Discussion Papers In Economics
And Business

Market Efficiency, Asymmetric Price Adjustment and Over-Evaluation:
Linking Investor Behaviors to EGARCH
Tatsuyoshi Miyakoshi
Yoshihiko Tsukuda
Junji Shimada

Discussion Paper 07-30

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANMarket Efficiency, Asymmetric Price Adjustment and Over-Evaluation:
Linking Investor Behaviors to EGARCH
Tatsuyoshi Miyakoshi
Yoshihiko Tsukuda
Junji Shimada

Discussion Paper 07-30
August 2007
この研究は「大学院経済学研究科・経済学部記念事業」
基金より援助を受けた、記して感謝する。
Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANMarket Efficiency, Asymmetric Price Adjustment and Over-Evaluation:
Linking Investor Behaviors to EGARCH*
by
Tatsuyoshi Miyakoshi a
Yoshihiko Tsukudab
Junji Shimada c
August 6, 2007

Keywords: investor behaviors; market efficiency; asymmetric response;
over-evaluation; EGARCHJEL Classification Number: G12; G14; C10aOsaka School of International Public Policy,Osaka Universityb

Graduate School of Economics, Tohoku UniversitycSchool of Management, Aoyama Gakuin University

Correspondence: Tatsuyoshi Miyakoshi, 1-31, Machikaneyama-machi, Toyonaka,
Osaka, 560-0043, Japan. tel:+81-6-6850-5638; fax:+81-6-6850-5656
E-mail: miyakoshi@osipp.osaka-u.ac.jp*Earlier versions of this paper were presented at 2006 Western Economic Association
Conference (in San Diego), 2006 Pacific Basin Finance, Economics, and Accounting
Conference (in Taipei), 2007 Asia-Pacific Economic Association Conference (in HongKong). We thank Mark Spiegel, Yao Lee, Leo Chan, Satoshi Koibuchi, Yushi Yoshida,
Henry Wan and participants for useful suggestions. The first and second authors’ researchwere respectively supported by Grant-in-Aid 16530204 and Grant-in-Aid 18530153 from
the Ministry of Education, Culture, Sport, Science, and Technology of Japan.

Abstract:

The paper incorporates a partial asymmetric price adjustment model for individual
investors action into an EGARCH model, clarifies the relationship between the priceadjustment speed, the market efficiency and asymmetric price adjustment, and measuresover (under)-evaluation of stock value. The stock price does not fully adjust to the market
value of stocks hoped by investors and does not adjust symmetrically in upturn anddownturn, if and only if the market is not efficient, and moreover the market value
generally diverts from the fundamental value of stocks even though the market is efficient.
As an operational example, the Tokyo stock market is found to be inefficient during
1980-2005. The speed of price adjustment is asymmetric in the 80s but symmetric in the
90s and 2000s. The over-evaluation of the market value is remarkably observed in the
80s but not in the 90s and 2000s.

11.

Introduction.
Figure 1 shows that that during 1980 and 1990 the TOPIX (Tokyo Stock
Exchange Price Index) goes up and down around the particular trend, and so we are
induced to think that (i) the stock prices in upturn adjust to its trend more quickly than theprice in downturn and (ii) sometime approaching to its trend instantaneously. However,even after adjustment toward a trend, (iii) the market value of stock is still evaluated overits fundamental value. These facts seem to support three propositions: (i) the asymmetricstock price adjustment by Koutmos (1998, 1999), the market efficiency by Fama (1976)
and the over (under)- evaluation of stock value by Lee (1999). Few previous papers
explain three propositions, being based on the investor behaviors. That is, how does theinvestor behave and as a result, how are the market price, the market value (or intrinsic
value), and the fundamental value decided? Can the previous model answer thisquestion?
The understanding of the investor behaviors related to three propositions can leadto the better interpretation for them. However, previous papers for the stock priceformation are silent about the market processes that might deliver the hypothesis on
investor behavior. However, there are, so far, some hypotheses on the investor behaviors.
The first idea was formalized by Grossman and Stiglitz (1980). In their theoretical model,
the market price does not fully incorporate all knowable information because informed
investors make returns by exploiting deviations of prices from security values.1 Second,
without explicitly characterizing investor behaviors, Busse and Green (2002) empiricallyfind that news reports about individual stocks on the financial television network CNBC
are incorporated into stock prices within few minutes. They shed light on the degree of
efficiency and conclude that the market is efficient enough that a trader cannot generate
profits based on widely disseminated news unless he acts almost immediately. Third,
Amihud and Mendelson (1987) and Koutmos(1998, 1999) developed and linked thepartial adjustment price model for investor behavior to an ARCH-type model. The stock
price is adjusted to ‘market values’ (they use ‘intrinsic value’ for it) by portfolio mangers
and the adjustment speed is different depending on whether the stock price is over orunder the market value.2
Later, Busse and Green (2002) found that the small profits available to very shorthorizon traders (i.e., informed investors) are consistent with compensation forcontinuously monitoring information sources, supporting the theory by Grossman andStiglitz (1980).
2 Koutmos (1998, 1999) also offered an empirical new finding that the adjustment speed
is faster when the stock price is over the intrinsic value. Motivated by his new finding,
12

The purpose of this paper is twofold. First, we provide the investor behaviors
with three propositions like the market efficiency in the framework of a partial
adjustment price model, the asymmetric stock price adjustment, the over(under)-evaluation of market value of stock, and finally link this partial price adjustment
model for investor behavior to an EGARCH model to test these propositions. Second, weshow an operational example of the model, by examining whether the Tokyo Stock
Market is efficient, the stock price in this market adjusts asymmetrically, and the market
value of stock is over (under)-evaluated against the fundamental value.
Where is our paper located, compared with the previous papers? First, thedefinition of market efficiency by Fama (1976) is based on informational concept. It ischaracterized as one in which security prices fully reflect all available information. In theframework of empirical researches, it is defined that the expectation of the returns
conditional on the previous information is constant, i.e. E{Rt | I t -1 } = constant, which is a
so-called weak-form of the market efficiency. Different from our framework, the concept
of Fama’s market efficiency was silent and slippery, however, about the market processes
that might deliver the hypothesis on investor behavior. Then, the concept of the market
efficiency should tell about the investor behavior to lead well-understanding. Second,
Koutmos (1998, 1999) offered an empirical new finding that the adjustment speed isfaster when the stock price is over the market value. Motivated by his new finding of theasymmetric stock price adjustment, several papers including Pagan and Soydemir (2001),
Bang and Shin (2003), and Nam et al. (2003, 2005) have got the same finding as
Koutmos (1998, 1999).However, Koutmos’s empirical framework is based on drastic
approximation more than our framework and then should be developed to confirm theempirical findings. Third, the market value (the stock price) is over(under)-valued against
the fundamental value. The fundamental value is determined by the fundamental factors
underlying the asset. Cheung and Ng (1998) and Gjerde and Saetterm (1999) related its
value to the macroeconomic variables. Hess and Lee (1999) and Lee (1999) related its
value to other financial variables such as rates of return, dividends, and earnings, while
there is no explicit theoretical explanation how the fundamental value is decided. In our
approach its value is exogenously given by the Dividend Discount Model with a constant
growth (which equals the growth rate of GDP).
The organization of the paper is as follows. Section 2 sketches a partial
adjustment price model by Amihud and Mendelson (1987) and Koutmos (1998, 1999) ,Pagan and Soydemir (2001), Bang and Shin (2003), and Nam et al. (2003, 2005) haveapplied Koutmos model and got the same finding as Koutmos (1998, 1999).

3

provides the mechanism how the fundamental value is decided exogenously, defines theasymmetric price adjustment, the market efficiency, over (under)-evaluated market value,and links the model to an EGARCH. Section 3 provides the estimation results of theTokyo Stock Market during 1980-2005 as an operational example. Section 4 gives
concluding remarks. Appendix gives proofs of propositions.

2. The Model
2. 1 Partial Asymmetric Adjustment Price Model and Market Efficiency
We follow a partial adjustment price model by Amihud and Mendelson (1987)
and Koutmos (1998, 1999) with some modifications. The model distinguishes theunobserved market value of stock (Vt ) from the observed stock price (P t ), both are
expressed in natural logarithms. 3 The process of market value follows a random walkprocess with drift:
Vt = a + Vt −1 + u t , u t | I t −1 ~ N (0, σ ut2 ), t = 1,..., T,(1)where a is constant and I t -1 denotes the information set of the time t-1. The market valueof stock is a value which investors in the market hope. This market value is different
from the so-called ‘fundamental value’ of stock. We assume that the disturbance term (ut)has the EGARCH process proposed by Nelson (1991):
log σ u2t = α 0 + α1 zt - 1 + α 2 (|z t - 1|- E(|z t - 1|)) + α 3 log σ u2t - 1
(2)

where
z t =ut / σ ut ~ N (0, 1) .

The partial asymmetric adjustment price process of Pt represents that adjustment
costs (θ + , θ −) are asymmetric in upturn and downturn markets:

Pt − Pt −1 = (1 − θ + )(Vt − Pt −1 ) + + (1 − θ − )(Vt − Pt −1 ) − , - 1 < θ + , θ − < 1,(3)

where (Vt - Pt-1)+ = max{ Vt - Pt-1, 0}, and (Vt - Pt-1)- = min{ Vt - Pt-1, 0}. If θ + = θ − (= θ),equation (3) reverts to the basic partial adjustment price process proposed by Amihud and
Mendelson (1987;p.536). Koutmos (1998; p.280, 1999; p.86) formulated the asymmetricadjustment to market value in (3). After the market value Vt is recognized at t, the stock
price Pt-1 is adjusted by (3) to Pt. Our adjustment process is based on the idea that priceapproaches its market value in competitive markets, so that the value can be estimated
Based on an idea by Black (1986;p.533), Amihud and Mendelson (1987;p.536) andKoutmos (1998; p.280, 1999; p.86) call this market value to be the intrinsic value.

3

4

from price directly.
The model consisting of (1),(2) and (3) is called a partial asymmetric adjustment
price model. As explained Koutmos(1998;p278), the partial price adjustment, θ + ≠0, θ −≠ 0 in (3), is attributed to the following costs which slow down the adjustment of asecurity’s price toward its market values.: 1) the cost of acquiring and processing
information; 2) the attempts of market specialists to create orderly markets and assure
price continuity; 3) the particular institutional market mechanism by which securities aretraded. Moreover, the possibility of the asymmetric adjustment (θ + ≠ θ −) is thatinvestors have a higher aversion to downside risk, so they react faster to bad news. The
use of stop-loss orders is an example of such aversion. Also, portfolio managers feel they
are penalized more if they underperform in a falling market than in a rising market. As
such, they are quicker to react to bad news. Similarly, for market makers, i.e., dealers andfloor specialists, the cost of not adjusting prices downward is higher than the cost of not
adjusting prices upward. Market specialists, who are required to maintain price continuity,
will find it easier and less costly to do so in a rising market than in a falling market, as thelatter involves building up inventory with overpriced securities.
Thus, Amihud and Mendelson (1987) and Koutmos (1998, 1999) provide theinvestor behaviors for the asymmetric partial adjustment price process, while in thisprocess they investigated neither the market efficiency, the asymmetric partial adjustment( which are analyzed in Section 2.1) and nor the over (under) –evaluation of stock, which
is given in Section 2.2. Also, they could not link this model to EGARCH model to test
the validity of the model in empirical sense, which is done in Section 2.3.
First, we define the market efficiency in the process as follows:
Definition 1: (i) The market is said to be efficient if θ + = θ − = 0 in (3) and inefficient
otherwise. (ii) The stock price adjustment is said to be asymmetric if θ +≠θ − andsymmetric otherwise.
This definition of market efficiency indicates that the speeds (1-θ + ) and (1- θ −)of price adjustment of both positive and negative discrepancy for the market value are
equal to unity in (3), namely the stock price instantaneously and fully adjusts to themarket value due to no adjustment costs (θ += θ −=0):
Pt − Pt −1 = (Vt − Pt −1 ) + + (Vt − Pt −1 ) − = Vt − Pt −1 , and hence Pt = Vt .

(4)

When 0 < θ +, θ − <1, the stock price partially adjusts for the market value. However,when θ +, θ − < 0, the stock price overshoots the market value. We test the null hypothesis

5

of market efficiency against the inefficiency of market as shown in (3):H0 : θ+ = θ− = 0 vs H1 : θ+ ≠ 0 or θ−

≠ 0.

(5)

The symmetric adjustment speed can be tested as
H0 : θ+= θ - vs H1 : θ+ ≠ θ - .

(6)Second, the digit of the asymmetric partial adjustment price model can beunderstood by the following proposition.

Proposition 1: (i) When the previous stock price Pt-1 is under (over) the market value Vt
at the present period t, the stock price Pt at the present period is adjusted to increase(decrease):
Vt -Pt-1 ⋚ 0 ⇔ Rt ⋚ 0(7)(ii) The market value Vt is computed by using the stock price Pt , return Rt and theestimated adjustment speed (1-θ + ) or (1-θ − ):Pt − Vt =R t (1−(where ξt ≡ 1 −θ

−

1

ξt

)

(8)

) + (θ −θ ) D , D = 1 for V − P ≥ 0, and D = 0 otherwise .
−

+

+
t

+
t

t

t-1

+
t

(iii) The market value V̂t in level can be expressed by stock price P̂t in level, returnand the estimated adjustment speed as follow:
ˆ =
V
t

P̂t
ˆ ≡ exp(V ) and Pˆ ≡ exp(P ), where Vtttt
exp{R t (1 − 1/ ξt )}Proof of Proposition 1:(9)

Rewrite the adjustment process (3) as

R t = (1 − θ ) ( Vt − Pt-1 ) D +t + (1 − θ − ) ( Vt − Pt-1 ) (1 − D +t )+={ (1 − θ ) + (θ − θ ) D } ( V − P )
−

−

+

+
t

t

t-1

= ξ t ( Vt − Pt-1 )

6

(10)

(

where D+t = 1 for Vt − Pt-1 ≥ 0, and D+t = 0 otherwise , and ξt ≡ 1 −θ−

) + (θ −θ ) D .
−

+

+
t

Then, the relation of (7) in (i) holds. On the other hand, (7) yields to Pt-1 +R t / ξt = Vt . By
using this relation, we get the (8) in (ii): Pt − Vt =Pt − Pt-1 −Rt

ξt= Rt (1 −1ξt) . We can express

the fundamental value by using (8) and then express its value in level by the stock price,
return and the estimated ξt as shown in (9) of (iii).

■

After the market value Vt hoped by investors is found at the beginning of period t
and compared with the previous stock price Pt-1 (i.e., Vt -Pt-1), the stock price Pt at period t
is adjusted to increase when Vt -Pt-1>0 and then Rt(=Pt -Pt-1)>0, as shown in (3) and (7).
However, the adjustment speed, 0<(1-θ + ), is less than one (i.e., partially), so that Pt isstill Pt<Vt shown in (3) and (8) after adjustment. Moreover, the market priceVt is
computed by using stock price Pt , return Rt and parameters θ + and θ −. The eq.(9) is alevel format of Vt .

2.2. Over (under) –evaluation of the market value
Is the market value of stock over- (under-) evaluated, compared with thefundamental value? We provide the fundamental value of stock by using a dividend
discounted model with a constant growth rate λ, which decides the fundamental value Stof stock at period t and t-1 as follows:∞
D0 (1 + λ ) j
D1 (1 + λ ) j
1+ λ
1+ λ
St ≡ ∑=() D1 , St −1 ≡ ∑
=(
)D
j
j
ρ −λ
ρ −λ 0
j =1 (1 + ρ )j =1 (1 + ρ )∞(11)where ρ is a discount rate and D1, D0 are a dividend at t and t-1, then the dividend grows
at the rate λ (ρ>λ) and D1 = D0 (1 + λ ) . A dividend discounted model determines thefundamental value to equal the discounted value of dividend stream with a constant
growth rate λ. We assume that λ equals the growth rate g of GDP per day (which is aconverted rate from per year basis to per day basis). Therefore, the growth rate η of thefundamental value is expressed by the growth rate λ of dividends and then the GDP
growth rate g as follows:

7

St
D
−1 = 1 − 1 = λ = g
St −1
D0

(12)

Then, given S0, we can compute the fundamental value St as :St = S0 (1 + g )t

(13)

Here, we can define the over (under)-evaluation of market value.

ˆ >S in (9) andDefinition 2: The market value of stock is said to be over-evaluated if Vtt(13), under-evaluated if V̂t < St , and normal otherwise.

This definition evaluates the value of stock in level as well as usual. This definition is
first characterized by the comparison between market value and fundamental value.
Second, since the stock price includes the noise like the adjustment delay to the market
value, we did not use the stock price in the definition. Hess and Lee (1993) and Lee(1999) related its value to other financial variables such as rates of return, dividends, andearnings. However, in our idea, the fundamental value is exogenously given by thedividend discount model.
Koutmos (1998, 1999) only incorporates the asymmetric effects to the partial priceadjustment model of Amihud and Mendelson (1987), though he did deal with neither
market efficiency nor over (under) –evaluation of stock prices based on the investor
behavior.
If we can get the parameters θ+ and θ- ( i.e., ξt ) in (5),(6), (8),(9) and (13), we caninvestigate whether the market is efficient, the stock price adjustment is asymmetric, themarket value of stock is over(under)-evaluated. Let us estimate the parameters θ+ and θin the next section

2. 3

The Reduced Model
We have an autoregressive process for the returns.

Proposition 2: (i) The return process consisting of (1) and (3) has the following

8expression
Rt = a + θ + R *t -+1 + θ − R *t -−1 + u t ,*(14)

where Rt = ξ t−1 Rt , Rt∗+−1 = ξt−−11Rt+−1 (Rt+−1 ≡ Max{Rt −1 ,0}), Rt∗−−1 = ξt−−11Rt−−1 ( Rt−−1 ≡ Min{Rt −1 ,0}) ,*and ξ t = 1 − θ + ( ≡ ξ + for all t) for R +t , and ξ t = 1 − θ − ( ≡ ξ − for all t) otherwise.

−1

Proof of Proposition 2: Noting the facts Vt = ξ t Rt + Pt -1 from (8), 0 < ξt <1 from (3),hence Vt − Vt -1 = ξ t−1 Rt − ξ t−-11 Rt −1 + Pt -1 − Pt -2 = a + u t . Considering the lastequation, we haveξt−1 Rt = a + (1 − ξt −1 )ξt−−11 ( Rt+−1 + Rt−−1 ) + u t(15)

where Pt-1 − Pt-2 =R t-1 =R +t-1 +R −t-1 . Considering (7), R +t-1 =R t-1 ≥ 0 in (15) corresponds(

uniquely to Vt-1 − Pt-2 ≥ 0 in (10) which decide uniquely the ξt-1 (≡ 1 −θ−

) + (θ −θ ) D ).
−

+

+
t-1

Then, the R +t-1 =R t-1 ≥ 0 can decide the ξt-1. As a symmetrical to this, theR −t-1 =R t-1 ≤ 0 can decide the ξt-1. The return process has the expression in (14).

■The model (14) with (2) is an EGARCH model. Equation (14) is alternatively
expressed as
Rt = ξ t (a + β + R +t -1 + β − R −t -1 ) + ε t ,(16)

where β + = (ξ + ) −1θ + , β - = (ξ − ) −1θ − and εt = ξ t u t .4 The process of Rt in (16) isapparently similar to that of Rt*

in (14). However, except for the case of θ + = θ −, theconditional expectation of εt is not zero and the process of {εt} is serially dependent: see

4

If θ + = θ − (= θ) and then ξt=1-θ, equation (16) reduces to
Rt = a(1 - θ ) + θ R t -1 + (1 − θ )u t .
9Lemma 1 of Appendix in Tsukuda, Miyakoshi and Shimada (2006). The conditionalvariance of εt does not follow an EGARCH process unlike to that of ut. Then, the joint
density function based on equation (16) and (2) is expressed as follows.
Proposition 3: The joint density function of {R1, . . . , RT} is given by⎫⎪⎧⎪(ξ +σut )-1 φ((Rt − ξ +µt )/ξ +σut )⎪
⎪
⎨
⎬
Π
+⎪
⎪
Φ(µt/σut )t ∈T ⎪⎪⎩⎭-1⎧ (ξ σ ) φ((R − ξ µ )/ξ σ )⎪
⎫
⎪
ut
t
t
ut ⎪
. Π ⎪
⎨
⎬
⎪
Φ(-µt/σut )t ∈ T− ⎪⎪⎪⎩⎭pdf(R1, . . . ,RT|ω) =(17)

where µ t = a + β + R +t -1 + β − R −t -1 , T+ = {t | Rt ≥ 0, t ∈ N }, T- = {t | Rt < 0, t ∈ N }, N ={1, . . . , T}, and ω = {a, θ+, θ− , α0, α1, α2, α3} is a vector of unknown parameters,and Φ and φ respectively denote the distribution and density functions of the standard
normal distribution.
Proof of Proposition 3:⎧ξ + µ t + ξ + u t
Rt = ⎨ −−⎩ξ µ t + ξ u t

From (16), the return process is rewritten as
for R t ≥ 0
for R t < 0

.

(18)

The conditional density of Rt given It-1 is written by⎧⎪ +⎪⎪ ( ξ σ ut )-1 φ((R t − ξ + µ t )/ ξ + σ ut )⎪⎪forR t ≥ 0
Φ( µ t / σ ut )⎪⎪pdf(R t ; ω |I t-1 ) = ⎪⎨⎪⎪⎪⎪( ξ - σ ut )-1φ((R t − ξ - µ t )/ ξ - σ ut )⎪⎪for R t < 0⎪⎪Φ(-µ t / σ ut )⎩(19)

Substituting equation (17) into the following relation
pdf(R 1 ,..., R T | ω ) = Π pdf(R t ; ω | I t -1 ) ,

(20)

t ∈T

the required joint density in (17) is obtained.

■

The model of (16) is estimated by the maximum likelihood estimation method using thejoint density of (17).Thus, we get the parameter set of ω including (θ+ , θ−), which

10provides the market efficiency, asymmetric price adjustment, and over(under)-evaluationof stock value.
Equation (7) in Koutmos (1998, pp. 280) takes too rough approximation and thenshould be developed as equation (16) in this paper. That is, since the conditional
expectation of εt is not zero and {ε t }is serially dependent, then ε t can not follow the
GARCH model. Though Koutmos (1998) finds empirically useful facts about the stock
market movements, the partial asymmetric adjustment model does not logically induce
the Threshhold GARCH model which is used for his empirical studies. Extending
Koutmos (1998), equation (4) of Koutmos (1999, pp. 86) introduces an error term in theasymmetric price adjustment process. However, the error term in equation (4) causes
discrepancy of stochastic orders between ut and ετ in his equations (4) and (5) because εtis expressed as difference of ut and ut-1. In other words, if ε t is an I(0) process, then ut
becomes an I(1) process.

3. An Illustrative Example
We investigate the Tokyo Stock Market for illustrating how our model works. The
daily closing stock price data of the Tokyo Stock Exchange Price Index (TOPIX) arepurchased from the Data Base of Nomura Research Institute, JAPAN. Figure 1 indicatesthe data of stock prices in natural logarithms and returns from January 4, 1980 to
December 2, 2005. It shows the up-trend to the end of 80s, but the down-trend in 90s and2000s. The returns move mildly in the former period, while they greatly fluctuate in thelatter one. Based on these visual observations, the sample period is divided into the two
sub-periods: the first is from January 4, 1980 to the end of 80s, the second is from the
beginning of 1990 to December 2, 2005.
The estimation results of the parameters in the model (16) with (2) are shown in
Table 1. The estimates of the drift term in equation of (16) are positive in the firstsub-sample but zero in the second, supporting the up-trend of stock prices in the 80s andmostly the down-trend in the 90s and after. The estimates of α1 and α3 reveal asymmetricvolatility and variance persistence, supporting the stylized facts for stock pricemovements. The estimates of θ+ and θ - are positive and significant at 1% level for eachsub-sample period. By using estimated θ+ and θ -, we can estimate Vt due to (8) or
Proposition 1-(ii) and can find how much the stock prices (i.e., closing price of the day)are over (under) market value hoped by the investors.
The first main interests of this study are market efficiency and asymmetric price11

adjustment speeds to market values. Figure 2 provides the visual inspection, showing that
|Vt-Pt-1| in the first period is larger than that in the second period. The adjustment speed(1-θ+ , 1-θ -) in the first period is slower (smaller) so that the market is less efficiency inthe first period than the second period. Figure 3A shows the plotted data of Vt-Pt-1 and Rt
in the first period. The coefficients of both variables imply (1-θ+) in the first quadrant+and (1-θ ) in the third quadrant, which show (1-θ ) < (1-θ ). That is, the adjustment
speed is asymmetric as found out by Koutmos (1998) and others. However, Figure 3B inthe second period shows symmetric adjustment speed, (1-θ+) = (1-θ -). Table 2 shows thestatistical test that the null hypothesis of market efficiency (H1: θ+=θ -=0) is rejected inboth sub-samples. Thus, the market is inefficient in the sense of adjustment speed. The
null hypothesis of symmetric adjustment speeds (H2: θ+ = θ -) in equation (3) is rejectedin the first sub-sample but not in the second.
How do we interpret the findings? In our framework, the observed values of stock
price (i.e., closing prices of the day) are adjusted towards the market values. The
adjustment speeds are (1-θ+) if Vt − Pt -1 ≥ 0 (or equivalently R t ≥ 0 from Proposition
1-(i)) and (1- θ−) if Vt − Pt -1 < 0 ( R t < 0). In particular, in the first sub-sample the speed is(1-0.289) for the case of under-market value (then, positive return), and (1-0.216) forover-market value (then, negative return). The adjustment speeds are significantly
different in this period. Koutmos (1998, pp. 285) finds the asymmetric adjustment speeds
in many stock markets and argues as “….One possibility is that investors have a higher
aversion to downside risk, so they react faster to bad news ( Vt − Pt -1 < 0). The use ofstop-loss orders is an example of such aversion. Also, portfolio managers feel they are
penalized more if they under-perform in a falling market than in a rising market….”[ INSERT Figures 1, 2, 3, 4 and Tables 1, 2]

The second main interest of this study is whether the market values Vˆt in thelevel bases are evaluated over (under) the fundamental values St . Based on Definition 2,
the market value and the fundamental value in level bases are illustrated in Figure 4. Bothvalues are computing by using (9) with the estimates in Table 1 and by using (13) with
g=const per daily bases in each year (while it is changeable over the years)
and S0 =Vˆ Jan/5/80 equal to the estimated market value 460 at January 5, 1980.5 As shown in
For example, the operating days of the market is about 280 days per year in Japan.
Then, the average growth rate per year of GDP in 80 is 0.075 and the growth rate per512Figure 4, the market value in 80s is obviously evaluated over the fundamental value,
while in 2002-2004 it is evaluated mostly under the fundamental value.

4.

Concluding Remarks
We have defined and analyzed three propositions such as the market efficiency inthe framework of an asymmetrical partial price adjustment model of the revised versionof Koutmos for investor behavior, the asymmetric stock price adjustment, and the over(under) -evaluation of market value of stock, and finally linked the model to an
EGARCH model to test these propositions. The understanding of the investor behavior
related to three propositions can lead to the better interpretation for them. In addition, we
add new view points of market efficiency and over (under)-evaluation to his model anddevelop its model. Second, as an operational example of the model, we pick up the Tokyo
Stock Market. We find that the Tokyo stock market is inefficient during 1980-2005 in thesense that the adjustment speeds to the market values are less than 1. In the 80s, the stock
price adjustment is faster when the price is above the market value than when the price isbelow it: the adjustment cost is cheaper when the stock price is above the market value
and then the adjustment speed is faster. The market value in 80s which investors hope isevaluated over the fundamental value, while it is relatively weekly over-evaluated in 90s
and 2000s.
There have never been trials to investigate what kinds of investor behaviors lead tothe market efficiency, partial asymmetric adjustment price and over (under)-evaluation ofmarket value, by incorporating it into the ARCH-type model.

operating day of the market is 0.000258=EXP((1/280)*LN(1.075))-1. For 1990, that is
0.000265 =EXP((1/280)*LN(1.077))-1 where that for GDP per year is 0.0773. Data source
for GDP is from the annual estimates of GDP by Cabinet Office of Japan(Base-year=1995; calendar year; nominal GDP; http://www.esri.cao.go.jp/jp/sna/qe052-2/ritu-smcy0522.csv).
13

References
Amihud, Y., and Mendelson, H.(1987), "Trading Mechanisms and Stock Returns:

An Empirical Investigation", The Journal of Finance, 42,533-553.
Bahng, J.S. and S.Shin (2003),"Do stock price indices respond asymmetrically?
Evidence from China, Japan, and South Korea", Journal of Asian Economics, 14,
541–563
Black, F.(1986), "Noise",The Journal of Finance, 41,529-543.
Busse, J.A.and Green, T.C.(2002), " Market efficiency in real time",Journal of Financial
Economics, 65, 415–437.
Cheung, Y. W., & Ng, L. K. (1998), " International evidence on the stock market andaggregate economic activity", Journal of Empirical Finance, 5, 281–296.
Fama, E.F. (1976), Foundations of Finance, Basic Books, New York.
Gjerde, O., & Saettem, F. (1999) , "Causal relations among stock returns andmacroeconomic variables in a small open economy", Journal of International
Financial Markets, Institutions, and Money, 9, 47–61.
Grossman, S.and Stiglitz, J.(1980), " On the impossibility of informationally efficient
markets", American Economic Review, 70, 393–408.
Hess, P. J., & Lee, B. S. (1999), "Stock returns and inflation with supply and demand
disturbances", Review of Financial Studies, 12, 1203–1218.
Koutmos,G..(1998), "Asymmetries in the conditional mean and the conditional variance:
evidence from nine stock markets",Journal of Economics and Business 50, 277-290
Koutmos(1999), "Asymmetric Price and Volatility Adjustments in Emerging Asian Stock
Markets", Journal of Business Finance and Accounting, 26,83-101.
Lee, B. S. (1998), " Permanent, temporary and non-fundamental components of stock
prices ", Journal of Financial and Quantitative Analysis, 33, 1–32.
Nam, K., C. S. Pyun and S. W. Kim (2003), "Is asymmetric mean-reverting pattern instock returns systematic? Evidence from Pacific-basin markets in the shorthorizon", International Financial Markets, Instituitions and Money, 13,481-502.
Nam, K., Washer,K.M. and Chu, Q.C.(2005), "Asymmetric return dynamics and technicaltrading strategies",Journal of Banking & Finance, 29, 391-418.
Nelson, D. B. (1991), "Conditional Heteroskedasticity in Asset Returns:A New
Approach", Econometrica, 59, 347-370.
Pagan, J. A. and Soydemir, G. A. (2001), "Response asymmetries in the Latin American
equity markets", International Review of Financial Analysis, 10, 175–185.
Tsukuda,Y., Miyakoshi,T.and Shimada,J.(2006),"A Partial Asymmetric Price, Adjustment
Model and Fama’s Market Efficiency ", TERG Discussion Paper Series, No.209.

14

Figure 1. TOPIX and Its Returns
TOPIX (in natural logarithms)

8
7.5
7
6.5
6
Jan-80Jan-85Jan-90%Jan-95Jan-00

Jan-05Jan-00

Jan-05Return

20
10
0
-10
-20
Jan-80Jan-85Jan-90Jan-95Figure 2. Adjustment of the stock prices Pt towards market value Vt:
Market Efficiency
Vt - Pt
0.1

0.05

0

-0.05

15

Jan-04Jan-02Jan-00Jan-98Jan-96

Jan-94Jan-92Jan-90Jan-88Jan-86

Jan-84Jan-82Jan-80-0.1

Figure 3. Adjustment of the stock prices based on Vt-Pt-1:
Asymmetric Adjustment Price
A. 1980-1990
0.1

Rt

0.05

Vt-Pt-1
0
-0.1

-0.05

0

0.05

0.1

-0.05

-0.1

B. 1990-2005
0.1
Rt

0.05

Vt-Pt-1
0
-0.1

-0.05

0

0.05

-0.05

-0.1

16

0.1

Figure 4. Over(under)-evaluation of market value in level
3000
2500
2000
1500
1000
500

Jan-04Jan-02Jan-00Jan-98Jan-96

Jan-94Jan-92Jan-90Jan-88Jan-86

Jan-84Jan-82Jan-800Market Value Vˆt () and Fundamental Value St ()17Table 1. Estimates of Parameters
Periods

1/4/80 –12/ 28/89

1/4/90 – 12/2/05

a0.055*

0.006(6.45)(0.45)

0.288*

0.104*(12.02)(6.69)

0.216*

0.110*(5.94)(5.27)-0.144*-0.092*(-8.55)(-9.01)0.932*

0.965*(94.47)(163.47)

θ+

θ−

α1

α3

Notes: The numbers in parentheses denote t-statistics. The asterisk "*" issignificant at 1% level.

Table 2. Testing Results
Null Hypothesis

Distribution

1/4/80 –12/ 28/89

1/4/90 –12/2/05

H1: θ+ = θ−=0

χ2(2)161.36*49.86*

H2 : θ+ = θ−

χ2(1)

8.33*

0.08

Note : The asterisk "*" is significant at 1% level. The critical values of χ2(2) andχ2(1) distributions are respectively 9.21 and 6.34 at 1% level.

18