1. Lecture 10: Testing Market Efficiency
The following topics will be covered:
Different forms of MEH
Random walk tests
Variance ratio tests
Autocorrelation
Also, review “economic-tricks” on:
Asymptotic distribution
Maximum likelihood estimator: efficient estimator
Method of moment estimator: consistent estimator
Least square estimator
L10: Market Efficiency Tests

2. Efficient Market Hypothesis
Reference: Fama (1970, 1991), CLM Ch 1.5
Definition: asset prices fully reflect available information, to the extent that no economic profits can be made by trading on the information (see CLM page 20)
Three forms:
Past price, return, or volume
Sequences and reversals, runs, variance ratio, technical analysis, momentum and contrarian
Publicly announced news
Event studies, accounting stock-selection models
Private information
Insider trading, mutual/hedge fund performance*
L10: Market Efficiency Tests

3. Martingale Hypothesis
E[Pt+1|Pt, Pt-1,…]=Pt or, equivalently, E[Pt+1-Pt|Pt, Pt-1,…]=0
If Pt represents one’s cumulative wealth at date t from playing some game, then a fair game is one for which the expected wealth next period is simply equal to this period’s wealth.
Another aspect is that nonoverlapping price changes are uncorrelated at all leads and lags.
Martingale is considered as a necessary condition for an efficient market
Does the hypothesis consider risk? No
By considering risk, asset returns should be positive. Thus the martingale property is not necessary nor sufficient
Risk-adjusted Martingale
L10: Market Efficiency Tests

4. L10: Market Efficiency Tests
Issues
Joint Hypothesis Problem
any test of market efficiency must assume an asset pricing paradigm. If we assume a wrong asset pricing model, it may lead to false rejection of acceptance of market efficiency. Alternatively, the rejection of a joint-hypothesis test may either be due to market inefficiency or a wrong asset pricing model used.
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5. L10: Market Efficiency Tests
Testing Weak-form EMH
Which of the following does weak-form EMH imply?
f(rt+k| rt, It ) = f(rt+k|It), or Cov[g(rt+k),h(rk)] = 0 for any g, h
E(rt+k|rt) = u, or Cov[rt+k, h(rt)] = 0 for any h
Or a simple put as Cov(rt+k, rt) = 0
Alternatively, consider stock price Pt+1 = u + Pt + et+1
Random Walk 1 (iid increments): et iid (strongest)
Random Walk 2 (independent increments):
Cov[g(et+1), h(et)]=0
Or (weaker) Cov[et+1, h(et)] = 0
3. Random Walk 3 (uncorrelated increments): Cov(et+1, et)=0
(weakest), but Cov(et+12, et2) ne 0
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6. Early Nonparametric Tests
Early tests (for iid):
Spearman rank correlation test, Speamn’s footrule test, Kendall τ correlation test
Sequences and Reversals
Runs
See CLM 2.2
Nonparametric tests, using signs of returns, no distributional assumption for returns required
Can be used to test both RW1(iid) and RW2 (independence)
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7. Sequences and Reversals
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8. L10: Market Efficiency Tests
Runs
Use the number of consecutive positive and negative returns
versus
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9. Tests of RW2: Independent Increments
Testing for independence without assuming identical distributions is quite difficult.
Filter rule
An asset is purchased when its price increases by x%, and short (short) when its price drops by x%
Compare the profit of this dynamic trading portfolio with that of a buy-and-hold portfolio
Need consider transaction costs
Technical analysis/charting
Filter rule is an example
Trading on patterns
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10. Test of Serial Correlations (RW3)
Under RW3, the increments of the random walk are uncorrelated at all leads and lags.
Therefore, to test RW3, look at the returns and construct tests based on:
Autocorrelations at a given order
Joint test of autocorrelations at multiple orders (Box-Pierce test, Ljung-Box test).
Variance ratios (linear combinations of the autocorrelations).
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11. Autocorrelation Coefficients
With a covariance-stationary time series of continuously compounded returns, we can define the
kth order autocovariance, γ(k)
kth order autocorrelation, ρ(k)
Sample counterparts:
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12. Sampling Theory for Autocorrelations
If rt is iid (RW1), and finite first 6 moments,
Negative bias (E(ρ) is negative) in sample autocorrelations
This is follows because of the estimation procedure.
You have to estimate the sum of the cross products of deviations from a mean (that is itself estimated).
Deviations from the sample mean are zero by construction so positive deviations must eventually be followed by negative deviations.
When you multiply these deviations together, the result is a negative bias.
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13. Asymptotic Distribution
If rt is iid (RW1), and finite first 6 moments, sample autocorrelations are asymptotically ( T  ∞ ) normal:
Joint tests:
Box-Pierce Statistic
Ljung-Box Statistic
Can be extended beyond RW1
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14. L10: Market Efficiency Tests
Variance ratio test
Intuition
Under the RW null VR(2) = 1
With positive (negative) first-order autocorrelation VR > (<) 1.
To Generalize,
Why?
VR(q) is a particular linear combination of ρ(k)
Linearly declining weights
Under all three RW nulls, VR(q) = 1, but the asymptotic distributions for sample VR(q) are different
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15. L10: Market Efficiency Tests
Under RW1
We estimate
Variance ( ) estimated using non-overlapping data:
Asymptotic distributions for sample variances:
Question: how about asymptotic distributions for
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16. Results from Hausman’s Specification Test
θe : asymptotically efficient estimator; θc : consistent estimator
Among all consistent estimators, the efficient estimator has the lowest variance
Hauseman (1978): Cov [θe , θc - θe ] = 0
Otherwise, let Cov [θe , θc - θe ] = γ, there exists w such that,
Var [ θe + w (θc - θe )] < Var (θe)  contradicts efficiency of θe
Applied to
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17. L10: Market Efficiency Tests
Delta Method
How about ?
Take 1st order Taylor expansion:
Therefore,
Delta method is discussed on page 118, Greene (2000)
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18. Generalization: VD(q) and VR(q)
Data is nq+1 observations of log prices {p0,…,pqn) where q is an integer greater than 1. Consider the following estimators:
Asymptotic distributions under RW1:
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19. L10: Market Efficiency Tests
Refinements
Using overlapping observations to estimate q-period variance:
Bias adjustment:
(nq)1/2VD(q)  N( 0, 2(2q-1)(q-1)/(3q) σ4 )
(nq)1/2 [VR(q) -1] N( 0, 2(2q-1)(q-1)/(3q) )
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20. L10: Market Efficiency Tests
Testing RW3
Under RW3, rt no longer iid. heteroskedasticity.
Properties that still hold:
VD(q)  0, VR(q)  1
And,
Further, sample autocorrelations at different orders are uncorrelated.
Therefore, variance of VR(q) remains of the form:
Properties that no longer hold:
Asymptotic variances of sample autocorrelations
Asymptotic variances of VR(q)
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21. L10: Market Efficiency Tests
Long-Horizon Returns
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22. L10: Market Efficiency Tests
Empirical Evidence
Autocorrelations
Daily ( ) equal-weighted CRSP index has a first-order autocorrelation of 35.0% (with a standard error of 1.11%). Implies that 12.3% of the daily variation is explainable by lagged return (page 66 CLM).
Box-Pierce Q statistic for 5 autocorrelations has value The 99.5-percentile for 25 is 16.7.
Weekly and monthly returns exhibit similar patterns for the indexes
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23. L10: Market Efficiency Tests
Empirical Evidence
Variance Ratios
As the autocorrelations suggest the variance ratios are greater than one.
The equal-weighted index has VR’s that are highly significant, larger in the 1st half of the sample (a common pattern). VR’s increase in q suggesting positive serial correlation for multiperiod returns.
VR’s of the value-weighted index are greater than one but insignificant in full sample and both subsamples. Suggests that firm size is an interesting issue.
Rejection of RW stronger for smaller firms. Their returns more serially correlated.
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24. L10: Market Efficiency Tests
Empirical Evidence
Individual Securities
Variance ratios suggest small negative serial correlations.
Insignificance likely due to fact that with so much nonsystematic risk any predictable components are hard to find.
L10: Market Efficiency Tests

25. Evidence of Cross-Correlation
The contrast with the indexes is suggestive: large positive cross-autocorrelations across individual securities across time
In addition to evidence of significant autocorrelations, there are also evidence of significant cross-autocorrelations (account for a half of the return predictability). This is another source of return predictability.
Lo and MacKinlay (1990) argue that cross-autocorrelation is the main source of profits for short-term contrarian strategies. Therefore, contrarian profits may not necessarily be evidence of market overreaction.
Notations:
Rt : vector of returns; E( Rt ) = u
k-th order autocovariance Matrix: Γ(k) = Cov[ Rt-k , Rt ]
k-th order autocorrelation matrix: Ŷ(k)
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26. Evidence from Long-Horizon Returns
Negative serial correlation in multi-year index returns
Fama and French (1988), Poterba and Summers (1988)
There is a substantial mean revision in stock market prices at longer horizons
Caveat: small sample size makes inference less reliable
Only 12 nonoverlapping five-year returns
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27. Economic-Trick Review (1) Asymptotic Distributions
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28. (2) Maximum Likelihood Estimator
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29. L10: Market Efficiency Tests
MLE Example
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30. L10: Market Efficiency Tests
Properties of MLE
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31. (3) Consistent Estimator: MOM
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32. L10: Market Efficiency Tests
MOM
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33. L10: Market Efficiency Tests
MOM Estimator of N(μ,σ2)
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34. (4) Assumptions of Linear Regression Models
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35. (5) Least Square Estimation
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36. Generalized Least Squares
When Ω is unknown, feasible generalized least squares (FGLS) approach can be used. To be specific, we can assume a specific form of variance-covariance matrix, either autocorrelation or heteroscedasticity, then estimate it. See page 465 to 470 of Greene (2000).
There are other ways to estimate beta here, such MLE and MOM.
SAS Procedure: Proc Model
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37. L10: Market Efficiency Tests
Exercises
2.4; 2.5 CLM
Use monthly data to make Table 2.8 and 2.9, page 75, CLM
Exercises regarding MLE, MOM and GLS
L10: Market Efficiency Tests