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(i) All agents act as if they have an equilibrium (valuation) model of return

determination).

(ii) Agents process all relevant information in the same way, in order to

equilibrium returns (or fundamental value). Forecast errors and hence exc

are unpredictable from information available at the time the forecast is m

(iii) Agents cannot repeatedly make excess profits.

profits depends on correctly adjusting returns, for risk and transactions cos

(iii) is best expressed by Jensen (1978):

a market is efficient with respect to an information set Q, if it is impossible to make e

profits by trading on the basis of 52,. By economic profits we mean the risk adjuste

return, net of all costs.

5.4 TESTING THE EMH

This section provides an overview of some of the test procedures used in as

EMH. It is useful to break these down into the following types:

( 9 Tests of whether excess (abnormal) returns qkl = Rjl+l - EPRi,+l are i

of information Qf available at time t or earlier. To test this propositiop

+ Yâ€™Qr +

= EPRit+l

Rir+l ˜lp+1

where E,PRi,+l = equilibrium returns. If information Qr adds any

explanatory power then Rjf+l - E,PRit+l is forecastable. This type of test

to as a test of informational efficiency and it requires an explicit repres

the equilibrium asset pricing model used by agents. These tests are discu

next chapter.

(ii) Tests of whether actual â€˜trading rulesâ€™ (e.g. buy low, sell high) can earn s

or above average profits after taking account of transaction costs and

to cover the general (systematic) riskiness of the assets in question.

usually involve â€˜experimentsâ€™ which mimic possible investor behaviou

are discussed in Chapter 8.

(iii) Tests of whether market prices are always equal to fundamental value.

use past data and try and calculate fundamental value (or the variance

mental value) using some form of DPV calculation. They then test to s

actual prices equal the fundamental value or more precisely whether th

in actual prices is consistent with that dictated by the variability in fun

These volatility tests are discussed in the next chapter.

In principle the above tests are not mutually exclusive but in practice it is po

results from the different type of tests can conflict and hence give different

concerning the validity of the EMH. In fact in one particular case, namely that

bubbles (see Chapter 7), tests of type (i) even if supportive of the EMH can n

(as a matter of principle) be contradicted by those of type (iii). This is because

bubbles are present in the market, expectations are formed rationally and fore

are independent of $2, but price does not equal fundamental value.

5.4.1 Tests of the RE Axioms using Survey Data

In the course of this book a large number of tests of increasing complex

presented. The EMH consists of the joint hypothesis of a particular equilibrium

Therefore our joint hypothesis is reduced to a test only of the informational

assumptions. Our results will be valid regardless of the equilibrium model used

Although tests using survey data appear to avoid a key problem area in testin

(i.e. which equilibrium model to use) nevertheless such tests have their ow

difficulties.

Survey data are sometimes available on individual agentsâ€™ expectations of

variables (e.g. of future inflation, exchange rates or interest rates). This may

form of quantitative information collected on an individualâ€™s expectations, fo

he may reply that â€˜interest rates will be 10 percent this time next yearâ€™. This i

for each individual i provides a time series of his expectations Z;f+j. Using p

can directly calculate the forecast error Ejt+l = Zir+j - Z;f+j for each individu

time periods. We do not need to know the precise model the individual uses

Z i t + j yet we can test for informational efficiency by running the regression:

= 1. If H o is not rejected then f

and testing the null H o : /30 = /32 = 0 and /31

the forecast error is zero on average

and is independent of any information At available at time t. The limited info

A r (C 52,) consists of any variables known at time t or earlier (e.g. past interest

prices, exchange rates). For the forecast error Eir+l to be independent of info

time t, we also require Eit+l to be serially uncorrelated. Standard test statistics ar

to test the latter proposition (e.g. Box-Pierce Q statistic, Lagrange multiplier

Frequently, survey data on expectations are only available â€˜in aggregateâ€™,

a sample of individuals (i.e. the figures are for the average forecast for any p

for all participants in the survey) and clearly this makes the interpretation of

more problematic. For example, if only one person in a small sample of

exhibits behaviour that violates the information efficiency assumptions, this m

in a rejection of the RE axioms. However, under the latter circumstances m

would argue that the information efficiency was largely upheld. Indeed, even

are survey data on individualâ€™s expectations it is always possible that individu

reveal their â€˜trueâ€™ expectations, that is the forecasts they would have made in an

world situation (e.g. by backing their hunch with a large $ investment). In other

survey data might reject the information efficiency assumption of RE because p

in the survey had little incentive to reveal their true forecasts, since they lose

such forecasts are erroneous. Another problem is that participants in a survey

typical of those in the market who are actually doing the trades and â€˜making t

(i.e. those who are â€˜on the marginâ€™ rather than intramarginal). Finally, althoug

econometric techniques available (such as instrumental variables estimation)

for random errors of measurement in the survey data, such methods cannot

mismeasurement based on an individualâ€™s systematic inaccurate reporting of

expectations.

require one to impose some restrictive assumptions, which may invalidate the

consideration.

The applied work in this area is voluminous and the results are not really

the subject matter of this book (see Pesaran (1987) for an excellent survey). H

is worth briefly illustrating the basic methodology. For example, Taylor (198

monthly categorical data from UK investment managers into quantitative e

series for expected annual price inflation , P : + ˜ annual wage inflation , M . â€™ ˜ + ˜

˜,

percentage change in the FTA all share index ,ft+12 and the US Standard

composite share index , ˜ , + 1 2 . The axioms of RE imply that the forecast error

pendent of the information set used in, making the forecast. Consider the regr

+

= Bâ€™A,

- r $+12)

(&+l., El

for x = p . CIâ€™, f.s and where A, is a subset of the complete information set. I

mational efficiency (orthogonality) property of RE holds, we expect B = 0. If

no measurement error in x:+12then E , is a moving average error of order 1

OLS yield consistent estimates of B because A, and E , are uncorrelated asy

but the usual formula for the covariance matrix of B is incorrect. However

residuals can be used to construct a consistent estimate of the variance-covaria

(White, 1980) along the lines outlined in Chapter 20 (i.e. the GMM-Hansen

adjustment).

The results of this procedure are given in Table 5.1 for the information

( ˜ ˜ - 1 x,-2). For the price inflation, wage inflation and the FT share index, th

,

errors on the own lagged variables indicate that all of these variables taken i

are not significantly different from zero. This is confirmed by the Wald test W

indicates that the two RHS variables in each of the first three equations are

significantly different from zero. For the S&P index the lagged values are si

Orthogonality Regressions with Small Information Sets 1981(7)- 1985(7

Table 5.1

Least Squares with Adjusted Covariance Matrix(aâ€™

Estimated Equation SEE

R

0.06 1.131

0.20 1.891

0.07 11.519

0.21 24.17

(a) Râ€™ is the coefficient of determination, SEE the standard error of the equation; W ( 2 ) is a Wal

for the coefficients of the two lagged regressors to be zero and is asymptotically central chi

the null of orthogonality, with two degrees of freedom: figures in parentheses denote estim

errors or for W ( 2 ) marginal significance levels.

Source: Taylor (1988), Table 1 . Reproduced by permission of Blackwell Publishers

The Efficient Markets Hypothesis

Orthogonality Regressions with Small Information

Table 5.2

Estimated Equation

Pr+12 = 0-55OrP;+;,, 1.315 - 0.399˜1-1 0.488˜r-2 +

+

(0.202) (1.122) (0.286) (0.270)

Wr+12 = O.O21rWF+12 6.151 0.006w,-l + O.185˜,-2

+

+

(0.144) (1.712) (0.075) (0.122)

0.199fr-1 0.124fr-2

f r + 1 2 = 0.473rf:+l2 20.066 +

+

+

(0.340) (6.925) (0.125) (0.175)

Sr+12 = - O.725rSF+;,, 62.658 - 0.614˜r-1- O.154˜,-2

+

(0.468) (16.716) (0.179) (0.260)

(a) Instruments used for the expectations variable were p r , w r , ft and s r ; H(3) is H

square with three degrees of freedom for three valid overidentifying instrument

Source: Taylor (1988), Table 2. Reproduced by permission of Blackwell Publishers

to be unity. There is also a non-zero

we do not expect the coefficient on

between and the error term and hence an instrumental variable estimator

Taylor uses p , , 5 , wr, s, as instruments for the expectations variables ,itq

results using the IV estimator are given in Table 5.2. The results are similar

Table 5.1, except for the FT share price index f1+12. Here the GMM estimato

that the forecast error for the FT share price index is not independent of the info

(W(2) = 46.9). This demonstrates that when testing the axioms of RE, correc

may require careful choice of the appropriate estimation technique. Taylor

above exercise using a larger information set A * = ( p , - , , w+,, f,-,,s+,); j =

this extended information set the GMM estimator indicates that the orthogonality

is decisively rejected for alE four variables.

Surveys of empirical work on direct tests of the RE assumptions of unbias

informational efficiency using survey data, for example those by Pesaran (

Sheffrin (1983) tend frequently to reject the RE axioms (for recent results see

Batchelor and Dua (1987), Cavaglia et a1 (1993), Ito (1990) and Frankel and Fr

At this point the reader may feel that it is not worth proceeding with the RE a

If expectations are not rational why go on to discuss models of asset prices t

rationality? One answer to this question is to note that tests based on survey d

definitive and have their limitations as outlined above. Indirect tests of RE bas

on returns or prices that are actually generated by â€˜real worldâ€™ trades in the ma

therefore provide useful complementary information to direct tests based on s

5.4.2 Orthogonality and Cross-Equation Restrictions

The use of survey data means that the researcher does not have to postulate

model to explain expected returns. If survey data are not available, the null hy

efficiency may still be tested but only under the additional assumption that the e

pricing model (e.g. the CAPM) chosen by the researcher is the one actually used

participants and is therefore the â€˜trueâ€™ model. To illustrate orthogonality and

cross-equation restrictions in the simplest possible way let us assume that an e

pricing model for Zr+l may be represented as:

+ yâ€™x,

EPZ,+l = yo

where xf is a set of variables suggested by the equilibrium pricing model.

informational efficiency (or orthogonality), conditional on the chosen equilibri

involves a regression

+

+ +

&+l = Yo &A, Er+l

Yâ€™Xr

The orthogonality test is Ho: 8 2 = 0. One can also test any restrictions on

suggested by the pricing model chosen. The test for 8 2 = 0 is a test that the de

of the equilibrium pricing model (i.e. x,) fully explain the behaviour of Zr+l

the RE (random error) or innovation &,+I). Of course, informational efficien

tested using different equilibrium pricing models.

on the first moment of the distribution, namely the expected value of & , + I . H

E ˜ + I is not homoscedastic, additional econometric problems arise in testing Ho

of these are discussed in Chapter 19.

Cross-Equation Restrictions

There are stronger tests of â€˜informational efficiencyâ€™ which involve cross-equat

tions. A simple example will suffice at this point and will serve as a useful i

to the more complex cross-equation restrictions which arise in the vector aut

(VAR) models of Part 5. To keep the algebraic manipulations to a minimum

a one-period stock which simply pays an uncertain dividend at the end of p

+

(this could also include a known redemption value for the stock at t 1). T

valuation formula determines the current equilibrium price:

Pt = SEtDt+I = 6D;+1

where 6 is the constant discount factor. Assume now an expectations generatin

for dividends based on the limited information set At = (Or,Dt-l):

with E(vt+lIAt)= 0, under RE. It can now be demonstrated that the equilibri

model (5.37) plus the assumed explicit expectations generating equation (5.3

assumption of RE; in short the EMH implies certain restrictions between the

of the complete model. To see this, note that from (5.38) under RE

+ Y2Dt-1

D;+l = YlDt

and substituting in (5.37):

+ SY2Dt-1

pt = b l D t

We can rewrite (5.40) as a regression eq˜ationâ€˜˜):

where 771 = 6 ˜ 1 , 1 1 2= Sy2. A regression of Pt on (Dt,Dt-l) will yield coefficien

771 and 772. Similarly, the regression equation (5.38) will yield estimates 773 an

= y1 and = y2. However, if (5.38) and (5.40) are true then we k

where 773 774

The values of (yl,y2) can be directly obtained from the estimated values of

while from (5.43) S can be obtained either from n1/113 or n2/n4. Hence in

obtain two different values for S (i.e. the system is â€˜overidentifiedâ€™).

We have four estimated coefficients (i.e. 771 to 774) and only three underlying

in the model (61, y1, y2). There is therefore one restriction (relationship) amo

An intuitive interpretation of the cross-equation restrictions is possible. I

below that these restrictions do nothing more than ensure that no supernormal

earned on average and that errors in forecasting dividends are independent of i

at time t or earlier. First, consider the profits that can be earned by using ou

equations (5.41)and (5.42).The best forecast of the DPV of future dividends

by V = SDf+, and using (5.42)

Usually, the realised price will be different from the fundamental value given

because the researcher has less information than the agent operating in the m

Ar c Q). The price is given by (5.41)and hence excess returns or profits are

+ + n4Dt-1)

PI - Vt = ( ˜ 1 D t n2Dr-1) - 6(˜3Dt

For all values of (Q, Dt-l), profit will be zero only if

but this is exactly the value of S which is imposed in the cross-equation restricti

Now consider the error in forecasting dividends:

where we have used (5.42)and the equilibrium model (5.37).Substituting f

(5.41)gives:

+ (774 - nz/W,-1 + vr+1

Dr+1 - q + 1 = (773 - n1/J)Q

Hence the forecast error can only be independent of information at time t (

Dr-l) if 8 = nl/n3 = n2/1r4.

By estimating (5.41)plus (5.42) without the restrictions imposed and then re

with the restrictions (5.43) imposed, a suitable test statistic can be used

validity of EMH for the given equilibrium model, under the specific expectat

ating equation (5.37). These tests of cross-equation restrictions are very preva

EMH/RE literature and are frequently much more complex algebraically than

example above, as we shall see in Part 5.However, no matter how complex, su

tions merely ensure that no abnormal profits are earned on average and that fore

are orthogonal to the information set assumed.

One additional problem with the above test procedure is that it is conditio

specific expectations-generating equation chosen for Q + l . If this is an incorrec

tation of how agents form expectations then the parameters y1 and y 2 ( ˜ 3774 ,

to be biased estimates of the true parameters and the cross-equation restriction

the estimated parameters may not hold. This concludes our overview of the ty

used to assess the EMH and it remains to mention briefly some conceptual lim

the EMH.

costless, is a very strong one. If prices â€˜always reflect all available relevant in

which is also costless to acquire, then why would anyone invest resources i

information? Anyone who did so would clearly earn a lower return than

costlessly observed current prices, which under the EMH contain all relevant i

As Grossman and Stiglitz (1980) point out, if information is costly, prices cann

reflect the information available. They also make the point that speculative mar

be completely efficient at all points in time. The profits derived from speculat

result of being faster in the acquisition and correct interpretation of existin

information. Thus one might expect the market to move towards efficiency a

informedâ€™ make profits relative to the less well informed. In so doing the sm

sells when actual price is above fundamental value and this moves the price c

fundamental value. However, this process may take some time, particularly if

unsure of the true model generating fundamentals (e.g. dividends) which may al

time. If, in addition, agents have different endowments of wealth and henc

market power in influencing price changes, they may not form the same expec

particular variable. Also irrational traders or noise traders might be present an

rational traders have to take account of the behaviour of the noise traders. It

possible that prices might deviate from fundamental value for substantial p

these issues are discussed further in Chapter 8. Recently, much research has

on the nature of sequential trading, the acquisition of costly information and the

of noise traders. These models imply that regression tests of the orthogonali

unbiasedness properties of RE are tests of a rather circumscribed hypothesis, n

where the true model is assumed known at zero cost and where expectations

on predictions from this true model.

SUMMARY

5.5

Consideration has been given to the basic ideas that underlie the EMH in bo

and mathematical terms and the main conclusions are:

The outcome of tests of the EMH are important in assessing public policy

as the desirability of mergers and takeovers, short-termism and regulation o

institutions.

The EMH assumes investors process information efficiently so that persisten

profits cannot be made by trading in financial assets. The return on

comprises a â€˜paymentâ€™ to compensate for the (systematic) risk of the po

information available in the market cannot be used to increase this return.

The EMH can be represented in technical language by stating that ret

martingale process.

Tests of the informational efficiency (RE) element of the EMH can be undert

survey data on expectations. In general, however, tests of the EMH require

equilibrium model of expected returns. Tests often involve an analysis o

informational efficiency (RE) or an inappropriate choice of the model for e

returns or simply that the EMH does not hold in the â€˜real worldâ€™.

ENDNOTES

1. LeRoy (1989) favours a definition of the EMH as constituting the propo

returns follow the fair game property and that agents have rational expect

practical purposes, his definition is not that different from the one adopte

2. Equation (5.32), which uses the superscript â€˜ p â€™ to represent the forecas

market participants v:+˜, makes it clear that these forecast errors only obe

erties of conditional mathematical expectations if agents actually do u

model of the economy (i.e. then E;R,+I = E,R,+I).

3. There are some rather subtle issues in developing these cross-equation

and a simplified account has been presented in the text. A more complete

of the issues in this section is given below. The researcher is unlikely to h

information set that is available to market participants, Ar c 52,. This i

equation (5.40) has an error term which reflects the difference in the inf

- E(

sets available, that is wr+l = [E(Dr+11Qr) D r + l l A r ) ]To see this, no

.

stock price is determined by the full information set available to agents:

The econometrician uses a subset Ar = (Or,Q-1) of the full information

cast dividends:

+ +

Dt+l = n3Dr r4Dr-1 vr+l

where

and qr+l is the true RE forecast error made by agents when using the full

set, Q r .Note that E(w,+l(A,)= 0. To derive the correct expression for (5

(I), (2) and (3):

+

pt = SEr(Q+1lQr) = SE(Dr+11Ar) 6wr+l

which is equation (5.45) in the text.

The forecast for dividends is based on the full information set availabl

(although not to the econometrician) and using (2) and (1) is given by:

However, substituting for P, from (5) and noting that w,+l = Vr+l - qt+l

Hence equation (8) above, rather than equation (5.47)â€™ is the correct

However, derivation of (5.47) in the text is less complex and provides th

required at this point.

6

I

Empirical Evidence on Efficienc

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