Weak Form Efficiency Tests

Abstract

While using standard tests of weak form market efficiency along with the more recent DELAY test, this report examines if the returns of six selected stocks and two decile indices follow a random walk which would evidence the non-predictability of future stock returns by historical prices which is a necessary condition for the weakest form of market efficiency. The evidence of four different measurement tests suggests that except of one stock all stocks and indices drift away from the weak form market efficiency hypothesis.

The efficient market hypothesis (EMH) is one of the most common theories in modern finance. In an efficient market prices are supposed to fully reflect all available information (Fama (1970)). The theory is widely used and has therefore been frequently tested by academics. One of these tests was conducted by Fama (1970) in which he distinguished among three different forms of market efficiency: the weak, semi-strong, and strong form. The weak form of market efficiency hypothesis assumes that "stock prices already reflect all information that can be derived by examining market trading data such as the history of past prices." (Bodie et al., 2005, p. 373). The semi-strong form market hypothesis comprises all publicly available information which additionally includes fundamental data on the company and macro-economic factors. The strong form market hypothesis stipulates that stock prices reflect all publicly and privately available data.

A market is said to be weak form efficient if future stock price returns cannot be predicted by the examination of the past returns. In order to fulfil this condition the distribution of stock prices needs to follow a random walk model. Campbell, Lo, and MacKinlay (1997) examine three different random walk models: RW1 implies that returns are independent and identically distributed, RW2 allows for no identical distribution over time, RW3 relaxes the independence assumption and allows dependent but uncorrelated increments. Fama (1970) gives evidence supporting the assumption that the random walk model is an indication for the weak form EMH but not vice versa.

The following report is testing the weak form efficient market hypothesis for three selected NYSE stocks (Advanced Micro Devices, Black & Decker, and Energen) and three selected NASDAQ stocks (Clean Harbors, CoBiz Financial, and Coca-Cola Bottling) as well as two decile indices (the NYSE/AMEX/NASDAQ index capitalisation- based Deciles 1 and 10) representing the largest and the smallest 10% of NYSE/AMEX/NASDAQ companies by market capitalisation.

The remainder of the report is divided into five main areas. Section I provides a brief overview of the selected companies. Section II describes the data and methodology applied to conduct the test of the weak form efficient market hypothesis. Section III discusses descriptive statistics. The results are dealt with in Section IV. The report ends with some concluding remarks in Section V.^[1]

I. Companies overview

A. Advanced Micro Devices Inc. (AMD)

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Advanced Micro Devices designs, manufactures and markets industry- standard semiconductor products. The Group operates through two business segments including Computing Solutions and a Graphic segment.

B. The Black & Decker Corp. (BDK)

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Black & Decker manufactures and markets power tools and accessories, hardware and home improvement products and technology based fastening systems. Its operation includes the three segments: Power Tools and Accessories, Hardware and Home Improvement as well as Fastening and Assembly Systems.

C. Energen Corp. (EGN)

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Energen Corporation acquires, develops, explores and produces oil, natural gas and natural gas liquids in the continental United States. The operations are carried out through its subsidiaries with the following divisions: Natural Gas Distribution and Oil and Gas Operations.

D.Clean Harbors Inc. (CLHB)

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Clean Harbors provides a wide range of environmental services and solutions to a diversified customer base. Its two business segments are Technical and Site services.

E. Cobiz Financial Inc. (COBZ)

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Cobiz Financial (formerly known as Cobiz) provides banking products and services to small and medium-sized businesses. The Group operates through CoBiz Bank, NA a wholly-owned subsidiary.

F. Coca Cola Bottling Company Consolidated (COKE)

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Coca-Cola Bottling Company Consolidated manufactures, markets and distributes carbonated and non-carbonated beverages, primarily products of The Coca-Cola Company. The products include carbonated soft drinks, teas, juices, isotonics and bottled water.

II. Data and Methodology

A. Data

The empirical analysis in this report involves daily as well as monthly returns for the six selected stocks and two decile indices over the period January 2002 to December 2006, comprising 1258 daily observations and 59 monthly observations. Stocks are chosen by criteria of absence of missing values over the period. The Standard and Poor's 500 Index (S&P 500) serves as a proxy for the market index. The index includes 500 actively traded US stocks which represent all market sectors proportionately and are weighted according to their market capitalisation. Returns are calculated as compounded or logarithmic (log) returns throughout the report. In Section III the descriptive statistics for arithmetic returns are also reported to compare the difference in results.

Log-returns are calculated as

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and simple or arithmetic returns as

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where Pt and Pt-i are stock prices or index levels at time t or t -1, respectively.

The Center for Research in Securities Prices (CRSP) database is the source for all daily and monthly price data for the six selected stocks, the two decile indices as well as the market index.

B. Methods

Campbell, Lo, and MacKinlay (1997) distinguish among three groups of tests for random walks (RW). The strongest form of RW, called Random Walk 1 (RW1), assumes that the increments are both linearly, and non-linearly uncorrelated(independent) and have the same probability distribution throughout time (identically distributed). Under RW1, it is impossible to predict future prices and volatility (Worthington and Higgs (2006)). Random Walk 2 (RW2) is the semi-strong form of RW, not requiring the increments to have the same probability distribution in time but still holding the assumptions of their independence. Finally, the weakest form of RW is Random Walk 3 (RW3). It relaxes the assumption of increments' independence and only requires the increments to be (linearly) uncorrelated.

The report examines through various tests the presence of RW in stock price behaviour.

B.l. Autocorrelation Tests

In order to test if an individual time series follows a random walk the data is checked for serial correlation which describes the correlation between two observations of the same series at different dates. The weakest version of a random walk, RW3, supposes that the increments or first-differences of the level of the random walk are uncorrelated at all leads and lags. RW3 can be tested by testing the null hypothesis stating that the autocorrelation coefficients of the first-differences are all zero at various lags. The sample autocorrelation at lag k is given by:

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where pk is the autocorrelation at lag k ; rit is the log-return on stock i at time t; and rit-k is the log-return onstock i at time t -k. The serial correlation ispositive if pk is also positive. The serial correlation is negative if pk is also negative. Equation (3) assumes covariance-stationarity of returns (i.e. their mean and variance are constant, while the covariance between lags should depend only on distance between lags, and not on shifts throughout the time). To reject the null hypothesis of no serial correlation in return series pk needs to be significantly different from zero.

The output of the autocorrelation test contains the coefficients for up to 12 lags for all six selected stocks as well as the two decile indices. Not only the statistical significance of each individual autocorrelation coefficient is tested but also the Ljung- Box test (cp. Campbell, Lo and MacKinlay (1997, p. 47)) is employed which tests the joint hypothesis stating that all the values of the autocorrelation coefficients up to the tested lag are simultaneously equal to zero. The test statistics for 3 and 12 lags are computed.

B.2. Variance Ratio Tests

The variance ratio test examines RW1 basing on the assumption that the variance of RW increments is a linear function of the time interval over which they are computed (Griffin et al, 2007). Thus, variance over the period of q equal intervals should be equal to the variance over one such interval times q. The variance ratio, VR(q), indicates if returns are uncorrelated and can be defined as follows:

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where cr2(q) is the unbiased estimator of 1/q of the variance of the q th difference of the logged security return [Abbildung in dieser Leseprobe nicht enthalten] and [Abbildung in dieser Leseprobe nicht enthalten] is an unbiased estimator of the variance of the logged return [Abbildung in dieser Leseprobe nicht enthalten].

The null hypothesis of a random walk with uncorrelated increments is that [Abbildung in dieser Leseprobe nicht enthalten]. Positive autocorrelation is indicated by variance ratios significantly above one whereas negative serial correlations are indicated by variance ratios significantly below one. The later is known as a mean-reverting process. Lo and MacKinlay (1988) and Campbell, Lo, and MacKinlay (1997) introduce heteroskedasticity consistent significance tests for the testing of the null hypothesis.

B.3. Runs Tests

The runs test is a common test for IID random walks (RW1) which tabulates and compares runs against its sampling distribution under the random walk hypothesis. The rejection of the random walk hypothesis would lead to the alternative hypothesis stating that the data is autocorrelated. The previousautocorrelation and variance ratio tests provide insufficient evidence for a complete assessment of the weak form EMH so the runs test is a valuable supplement to the obtained results. The runs test examines the prevailing patterns in time series. A run is defined as a sequence of consecutive positive and negative (price) returns. The null hypothesis of the IID random walk (RW1) can be rejected if the observed number of runs is significantly different from the expected number of runs. A detailed description of the construction of a runs test can be found in Campbell, Lo and MacKinlay (1997, pp. 38-41).

B.4. Griffin-Kelly-Nardari DELAY Tests

The Griffin-Kelly-Nardari DELAY test is employed to assess the impact of past market-wide information on current returns. The Standard & Poor's (S&P) 500 index serves as a proxy for the market portfolio. For each of the six selected stocks and the two decile indices the restricted and unrestricted models are estimated over the period January 2002 to December 2007. The unrestricted model is

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where [Abbildung in dieser Leseprobe nicht enthalten] is the log-return on stock i at time t; rmt is the log-return on the S&P 500 index at time t; rmt-n is the lagged market return ; βλ¡ is the coefficient on the lagged market return; and λ is the lag (λ = 1, 2, 3, 4 for daily data and 1, 2, 3 for monthly data). The restricted model constraints the coefficients on the lagged market returns to zero which results in the following equation:

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DelayGKN is calculated as the difference between the adjusted R2s from the regressions (3) and (4) as follows:

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The model uses adjusted R2s to control that the higher explanatory power is not only due to a higher number of explanatory variables.

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