Weak Form Market Efficiency Hypothesis Testing –
A Price Asymmetry Viewpoint
Sheng-shyr Cheng, Associate Professor, Business Administration, Chang Jung Christian University
Hwey-chyi Lee, Associate Professor, Department of Finance, Chang Jung Christian University
ABSTRACT
Symmetric attitude towards risk is a basic assumption in the theory of finance. This point of view is challenged.
The purpose of this research is to extend those studies done by Roberts (1959), Keane (1983), Fama (1970, 1976, 1990),
Danthine (1977), and Duffy-Deno (1996) and investigate two aspects of a price adjustment to new information, namely
the speed and quality (or direction and magnitude) of the adjustment. In this paper, a security pricing model with the
consideration of price change asymmetry is derived first, based on Fama’s weak form efficiency hypothesis and
martingales models. To achieve research purpose, weekly adjusted stock price data of publicly traded firms in Taiwan’s
high technology industries were collected from TEJ data bank. Sample period is from 1999/10/22 to 2004/10/22. In
total, 135 stocks were analyzed. Changes in stock price were used as dependent variable in multiple regressions, while
one to five periods of lags of changes in stock prices were the independent variables. Empirical results suggest that
efficient market hypothesis cannot be rejected for those stocks are heavily traded or are major components of Taiwan
Market Index. However, efficient market hypothesis is rejected for those stocks which are issued by secondary
companies.
Keywords: Efficient market hypothesis, weak-form efficiency, price asymmetry
INTRODUCTION
There exists a large literature that defines and models prices set in efficient markets (Fama 1970, 1976, 1990;
Keane 1983; Lo 1995；Timmermann & Granger 2004). Ultimately, the consequence of the efficiency of a market is that
prices always fully reflect all available information. Fama (1970) was the first scholar who defined three types of
efficient markets. The first one is weak form market efficiency in which the information subset of interest is past price
histories. Second, semi-strong market efficiency concerns the speed of price adjustment of other obviously publicly
available information. Finally, strong form market efficiency concerns whether any investor has monopolistic access to
any information relevant for the formation of prices have appeared. This study will focus on the weak form market
efficiency hypothesis testing.
Roberts’ (1959) paper is one of the earliest papers on weak form market efficiency. He found that weekly changes
of the Dow Jones Index behaved very much as if they had been generated by a simple chance model. The most
important contribution of his paper is to show that the accumulation of random shocks creates apparent patterns, similar
to the behavior of actual stock prices. He made the point that changes in stock prices caused by the arrival of new
information should be random, so the level of stock prices follows a random walk.
Symmetric attitude towards risk is a basic assumption in the theory of finance. Also, there exists symmetric random
walk property. In this paper, the symmetric point of view is challenged. Many previous studies on weak form market
efficiency focused only on the direction of level of price movement, however. Financial theories maintain that only the
patterns of the past stock prices need be studied (Roberts 1959). Few studies tried to relate runs of price changes
(Roberts 1959) and magnitude of the adjustment (Keane 1983) into empirical hypotheses settings. Hence, Roberts (1959)
invited scholars to analyze price changes as well as price levels.
The purpose of this study is to extend those studies done by Roberts (1959), Keane (1983), Fama (1970, 1976,
1990), Danthine (1977), and Duffy-Deno (1996) and investigate two aspects of a price adjustment to new information,
namely the speed and quality (or direction and magnitude) of the adjustment. According to Fama’s weak form efficiency
hypothesis and martingales models, a security price change model with the consideration of price change asymmetry is
derived. Empirical multiple regressions are also established, in which the asymmetric movements of past stock prices
changes in the information set are used as independent variables. To achieve research purpose, weekly adjusted stock
price data of publicly traded firms in Taiwan’s high-tech industries were collected from TEJ data bank. Sample period is
from 1999/10/22 to 2004/10/22. Empirical findings of this study would improve our understanding of efficient market
literature. Hence, this study has both theoretical and empirical contributions to this topic.
RELATED LITERATURE
The history of security prices has probably been widely and intensively studied. The followings will, however,
focus on research related to this study only. For example, Cowles & Jones (1937) used the sum of directions of
successive changes of returns to test whether stock prices move randomly. They defined an indicator, I, to reflect stock’s
up or down movements. That is,
It = {1, if rt = pt - pt-1 > 0,
{0, if rt = pt - pt-1≦0,
where r1, r2,…, rn+1 are known. And define
Yt= It It+1 + (1 - It)(1 - It+1).
Then, sum of Yt equals 1 is the total number of price moves up during two time periods. Similarly, sum of Yt equals
0 represents the total number of price moves in different directions during two periods. Cowles & Jones (1937) showed,
if a stock moves randomly, then (pt - pt-1) will be a random variable with an independent and identical distribution.
Researchers could use the ratio of the total numbers of up movements and down movements to test whether stock price
moves randomly. Cowles & Jones (1937) showed that the above ratio would follow a normal distribution with mean 1
and variance 12/n.
Mood’ (1940) approach slightly differed from that of Cowles & Jones (1937). Mood used the size of runs to
indicate the randomness of a stock price. When the size of runs (kn) is either too small or too big, a stock price would
not have randomness. Mood has showed that if a price moved either up or down with the same probability (0.5), then
one could get a statistic tn=(2kn+1-n)/(√n) which follows a standard normal distribution. Therefore, one could use
Mood’s tn statistic to test the randomness of stock prices.
Roberts (1959) criticized financial theories focused only on the patterns of the past stock prices and made the point
that changes in stock prices caused by arrival of new information should be random, so the level of stock prices follows
a random walk. He showed that the accumulation of random shocks creates apparent patterns, similar to the behavior of
actual stock prices. Roberts also invited scholars to analyze price change as well as price level movements. In addition,
he argued that multiple regressions would be sounder if based on independent changes rather than dependent levels.
This study will response to Roberts’s call and address a question which he once posed: How well do price change
patterns apply to individual securities?
The concept of market efficiency is closely related to the process of price formation. In order to make the concept
testable, Fama (1970, 1976) used expected return theory to illustrate what he meant about market efficiency. What
expected return theory says is that
(1) E[Pt+1|Ωt ]= [1 + E(R t+1|Ωt )] * Pt ,
where E is the expectation operator; Pt is the price of security at time t; Pt+1 (a random variable) is its price at t+1;Ω
is a general symbol for whatever set of information is assumed to be fully reflected in the price at time t; R (a random
variable as well) is the proper discount rate. Both random walk and martingale models are two special cases of expected
t
return theory.
Fama (1970) was the first scholar who defined three types of efficient markets. The major difference among three
types of efficient markets is the contents in the information set. The first one is weak form market efficiency in which
the information subset of interest is past price histories. To test whether weak form market efficiency hypothesis holds,
researchers can conduct serial correlation tests, runs test, or distribution tests for an individual security. One could also
follow CAPM arguments to test multiple security expected return model (Fama 1970). Or, with full use of past price
data as well, one can replace random walk models by the more appealing martingale models (Sharpe 1970; Danthine
1977).
Martingales are one of the central tools in the modern theory of finance (Neftci 2000, ch. 6). A process {St, tε[0,
∞] } is a martingales with respect to the family of information set Ωt, and with respect to the probabilityΡ, if for all t>0,
a. St is known, given Ωt.
b.Unconditional forecast are finite: E|St|<∞.
c. And if E[ST|Ωt ]= St, for all t<T, with probability 1. That is, the best forecast of unobservable future values is the last
observation on St .
According to the definition, E[St+u- St|Ωt ]= 0, u>0. This means that martingales are random variables whose future
variations are completely unpredictable given the current information set. The best forecast of the changes in random
variable over an arbitrary interval is zero. In other words, the directions of the future movements in martingales are
impossible to forecast. These properties of martingales are very important and will be used to test weak form market
efficiency hypothesis.
Danthine’s (1977) model is an application of (sub-) martingale. He wanted to know (1) Under which conditions is it
feasible to describe equilibrium in efficient spot commodity markets in terms of expected returns? (2) Is the expected
return assumption necessary to test market efficiency and what alternative can be proposed? Following Fama (1970),
Danthine conducted a test of efficiency checking the impossibility of various trading system. Based upon the assertion
that the zero expected net profit rule translates itself into a relationship between any price and the expectation of its next
period realization that can be described as E[e -rt Pt|Ωt-1 ]=Pt-1, a conditional expectation on the information available in
t-1, the discounted present value of Pt is the last observation on Pt. Danthine showed that if a stock price exists a pure
martingale property, then E[Pt+1- Pt|Ωt ] =0, and cov(Pt+1 - Pt, Pt - Pt-1) =0.
It would be an interesting research combining Cowles & Jones (1937), Mood (1940) and Roberts (1956) arguments
about the different directions of stock price movements, and Danthine’s (1977) martingale model to test weak form
market efficiency hypothesis for individual securities. Before showing how to form this test, one more piece of
literature needs to be brought up. Duffy-Deno (1996) was interested in how the speed and magnitude of oil prices
transmit from wholesale level to retail level. He first separated wholesale oil price movements into two different
patterns. One pattern shows the wholesale oil price goes up if △ Pupt-1= Pt-Pt-1 if Pt-Pt-1 > 0. The other pattern shows
wholesale oil price goes down if △ Pdownt-1=Pt-Pt-1, if Pt-Pt-1 <0. Then, regress changes of retail price on lags of
wholesale up prices and down prices. In this approach, Duffy-Deno (1996) could check how retail oil price responded
differently to two different wholesale price changes and whether there are symmetric effects between up and down
movements. Similar approach will be applied to check how individual security price would change differently.
MODEL SETTING
According to Fama (1970), efficient market would result from an ideal world in which (a) there are no transaction
costs, (b) all relevant information is costlessly available to all market participants, and (c) all agree on the implications
of current information for the current price and the distributions of future prices. As a result of (a) and (b), it is claimed
that all information potentially leading to expected profits is exploited up to the point where these profits do not exceed
the ‘normal’ returns level available in other places (Danthine 1977). But though transactions costs, information that is
not freely available to all investors, and disagreement among investors about the implications of given information are
not necessary sources of market inefficiency, they are potential sources. Measuring these effects on the process of price
formation is the major goal of empirical study on market efficiency (Fama 1970).
Testing efficient market hypothesis is an empirical issue. As Fama (1970) has pointed out, the martingale model is
based on two hypotheses: (a) the efficient utilization of information, and (b) the possibility of expressing market
equilibrium in terms of expected returns. Following Fama’s efficient market hypothesis (1970; 1976), a security price at
time t can be expressed as eq. (1) if information set at time t can fully reflect past information,
(1) E[Pt+1|Ωt ]= [1 + E(R t+1|Ωt )] * Pt ,
where E is the expectation operator, Ωt is the information set at time t, and R is the proper discount rate.
As discussed in Danthine (1977), LeRoy (1976, 1989), Huang (1985) and Neftci (2000), if a security market can
be equilibrium and for sure be a fair game, then equation (2) holds,
(2) E[Pt+1|Ωt ]= Pt.
The best forecast of unobserved future values is the last observation on Pt. In other words, the best forecast of the
changes in random variable over an arbitrary interval is zero,
(3) E[Pt+1- Pt|Ωt ] =0.
The directions of the future movements in martingales are impossible to forecast.
If eq. (2) correctly describes the equilibrium condition for P t, it is possible to obtain a testable implication of
efficiency in terms of the observed security past price series. That is, when Pt is a pure martingale, eq. (2) or (3) can be
rewritten as (Danthine 1977),
(4) E[Pt+1 - Pt|Pt - Pt-1 ] =0. And
(5) cov(Pt+1 - Pt, Pt - Pt-1) =∫(Pt - Pt-1)E(Pt+1 - Pt|Pt - Pt-1)Ψ(Pt - Pt-1)d(Pt - Pt-1)=0,
provided a stationary distributionΨexists. Therefore, it is possible that one can look at a simple adjustment of the price
series for which pure martingale condition holds and test the zero serial correlation for the sequence of these price
changes. One possible extension of Danthine’ (1977) expression of past price difference is,
(6) E[Pt+1 - Pt|Ωt] = E[ΔPt+1|(Pt -1- Pt-2), ( Pt-2 - Pt-3),…, ( Pt-r+1 - Pt-r) ]
= E[ΔPt+1|ΔPt-1, ΔPt-2, ΔPt-3 ,…, ΔPt-r+1].
Extend the runs test suggested by Hagerman & Richmond (1973) and Duffy-Deno (1996) approach to separate
individual security price movements into two different patterns. That is, we define two dummy variables. One is
△ Pupt-1= Pt-Pt-1 if Pt-Pt-1 > 0, otherwise △ Pupt-1=0. The other dummy variable is △ Pdownt-1=Pt-Pt-1, if Pt-Pt-1 <0, otherwise
△ Pdnt-1=0. Therefore, eq. (6) now becomes eq. (7) in a multiple regression format
(7)
where Ϊi are regression coefficients to be estimated, r and s are the lags of up and down periods, respectively. And
εt is the regression error term and its distribution is assumed to follow (5). It is not necessary to requireεt to follow an
independent and identical distribution. Nor it is required to be a normal distribution.
It is now possible to setup hypothesis testing for weak form market efficiency. It is expected that, under market
efficiency hypothesis, allΪi are to be zeros. If not, the underlying stock price is not informationally efficient. For those
individual stocks are not informationally efficient, further hypotheses are possible to conduct. Following Duffy-Deno
(1996), one can test whether the number of up movement r equals the number of down movement s. If r is not equals to
s, then there exists a timing asymmetry. Second, one can test magnitude asymmetry. That is, one tests whether

r
i  0 α1,i=

s
j  0 α2,j
or not. Third type of price asymmetry test is called pattern asymmetry. One can test whether
α1,i=α2,j when r equals s. The hypothesis of weak form market efficiency or pure martingale will be tested with weekly
price movement data.
EMPIRICAL RESULTS AND DISCUSSION
This study uses individual securities of publicly traded firms in Taiwan’s high technology industries. There are
several reasons to do so. First, daily trading volume of these high-tech stocks accounts for more than 80% of the whole
Taiwan capital market. The frequency of turnover is high. Information of past prices and volumes are very freely
available to all investors. Second, these high-tech stocks cover several sub-industries, such as semi-conductor
manufacturing, printed circuit board (PCB) manufacturing, passive devices manufacturing, computer and information
manufacturing, optical industry and network industry. Firms of up stream and down stream make a whole and complete
representation of a specific industry. Also in each industry, there are primary leading firms and secondary firms. It is
possible to compare the price movements of these firms and tell how they change in different formats. Third, the
numbers of high-tech companies are growing. In any specific industry, there are mature firms and young firms. So, it is
possible to compare the price movements of these young and ‘old’ companies. Finally, trading histories of most firms
are long enough to conduct time series study.
Samples used in this study were collected from TEJ data bank. Sample period is from 1999/10/22 to 2004/10/22.
They are weekly adjusted prices. In this 5-year period, each stock has more than 250 data points. In total, 135 stocks
with complete trading history were chosen and analyzed.
The first task needs to be done is to decide the numbers of lags in up and down variables in equation (7). Initially, arbitrarily
long lags for both up and down variables were included (Duffy-Deno 1996). Following Conrad & Kaul (1988),
maximum of 5 lags for both up and down were chosen. The results of this procedure were verified by initially
specifying a one week lag and then adding lags if singly or jointly significant. In total, there are more than 30 different
model specifications waited to be tested. Then working backwards, t- and F-tests were used to confirm if any of the lags
individually or as a group could be eliminated as statistically insignificant. A 5% level of significance was employed in all tests.
A summary of multiple regression results of equation (7) is listed in Table 1. There are 71 stocks do not reject weak
form market efficiency hypothesis (a listing of these firms is available upon request), while the remaining 64 stocks
reject the null hypothesis. Since individual stocks cover all six major industries, it is hard to draw a clear line to
conclude which industry is more likely to support market efficiency hypothesis. However, if we compare and match
firms in each industry, it is possible to see that primary leading firms are more likely to support the weak form market
efficiency hypothesis than secondary firms. For example, the price changes of two worldwide recognized foundry firms,
UMC (2303) and TSMC (2330), follow what weak form market efficiency hypothesis is expected. This should be
closely related to their information which is freely available to all investors and agreement among investors about the
implications of given information. Another example comes from motherboard sub-industry. Elitegroup (2331), Gigabyte
(2376) and MSI (2377) are the three major players in this sub-industry and their stock price movements support weak
form market efficiency hypothesis. While USI (2350), BIOSTAR (2399), Shuttle (2405), and Chaintech (2425) are the
secondary firms in the same sub-industry. Their stock price movements reject weak form market efficiency hypothesis.
In computer industry, Compal (2324), INVENTEC (2356), ASUSTEK (2357), and QCT (2382) are the primary firms
and their stock price movements support market efficiency hypothesis. Stock price changes of secondary firms such as
Twinhead (2364), Arima (2381), and Innovace (2387) do not support market efficiency hypothesis. More examples can
be pulled out from other sub-industries such as CRT (Cathode Ray Tub), PCB manufacturing and network industry.
Since primary firms are usually the ‘old’ firms, therefore, stock price changes of ‘old’ firms are more likely to support
weak form market efficiency hypothesis than young firms in the same sub-industry.
Although it is generally true that primary and/or ‘old’ firms are more likely to show weak form market efficiency
than secondary firms do in the same sub-industry. All seven software companies such as Systex (2343), Eten (2432),
Ares (2471), and Ulead (2487) reject the market efficiency hypothesis. Potential sources of market inefficiency may be
due to transactions costs, information that is not freely available to all investors and disagreement among investors
about the implications of given information.
Table 1 Summary of market efficiency hypothesis testing- number of firms
Not reject Ho
Reject Ho
Symmetric firms
High-tech industry
obs.
obs.
Code and pattern (lags of up and down)
Semi-conductor
15
8
2388(5,5), 3041(5,5), 2351(4,4),
manufacturing
2434(5,5), 2470(5,5)
PCB manufacturing
8
5
2467(5,5)
Passive devices
2
6
2492(5,5)
manufacturing
Computer and
27
26
2487(5,5), 2399(4,4), 2378(3,3),
information
2414(5,5), 2438(5,5), 2476(5,5)
Optical industry
15
13
Network industry
4
6
Total Obs=
71
64
13
Table 2 is a summary of results of symmetry tests for 13 stocks which exhibit potential symmetric patterns. In other
words, these 13 stocks show timing symmetry, while other 51 stocks do not. The hypothesis of timing symmetry is
rejected for most asymmetric stocks.
Following Duffy-Deno (1996), two more (a)symmetric tests were conducted for those 13 stocks. As also seen in
Table 2, only one stock (2467) rejects the hypothesis of zero magnitude symmetry when prices had gone up. However,
there are 3 stocks reject the hypothesis of zero magnitude symmetry when prices had gone down. When testing whether
past price changes fully (completely) adjusted or not (Σα up-i=1 or Σα dn-i=1), all 13 stocks reject the hypothesis of fully
adjustment, no matter prices had gone up or down. This may indicate that more lags of past price changes should be
included in the regressions. Finally, the hypothesis of equal magnitude effect of up and down can not be rejected in all
most 13 cases. This indicates that the effects of up and down cancelled out each other, as many efficient market
believers often argued. Based on the results of symmetric tests, it is possible that these 13 stocks with timing symmetry
could end up with zero effects on price changes.
Table 2 Summary of symmetry Tests
Magnitude symmetry tests
Pattern symmetry tests
Σαup-i=0 Σαup-i=1 Σαdn-i=0 Σαdn-i=1 Σαup-i=Σαdn-i αup-1=αdn-1 αup-2=αdn-2 αup-3=αdn-3 αup-4=αdn-4
2388
***
***
**
3041
***
**
***
*
2351
*
***
***
*
***
2434
***
***
*
*
**
2470
***
***
***
***
2467
**
***
***
*
*
***
2492
***
**
***
***
2487
***
**
***
**
2399
***
***
*
***
2378
***
***
**
**
2414
***
*
***
***
2438
***
***
*
2476
***
***
**
Note: *, ** and *** indicate hypotheses were rejected at 10%, 5% and 1% level, respectively.
Code
αup-5=αdn-5
***
**
***
**
**
***
***
**
**
***
The second half of Table 2 shows the results of the hypothesis testing of pattern symmetry. It can be seen that this
hypothesis is more likely to be rejected as the lagged periods get longer. There are about 50% chance of rejecting
pattern symmetry hypothesis if lags are 4 periods. The probability of rejecting the same hypothesis is increasing to 70%
if lags are 5 periods. This may indicate that there are asymmetric effects of past price changes between up and down
patterns. One could investigate these findings more thoroughly, especially based on prospect theory proposed by
Kahneman and Tversky (1979). That is, future research could incorporate prospect theory into asset pricing and study
how price changes would respond to new information if one thinks price changes in the past as moveable reference
points affecting investors’ attitude toward risk (Barberis & Huang 2001; Barberis, Huang & Santos 2001).
CONCLUSIONS
There is a large literature that defines and models prices set in efficient markets. In this paper, only weak form
market efficiency hypothesis was tested. The purpose of this study is to extend research done by Roberts (1959), Keane
(1983), Fama (1970, 1976, 1990), Danthine (1977), and Duffy-Deno (1996) and investigate the speed and quality (or
direction and magnitude) adjustments of stock price to new information.
Symmetric attitude towards risk is a basic assumption in the theory of Finance. Also, there exists symmetric
random walk property. In this paper, the symmetric point of view is challenged. According to Fama’s weak form
efficiency hypothesis and martingales models, a security price model with the consideration of price change asymmetry
is derived first. Based on Danthine’s (1977) argument about pure martingale and fair game concept of Fama (1970), an
empirical regression setting is proposed and then tested. Data used in regression analysis were weekly data taken from
TEJ data bank. Sample period is from 1999/10/22 to 2004/10/22. Empirical results from publicly traded companies in
Taiwan’s high-tech industries during sample period suggest that weak form efficient market hypothesis can not be
rejected by more than half of 135 sample firms, but rejected by the other half sample firms.
A major empirical finding of this study is that if we compare and match firms in each sub-industry, we can see
primary leading firms be more likely to support the weak form market efficiency hypothesis than secondary firms.
Since primary leading firms are usually the ‘old’ firms, therefore, stock price changes of those ‘old’ firms are more
likely to support weak form market efficiency hypothesis than young firms in the same sub-industry.
Empirical findings of Table 2 may reveal that the accumulative effects of up and down may cancel out each other
(Σαup-i=Σαdn-i), keeping the changes in prices remaining symmetric. But changes in the value of more lags on future
price change may not be the same (αup-4=αdn-4 and αup-5=αdn-5). One could investigate these findings more thoroughly,
especially based on prospect theory proposed by Kahneman and Tversky (1979). That is, future research could
incorporate prospect theory into asset pricing and study how price changes would respond to new information if one
thinks price changes in the past as moveable reference points affecting investors’ attitude toward risk (Barberis &
Huang 2001; Barberis, Huang & Santos 2001).
Since all 13 symmetric stocks reject the hypothesis of fully adjustment, no matter prices had gone up or down, one
may consider adding more lags of past price changes in the regressions. And, more stocks should be analyzed, if
possible, to confirm the hypothesis of asymmetric holds of not.
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