# Weak Form Market Efficiency Hypothesis Testing

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## Abstract

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 to 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. Each stock has all gone through more than 30 different model specifications. 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.

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Keywords: Efficient market hypothesis, weak-form efficiency, price asymmetry

## Weak Form Market Efficiency Hypothesis Testing -

## A Price Asymmetry Viewpoint

## Introduction

There is a large literature that defines and models prices set in efficient markets (Fama, 1970, 1976, 1990; Keane, 1983; Lo, 1995ï¼›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 Roberts' paper is to show that the accumulation of random shocks creates apparent patterns, similar to the behavior of actual stock prices. He makes 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. Finally, Roberts (1959) is an illustration of the random walk model for stock prices.

Many previous studies on weak form market efficiency focused only on the direction of level of price movement, however. Roberts (1959) once criticized, financial theories maintain that only the patterns of the past stock prices need be studied, because successive levels of stock prices can reveal an appearance of pattern or trend. 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. In addition, he argued that multiple regressions would be sounder if based on independent changes rather than dependent 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 to 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,

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It = {1, if rt = pt - pt-1 > 0,

{0, if rt = pt - pt-1â‰¦0,

where r1, r2,â€¦, 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 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 differs 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)/(âˆšn) which follows a standard normal distribution. Therefore, one could use Mood's tn statistic to test the randomness of stock prices.

Roberts' (1959) paper is one of the earliest papers on weak form market efficiency. He cited a research done by Kendall (1953) to make his point, "weekly changes of the Dow Jones Index behaved very much as if they had been generated by a chance model (say a distribution with mean 0 and variance 5). He also criticized financial theories focused only on the patterns of the past stock prices, because successive levels of stock prices can reveal an appearance of pattern or trend. He makes 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. In other words, the most important contribution of Roberts' paper is to show 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+1âˆ£Î©t ]= [1 + E(R t+1âˆ£Î©t )] * 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;Î©t 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 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 with 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, tÎµ[0,âˆž] } is a martingale with respect to the family of information set Î©t, and with respect to the probabilityÎ¡, if for all t>0,

a. St is known, given Î©t.

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Examples of our workb.Unconditional forecast are finite: Eâˆ£Stâˆ£<âˆž.

c. And if E[STâˆ£Î©t ]= 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- Stâˆ£Î©t ]= 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 Ptâˆ£Î©t-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. Making explicit the conditional nature of the expectation and the information set, E[Ptâˆ£Î©t-1, Pt-1, Pt-2, â€¦ ]â‰§Pt-1. The usefulness of this formulation is to transform price movements into a sub-martingale and a testable regression format

E[Ptâˆ£Î©t-1, Pt-1, Pt-2, â€¦ ] =Î±+ Pt-1,

whereÎ±is a constant and the covariance (Pt+Ï„- Pt+Ï„+1, Pt - Pt-1)=0, for anyÏ„â‰ 0. Danthine showed that if a stock price exists a pure martingale property, then E[Pt+1- Ptâˆ£Î©t ] =0, and cov(Pt+1 - Pt, Pt - Pt-1) =0.

It would be an interesting research by combining Cowles & Jones' (1937), Mood's (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 â-³Pupt-1= Pt-Pt-1 if Pt-Pt-1 > 0. The other pattern shows wholesale oil price goes down if â-³Pdownt-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+1âˆ£Î©t ]= [1 + E(R t+1âˆ£Î©t )] * Pt ,

where E is the expectation operator, Î©t 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+1âˆ£Î©t ]= 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- Ptâˆ£Î©t ] =0.

The directions of the future movements in martingales are impossible to forecast.

If eq. (2) correctly describes the equilibrium condition for Pt, 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 - Ptâˆ£Pt - Pt-1 ] =0.

and

(5) cov(Pt+1 - Pt, Pt - Pt-1) =âˆ«(Pt - Pt-1)E(Pt+1 - Ptâˆ£Pt - Pt-1)Î¨(Pt - Pt-1)d(Pt - Pt-1)=0,

provided a stationary distributionÎ¨exists. 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 expression of past price difference is as eq. (6),

(6) E[Pt+1 - Ptâˆ£Î©t] = E[Î”Pt+1âˆ£(Pt -1- Pt-2), ( Pt-2 - Pt-3),â€¦, ( Pt-r+1 - Pt-r) ]

= E[Î”Pt+1âˆ£Î”Pt-1, Î”Pt-2, Î”Pt-3 ,â€¦, Î”Pt-r+1].

This study extends the runs test suggested by Hagerman & Richmond (1973) and Duffy-Deno's (1996) approach to separate individual security price movements into two different patterns. That is, we define two dummy variables. One is â-³Pupt-1= Pt-Pt-1 if Pt-Pt-1 > 0, otherwise â-³Pupt-1=0. The other dummy variable is â-³Pdownt-1=Pt-Pt-1, if Pt-Pt-1 <0, otherwise â-³Pdnt-1=0. Therefore, eq. (6) now becomes eq. (7) in a multiple regression format

(7)

where Î±i are regression coefficients to be estimated, r and s are the lags of up and down periods, respectively. AndÎµt is the regression error term and its distribution is assumed to follow (5). It is not necessary to requireÎµt 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. Under the assumptions of (a) there are no transaction costs, and (b) all relevant information is costlessly available to all market participants, all past price movements 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). It is expected that, under market efficiency hypothesis, allÎ±i 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), there are three types of price asymmetry can be tested. First, 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 Î±1,i=Î±2,j or not. Third type of price asymmetry test is called pattern asymmetry. One can test whether Î±1,i=Î±2,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, 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 (see Appendix). 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 (see Appendix). 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.

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 (up, down)

Semi-conductor manufacturing

15

8

2388(5,5), 3041(5,5), 2351(4,4), 2434(5,5), 2470(5,5)

PCB manufacturing

8

5

2467(5,5)

Passive devices manufacturing

2

6

2492(5,5)

Computer and information

27

26

2487(5,5), 2399(4,4), 2378(3,3), 2414(5,5), 2438(5,5), 2476(5,5)

Optical industry

15

13

Network industry

4

6

Total Obs=

71

64

13

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 2 lists 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 cases.

Table 2 Summary of symmetry Tests

## Code

Magnitude symmetry tests

Pattern symmetry tests

Î£Î±up-i=0

Î£Î±up-i=1

Î£Î±dn-i=0

Î£Î±dn-i=1

Î£Î±up-i=Î£Î±dn-i

Î±up-1=Î±dn-1

Î±up-2=Î±dn-2

Î±up-3=Î±dn-3

Î±up-4=Î±dn-4

Î±up-5=Î±dn-5

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.

Following Duffy-Deno (1996), two more asymmetric tests were conducted for those 13 stocks with timing symmetry. As seen in Table 2, most stocks can not reject the hypothesis of zero magnitude symmetry. But all 13 stocks reject the hypothesis of fully adjustment when prices up (Î£Î±up-i=1). And the hypothesis of fully adjustment when prices down (Î£Î±dn-i=1) is rejected as well. However, the hypothesis of equal magnitude effect of up and down can not be rejected in most 13 cases. Based on the results of asymmetric tests, it is possible that these 13 stocks with timing symmetry could end up with zero effects on price changes.

Pattern symmetry hypothesis is rejected in about half of the 13 stocks. Especially when the lagged periods are longer, say lag 4 and 5.

## 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 to investigate the speed and quality (or direction and magnitude) adjustments of stock price to new information.

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.