L10: Market Efficiency Tests1 Lecture 10: Testing Market Efficiency The following topics will be...

L10: Market Efficiency Tests 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


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*


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


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.


Testing Weak-form EMH

• Which of the following does weak-form EMH imply?1. f(rt+k| rt, It ) = f(rt+k|It), or Cov[g(rt+k),h(rk)] = 0 for any g, h

2. E(rt+k|rt) = u, or Cov[rt+k, h(rt)] = 0 for any h

3. Or a simple put as Cov(rt+k, rt) = 0

• Alternatively, consider stock price Pt+1 = u + Pt + et+1

1. Random Walk 1 (iid increments): et iid (strongest)

2. 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, et

2) ne 0


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)


Sequences and ReversalsAssume ty follows an IID random walk without drift. Denote It as the following:

0

0

0

1

1

1

ttt

tttt ppr

ppr

if

ifI

The number of sequence is Ns and the number of reversal is Nr.

n

tts YN

1

, where )1)(1( 11 ttttt IIIIY

ss NnN

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,

10

1

yprobabilit

yprobabilit

with

withI t

)()0Pr( tr

)1(2

)1( 22

CJ


Runs

• Use the number of consecutive positive and negative returns

• 1001110100 versus 0000011111


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


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).


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:

],[)( ktt rrCovk

)(

],[],[)(

t

ktt

rr

ktt

rVar

rrCovrrCovk

ktt

T

ttT

T

tTktTt

rT

r

kk

rrrrT

k

1

1

1

)0(ˆ)(ˆ

)(ˆ

))((1

)(ˆ


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.

.)(

0)()](ˆ),(ˆ[

)()1(

)](ˆ[

2

22

2

otherwiseTO

lkifTOT

kTlkCov

TOTT

kTkE


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

)1,0(~)(ˆ NkTa

2

1

2 ~)(ˆˆm

m

km kTQ

2

1

2' ~

)()2(ˆ

m

m

k

mkT

kTTQ


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

)1(1

][2

],[2][2

][2

][

][2

)]2([)2(

1

1

t

ttt

t

tt

t

t

rVar

rrCovrVar

rVar

rrVar

rVar

rVarVR

)(121][

)]([)(

1

1

kq

k

rqVar

qrVarqVR

q

kt

t


Under RW1• We estimate

• Variance ( ) estimated using non-overlapping data:

• Asymptotic distributions for sample variances:

• Question: how about asymptotic distributions for

2b

)4,0(~)ˆ(2

)2,0(~)ˆ(2

422

422

Nn

Nna

b

a

a

22 ˆˆ ba 2

2

ˆ

ˆ

a

b

2))1(( arVAR 2))2(( brVAR


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

22 ˆˆ ba

)2 N(0, ~ )ˆˆ(2

)ˆ2()ˆ2()]ˆˆ(2[

422

2222

a

ab

abab

n

or

nVarnVarnVar


Delta Method• How about ?

• Take 1st order Taylor expansion:

• Therefore,

• Delta method is discussed on page 118, Greene (2000)

2

2

ˆ

ˆ

a

b

2

222

222

4

2

2

2

)ˆ(1

)ˆ(ˆ

ˆ

baa

b

N(0,2)~ 1)-ˆ

ˆ(2

2442

)ˆ2,ˆ2(2

)ˆ2(1

)ˆ2(1

)ˆ

ˆ2(

2

2

224

24

242

2

a

a

b

babaa

b

n

or

nnCovnVarnVarnVar


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:


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) )


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)

)(ˆ121)(1

1

_

kq

kqVR

q

k

a

k k

m k for , 0)ˆˆ( mkE

)(ˆ q


Long-Horizon Returns

Basic Structure (page 56-57, CLM):

ttt ywp ; ttt ww 1 ; 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 11 tttttt yyppr

qtt

q

kkt

q

jjtt yyqrqr

1

0

1

0

)(

)(2)0(2)]([ 2 qqqrVar yyt , where γ is the autocovariance function of y.

][

)]([)(

t

t

rqVar

qrVarqVR

We can show: ][

][1

][

)]([)(

pVar

yVar

rqVar

qrVarqVR

t

t


Empirical Evidence

• Autocorrelations– Daily (1962-1994) 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 263.3. The 99.5-percentile for 2

5 is 16.7.– Weekly and monthly returns exhibit similar patterns for the

indexes


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.


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.


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)


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


Economic-Trick Review (1) Asymptotic Distributions

See Greene (2000, Chapter 4, Statistical Inference). If ,1,0/)( Nxn d

n then approximately, or asymptotically, ,/, 2 nNx a

n which we write as

,/, 2 nNx an

Extending this definition to the multivariate case, suppose that nθ is an estimator of the parameter vector θ . The asymptotic distribution of the vector

nθ is obtained from the limiting distribution: ,,0)ˆ( VNn d

n which implies that

Vn

Na 1

,~ˆ nθ

The covariance matrix of the asymptotic distribution is the asymptotic covariance matrix, denoted as

Vn

VarAsy n

1]ˆ[.


(2) Maximum Likelihood EstimatorThis 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:

)(),()(1

X|θθxθ,x,...,x,x in21 Lffn

i

n

i

fL1

)(ln)(ln θ,xX|θ i

The necessary condition for maximizing )(ln X|θL is:

0)(ln

θ

θL -- known as “likelihood equation”


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

i

ixnnL

12

222 ]

)([

2

1ln

2)2ln(

2),(ln

0)(1ln

12

n

iix

L

0)(2

1

2

ln

1

2

422

n

iix

nL

n

n

iiML xx

n

1

1

n

iniML xx

n 1

22 )(1


Properties of MLE

M1: Consistency: MLp ˆlim

M2: Asymptotic normality: ,)(,ˆ 1 INaML , where ],/ln[)( 2 LEI

M3. Asymptotic efficiency: ML is asymptotically efficient and achieves the Cramer-Rao 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 )ˆ( MLc .

Asymptotic Covariance Matrix of MLE: ,])(ln

[)]([ 12

1

L

EI also known as

the information matrix. It is evaluated at ML .

Suppose ]),|[,,( 11122 xyExfy FIML estimation: estimating the joint distribution f(y1,y2) LIFL estimation: estimating y1 first, then maximize a conditional log-likelihood function

using the estimates from step1:

n

iii xxyfLnL

1

^

122212 )]),(,,|(


(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 ),,( 1 kxf with finite moments ][ kxE .

The sample consists of n observations, nxx ,,1 . The kth “raw” or uncentered moment is

n

i

kik x

nm

1

1

By sustituting kii xz , we obtain the following results. By (3-14) and Theorem 4.1,

][][ kikk xEmE

)(1

][1

][ 22 kk

kik n

xVarn

mVar

][lim kikk xEmp

],0[)( 22 kk

dkk Nmn


MOM ][1 ixE

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.

0),...,(

.

.

.

0),...,(

0),...,(

1

122

111

kkk

k

k

g

g

g


MOM Estimator of N(μ,σ2)

In random sampling from ],[ 2N ,

][lim1

lim 11

i

n

ii xEmpx

np

2222

1

2 ][lim1

lim

i

n

ii xVarmpx

np

Note: the variance of MOM estimator of μ is n/2 . The variance of MOM estimator of 2 is whatever listed in Greene’s book.


(4) Assumptions of Linear Regression Models

(1) Linear functional form of the relationship (2) X is an nxK matrix with rank K (3) )|( XE = 0

(4) 2)|( XVar -- Spherical disturbances (5) nonstochatic regressors

(6) normality: Xε | =N[0,σ2I]


(5) Least Square Estimation

iiy βx 'i

This is an unknown relation between x and y. In other words, βxx 'ii ]|[ iyE

The estimate of ]|[ ixiyE is denoted bx 'i

^

iy

ii ey bx 'i

One way to estimate this is the least square estimator: )()'(min 00

βXβyXβy

0

We get yX'XbX'

yX'XX'b 1)(

For finite samples, b follows any distribution with a mean of β and variance of 12 )( XX' . For large samples, when the regression model has normally distributed

disturbance, b has the minimum variance.


Generalized Least Squares

yXXX 111^

')'( 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


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 Tests1 Lecture 10: Testing Market Efficiency The following topics will be...

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Transcript of L10: Market Efficiency Tests1 Lecture 10: Testing Market Efficiency The following topics will be...