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Efficient Market Theory

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In this paper, the random walk hypothesis based on monthly stock market returns is tested by comparing variance ratios. The random walk model is not rejected for the entire sample period (1926-2007) and for its subperiods when using value-weighted stock returns. However, the model is rejected on equal-weighted stock returns when using low aggregation values q. Generally, the random walk model is not rejected when using monthly data.

It was widely believed that the market is efficient in reflecting all available information that no abnormal returns can be earned by studying the historical prices. However, some recent papers suggest that stock prices are partially predictable. In this paper, I test the weak form efficiency of the market using the variance ratio test by Lo and Mackinlay (1988). And the result yields that the market generally follows the random walk for monthly stock returns. The rest of the paper is organized as follows. In section 1, the previous main findings on the Efficient Market Hypothesis are reviewed. The data and sample period are described in section 2. The main results are given in section 3, where tests for both homoscedastic and heteroscedastic random walks are conducted. Section 4 is the conclusion of this paper.

1. Related literature on Efficient Market Hypothesis

According to Eugene Fama’s (1970) survey article, “Efficient Capital Market”, the vast majority of academic financial economist failed to reject the efficient market hypothesis in their studies. Malkiel (2003) in his recent study “the Efficient Market and Its Critics” also concludes that “the stock markets are far more efficient and far less predictable”. However, Kiem and Stambaugh (1986) suggest that stock returns contain predictable components that are statistically significant. Furthermore, Lo and Mackinley (1988) state in their study that stock prices do not follow random walk by comparing variance estimators, they also find a positive autocorrelation for weekly stock returns.

2. Data and Sample Period

Since the test is based on asymptotic approximations, a large number of data is chosen for this study. The sample period is from 1926 to 2007, and a total of 984 monthly observations are derived from the CRSP index. Two different indexes are used in this study: equalweighted CRSP index and value-weighted CRSP index. Also, observations from both index are divided into two subperiods. The first subperiod starts from 1926 and ends in 1964; the second subperiod starts from 1965 and ends in 2007.

3. Empirical Results

Table 1 provides the results based on the methodology of variance ratio test by Lo and MacKinlay (1988). It demonstrates that variance ratio estimates and test statistics of the Random Walk Hypothesis for the entire study period and two subperiods.

Table 1 Market Index results for one-month base observation period Number nq of Time period A. Equal-weighted CRSP index 1/1/1926-12/1/2007 1/1/1926-12/1/1964 1/1/1965 – 12/1/2007 B. Value-weighted CRSP index 1/1/1926-12/1/2007 1/1/1926-12/1/1964 1/1/1965 – 12/1/2007 984 468 516 1.0952 (1.6822) 1.1218 (1.5087) 1.0490 (0.9310) 1.0776 (0.7421) 1.1146 (0.7687) 1.0242 (0.2493) 1.1013 (0.6238) 1.1541 (0.6666) 1.0403 (0.2702) 1.2361 (0.9853) 1.4067 (1.1904) 1.0077 (0.0361) 984 468 516 1.1691 (2.4694) 1.1639 (1.6903) 1.1888 (3.8454) 1.2157 (1.9250) 1.2182 (1.3120) 1.235 (2.5590) 1.1484 (0.8588) 1.1445 (0.5980) 1.2105 (1.4532) 1.2328 (0.9219) 1.3337 (0.9470) 1.1248 (0.5810) base observations Number q of base observations aggregated to form variance ratio 2 4 8 16

For the equal-weighted CRSP index, the random walk null hypothesis is only rejected with an aggregation value q equal to 2. Since a variance ratio with q of 2 is approximately equal to 1 plus the first-order autocorrelation coefficient estimator of monthly stock returns, the result 1.1691 indicates that the first-order autocorrelation for monthly returns is around 17 per cent.

And the test statistic suggests rejections of the Random Walk Hypothesis at 5 per cent of significance. As the value of q increases, the test statistics decreases, and this leads to the fail of rejections of the random walk hypothesis. For the value-weighted CRSP index, the null hypothesis is not rejected at the significance level of 5 per cent for all different value of q. Since the value-weighted index put more weight on the large capital firms, and the Random Walk Hypothesis is more likely to hold for large capital stocks, the empirical finding in this paper is consistent with such theory that the random walk is not rejected for the valueweighted CRSP index. For the equal-weighted index, the hypothesis is rejected due to the fact that the index put more weight on the small capital socks relatively to the value- weighted index. And it is more likely to find some predictability in small capital firms. This result is consistent with Lo and Mackinlay’s (1988) finding that “the rejection of the random walk hypothesis is much weaker for the value-weighted index”.

For both indexes, the variance ratios for their subperiods are quite stable and consistent with the variance ratio of the entire sample period. The Random Walk Hypothesis is not rejected for both subperiods for the value-weighted index. However, for the equal-weighted index, the null hypothesis is not rejected for the first subperiod from 1926 to 1964. But for the second subperiod from 1965 to 2007, the random walk hypothesis is rejected when the aggregation value q equals to 2 and 4. The rejection of RWH in this subperiod (1965-2007) is probably due to the fact that the US economy fluctuated more often in this period. The OPEC recession, the market crash of October 1987, the 1990-91 recession, the dot-com bubble and the recent financial crisis may all lead to some extend of irrationality of the market. For both indexes, all of the variance ratios are greater than 1 and a pattern that the ratios increases as the aggregation value q increases while the test statistics generally decreases as q increases can be found. Also, the returns are positively correlated for both indexes.

This is consistent with Lo and Mackinlay’s (1988) finding but differ from Fama’s (1987) finding that a negative serial correlation can be found in long holding period returns. In this paper, only monthly returns are tested, and generally, the Random Walk Hypothesis is not rejected for both equal-weighted and value-weighted index. This is consistent with previous studies that the hypothesis is generally not rejected when using monthly returns. However, according to Lo and Mackinlay (1988), it is much more likely to reject the random walk when using a base observation period of one week. In their study, the Random Walk Hypothesis is rejected at all usual significance levels for the whole sample period from 1962 to 1985 using weekly data.

4. Conclusion

Generally, the Random Walk Hypothesis is not rejected for monthly stock market returns by using the Lo and Mackinlay (1988) variance ration test. The test results for the valueweighted CRSP index indicate that none of the test for heteroscedasticity at any different value of q is significant. Therefore, the RWH cannot be rejected for the value-weighted index. For the equal-weighted CRSP index, the RWH is rejected only when the aggregation value q equals to 2. Such difference in the two indexes indicates that the rejections of the Random Walk Hypothesis may largely due to the predictability of small capital stocks.


1. Lo, A. W. and A. C. MacKinlay, 1988, Stock Market Prices Do Not Follow Random Walks: Evidence from a Simple Specification Test, The Review of Financial Study 1, 41-66. 2. Fama, E., 1970, “Efficient Capital Markets: A Review of Theory and Empirical Work,” Journal of Finance 25, 383-417 3. Keim, Donald B. and Robert T. Stambaugh.1986. “Predicting Returns in Stock and Bond Markets.” Journal of Financial Economics. December, 17, pp. 357–90. 4. Malkiel, Burton G. 1973. A Random Walk Down Wall Street. New York: W. W. Norton & Co.

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