earthquake as a representative catastrophe to examine
these implications, and provide support for the model. Finally, we ﬁnd that the stock price of aﬁrm disclosing opaque information tends to covariate with the market, and hence suﬀers more
serious damage from the earthquake, which is consistent with the model.
Keywords: Financial reporting opaqueness, Stock price synchronicity, Stock crash, the Great
East Japan Earthquake, Limited attention
JEL classification: G12, G14, M41

1Introduction

We examine the eﬀects of opacity to investors, in the sense that they cannot understand what
a ﬁrm intrinsic value really is, on stock price synchronicity and crash risk. Since Roll (1988)
found that stock prices mostly do not co-move with the whole market and argued that informed
traders acting on private information play an important role, there has been considerable researchconcerning the determination of stock price synchronicity. Morck, Yeung and Yu (2000) showthat stock returns are more synchronous in poor economies than in rich economies. They statethat the presence of public investor property rights causes this diﬀerence.
Jin and Myers (2006) argue that it is not enough that the reason for high R2 s can be
attributed to just poor protection of investors. They investigate the relation between R2 andopaqueness (lack of transparency) for 40 stock markets from 1990 to 2001 and ﬁnd theoreticaland empirical evidence that more opaque markets have higher R2 s, leading to their being more
likely to crash. Based on a model by Jin and Myers (2006), Hutton, Marcus and Tehranian(2009) test the ﬁrm-level relation between R2 and opaqueness using the U.S. ﬁrms from 1991 to
2005. Using abnormal accruals (i.e., earnings management) as a proxy for opacity, they ﬁnd that
more opaque ﬁrms have higher R2 s and tend to experience stock crashes. This result suggeststhat R2 s and crash risks are driven by accounting opaqueness.
Researchers in accounting and ﬁnance have used various kinds of measures to indicate accounting opaqueness. For example, Bhattacharya, Daouk and Welker (2003) use earnings aggressiveness (i.e., the opposite concept of accounting conservatism), loss avoidance, and earnings
smoothness as proxies for diﬀerences in accounting opacity and explain the association between
accounting opacity and cost of capital and trading volume world-wide. Francis, LaFond, Olsson
and Schipper (2004) examine the link between the cost of equity capital and accounting opacity
by using the following earnings attributes: accruals quality proposed by Dechow and Dichev(2002), persistent, predictability, smoothness, value relevance, timeliness, and conservatism.
Thus, empirical research so far has basically used reported earnings to quantify accounting opacity1 .
Although we also use reported earnings-based opacity measures and examine the relation
between opacity, R2 , and crash risk, those measures have potential weaknesses. First, as Hutton
et al. (2009) point out, the relation between discretionary accruals and both R2 and crash risk
essentially disappears in the post-Sarbanes-Oxley Act (SOX) years. They interpret this result as
1

Dechow, Ge and Schrand (2010) provide comprehensive review of accounting earnings-based accounting qual-

ity literature.

1

indicating that earnings management has substantially subsided due to greater monitoring andscrutiny of accounting practices after SOX, and then, earnings-based opacity measures could nolonger be used as a proxy for opaqueness after SOX. Because many developed countries including Japan have recently crafted regulations similar to SOX, we need to employ an alternative
measure of opaqueness to replace it.
Japanese listed ﬁrms have a distinctive ﬁnancial reporting system in that they report actual
earnings for each year. In addition, almost all of them release point-estimate of management
earnings forecasts for each following year. Utilizing this unique setting, we use the precisionof management earnings forecasts as the opacity measure. It is reasonable that regardless ofwhether the causes of less precise earnings forecasts are intentional biases or major changes inthe business environment, ﬁrms reporting less precise earnings forecasts are regarded as being
more opaque by investors, who form their expectations on the earnings forecasts, especially inuncertain situations.
We ﬁrst develop a simple approach to valuing stocks in the presence of management earnings
forecasts and market-wide information under “usual” conditions. Our model predicts that (1)stock crashes are likely to be observed in ﬁrms whose management reports opaque forecastsand (2) stock price synchronicity increases with management-forecast-based opacity. These
predictions are conﬁrmed empirically.
We next develop a model of describing investor attention and allocation under “unusual”
conditions in which investor attention is severely limited due to an unexpected catastrophic
shock, and provide predictions about the impact of the limited attention on stock return comovement and individual stock crashes. This model implies that after a catastrophe, (3) stock
prices become more synchronous and (4) crash risks become higher in the market. In addition, our model predicts that (5) opaque ﬁrms are more prone to stock price collapse aroundan unexpected catastrophe. We use the Great East Japan Earthquake (GEJE) as a representative catastrophe to examine these implications from (3) to (5), and provide evidence that is
consistent with our model.
This study contributes to the extant literature in several ways. First, while Jin and Myers(2006) developed an analytical model that explains that opaqueness increases stock price synchronicity by shifting ﬁrm-speciﬁc risk to managers, they show the relation between ﬁrm-level
opaqueness and stock crashes anecdotally. They nearly describe that a greater frequency of large,
negative, ﬁrm-speciﬁc returns occurs in countries where ﬁrms are more opaque to investors. We
developed a simple model that explains the relation between ﬁrm-level opaqueness and its stock2

crash risk, and then, show empirical evidence that the more opaque the ﬁrms, the more prone
they are to crash.
Second, recent years have seen an increase in studies investigating the determinants of crashrisk (e.g., Chen et al., 2001; Hutton et al., 2009; Kim et al., 2011a,b; An and Zhang, 2013).
This research provides surprising results that larger ﬁrms are more crash-prone than smallerﬁrms. This seems counter intuitive and casts doubt on the ability of the measures used in priorliterature to accurate stock crash risk. We regard the frequency with which weekly returns
exceed a certain threshold and severity of stock crashes (i.e., minimum weekly returns) during
the year as a measure of crash risk. We show that our crash risk measures are intuitively
plausible. For example, larger and more proﬁtable ﬁrms are not likely to be experience stock
crash. In contrast to previous studies, we use such intuitively plausible measures and provide
additional empirical evidence to the growing literature explaining the relation between opacity
and crash risks.
Third, Peng and Xiong (2006) developed a model showing the learning process of a representative investor who has limited attention. They view this representative investor as one ofmany retail investors in the stock market and show the eﬀect of limited attention on return comovement by some investors in the market. Unlike Peng and Xiong (2006), we theoretically andempirically show that when all investors regardless whether they have philological bias or not,face limited attention due to exogenous shock, stock prices become synchronous with the market
and the frequency and severity of individual stock crashes become higher. We believe that thispaper has implications for situations in which any exogenous shock, such as an earthquake, atsunami, or exogenous ﬁnancial crisis occurs.
Finally, our study contributes to growing literature on the linkage between unexpected shocks
and ﬁnancial reporting. For example, Francis et al. (2013) and Watts and Zuo (2012) show thatﬁrms with more conservative ﬁnancial reporting experience less negative “long-run” stock returns
during ﬁnancial crises. However, there is little evidence documenting what factors aﬀect “shortrun” stock returns when unexpected shocks actually happen. We develop the theory that stock
price synchronicity is a key factor explaining cross-sectional diﬀerence in stock returns in theevent of shocks, and then that synchronicity is positively associated with opacity. As a result,
we predict that opaque ﬁrms tend to have large negative returns after the shock has occurred.
Ample support for this prediction is found in the data relating to a representative unexpected
shock, the GEJE. This study demonstrates the importance of transparency when unexpected
shocks have occurred.

3

The paper proceeds as follows. The next section develops theoretical predictions. In Section
3, we present the research design and data description. Section 4 reports the empirical results
using data from Japan. Finally, Section 5 presents the summary and conclusion.
2

Model and Hypotheses

Our model is partly based on Peng and Xiong (2006). Their model focused on investors’ limited
attention and overconﬁdence, and concluded that these two factors increase the return comovement (i.e., high synchronicity). We focus on opacity, investors’ limited attention, crash risk, andsynchronicity. First, we clarify what eﬀects the opacity of ﬁnancial information has on the crashrisk and the synchronicity by using the Bayes rule. Second, we clarify what eﬀects investors’
limited attention has on the synchronicity by revised Peng and Xiong (2006) model. Third, weclarify that the stock price of less opaque ﬁrms is less synchronous and less likely to crash even
where investors’ attention is limited.
2.1

The eﬀects of opacity on crash risk and synchronicity

We assume that all investors are risk-averse. Let θi be the value of ﬁrm i. Investors do not know
θi , but know its probability distribution θ̃i . We assume that θ̃i has a normal distribution with2 .
a mean of µ0,i and a variance of σ0,i2
θ˜i ∼ N(µ0,i , σ0,i
) (i = 1, 2, . . . , n)

The probability density function of θ̃i is}{(θi − µ0,i )21p(θi ) = √
exp −
2
2σ0,i
2πσ0,i

The investor updates her belief about θ̃i based on ﬁrm speciﬁc-information ỹi and market
information m̃.
The manager knows θi and adds his intentional bias ε̃i to it to make noisy speciﬁc-informationỹi . We assume that ε̃i has a normal distribution with a mean of 0 and a variance of σi2 , andthat the investor knows the distribution.
ỹi = θi + ε̃i , ε̃i ∼ N(0, σi2 )(i = 1, 2, . . . , n)Hence ỹi has a a normal distribution with a mean of θi and a variance of σi2 . We further
assume that ε̃i and ε̃j are independent across diﬀerent ﬁrms (i 6= j).
ỹi ∼ N(θi , σi2 ) ,4ε̃i ⊥ ε̃j ( i 6= j)

The conditional probability density function of ỹi given θi is{}1(yi − θi )2p(yi | θi ) = √
exp −
2σi2
2πσi
Assuming that the market-wide information m̃ consists of the sum of the ﬁrms’ value in themarket and the error term ε̃m . We assume that ε̃m has a normal distribution with a mean of 0
2 , and that the investor knows the distribution.
and a variance of σm

m̃ =

n
∑

θi + ε̃m = θi +

i=1

∑

2
θj + ε̃m , ε̃m ∼ N(0, σm)

j6=iHence m̃ has a normal distribution with a mean of∑n2i=1 θi and a variance of σm . Assumingthat ε̃m and ε̃i are independent (i = 1, 2, . . . , n), m̃ and ỹi are independent.
2m̃ ∼ N(θi + θ−i , σm
)

where

θ−i ≡

∑

θj

j6=i

ε̃m ⊥ ε̃i , m̃ ⊥ ỹi(i = 1, 2, . . . , n)

The conditional probability density function of m̃ given θi is{}1(m − (θi + θ−i ))2p(m | θi ) = √
exp −
2
2σm
2πσm
Since ỹi and m̃ are independent. The joint conditional probability density function of ỹi andm̃ isp(yi , m | θi ) = p(yi | θi )p(m | θi )
{
[
]}
(
)2
1
1
1 (yi − θi )2 (m − (θi + θ−i ))2
exp −
+
= √
2
σi σ m2σmσi22πApplying the Bayes rule, the conditional probability density function of θei given the ﬁrmspeciﬁc information yi and the market-wide information m isp(θi )p(yi , m | θi )−∞ p(θi )p(yi , m | θi )dθi{}
1
∝ exp − ( 1 ) (θ − B)22 Ap(θi | yi , m) = ∫ ∞1
1
1+ 2 + 22σm σ0,iσi)(1m − θ−i µ0,i
yi
B= 1
+
.
+ 2
2
σm
σi2
σ0,i
+ σ12 + σ12
σ2where A =

i

m

0,i

5

Appendix 1 provides the proof. Hence,(θ̃i | yi , m ∼ N wy,i yi + wm,i (m − θ−i ) + w0,i µ0,i ,

1
py,i + pm + p0,i

)
(1)

py,i
p0,i
pm
, wm,i =, w0,i =py,i + pm + p0,i
py,i + pm + p0,i
py,i + pm + p0,i111py,i = 2 , pm = 2 , p0,i = 2
σm
σi
σ0,i

where wy,i =where py,i , pm and p0,i are respectively the precision of the ﬁrm speciﬁc information, themarket-wide information and the investor’s prior belief, and py,i + pm + p0,i is the precision of
investors’ posterior belief about the ﬁrm value given the ﬁrm speciﬁc information yi and themarket-wide information m.
From Equation (1), the lower the precision of the ﬁrm speciﬁc information is, the higher the()
conditional variance of investors’ posterior belief Var θ̃i | yi , m = py,i +p1m +p0,i is. That means
higher crash risk. Therefore we have the ﬁrst hypothesis.
Hypothesis 1. The higher the opacity of information that a ﬁrm discloses, the higher the crashrisk is.
()
Focusing on Var θ̃i | yi , m = py,i +p1m +p0,i , the lower precision of ﬁrm-speciﬁc information
yi means that it has a weaker impact on, and the market-wide information m has a stronger()
impact on the conditional variance Var θ̃i | yi , m . If the ﬁrm-speciﬁc information is completely
unreliable (pyi → 0), the conditional variance is explained only by the precision of market()
wide information and the investor’s prior belief, and the conditional expectation E θ̃i | yi , mconverges to a value that the ﬁrm-speciﬁc information does not have any impacts on.
()
E θ̃i | yi , m =

p0,i
pm
1
(m − θ−i ) +µ0,i , Var(θ̃i | yi , m) =pm + p0,i
pm + p0,i
pm + p0,iTaken together, concerning a ﬁrm which discloses opaque information, the investor’s expectation about the ﬁrm value becomes highly covariated with the market-wide information. Hencethe stock price has higher synchronicity. Therefore we have the second hypothesis.
Hypothesis 2. The more opaque information a ﬁrm discloses, the more synchronous the stock
price is.
2.2

The eﬀects of catastrophe on crash risk and synchronicity

In this section, we investigate an investor’s decision making under the situation after a catastrophic event. The time-line is as follows. First ﬁrm values θi (i = 1, 2, . . . , n) that an investor
6does not know are realized. Second the investor forms a portfolio based on the prior belief about
the ﬁrm value θ̃i , the ﬁrm-speciﬁc information ỹi (i = 1, 2, . . . , n), and the market-wide information m̃. Third the ﬁrm-speciﬁc information yi (i = 1, 2, . . . , n) and the market-wide information
m are realized. Finally the investor liquidates her portfolio.
Since from Equation (1), the posterior conditional expectation given yi and m is()
E θ̃i | yi , m = w0,i µ0,i + wy,i yi + wm,i (m − θ−i ) ,the prior conditional expectation can be written as()
E θ̃i | ỹi , m̃ = w0,i µ0,i + wy,i ỹi + wm,i (m̃ − θ−i ) ≡ αi + ci ỹi + ki m̃
αi = w0,i µ0,i − wm,i θ−ici = wy,iki = wm,iwhere ki is the impact of the market-wide information m̃ on, and ci is the impact of the ﬁrmspeciﬁc information ỹi on the prior conditional expectation of the investor.
∑
Let {w1, w2 , . . . , wn } be the investor’s portfolio( ni=1 wi = 1). The variance V of her portfolioisV ≡ Var

=

( n
∑

( n
∑

)
wi E(θ̃i | ỹi , m̃)

i=1

i=1

wi (αi + ki m̃ + ci y˜i )

i=1

)2

wi ki

= Var

)

( n
∑

Var(m̃) +n∑

wi2 c2i Var(ỹi ) =

( n
∑

i=1

= Var m̃

n
∑
i=1

)2
wi ki

(

2
σm
+

i=1

n
∑

wi ki +n∑)wi ci ỹii=1wi2 c2i σi2

i=1

2 = Var(m̃) and σ 2 = Var(ỹ ).
where σmii

Since the investor is risk-averse, her problem is to minimize the variance of her portfolio’s
2 and σ 2 (i = 1, 2, . . . , n)value given a certain level of the expected value. When she knows σmiand has the ability enough to process all the available information immediately, her problem
can be written as
min V =

( n∑wi(s.t.)

i=1

n
∑

)2
wi ki

2
σm
+

n
∑

wi2 c2i σi2

i=1

(

)
wi E E(θ̃i | ỹi , m̃) = const.

i=1

However, consider a situation after a catastrophic shock(e.g., a big earthquake, a tsunami,etc). She must make her decision quickly to avoid a big loss. Her attention is severely limited.
2 and σ 2 (i = 1, 2, . . . , n) that she
In this situation, she cannot get the exact information of σmi

7would have in an ordinary situation, and has to estimate the distributions of m̃ and y˜i and get
2
2 , respectively.
the subjective variances σs,mand σs,iAssuming that an investor believes that the more attention she pays to the market or the2
speciﬁc ﬁrms, the more exact information she can get about m̃ and ỹi (i.e., the less σs,mand
2 are).
σs,i2
2 . Concerning an event E that occurs
We use an information theory to formalize σs,mand σs,iwith probability p, the information value I(E) that E occurred is deﬁnes as2
I(E) = −a log2 p(a > 0).

(2)

The message of Equation (2) is that the more probability p that an event E occurs, the lessinformation value I(E) that it occurred.
2 is right (σ 2 = σ 2 ), p the probability
Let E be an event that an investor estimation about σs,is,ii2 = σ 2 , and I(E) the information value that σ 2 = σ 2 . Let λ be the amount of thethat σs,i
i
i
s,i
i

investor’s attention to ﬁrm i divided by the total amount of her attention. The more attention
she pays to ﬁrm i, the higher p is. Hence, we use λi as a proxy for p. On the other hand, the2 means that the investor believes the uncertainty of ỹ is small. In that case, even if
small σs,ii

the event E occurs, the investor does not regard the information that E occurred as valuable.
2 as a proxy ofThat is, the information value I(E) is small for the investor. Hence we use σs,iI(E). Further we use a natural logarithm for simplicity of calculation instead of a logarithm
whose base is two.
2
σs,i
= −ai ln λi(ai > 0, i = 1, 2, . . . , n)2
Similarly we specify the subjective variance σs,m
as
2
σs,m
= −am ln λm(am > 0)where λm is the amount of the attention to the market divided by the total amount of her
attention. She estimates ai and am based on the precision of the ﬁrm speciﬁc information andthe market-wide information, respectively, that she got before.
2 depending upon λ when a = 1 andFigure 1 illustrates the curves of subjective variance σs,ii
i

2. Given a certain level of her attention λi , the smaller ai is, the smaller the subjective variance
2 is. This means that a ﬁrm with smaller (larger) a discloses more transparent (opaque)σs,ii
2 can be illustrated as well.
information3 . The curve of σm
2
3

The first seminal paper is Shannon (1948).
Similarly, as to the market-wide information, the smaller am is, the more precise the market-wide informationis.

8

2 are decreasing in λ and λ respectively, there exist certain levels of her
2and σs,i
Since σs,mmi
2 (Λ∗ ) = σ 2 and σ 2 (Λ∗ ) = σ 2 respectively. Since she does notattention Λ∗m and Λ∗i where σs,m
m
m
s,i
i
i
2 and σ 2 , she does not know Λ∗ and Λ∗ . In such situation, she makes decision by twoknow σmmiisteps. The ﬁrst step is to decide how she distributes her attention to the market and the speciﬁc
ﬁrms (to decide [λm , λ1 , · · · , λn ]). The second step is to decide how she distributes her wealthto the speciﬁc ﬁrms (to decide [w1 , · · · , wn ]). Hence her problem can be written as()
minmin Vswi

where Vs =(s.t.)n∑( n∑λm ,λi

)2

n
∑

2
σs,m
+

wi ki

i=1

(

2
wi2 c2i σs,i

i=1

n
)
∑
wi E E(θ̃i | ỹi , m̃) = const , 0 < λi < 1, 0 < λm < 1, λm +λi = 1.

i=1

i=1

Vs can be regarded as a subjective variance of her portfolio.
The ﬁrst step of her decision is to solve
min Vs =λm ,λi

=−

( n
∑

( n
∑

)2
2
σs,m
+

wi ki

i=1
)2

wi ki

n
∑

2
wi2 c2i σs,ii=1

am ln λm −

n
∑

i=1

wi2 c2i ai ln λi

i=1(s.t.) 0 < λm < 1 , 0 < λi < 1 , λm +n∑λi = 1.

i=1

We use the Lagrangian multiplier method.
L=−

( n
∑

)2
wi ki

am ln λm −i=1
∑n
( i=1 wi ki )2 amn∑(wi2 c2i ai ln λi − ν

i=1

1 − λm −

n
∑

)
λi

i=1

∂L
=−
+ν =0
∂λm
λm
∂L
w2 c2 ai
= − i i + ν = 0 (i = 1, 2, . . . , n)
∂λi
λi
n
∑
∂L
= 1 − λm −λi = 0∂ν
i=1From the three equations above,∑( ni=1 wi ki )2 am
wn2 c2n an
w12 c21 a1
=
·
·
·
=
=
≡ κ∗
λ∗m
λ∗1
λ∗n
∑
λ∗m +
λ∗i = 19

(κ∗ 6= 0)(3)

The solutions are
λ∗m =

(

∑n

2

w12 c21 a1
w2 c2 an
, · · · , λ∗n = n n∗
∗
κ
κ
( n
)2
n
∑
∑
where κ∗ =wi ki am +wi2 ci ai .
i=1 wi ki )κ∗am, λ∗1 =

i=1

(4)

i=1

Since 0 < λ∗m < 1 and 0 < λ∗i < 1, λ∗m and λ∗i are the interior solutions and are the functionsof wi (i = 1, · · · , n).
λ∗m = λ∗m (w1 , · · · , wn ), λ∗1 = λ∗1 (w1 , · · · , wn ), · · · , λ∗n = λ∗n (w1 , · · · , wn )From Equation (3), we can derive∑∑n2 2( ni=1 wi ki )2 am
i=1 wi ci ai
∑
.
=
n
∗
λ∗m
i=1 λi(5)

Appendix 2 provides the proof.
For suﬃcient large n,
( n
∑

)2
wi ki

am >

i=1

n
∑

wi2 c2i ai

(6)

i=1

Appendix 3 provides the proof4 .
From Equation (5) and (6), we have
λ∗m >

n
∑

λ∗i

(7)

i=1

Equation (7) means that the investor pays more attention to the market-wide information
than to all the ﬁrm-speciﬁc information disclosed by the ﬁrms in the market. That leads tohigher synchronicity in their stock prices. Therefore we have the third hypothesis.
Hypothesis 3. After a catastrophe (e.g., an big earthquake, a tsunami) where investors’ attention is severely limited, the stock prices become synchronous.
Since soon after a catastrophe bad news is pervasive over the market , we also hypothesizethat the crash risk becomes higher where investors’ attention is severely limited.
4

We assume that the more firms are there in the market, the smaller portion of her wealth is distributed tofirm i (i = 1, 2, . . . , n). That is∀i ∈ {i = 1, 2, . . . , n}, lim wi = 0.
n→∞10Hypothesis 4. After a catastrophe (e.g., an big earthquake, a tsunami) where investor attention
is severely limited, the crash risk becomes higher in the market.
The more ﬁrms are there are in the market, the more likely it is that Equation (6) holds.
Appendix 5 provides the proof. Therefore we have a corollary.
Corollary 1. The more ﬁrms there are in the market, the more attention an investor pays tothe market-wide information than the ﬁrm speciﬁc information.
2.3

The eﬀects of transparency on crash risk after a catastrophic shock

The second step of an investor’s decision is to solve
( n
)2
n
∑
∑
min Vs = −wi ki am ln λ∗m (w1 , · · · , wn ) −wi2 c2i ai ln λ∗i (w1 , · · · , wn )wi

i=1

(s.t.)

n∑wi E(θ̃i | y, m) = const ,

i=1
n
∑

i=1

wi = 1.

i=1Since λ∗m and λ∗i (i = 1, 2, . . . , n) functions of wi (i = 1, 2, . . . , n), the solution of her problem
can be written as[w1∗ , · · · , wn∗ , λ∗m (w1∗ , · · · , wn∗ ), λ∗1 (w1∗ , · · · , wn∗ ), · · · , λ∗n (w1∗ , · · · , wn∗ )] .
Now we compare synchronicity between a transparent ﬁrm and an opaque ﬁrm. Let ﬁrm 1
be the transparent ﬁrm (a1 = 1) and ﬁrm 2 be the opaque ﬁrm (a2 = 2).
2σs,i= −ai ln λi(i = 1, 2 ; a1 = 1, a2 = 2)

For simplicity, we assume that the investor’s unknown variances are the same between thetwo ﬁrms (σ12 = σ22 ≡ σ 2 ). Let Λi be the amount of her attention to ﬁrm i where the subjective
2 (Λ ) = σ 2 (i = 1, 2)). Since she does not know σ 2 ,variance is equal to the unknown variance (σs,iishe does not know Λi (i = 1, 2). Figure 2 shows the curves of the two ﬁrms and contrasts themagnitude relations of the widths between [0, Λ1 ] and [0, Λ2 ], and between [Λ1 , 1] and [Λ2 , 1]. In
Figure2, the curve of ﬁrm2 is located above that of ﬁrm 1.
Since the range of [Λ1 , 1] is wider than that of [Λ2 , 1], it is more likely that Λ1 < λ∗1 (w1∗ , · · · , wn∗ ) <1 than that Λ2 < λ∗2 (w1∗ , · · · , wn∗ ) < 1. On the other hand, since the range of [0, Λ2 ] is wider thanthat of [0, Λ1 ], it is more likely that 0 < λ∗2 (w1∗ , · · · , wn∗ ) < Λ2 than that 0 < λ∗1 (w1∗ , · · · , wn∗ ) < Λ1 .
In the case that Λi < λ∗i (w1∗ , · · · , wn∗ ) < 1, the investor pays more attention than is required
2 (Λ ) with σ 2 . It can be considered that all the speciﬁc information about ﬁrm
to equalize σs,ii
i

11

i is reﬂected in her decision since she pays more attention to ﬁrm i than Λi . Even though theamount of attention paid to the ﬁrm is small, as far as she pays more attention than Λi , herbelief about the ﬁrm value does not tend to covariate with the market. Hence, the stock priceof the ﬁrm is less synchronous and less likely to crash even after a catastrophic shock.
In the case that 0 < λ∗i (w1∗ , · · · , wn∗ ) < Λi , the investor pays less attention than is required to
2 (Λ ) with σ 2 . She could get more ﬁrm speciﬁc information by paying more attention
equalize σs,ii
i

to ﬁrm i. However she pays more attention to the market since the cost of paying attention to
ﬁrm i, ai , is high. Hence her belief about the ﬁrm value tends to covariate with the market,
and the stock price of the ﬁrm is more synchronous and more likely to crash after a catastrophic
shock. Therefore we have the ﬁfth hypothesis.
Hypothesis 5. The stock price of a ﬁrm that discloses opaque information is more likely tocovariate with the market, and hence is more likely to crash after a catastrophe.
3

Data, Variable Measurement, and Research Design

3.1

Measuring ﬁnancial reporting opacity using management earnings forecasts

Firms listed on the Tokyo Stock Exchange (TSE) have a unique practice of annual earnings
announcements. The TSE requires listed ﬁrms to report not only the actual earnings of eachyear and the current year but also the management earnings forecasts for the following year
at the earnings announcements so as to provide information that is useful in decision making.
Furthermore, these forecasts are released in the form of point estimates, because ﬁrms arerecommended not to issue range or qualitative estimates (e.g., TSE, 2006). Actually, almost
all ﬁrms report point estimates of the management earnings forecasts for the following year inaccordance with the TSE requirement. This fact implies that disclosure of management earnings
forecasts are eﬀectively mandated in Japan.
In Japan, market expectations substantially depend on earnings guidance by management
who are the most familiar with future prospects of ﬁrms. This is because a competing source of
earnings expectations, analyst forecasts, are not suﬃciently provided for a wide range of ﬁrms—
about two thirds of ﬁrms have zero coverage, and management earnings forecasts dominate, on
average, analysts forecast in accuracy for ﬁrms with coverage. In this situation, ﬁrms that areprovided forecasts with lower accuracy are more opaque to investors in the sense that they cannot
see what a ﬁrm’s intrinsic value really is. Since most ﬁrms provide management earnings forecasts
in Japan, management forecasts are generally accurate work well as a proxy for explaining the12cross-sectional diﬀerence of opacity.
We measure management forecast accuracy (ACCURACY ) for each ﬁrm-year observation
based on the absolute value of the diﬀerence between the initial management earnings forecastof period t made at the earnings announcement of period t − 1 and the actual current earnings,
deﬂated by the market value of equity as of the end of ﬁscal year t − 1:
ACCURACY j,t =f et−1j,t − ej,t
MVE j,t−1.

where ACCURACY j,t is management forecast accuracy for ﬁrm j of period t, f et−1
is thetmanagement forecast of period t earnings made at the earnings announcement of period t − 1,
et is the actual earnings in period t, and MVE t−1 is the market value of equity at the end ofﬁscal year t − 1.
We deﬁne our opacity measure using management forecasts (MF OPACITY ) as the sum of
the management forecast accuracy over the past three-year:

MF OPACITY j,t =3∑

ACCURACY j,t−k

(8)

k=1

The simple idea behind this measure is that ﬁrms that consistently provide lower accuracy
forecasts (i.e., higher value of MF OPACITY) are regarded as more opaque to investors.
In our framework, considering that fundamental ﬁrm value, θ, is an increasing function offorthcoming earnings and that y is management earnings forecasts, absolute value of singed
error, εy , reﬂects future ACCURACY. MF OPACITY is essentially identical concept to σy2 tothe extent that higher ACCURACY in the past presage higher ACCURACY in the future.
In Japan, the bias in management earnings forecast made at the earnings announcement in
the previous year has a strong autocorrelation structure (e.g., Kato et al., 2009; Shimizu, 2007).
Therefore, it seems reasonable that the past ACCURACY is a signal for the future ACCURACY
and MF OPACITY is a proxy for σy2 .
3.2

Measurement of stock price synchronicity

A stock volatility comprises the following two components: (1) those tied to market and/or
industry-wide information and (2) those tied to ﬁrm-speciﬁc information. Using weekly data,
we ﬁrst estimate the R2 for each observations for ﬁrm-year t from the expanded index model inEquation (9), which allows us to decompose total return variations into these two components.
rj,w = αj + β1,j rm,w−1 + β2,j ri,w−1 + β3,j rm,w + β4,j ri,w + β5,j rm,w+1 + β6,j ri,w+1 + εj,w ,13(9)

where rj,w is the stock return in week w of ﬁrm j in industry i, rm,w is the Tokyo Stock Price
Index (TOPIX) return in week w, and ri,w is the value-weighted of return of industry i, basedon two-digit NIKKEI Industrial Code, in week w. We correct for nonsynchronous trading by
including one lead and lag terms for the market and industry indexes, following Dimson (1979).
Given the bounded nature of R2 , we use a logistic transformation of R2 , which can range
from negative to positive inﬁnity (e.g., Morck et al., 2000; Gul et al., 2010). We deﬁne stock
price synchronicity as follow:(
Ψj,t = log

2
Rj,t
2
1 − Rj,t),where Ψj,t is our measure of synchronicity for ﬁrm in year t. Higher value of Ψ means more
synchronous.
3.3

Crash measure

Negative coeﬃcient of skewness (NCSKEW), down-to-up volatility (DUVOL), and Hutton et al.
(2009) measure that based on the number of the ﬁrm-speciﬁc returns exceeding certain standard
deviations below its mean value have become popular and widely used measures of crash risk(e.g., Chen et al., 2001; Kim et al., 2011a,b; An and Zhang, 2013). This research provides asurprising result that large ﬁrms are associated with a higher likelihood of a crash, suggestinglarge ﬁrms appear to be more crash-prone than small ﬁrms. This seems counter to intuitive.
However, previous paper does not seem to tell well convincing.
One possible reason is due to artifact of crash deﬁnition. With regard to Hutton et al.
(2009) measure, since size and standard deviation of stock are highly negatively correlated, thethreshold return that qualiﬁes as a crash is smaller in absolute value for larger ﬁrms. Thus, forlarge ﬁrms, Hutton et al. (2009) measure are likely to easily qualify as a crash even though thenegative returns are not so high in severity. This fact raises doubts regarding the ability of thesemeasures used in prior research to capture crash phenomenon.
Therefore, we focus on the diﬀerent aspects of stock crash and use the following new measures.
The ﬁrst is frequency of stock crash and the second is the severity. We deﬁne the frequencyof crash (CRASH x%j,t ) as indicator variable equal to one if ﬁrm j within year t experiences
one or more weekly return fall x% and equal to zero otherwise. For robustness, we report theresults using the threshold x, 15%, 20%, 25%, and 30%. Another aspect of stock crash, severity(SEVERITY j,t ), is deﬁned as minimum weekly return for ﬁrm j in year t.

14

3.4

Timeline and our research design

Figure 3 presents a timeline of when major variables is measured. Hypothesis 3 and Hypothesis
4 predict that stock prices become synchronous and stock crashes become higher, respectively,
after the GEJE that occurred on March 11. To test these hypotheses, we deﬁne year t as theperiod from week after Mach 11 in year t − 1 to week before March 10 in year t (52 or 53 weeks
each year) throughout all analyses.
To ensure that the accounting-based variables are known to investors before synchronicity
and crash in year t are measured, we match the accounting data for December ﬁscal year-end
in our year t − 2 to November ﬁscal year-end in our year t − 1 with the explained variables inyear t.
3.5

Sample selection

We obtained management forecasts data reported at the time of earnings announcement fromthe Nikkei NEEDS-BULK management forecasts database, and other accounting data from the
Nikkei NEEDS-FinancialQUEST. Market data, such as market prices, number of shares outstanding, and stock returns, is from the Financial Data Solutions NPM daily returns database5 .
In Japan, the Financial Services Agency requires listed ﬁrms to report the statement of cash
ﬂows from 2000. To calculate over the past three-year accruals directly from the statement ofcash ﬂows, our sample period begin with 2004. Our sample period ends in 2011, after which
accounting and stock market-based data are not available at the time of our data collection.
This study includes ﬁrm-year observations from 2004 to 2011 for the ﬁrms listing on Japanesestock markets for synchronicity and crash analysis. We require a minimum of 24 week data formeasuring stock price synchronicity and the frequency and severity of crash to avoid the problems
from small sample issues. In addition, we exclude ﬁnancial ﬁrms and ﬁrm-year with insuﬃcient
data to calculate both MF OPACITY and ACC OPACITY as well as several control variables.
The accumulative sample is 18,395 ﬁrm-years from 2004 to 2011 with the number of observations
from 1,702 in 2004 to 2,520 in 2011.
In regard to an event study for the GEJE, the sample observations meet the following sample
selection criteria; (1) non-ﬁnancial ﬁrms, (2) ﬁrms with stock returns around the GEJE, and (3)ﬁrms with suﬃcient data to both MF OPACITY and ACC OPACITY as well as several control
variables. These criteria yield a ﬁnal sample of 2,559.
5This returns database basically corresponds to the CRSP in the U.S.

15

4

Empirical Analysis

4.1

Financial reporting opacity and our crash measures

We begin by showing the relation between stock crashes and our management-forecast-based
opacity measure. In Panel A of Table 2, we highlight the distribution of stock crashes based on
four-thresholds for each MF OPACITY deciles. To construct this table, break points for the sortare computed on the basis of the annual rank of MF OPACITY. Panel A of Table 2 shows thatfor any threshold, the percentage of crash ﬁrms monotonically increases across MF OPACITY
deciles and chi-squared tests conﬁrm that stock crashes are signiﬁcantly concentrated in thehigh MF OPACITY portfolios. For example, the percentage of our sample that experienced atleast more than one −20% crash for the lowest MF OPACITY decile is 10.5%, while that for
the highest decile is 30.1%. The diﬀerence between those two groups of 19.7% is statisticallysigniﬁcant at a 1% level. In Hutton et al. (2009), 17.1% of their sample experience at least
one crash based on their criteria. This percentage is the nearest when we use −20% as a crash
threshold (i.e., 15.0% of our sample experience at least one crash based on our criteria). Hence,compared to Hutton et al. (2009), the −15% (−25% and −30%) crash threshold in our analysis
might be less (more) strict.
Panel B of Table 2 provides summary statistics of minimum weekly returns during the year
(i.e., SEVERITY ) for each MF OPACITY decile. Mean and median value of SEVERITY
decrease monotonically between the lowest and highest deciles. In addition, we can easily reject
the null hypothesis that the mean and median diﬀerences between SEVERITY for the lowest
and highest decile is equal to zero at 1% level (t-stat. = −23.3, z-stat. = −27.6). These results
suggest that more opaque ﬁrms, on average, tend to experience at least one large negative return.
In sum, results in Table 2 indicates that stock crashes are more frequent and more severe foropaque ﬁrms, which is consistent with Hypothesis 1.
Next, to test Hypothesis 4, we investigate whether the frequency and severity of individual
stock crashes become higher due to the GEJE. Figure 4 highlights the frequency of stock crashes
based on −20% threshold across the year. If Hypothesis 4 is correct, then we can observe anabnormally high frequency of stock crashes in the GEJE year (i.e., 2011). However, Figure
4 shows that abnormally high frequency of stock crashes is observed in 2008 rather than in
2011, although 2011 seems high. Recall that 2008 was the height of the Global Financial
Crisis. This crisis aﬀected not only the U.S. stock market but also the Japanese market. The
Nikkei Stock Average was 12,214 Japanese Yen on the day before the bankruptcy of Lehman
16

Brothers (September 12, 2008), the price on October 28, 2011 reached levels not seen since before
October 1982. Considering the great eﬀect of the crisis on Japanese stock market, abnormallyhigh frequency of stock crash in 2008 is not surprising.
During a crisis period, investors must do the same after a catastrophic event such as the
GEJE. As we describe in Section 2, such a situation leads to severely limited attention by
investors. Therefore, Hypothesis 3 and 4 apply to not only 2011 but also 2008. Stated diﬀerently,
if these two hypotheses are correct, stock prices should have become more synchronous during
2008 due to severely limited investor attention, and since 2008, stock crashes should be more
frequent and more severe during 2008, because the news spread in the Japanese market about
the Global Financial Crisis. As the result, the fact that an abnormally high frequency of stock
crashes is observed in 2008 is consistent with our model. In addition to 2008, it makes sensethat we observe a high frequency of stock crashes in 2007. This is because the crisis began in
December 2007 as deﬁned by the National Bureau of Economic Research. To mitigate omitted
variable bias arising from not including important regressors, we include a 2007 and 2008 dummy
as control variables when we conduct regression analyses to test Hypothesis 1 to 4.
Figure 5 gives year-by-year mean data for SEVERITY . The mean of SEVERITY for threeyears, 2007, 2008, and 2011 are well below the mean across our sample period (−13.3%). The
mean of SEVERITY for those year is −13.9%, −23.1%, and −14.1%, respectively. We believethat the results of 2007 and 2008 are due to the eﬀect of the Global Financial Crisis on theJapanese stock market. Given that there is a linkage between investors’ limited attention, stock
price synchronicity, and stock crashes, it represents the most convincing evidence. In fact, weshow that stock price synchronicity increases for 2007 and 2008 in the next section. Given the
wide-spread bad news across the markets, greater stock price synchronicity should lead to more
stock crashes as described in our model. Therefore, strong evidence shows that more frequentand severe of stock crashes are likely to be observed in 2007 and 2008, for supporting our model.
Besides 2007 and 2008, we observed abnormally low SEVERITY in 2011, which is consistent
with Hypothesis 4. This evidence suggests that after the GEJE, ﬁrms tended to experience atleast one severe large negative returns. Overall, the results of Figure 4 and 5 support Hypothesis
4 (i.e., the frequency and severity of individual stock crashes become higher after the GEJE).
The key question is whether the eﬀect of opacity and the GEJE on stock crashes remains
even after controlling for the known determinants of the frequency and severity of stock crashes.
This issue is investigated next in a regression framework. With regard to the analysis for thefrequency of stock crashes, we assume that the marginal probability of the frequency follows a17

logistic distribution and is given by
Prob(CRASH x%j,t = 1) =1,
1 + exp(−x0j β)(10)where CRASH x%j,t is an indicator variable equal to one if ﬁrm j within year t experiences one
or more weekly return fall x% (x ∈ {−15%, −20%, −25%, −30%}) and equal to zero otherwise,
xj is ﬁrm j’s K × 1 vector of explanatory variables including intercept, and β is K × 1 unknown
coeﬃcient vector. A higher level of x0j β implies a higher probability of stock crash during the year
t. Our main focus among explanatory variables is MF OPACITY j,t and D EARTHQUAKE j,t
for testing Hypothesis 1 and 4, respectively. The former is the sum of the management forecast
accuracy over the past three years, and the latter is an indicator variable equal to one year after
earthquake (i.e., 2011). Following Hutton et al. (2009), we include a discretionary-accruals-based
opacity measure (ACC OPACITY ), ﬁrm size (log(MVE )), book-to-market ratio (log(BM )),return on net operation asset (RNOA), and leverage (LEV ) as control variables. In addition,
we include a 2007 and 2008 dummy as control variables to avoid the omitted variables bias.
Detail deﬁnition for these control variables are provided in Table 1.
Regarding the severity of stock crashes, we estimate the following model:
SEVERITY j,t = β0 + β1 D EARTHQUAKE j,t∑γk Control variablek + εj,t ,+β2 MF OPACITY j,t +

(11)

k

where SEVERITY j,t is a minimum weekly return for ﬁrm j in year t. In this model, we include
the same control variables as in Equation (10).
Table 3 reports these two regression results for alternative speciﬁcations and dependent
variables. Regression 1 shows the results when we estimate Equation (10) with CRASH − 15%
as a dependent variable. The coeﬃcient on MF OPACITY is positive (= 1.564) and statisticallysigniﬁcant (p < 0.01). This result suggests that stock crashes are more frequent for ﬁrms withconsistently providing lower accuracy management forecasts, which is consistently Hypothesis
1. In addition, we ﬁnd that there is also a signiﬁcantly positive coeﬃcient on ACC OPACITY.
Overall, we ﬁnd that both mandatory-management forecast-based and discretionary-accrualsbased opacity measures can explain the cross-sectional variation in stock crash frequency. As arobustness check, we estimate Equation (10) using annual decile rank, scaled to range between
0 and 1, for MF OPACITY and ACC OPACITY, to mitigate outlier eﬀect and/or consider nonlinearity relation as shown in Regression 2. The pattern of coeﬃcients estimated in terms of thesign and signiﬁcance is similar to Regression 1. Hence, our results are a robust measurement of
those two variables.
18

A dependent variable of Regression 3, 4, and 5 is CRASH − 20%, CRASH − 25%, andCRASH − 30%, respectively. The coeﬃcient of MF OPACITY is positive and signiﬁcant at 1%
level in all three speciﬁcations. These results indicate that Hypothesis 1 is supported when weuse any four values as stock crash thresholds.
In regard to Hypothesis 4, the coeﬃcient on D EARTHQUAKE is positive and signiﬁcant
at 1% level in Regression 1 to 5. Considering the magnitude of the coeﬃcient, the economic
impact of the GEJE on the frequency of stock crashes appears to be high. For example, thecoeﬃcient of D EARTHQUAKE in Regression 3 is 1.061 and it implies that after the GEJE, the−20% crash frequency increase by x% even after controlling for other well-known determinantsof stock crashes. Hence, we conclude that Hypothesis 4 is supported.
To test the impact of opacity and the GEJE on the stock crash severity, we estimate Equation (11). Regression 6 of Table 3 shows the OLS estimator with t-statistics based on the
two-way cluster robust standard errors in parentheses following Petersen (2009). The coeﬃcientof MF OPACITY is negative and signiﬁcantly diﬀerent from zero. To conﬁrm the economics
impact of it, we estimate Equation 11 using annual decile rank, scaled to range between 0
and 1, for MF OPACITY as well as ACC OPACITY. By this coding scheme, the coeﬃcient on
MF OPACITY can be interpreted as the diﬀerence in minimum returns during the year between
the portfolio for the most opaque ﬁrms and that for the least opaque ﬁrms. Regression 7 showsthat the coeﬃcient on MF OPACITY is −0.033 and statistically signiﬁcant at 1% level. This result means that the diﬀerence in minimum return between the lowest and highest MF OPACITY
deciles is 3.3%. We believe that these diﬀerences are also economically signiﬁcant.
Turning to the coeﬃcient on D EARTHQUAKE, we ﬁnd that there is a signiﬁcantly negative
coeﬃcient on it (coef. = −0.025, p < 0.01) in Regression 6. It implies that minimum returns
decreased 2.5% in the GEJE year, even after accounting for other known determinants of stock
crash severity. The conclusion drawn from Regression 7 remains basically unchanged in term ofsign and magnitude by other coding scheme for MF OPACITY and ACC OPACITY.
The results of Table 3 strongly support Hypothesis 1 and 4. Overall, these results suggestthat stock crashes are more frequent and more severe for opaque ﬁrms after the GEJE.
4.2

Financial reporting opacity and stock price synchronicity

In this section, we test Hypothesis 2 and 3 related to stock price synchronicity. Panel A of Table
4 reports the results of the univariate relationship between R2 and MF OPACITY. To constructthis table, we sort the sample ﬁrms into deciles based on the annual rank of MF OPACITY, and19then, compute the average R2 for each portfolio. Panel A of Table 4 shows that R2 monotonicallydecreases across MF OPACITY deciles, which is inconsistent with Hypothesis 1. Given Roll
(1988)’s ﬁnding that ﬁrm size is an important factor of why R2 s diﬀer, this result is not surprising.
In line with our intuition, MF OPACITY is strongly correlated with ﬁrm size (log(MVE ))(ρ = −0.33, p < 0.01) for our sample, suggesting that opaque ﬁrms are concentrated in small
ﬁrms. Therefore, the negative correlation between MF OPACITY and R2 may be spurious dueto the ﬁrm size eﬀect on R2 .
To control ﬁrm size, we form 25 portfolios by independently sorting ﬁrms into ﬁve quintiles
based on their annual ﬁrm size ranking and ﬁve quintiles based on their annual MF OPACITY
ranking. Panel B of Table 4 reports mean values of R2 for each portfolios. Looking down eachcolumn, the result indicates that the larger ﬁrm tend to have higher R2 , which conﬁrm Roll
(1988)’s ﬁnding in our sample. Conversely, looking across each row, we observe higher opacity
is associated with higher R2 except for the lowest quintile and Q2 quintile. For Q2 quintile, we
also observe that the highest R2 is in the highest quintile. In sum, our management forecast
based opacity measure appears to independently aﬀect R2 even after controlling for ﬁrm size.
Figure 6 shows mean values R2 from Equation (9) for each year over our sample period.
Noteworthy is that the high R2 s are observed in 2007, 2008, and 2011. This evidence is consistent
with our model. In Section 2, we argued that stock prices become more synchronous wheninvestor attention is severely limited. As described in the previous section, investor attention isseverely limited in the three-years corresponding to the Global Financial Crisis and the GEJE.
Hence, the evidence on Figure 6 supports our model is consistent with Hypothesis 3.
Next, we conduct regression analysis to control for potential eﬀect of the various ﬁrm characteristics including ﬁrm size on the R2 . We estimate that the following model examines theeﬀect of opacity and the GEJE on stock price synchronicity:
Ψj,t = β0 + β1 D EARTHQUAKE j,t + β2 MF OPACITY j,t +∑γk Control variablek + εj,t , (12)kwhere D EARTHQUAKE j,t is an indicator variable equal to one year after earthquake (i.e.,2011) and equal to zero otherwise. According to the previous studies (e.g., Hutton et al.,2009), we control for discretionary-accruals-based opacity measure (ACC OPACITY ), ﬁrm size(log(MVE )), book-to-market ratio (log(BM )), the variance of weekly returns of industry index(log(VARIND)), skewness (SKEW ) and kurtosis (KURT ) of the ﬁrm-speciﬁc weekly returns,
return on net operation asset (RNOA), and leverage (LEV ). Detailed deﬁnition for these control
variables are provided in Table 1. In addition, we include a 2007 and 2008 dummy as control
20

variables to control the eﬀect of the U.S. sub-prime mortgage ﬁnancial crisis on the Japanesemarket. Hypothesis 2 predicts the coeﬃcient of MF OPACITY should be signiﬁcantly positive. With regard to Hypothesis 3, as long as the catastrophic shock increases the stock pricesynchronicity, the coeﬃcient on D EARTHQUAKE should be signiﬁcantly positive.
Table 5 reports the estimation results of Eq (12). Regression 3 shows that the coeﬃcient on
MF OPACITY is positive (= 0.221) and statistically signiﬁcant (p < 0.01) while coeﬃcient on
ACC OPACITY is negative (= −0.049) and insigniﬁcantly diﬀerent from zero at a conventional
level. The former result indicates that ﬁrms with opaque ﬁnancial reports in terms of management forecasts have stock returns that are more synchronous. On the other hand, the latter
result suggests that opacity based on earnings management has no explanatory power for stock
price synchronicity, which is inconsistent with Hutton et al. (2009).
As a robustness check, we transform two opacity measures to the annual decile rank, scaled
to range between 0 and 1, to mitigate the eﬀect of outliers. Regression 4 of Table 5 presents
estimates using ranked measures. We ﬁnd that the pattern of the estimated coeﬃcients onranked measures in terms of the sign and signiﬁcance is similar to Regression 3 in Table 5.
Hence, our results are robust to an alternative treatment of the opacity measures. Overall,these results strongly support our Hypothesis 2.
Regarding Hypothesis 3, the coeﬃcient on D EARTHQUAKE is positive and statisticallysigniﬁcant in both Regression 3 and 4 as expected. The implied value impact of R2 using theparameter from Regression 3, holding all other variables at their mean, after the GEJE (38.0%)
increases by around 8.6% as compared to R2 before the GEJE (29.5%). This impact is an
economically as well as statistically signiﬁcant amount. Hence, the results strongly support our
Hypothesis 3.
4.3

Financial reporting opacity and crash severity around the GEJE

In this section, we examine the association between management-forecast-based opacity and the
behavior stock price around the GEJE to test Hypothesis 5. Table 6 is reports the mean of stock
returns for each portfolios classiﬁed by MVE and MF OPACITY. The positive relation between
ﬁrm size and stock returns around the GEJE is intuitively appealing: i.e., large ships are lessaﬀected by a big wave compared to small ships. Therefore, it is important to control ﬁrm size
when we examine the stock returns around the GEJE.
To construct Table 6, we independently sort sample ﬁrms (i.e., 2,559 ﬁrm) into quintiles
based on rank of each variable, and then compute the average stock returns for each portfolio.
21

Panel A of Table 6 shows the mean of raw return on the GEJE day (day 0) for each portfolios.
We can point out that for all MVE quintiles, stock returns are likely to almost monotonicallydecrease across MF OPACITY quintiles and the diﬀerence in stock returns between the lowest
and highest MF OPACITY deciles is statistically signiﬁcant at 1% level.
These results suggest that MF OPACITY quintiles have predictive power for cross-sectional
stock returns on the day the quake struck. This is a surprising result for given the time of 2:46
p.m occurred the GEJE. Since the Japanese stock market close 3:00 p.m., the collapse of theprice occurred between 2:46 p.m. and 3:00 p.m. as shown in Figure 7. This ﬁgure plots theprice movement of the Nikkei 225 in the afternoon session, called the “Goba”, between 12:30
p.m. and 3:00 p.m. on March 11, 2011. MF OPACITY were able to predict the severity levelof “ﬂash” crash for just 14 minutes. Turning to MVE, looking down each column, we see that alargely monotonic and statistically signiﬁcant negative relation between MVE and stock returns,
which is inconsistent with our intuition. In contrast, in Panel B and C of Table 6 indicating
cumulative returns from days 0 to +l and from days 0 to +2, respectively, we observe a largely
monotonic and statistically positive relation between MVE and stock return. This result is inline with our inference.
With regard to the relation between MF OPACITY and stock returns around the GEJE,
looking across each row, we observe very large negative returns for extremely opaque ﬁrms for
all MVE quintiles in Panel B of Table 6. For example, the average CR[0; +1] of the lowest
MF OPACITY quintile is −9.5% while that of the highest MF OPACITY quintile is −17.4%.
The diﬀerence of −7.9% is statistically signiﬁcant at the 1% level. Portfolios with higher
MF OPACITY have a tendency for lower returns as shown in Panel B, which is similar to Panel Cof Table 6. In sum, Table 6 suggests that the predictive power of MF OPACITY for stock returns
around the GEJE is not due to the correlation between ﬁrm size and MF OPACITY. Rather,
given the fact that the average stock returns do not relate to MVE in three MF OPACITY
portfolios (i.e., Q2 MF OPACITY quintiles in Panel B and Q1 and Q2 MF OPACITY quintiles
in Panel C), MF OPACITY itself can explain cross-sectional variation in stock returns aroundthe GEJE.
Finally, the following model examines whether our inference from the analyses in Table 7
remains unchanged, even after controlling for the potential eﬀect of various ﬁrm characteristics
on stock returns;
Retj = β0 + β1 MF OPACITY j,t +

∑
k

22

γk Control variablek + εj,t ,(13)

where Retj ∈ {R[0]j , CR[0 : +1]j , CR[0 : +2]j }, R[0]j represents the raw return for stock j on
the day occurred the GEJE (day 0), and CR[0 : +k]j indicates cumulative return stock j fromday 0 to +k (k ∈ {+1, +2}) relative to day 0.
Table 7 reports the OLS coeﬃcient estimates and White (1980) heteroscedasticity consistentt-statistics in parentheses. Regression 1 shows the result without considering the two opacity
measures based on management forecast precision and abnormal accruals. The coeﬃcient of determination is 4.2% and slightly low as in event studies in accounting and ﬁnance. However, with
two opacity measures, the coeﬃcient of determination increase by 1.7% and 3.2% in Regression
2 and 3, respectively. This implies that opacity measures are important factors in explaining
the stock returns on the day of the GEJE. The results of the coeﬃcient on ACC OPACITY in
Regression 2 and 3 are noteworthy. While the coeﬃcient is not negative but positive and statistically signiﬁcant at marginal level in Regression 2, it is insigniﬁcant in Regression 3. Therefore,
increasing R-squared from Regression 1 to 2 and 3 is mainly due to MF OPACITY.
In both Regression 2 and 3, we observed that coeﬃcients on MF OPACITY are negativeand signiﬁcant at 1% level. For example, the coeﬃcient on rank(MF OPACITY) is −0.011 inRegression 3. This result means that the diﬀerence in stock returns on the GEJE day between
the lowest and highest MF OPACITY is 1.1%, even after controlling for other determinants. The
diﬀerence between these two groups grows larger as the event period lengthens. The diﬀerenceis 8.7% (13.2%) when we set event period from day 0 to day +1 (+2) as shown in Regression 6(9). These results suggest that MF OPACITY is economically important as well as statisticallysigniﬁcant. Moreover, we can point out that the coeﬃcient of determination with MF OPACITY
has increased 1.5 times than the coeﬃcient of determination without MF OPACITY for both
event periods. This result implies that MF OPACITY has stronger explanatory power for stock
returns around the GEJE than any other variables.
Overall, we interpret the evidence in Table 7 as suggesting that the stock price of ﬁrms
disclosing opaque management forecasts tend to co-move with the market around the GEJE
and, hence catastrophe-based collapse occurred. This is because investors fell into a crisis oflimited attention due to the earthquake and were likely to pay little attention to speciﬁc ﬁrms
and more attention to the market as a whole, and as a result, the stock price of such ﬁrms weremore covariated with the market than transparent ﬁrms. Management-forecast-based opacity
measures were able to predict stock returns when the unexpected catastrophic shock occurred.
We conclude that the results in this section strongly support Hypothesis 5.

23

5Conclusions

This study provides theory and empirical evidence to demonstrate that opacity of ﬁrms’ ﬁnancial
reports plays a role in inﬂuencing the distribution of stock returns. Developing a simple bayesianmodel in the presence of management earnings forecasts and market-wide information, we predictthat at normal time, (1) ﬁrms with opaque ﬁnancial reports have stock returns that are more
synchronous with the market and (2) stock crashes are more frequent and more severe for opaqueﬁrms. Using the precision of eﬀectively mandated management forecasts as a measure of opacity,
we obtain empirical results consistent with these predictions.
In addition, we extend Peng and Xiong (2006) model to examine how investors’ limited
attention arising from catastrophic event (i.e., an big earthquake and a tsunami) aﬀects stock
price synchronicity and stock crash in the market. Our model reveals that (3) investors arelikely to pay more attention to the market-wide information than to ﬁrm-speciﬁc information,
thus leading to higher synchronicity and (4) higher crash risk, and (5) the stock price of ﬁrms
disclosing opaque ﬁnancial reports tend to more co-move with the market around a catastrophic
event and, hence catastrophe-based collapse occurred. Using the GEJE as a representative
catastrophic event, ample support for these predictions is found in the data.
Our results have important implication for capital allocation. Since prior work shows thatless stock price synchronicity (i.e., lower R2 ) exhibit a better allocation of capital (e.g., Wurgler, 2000), our results suggest that the allocation eﬃciency improve through disclosing greatertransparent of ﬁnancial information and deteriorate around a catastrophic event. Besides, ourresults have implication for managers who wish to avoid a stock crash. Our results suggest that
improving ﬁnancial reporting reduce any collapse in a ﬁrm’s stock price.
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Economics 79 (2): 257–292. (2006).
Kato, K., D. J. Skinner; and M. Kunimura. “Management Forecasts in Japan: An Empirical Studyof Forecasts that Are Eﬀectively Mandated.” The Accounting Review 84 (5): 1575–1606. (2009).
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Morck, R., B. Yeung; and W. Yu. “The Information Content of Stock Markets: Why Do Emerging
Markets Have Synchronous Stock Price Movements?” Journal of Financial Economics 58 (1-2): 215–
260. (2000).
Peng, L.; and W. Xiong. “Investor Attention, Overconﬁdence, and Category Learning.” Journal of
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Shimizu, Y. “Continuity of Bias in Management Forecasts and Mispricing.” Security Analysts Journal
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26

Figure 1: Curves of subjective variances of the two ﬁrms
This figure illustrates that given a certain level of an investor’s attention, the smaller ai , the smaller the subjective varianceis.

27

Figure 2: Magnitude relations of the widths of the ranges
This figure illustrates that the range [Λ1 , 1] is wider than [Λ2 , 1] and that the range [0, Λ2 ] is wider than [0, Λ1 ].

28

Earnings announcement
day for fiscal year t − 2

Management forecastof year t − 2, f et−2
j,t−1 ,
is provided

Earnings announcement
day for fiscal year t − 1
Week before March 10of year t

Week after March 11of year t − 1

Actual earnings of year t − 1,ej,t−1 , is announced

the Great East Japan Earthquake
11 March 2011

year t

Crash Analysis

measuring CRASH x%j,tStock Price Synchronicity Analysis

measuring SEVERITY j,t
Ψj,t estimation

ACCURACY j,t−1 = (f et−2
j,t−1 − ej,t−1 )/MVE j,t−2
�3
k=1 ACCURACY j,t−k

MF OPAQUE j,t =Figure 3: Timeline for measurement of main variables

29

Event Study
Stock return around 3.11, Retj
Retj ∈ {R[0]j , CR[0 : +1]j , CR[0 : +2]j }100%!
90%!
80%!
70%!
60%!
50%!
40%!
30%!
20%!
10%!
0%!
2004!

2005!

2006!

2007!
0!

2008!

1!

2!

2009!

2010!

2011!

3"!

Figure 4: Yearly cross-sectional distribution stock crash based on CRASH−20%
This figure plots the cross-sectional means of SEVERITY over our sample period from 2004 to 2011. SEVERITY is defined
as a minimum weekly return during the year. We define year t as the period from week after Mach 11 in year t − 1 to week
before March 10 in year t

30

2004!
-10.0%!

2005!

2006!

2007!

2008!

2009!

2010!

2011!

-12.5%!

-15.0%!

-17.5%!

-20.0%!

-22.5%!

-25.0%!

Figure 5: Yearly cross-sectional means of SEVERITYThis figure plots the cross-sectional means of SEVERITY over our sample period from 2004 to 2011. SEVERITY is defined
as a minimum weekly return during the year. We define year t as the period from week after Mach 11 in year t − 1 to week
before March 10 in year t

45.0%!
42.5%!
40.0%!
37.5%!
35.0%!
32.5%!
30.0%!
27.5%!
25.0%!
2004!

2005!

2006!

2007!

2008!

2009!

2010!

2011!

Figure 6: Yearly cross-sectional means of R2
This figure plots the cross-sectional means of R2 estimated by Equation (9) over our sample period from 2004 to 2011. We
define year t as the period from week after Mach 11 in year t − 1 to week before March 10 in year t

31

The GEJE occurredat 2:46 p.m., March 11, 2011

10,375!

10,350!

10,325!

10,300!

10,275!

10,250!
12:30 PM!

1:00 PM!

1:30 PM!

2:00 PM!

2:30 PM!

3:00 PM!

Figure 7: Nikkei 225 during the Great East Japan Earthquake
This figure shows the price movement of the Nikkei 225 in the afternoon session, called the “Goba”, between 12:30 p.m.
and 3:00 p.m. on March 11, 2011. The GEJE occurred at 2:46:18 p.m. The price of Nikkei 225 was 10,360,21 Japanese Yen
before the GEJE, and collapsing by 1.02% between 2:46 p.m. and 3:00 p.m.

32

Dependent variables
Synchronicity Measure Ψj,t

Stock Crash Measure

CRASH x%j,tSEVERITY j,t
Event Period Returns

Retj

Independent Variables
Opacity Measure
MF OPACITY j,t

ACC OPACITY j,t

The GEJE Dummy

D EARTHQUAKE j,t

Control Variableslog(MVE )j,t−1log(BM )j,t−1
RNOAj,t−1
LEV i,t−1log(VARIND)j,t

SKEW j,t
KURT j,tΨj,t is our measure of annual synchronicity based on
R2 from Equation (9) for ﬁrm i in year t. Given thebounded nature of R2 , we use a logistic transformation of R2 , which can range from negative to positiveinﬁnity.
CRASH x%j,t is an indicator variable equal to one if
ﬁrm j within year t experiences one or more weekly
return fall x% (x ∈ {−15%, −20%, −25%, −30%})and equal to zero otherwise.
SEVERITY j,t is a minimum weekly return for ﬁrm
j in year t.
Retj ∈ {R[0]j , CR[0 : +1]j , CR[0 : +2]j }, R[0]j represents the raw return for stock j on the day occurred
the GEJE (day 0), and CR[0 : +k]j indicates cumulative return stock j from day 0 to +k (k ∈ {+1, +2})
relative to day 0.
MF OPACITY j,t is our opacity measure using management forecasts as the sum of the management forecast accuracy over the past three-year.
ACC OPACITY j,t is another opacity measure basedon discretionary accruals as the sum of absolute abnormal accruals over the past three-year. Normal accruals are estimated in cross-section for each two-digit
Nikkei Industrial Code and year combination using
Modiﬁed Jones model.
D EARTHQUAKE j,t is an indicator variable equal
to one year after earthquake (i.e., year 2011) andequal to zero otherwise.
log(MVE )j,t−1 is the natural log of the market valueof equity at the beginning of ﬁscal year t.
log(BM )j,t−1 is the natural log of book-to-market ratio measured at the beginning of ﬁscal year t.
RNOAj,t−1 is return on net operation assets for yeart − 1.
LEV j,t−1 is the book value of all liabilities divided
by total asset, measured at the beginning of the ﬁscal
year t − 1.
log(VARIND)j,t is the natural log of the varianceof the weekly industrial returns based on two-digit
Nikkei Industrial Code during the ﬁrm’s ﬁscal year t.
SKEW j,t is the skewness of the weekly returns forﬁrm j over the ﬁscal year t.
KURT j,t is the kurtosis of the weekly returns for ﬁrm
j over the ﬁscal year t.

Table 1: Variable Deﬁnitions

Panel A: Distribution of stock crash for each MF OPACITY deciles
CRASH − 15% CRASH − 20% CRASH − 25% CRASH − 30%
D1 (Less opacity)
20.5%
10.5%
4.9%
2.1%
D2
20.3%
10.2%
4.7%
1.9%
D3
23.6%
11.4%
5.2%
2.4%
D4
24.4%
11.8%
5.0%
2.2%
D5
25.3%
12.5%
6.1%
2.9%
D6
27.3%
12.7%
6.0%
2.8%
D7
30.6%
14.6%
6.4%
2.8%
D8
34.3%
15.8%
8.1%
3.7%
D9
39.1%
20.1%
11.2%
5.7%
D10 (More opacity)
52.7%
30.1%
16.2%
9.5%
All Firms
29.8%
15.0%
7.4%
3.6%
Chi-squared statistics
797.3∗∗∗
484.6∗∗∗
326.2∗∗∗
261.6∗∗∗
Panel B: Mean and median value of SEVERITY for portfolios formed by MF OPACITY
MeanMedian
D1 (Less opacity)−11.1%−9.0%
D2−11.3%−9.3%
D3−11.9%−10.0%
D4−12.0%−10.2%
D5−12.7%−10.9%
D6−12.9%−11.1%
D7−13.5%−11.7%
D8−14.0%−12.2%
D9−15.4%−13.0%
D10 (More opacity)−18.1%−15.5%All Firms−13.3%−11.3%
D10−D1−7.0%∗∗∗−6.5%∗∗∗

Table 2: Association between MF OPACITY and stock crash

34

35

Observations
Pseudo R2
Adjusted R2

Y earDummy2007
Y earDummy2008
log(MVE )log(BM )RNOA
LEVD EARTHQUAKE

rank(ACC OPACITY )rank(MF OPACITY )ACC OPACITY

MF OPACITY

Dependent Var.
Constant

18,3950.257918,3950.3085

18,3950.2835

18,3950.2572

Table 3: Regression results of stock crash on MF OPACITY and the GEJE

18,3950.2626

18,3950.3416

18,3950.3632

Regression 1
Regression 2
Regression 3
Regression 4
Regression 5
Regression 6 Regression 7
CRASH − 15% CRASH − 15% CRASH − 20% CRASH − 25% CRASH − 30% SEVERITY SEVERITY
1.752
1.102
0.095
−0.5610.484−0.189∗∗∗−0.188∗∗∗(0.91)(0.62)(0.030)(−0.19)(0.18)(−4.57)(−4.56)1.564∗∗∗
1.295∗∗∗
0.921∗∗∗
1.010∗∗∗−0.055∗∗∗(5.33)(4.36)(3.67)(4.38)(−7.83)3.668∗∗∗
3.846∗∗∗
3.752∗∗∗
3.820∗∗∗−0.160∗∗∗(8.25)(7.08)(7.48)(7.67)(−7.07)
1.159∗∗∗−0.033∗∗∗(8.54)(−8.26)
0.935∗∗∗−0.034∗∗∗(7.46)(−6.04)
1.131∗∗∗
1.302∗∗∗
1.061∗∗∗
0.947∗∗∗
0.608∗∗∗−0.025∗∗∗−0.032∗∗∗(8.07)(9.23)(9.51)(8.82)(4.56)(−6.36)(−8.44)
1.114∗∗∗
1.070∗∗∗
1.048∗∗∗
0.924∗∗∗
0.988∗∗∗−0.032∗∗∗−0.030∗∗∗3.707∗∗∗
3.721∗∗∗
3.744∗∗∗
3.451∗∗∗
3.156∗∗∗−0.126∗∗∗−0.125∗∗∗−0.188∗∗−0.175∗∗−0.169−0.183−0.260∗∗0.005∗∗∗
0.005∗∗∗−0.296−0.351∗−0.321∗∗−0.284∗∗−0.1860.007
0.010∗−1.116∗∗∗−1.287∗∗∗−1.326∗∗∗−1.859∗∗∗−1.567∗∗∗0.054∗∗∗
0.066∗∗∗
0.425∗
0.345
0.348
0.590∗
0.516−0.018∗∗−0.016∗

36

Q4
22.0%
26.4%
33.1%
38.8%
52.8%
34.6%

Table 4: Association between R2 and MF OPACITY

Panel B: Mean R2 for portfolios formed by MF OPACITY and ﬁrm size
Q1 (Less opacity)
Q2
Q3
Q1 (Small)
21.8%
21.3%
20.3%
Q2
26.7%
25.9%
26.6%
Q3
30.6%
31.6%
32.6%
Q4
35.7%
37.8%
37.7%
Q5 (Large)
50.0%
49.3%
50.2%
All Firms
33.0%
33.2%
33.5%

Panel A: Mean R2 for portfolios formed by MF OPACITY
D1 (Less opacity)
39.5%
D2
38.1%
D3
36.2%
D4
35.6%
D5
35.0%
D6
34.2%
D7
32.9%
D8
31.7%
D9
30.7%
D10 (More opacity)
27.8%
All Firms
34.2%

Q5 (More opacity)
21.4%
28.8%
35.2%
41.3%
56.1%
36.6%

All Firms
21.4%
26.9%
32.6%
38.3%
51.7%

Dependent Var.
Constant
MF OPACITYACC OPACITY
rank(MF OPACITY )rank(ACC OPACITY )D EARTHQUAKEY earDummy2007
Y earDummy2008
log(MVE )log(BM )RNOA
LEVlog(VARIND)
SKEW
KURT
Observations
Adjusted R2

Regression 1 Regression 2 Regression 3 Regression 4ΨΨΨΨ−7.609∗∗∗−7.922∗∗∗−7.929∗∗∗−8.002∗∗∗(−20.5)(−16.8)(−16.6)(−14.5)0.355∗∗∗
0.221∗∗∗(2.88)(3.75)−0.049−0.048(−0.96)(−0.85)0.153∗∗∗(2.60)−0.041(−0.79)0.388∗∗∗
0.411∗∗∗(6.89)(7.33)−0.096∗∗−0.102∗∗0.111∗
0.107
0.357∗∗∗
0.370∗∗∗
0.364∗∗∗
0.365∗∗∗
0.186∗∗∗
0.198∗∗∗
0.176∗∗∗
0.165∗∗∗−0.241∗−0.110−0.126−0.1420.505∗∗∗
0.444∗∗∗
0.489∗∗∗
0.470∗∗∗
0.270∗∗∗
0.270∗∗∗
0.254∗∗∗
0.249∗∗∗−0.123∗∗∗−0.126∗∗∗−0.099∗∗−0.098∗∗
0.004
0.003
−0.005−0.004
18,3950.3994

18,3950.4033

18,3950.422018,3950.4223

Table 5: Regression results of stock price synchronicity on MF OPACITY and the GEJE

37

Panel A: Mean R[0] for portfolios formed by MF OPACITY and ﬁrm size
Q1 (Less opacity)
Q2
Q3
Q4 Q5 (More opacity) Q5−Q1
Q1 (Small)
−0.6%
−0.5%
−1.2%
−1.5%
−1.5%
−0.9%∗∗∗
Q2
−0.7%
−0.8%
−0.7%
−1.1%
−1.6%
−0.9%∗∗∗
Q3
−0.9%
−1.0%
−1.1%
−1.4%
−1.8%
−0.9%∗∗∗
Q4
−1.3%
−1.1%
−1.4%
−1.8%
−2.1%
−0.8%∗∗∗
Q5 (Large)
−1.2%
−1.4%
−1.7%
−1.8%
−2.0%
−0.9%∗∗∗
Q5−Q1−0.6%∗∗ −0.9%∗∗∗ −0.5%∗∗ −0.2%−0.5%∗∗
Panel B: Mean CR[0 : +1] for portfolios formed by MF OPACITY and ﬁrm size
Q1 (Less opacity)
Q2
Q3
Q4 Q5 (More opacity) Q5−Q1
Q1 (Small)−9.5%−8.9% −11.7% −13.6%−17.4%−7.9%∗∗∗
Q2
−7.9%
−9.5%
−8.8% −12.7%−15.7%−7.9%∗∗∗
Q3−9.0% −10.2% −10.6% −12.2%−15.8%−6.8%∗∗∗
Q4
−9.9%
−9.2%
−9.9% −13.7%−15.8%−5.8%∗∗∗
Q5 (Large)
−7.5%
−7.3%
−9.2%
−9.7%
−12.0%−4.6%∗∗∗
Q5−Q1
2.0%∗
1.7%
2.5%∗∗
3.9%∗∗∗
5.4%∗∗∗
Panel C: Mean CR[0 : +2] for portfolios formed by MF OPACITY and ﬁrm size
Q1 (Less opacity)
Q2
Q3
Q4 Q5 (More opacity) Q5−Q1
Q1 (Small)−20.0% −19.5% −25.8% −28.4%−33.9% −13.9%∗∗∗Q2−18.8% −20.8% −21.3% −27.0%−33.3% −14.5%∗∗∗
Q3−19.7% −21.6% −23.8% −25.1%−32.6% −13.0%∗∗∗Q4−20.5% −20.7% −22.0% −27.4%−30.4%−9.9%∗∗∗
Q5 (Large)−18.2% −17.7% −20.0% −19.5%−23.8%−5.7%∗∗∗
Q5−Q1
1.8%
1.8%
5.8%∗∗∗ 8.8%∗∗∗
10.1%∗∗∗

Table 6: Association between stock returns around the GEJE and MF OPACITY

38

39

Observations
Adjusted R2

log(MVE )log(BM )RNOA
LEVrank(ACC OPACITY )rank(MF OPACITY )ACC OPACITY

MF OPACITY

Dependent Var.
Constant

2,559
0.05922,559
0.07402,559
0.10802,559
0.15462,559
0.1648

Table 7: Regression results of stock returns around the GEJE on MF OPACITY

2,559
0.0421

2,559
0.10542,559
0.16042,559
0.1731

Regression 1 Regression 2 Regression 3 Regression 4 Regression 5 Regression 6 Regression 7 Regression 8 Regression 9
R[0]
R[0]
R[0]
CR[0 : +1]
CR[0 : +1]
CR[0 : +1]
CR[0 : +2]
CR[0 : +2]
CR[0 : +2]]
0.009∗
0.019∗∗∗
0.027∗∗∗−0.380∗∗∗−0.284∗∗∗−0.242∗∗∗−0.631∗∗∗−0.473∗∗∗−0.389∗∗∗(1.74)(3.36)(4.81)(−14.7)(−10.2)(−8.36)(−16.9)(−11.9)(−9.42)−0.009∗∗∗−0.082∗∗∗−0.126∗∗∗(−5.39)(−9.47)(−11.0)0.006∗
0.037
0.037(1.84)(1.54)(1.14)−0.011∗∗∗−0.087∗∗∗−0.132∗∗∗(−9.86)(−13.3)(−14.2)0.001
0.005−0.010(0.73)(0.84)(−1.12)−0.001∗∗∗−0.001∗∗∗−0.002∗∗∗0.010∗∗∗
0.006∗∗∗
0.005∗∗∗
0.016∗∗∗
0.010∗∗∗
0.008∗∗∗
0.004∗∗∗
0.003∗∗∗
0.004∗∗∗
0.046∗∗∗
0.044∗∗∗
0.049∗∗∗
0.058∗∗∗
0.054∗∗∗
0.060∗∗∗
0.012∗∗∗
0.007∗∗
0.007∗∗
0.099∗∗∗
0.053∗∗∗
0.057∗∗∗
0.165∗∗∗
0.093∗∗∗
0.103∗∗∗
0.000
0.003∗∗
0.005∗∗∗
0.043∗∗∗
0.071∗∗∗
0.075∗∗∗
0.018
0.061∗∗∗
0.066∗∗∗

Appendix 1
p(θi )p(yi , m | θi )p(θi )p(yi , m | θi )dθi−∞p(θi |yi , m) = ∫ ∞∝ p(θi )p(yi , m | θi )
{
[
]}
(
)3
2
1 (yi − θ)2(m − (θi + θ−i ))11(θ − µ0,i )2+= √+2 exp − 222σi σm σ0,i
σi2
σm
σ0,i
2π
2

(yi − θi )2(m − (θi + θ−i ))(θi − µ0,i )2
+
+
2
2
2
σi
σm
σ0,i
1
1
1
= 2 (θi2 − 2yi θi + yi2 ) + 2 (θi2 − 2(m − θ−i )θi + m2 ) + 2 (θi2 − 2µ0,i θi + µ20,i )
σi
σm
σ0,i

(
) 
µ
m−θ−i
yi
+ σ0,iθi2 − 2 1 + 11 + 1
θi 


(
)
2 +
2
2


σ
σ
m
2
i
0,i


σm
σ2
σ2
i
0,i
1
1
1
]
[
2
=
+
+
)
(
2
2


σi2
σm
σ0,i
µ
yi
1
−i




+ m−θ
+ σ0,i2 + 1+ 1 + 1
σ2
σ2
σ2
i

2
σm

(

σ2
0,i

m

i

0,i

)2

yi
m
µ0,i
1
m − θ−iµ0,iyi+ 2 + 2 − 1
+ 2
1
1
2 +
2
σi2
σm
σ0,i
σ
σ
σ0,i
+
+
2
2
2
m
i
σm
σi
σ0,i

)
(
)2
(

1
1
m − θ−iµ0,i 
1yi
1+ 2 + 2++ 2=
θi − 1
2
1
1
2
2
σi
σm
σ0,i 
σi
σmσ0,i 
+ σ2 + σ2
σi2
m
0,i
(
)2
m
µ0,i
1
yi
m − θ−i
µ0,i
yi
+
+ 2
+ 2+ 2 + 2 − 1
2
σi
σm
σ0,i
σi2
σm
σ0,i
+ σ12 + σ12
σ2

+

m

i

0,i

Hence,
(

{
)3
]}
1
1[
1
2
p(θi | yi , m) ∝ √2 exp − 2 A (θ − B) + C
σy σm σ0,i
2π
{
}
{
]}
1
1[
2
2
∝ exp − A (θ − B) + C= exp − ( 1 ) (θ − B)
2
2 A
A=

1
1
1
+ 2 + 2
σi2
σm
σ0,i

(

)
yi
m − θ−i
µ0,i
B= 1
+
+ 2
2
σi2
σm
σ0,i
+ σ12 + σ12
σi2
m
0,i
(
)2
yi
m
1
m − θ−i
yi
µ0,i
µ0,i
+
C= 2+ 2 + 2 − 1
+ 2
2
σi
σm
σ0,i
σi2
σm
σ0,i
+ σ12 + σ12
σ2
1

m

i

Therefore,

0,i

{
]}
1[
2
p(θi | yi , m) ∝ exp − A (θ − B) + C
2
{
}
}
{
1
1
2
= exp − C exp − ( 1 ) (θ − B)
2
2 A
{
}
1
2
∝ exp − ( 1 ) (θ − B)

2 A

40

Appendix 2
From Equation (3),w12 c21 a1 = κ∗ λ∗1 , · · · , wn2 c2n an = κ∗ λ∗n∑∑ 2 2
wi ci aiκ∗ λ∗i∑ ∗ = ∑ ∗ = κ∗
λi
λi

(A1)

∑ 2 2
w c a
∑i ∗i i .
λi

(5)From Equation (3) and (A1),(∑2

wi ki ) am=
λ∗m

Appendix 3
We prove( n∑∀i ∈ {i = 1, 2, . . . , n},

)2
wi ki

am >

i=1

n
∑

wi2 c2i ai .

(6)

i=1

Let kmin be the minimum value of ki .
( n
)2
( n
)2
(
)2
n
∑
∑
∑
2
wi ki am ≥wi kmin am = kmin
wi am = kmin
am > 0
i=1

i=1

i=1

∑
2
2
am .
This means that the minimum value of ( wi ki ) am is a positive value kmin
On the other hand, let cmax be the maximum value of c2i ai .
n∑

wi2 c2i ai ≤i=1

Since limn∑n→∞ i=1n∑

wi2 cmax = cmax

n
∑

i=1

i=1

wi2 = 0 (Appendix 4 provides the proof) and

lim

n
∑

n→∞

wi2

n
∑
i=1

wi2 c2i ai > 0,

wi2 c2i ai = 0.

i=1

Taken together, for suﬃciently large n,
( n
∑

)2
wi ki

2
am ≥ kmin
am > cmax

i=1

n
∑

wi2 ≥

i=1

n
∑

wi2 c2i ai .

i=1

Appendix 4
We can write∀i ∈ n, wi =

1
+ εi
n

41

where

n
∑
i=1

εi = 0.

Since ∀i ∈ {1, 2, . . . , n},lim wi = 0,

n→∞

(
lim

n→∞

1
+ εi
n

)
= 0.

Hence,∀i ∈ {1, 2, . . . , n},lim εi = 0n→∞Using this,
lim

n→∞

n
∑

n (
∑
1

wi2 = lim

i=1

n→∞

i=1
n (
∑

n

)2
+ εi

)
1
2εi
2
+ εi
= lim
+
n→∞
n2
n
i=1
)
(
n
2∑
1
2
= lim
+
εi + εin→∞n n i=1n11∑εi + lim ε2i= lim+ 2 lim
n→∞
n→∞ n
n→∞ n
i=1
(
)∑
n
(
)(
)
1
1
= lim
+ 2 limlim εi + lim εilim εi
n→∞ n
n→∞ n
n→∞
n→∞
n→∞
i=1

=0



( n
∑

)2

Appendix 5
wi ki

am >

i=1

n
∑

wi2 c2i ai(i = 1, · · · , n)

(6)

i=1

We prove that the larger n is, the more likely Equation (6) holds. Let cmax be the maximum valueof c2i ai . The LHS of Equation (6) is
( n
∑

)2
wi ki

am ≥

i=1

( n
∑

)2
wi kmin

(
am =

kmin

n
∑

i=1

On the other hand,

i=1
n
∑

wi2 c2i ai ≤ cmax

i=1

Since from Appendix 4, limn∑n→∞ i=1

n
∑

wi2

i=1

wi2 = 0,lim

n→∞However, since wi2 c2i ai > 0,
lim

n→∞

n
∑

wi2 c2i ai ≤ 0

i=1

n
∑

wi2 c2i ai ≥ 0

i=1

42

)2
wi

2
am = kmin
am > 0.

(A2)

Taken together,
lim

n→∞

n
∑

wi2 c2i ai = 0

(A3)

i=1

From (A2) and (A3), the larger n is, the more likely Equation (6) holds.

43