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
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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*
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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
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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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Testing Weak-form EMH
•
Which of the following does weak-form EMH imply?
1.
2.
3.
•
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
1.
2.
3.
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
Random Walk 3 (uncorrelated increments): Cov(et+1, et)=0
(weakest), but Cov(et+12, et2) ne 0
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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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Sequences and Reversals
Assume yt follows an IID random walk without drift. Denote It as the following:
1 if rt  pt  pt 1  0
It  
0 if rt  pt  pt 1  0
The number of sequence is Ns and the number of reversal is Nr.
n
N s   Yt , where Yt  I t I t 1  (1  I t )(1  I t 1 )
t 1
Ns  n  Ns
For any pair of consecutive returns, a sequence and reversal are equally probable. Thus
the Cowles-Jones ratio CJ=Ns/Nr should be approximately equal to one.
When there is a drift in stock return,
1 with probability 
It  
0 with probability 1  

  Pr( rt  0)   ( )

 2  (1   ) 2
CJ 
2 (1   )
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Runs
• Use the number of consecutive positive and negative returns
• 1001110100 versus 0000011111
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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 buyand-hold portfolio
– Need consider transaction costs
• Technical analysis/charting
– Filter rule is an example
– Trading on patterns
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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 (BoxPierce test, Ljung-Box test).
• Variance ratios (linear combinations of the
autocorrelations).
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Autocorrelation Coefficients
• With a covariance-stationary time series of continuously
compounded returns, we can define the
– kth order autocovariance, γ(k)  (k )  Cov[rt ,rt k ]
– kth order autocorrelation, ρ(k)
Cov[rt ,rt  k ] Cov[rt ,rt  k ]
– Sample counterparts:  (k ) 

r r
Var (rt )
t
T
t k
1
( rt  rT )(rt  k  rT )

T t 1
ˆ (k )
ˆ
 (k ) 
ˆ (0)
ˆ (k ) 
1 T
rT   rt
T t 1
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Sampling Theory for Autocorrelations
• If rt is iid (RW1), and finite first 6 moments,
T k
O(T  2 )
T (T  1)
T k
2
Cov[ ˆ (k ),ˆ (l )] 

O
(
T
) if k l  0
2
T
 O(T  2 ) otherwise.
E[ ˆ (k )]  
• 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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Asymptotic Distribution
• If rt is iid (RW1), and finite first 6 moments, sample
autocorrelations are asymptotically ( T  ∞ ) normal:
a
ˆ (k ) ~ N (0,1)
T
• Joint tests:
– Box-Pierce Statistic
m
Qˆ m  T  ˆ 2 (k ) ~  m2
k 1
m
– Ljung-Box Statistic Qˆ m  T (T  2)
'
k 1
 2 (k )
T k
~  m2
• Can be extended beyond RW1
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Variance ratio test
• Intuition
VR (2) 

Var[rt (2)] Var[rt  rt 1 ]

2Var[rt ]
2Var[rt ]
2Var[rt ]  2Cov[rt ,rt 1 ]
2Var[rt ]
1   (1)
• Under the RW null VR(2) = 1
• With positive (negative) first-order autocorrelation VR > (<) 1.
q 1
 k
Var
[
r
(
q
)]
t
• To Generalize, VR (q) 
1  2 1   (k )

q 
k 1 
– Why?
– VR(q) is a particular linear combination of ρ(k)
– Linearly declining weights
qVar[rt ]
• Under all three RW nulls, VR(q) = 1, but the asymptotic
distributions for sample VR(q) are different
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Under RW1
• We estimate VAR(r (1))   a2 VAR(r (2))   b2
2
• Variance (  b ) estimated using non-overlapping data:
• Asymptotic distributions for sample variances:
a
2n (ˆ
2
a
  ) ~ N (0,2 4 )
2n (ˆ
2
b
  ) ~ N (0,4 4 )
2
2
a
• Question: how about asymptotic distributions for
2
2
2
ˆ

b
ˆ a  ˆ b
ˆ a2
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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 ˆ a2  ˆb2
Var[ 2n (ˆ b2  ˆ a2 )]  Var ( 2nˆ b2 )  Var ( 2nˆ a2 )
or
a
2n (ˆ  ˆ ) ~ N(0, 2 4 )
2
b
2
a
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ˆ
• How about
ˆ
2
b
2
a
Delta Method
?
• Take 1st order Taylor expansion:
ˆ b2
2 2
1
2
2
2
2
  4 (ˆ a   )  2 (ˆ b   )  2
2
ˆ a



• Therefore,
ˆ b2
1
1
2
Var ( 2n 2 )  4 Var ( 2nˆ a2 )  4 Var ( 2nˆ b2 )  4 Cov( 2nˆ a2 , 2nˆ b2 )
ˆ a 


 244  2
or
a
ˆ b2
2n ( 2 - 1) ~ N(0,2)
ˆ a
• Delta method is discussed on page 118, Greene (2000)
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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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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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Testing RW3
• Under RW3, rt no longer iid. heteroskedasticity.
• Properties that still hold:
– VD(q)  0, VR(q)
 1 q 1
_
a
 k
VR (q)  1  2 1   ˆ (k )
– And,
q
k 1 
– Further, sample autocorrelations at different orders are
uncorrelated. E( ˆ k ˆ m )  0 , for k  m
– Therefore, variance of VR(q) remains of the form:
• Properties that no longer hold:
– Asymptotic variances ˆk of sample autocorrelations ˆ k
– Asymptotic variances ˆ(q) of VR(q)
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Long-Horizon Returns
Basic Structure (page 56-57, CLM):
pt  wt  yt ; wt    wt 1   t ; yt= any zero-mean stationary process
wt is the “fundamental” component that reflects the efficient markets price, and yt is a
zero-mean stationary component (mean reversion) that reflects a short-term or transitory
deviation from the efficient-market price wt, implying the presence of “fads” or other
market inefficiency.
We have rt  pt  pt 1     t  yt  yt 1
q1
q 1
j 0
k 0
rt (q)   rt  j q    t k  yt  yt q
Var[rt (q)]  q 2  2 y (0)  2 y (q) , where γ is the autocovariance function of y.
Var[rt (q)]
VR (q) 
qVar[rt ]
Var[rt (q)]
Var[y]
VR (q) 
1
qVar[rt ]
Var[p]
W
e
c
a
n
s
h
o
w
:
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Empirical Evidence
• Autocorrelations
– Daily (1962-1994) equal-weighted CRSP index has a firstorder 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
263.3. The 99.5-percentile for  25 is 16.7.
– Weekly and monthly returns exhibit similar patterns for the
indexes
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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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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.
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Evidence of Cross-Correlation
• The contrast with the indexes is suggestive: large positive crossautocorrelations 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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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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Economic-Trick Review
(1) Asymptotic Distributions
See Greene (2000, Chapter 4, Statistical Inference).
d
If n( xn  ) /   
N0,1, then approximately, or asymptotically,
a
xn 

N ,  2 / n, which we write as


a
xn 

N ,  2 / n ,
Extending this definition to the multivariate case, suppose that θˆ n is an
estimator of the parameter vector θ . The asymptotic distribution of the vector
θˆ n is obtained from the limiting distribution:
d
n (ˆn   ) 

N 0, V , which implies that
a
ˆθ ~ N  , 1 V 
n
 n 


The covariance matrix of the asymptotic distribution is the asymptotic
covariance matrix, denoted as
1
Asy .Var [ˆn ]  V
n
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(2) Maximum Likelihood Estimator
This material is from Chapter 4, Greene 2000
The principle of maximum likelihood provides a means of choosing an asymptotically
efficient estimator for a parameter or a set of parameters. Each observation is considered
as a realization from a random sample:
n
f (x 1 , x 2 ,...,x n , θ)   f (x i ,θ)  L(θ | X)
i 1
n
ln L(θ | X)   ln f (x i , θ)
i 1
The necessary condition for maximizing ln L(θ | X) is:
 ln L(θ)
 0 -- known as “likelihood equation”
θ
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MLE Example
Likelihood for the Normal Distribution
In sampling from a normal distribution with mean  and variance  2 , the log-likelihood
function and the likelihood equations for  and variance  2 are
n
n
1 n ( xi   ) 2
2
ln L(  ,  )   ln(2 )  ln    [
]
2
2
2 i 1
2
 ln L
1 n
 2  ( xi   ) 0

 i 1
2
 ln L
n
1



 2
2 2 2 4
n
ˆ ML
1

n
ˆ
1 n
  ( xi  x n ) 2
n i 1
2
ML
x
i 1
i
n
 (x
i 1
i
  ) 2 0
x n
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Properties of MLE
M1: Consistency: p limˆML  
a
M2: Asymptotic normality: ˆML 

N  , I ( ) 1 , , where I ( )  E[ 2 ln L /  ],
M3. Asymptotic efficiency: ˆ is asymptotically efficient and achieves the Cramer-Rao
 

ML
lower bound for consistent estimators, given in M2, Theorem 4.2, (4-12), and (4-13).
M4: Invariance: The maximum likelihood estimator of   c( ) is c(ˆML ) .
 2 ln L( ) 1
]} , also known as
Asymptotic Covariance Matrix of MLE: [ I ( )]  { E[
 
the information matrix. It is evaluated at ˆML .
Suppose y2  f ( x2 , , E[ y1 | x1 ,1 ])
1
FIML estimation: estimating the joint distribution f(y1,y2)
LIFL estimation: estimating y1 first, then maximize a conditional log-likelihood function
n
^
using the estimates from step1: LnL   f ( y12 | xi 2 , 2 , ( xi 2 ,1 )])
i 1
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(3) Consistent Estimator: MOM
The idea is that to estimate K parameters, 1 ,..., k , we compute K statistics, m1, …, mk.
Consider random sampling from a distribution f ( x 1 ,, k ) with finite moments E[ x k ] .
The sample consists of n observations, x1 ,, xn . The kth “raw” or uncentered moment is
1 n k
mk   xi
n i 1
By sustituting z i  xik , we obtain the following results. By (3-14) and Theorem 4.1,
E[mk ]  k  E[ xik ]
1
1
Var [mk ]  Var [ xik ]  (  2 k   k 2 )
n
n
p lim mk   k  E[ xik ]
d
n (mk  k ) 

N[0, 2 k  k 2 ]
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MOM
1  E[ xi ]  
In general,  k will be a function of the underlying parameters. By computing K raw
moments and equating them to these functions, we obtain K equations that can be solved
to provide estimates of the K unknown parameters.
g is from the sample, gamma is the parameter. The following is called the K moment
equations.

g 1   1 (1 ,..., k )  0

g 2   2 (1 ,..., k )  0
.
.
.

g k   k (1 ,..., k )  0
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MOM Estimator of N(μ,σ2)
In random sampling from N[, 2 ] ,
1 n
p lim  xi  p limm1  E[ xi ]  
n i 1
1 n 2
p lim  xi  p limm2  Var[ xi ]   2   2   2
n i 1
Note: the variance of MOM estimator of μ is  2 / n
2
T
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33
.
(4) Assumptions of Linear Regression Models
(1)
(2)
(3)
(4)
(5)
Linear functional form of the relationship
X is an nxK matrix with rank K
E ( | X) = 0
Var( | X)   2 -- Spherical disturbances
nonstochatic regressors
(6) normality: ε | X =N[0,σ2I]
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(5) Least Square Estimation
yi  x'i β   i
This is an unknown relation between x and y. In other words, E[ yi | x i ]  x 'i β
^
The estimate of E[ yi | x i ] is denoted y i  x 'i b
yi  x'i b  ei
One way to estimate this is the least square estimator:
min (y  Xβ 0 )' (y  Xβ 0 )
β0
We get X' Xb  X' y
b  (X' X) 1 X' y
For finite samples, b follows any distribution with a mean of β and variance of
 2 (X' X) 1 . For large samples, when the regression model has normally distributed
disturbance, b has the minimum variance.
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Generalized Least Squares
^
  (X ' X ) X ' y
1
1
1
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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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
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