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

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
+ Y™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 ˜trading rules™ (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 ˜experiments™ 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
Survey data are sometimes available on individual agents™ expectations of
variables (e.g. of future inflation, exchange rates or interest rates). This may
form of quantitative information collected on an individual™s expectations, fo
he may reply that ˜interest rates will be 10 percent this time next year™. 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 ˜in aggregate™,
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 individual™s expectations it is always possible that individu
reveal their ˜true™ 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 ˜making t
(i.e. those who are ˜on the margin™ 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 individual™s systematic inaccurate reporting of
require one to impose some restrictive assumptions, which may invalidate the
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 . ™ ˜ + ˜
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

= B™A,
- r $+12)
(&+l., El

for x = p . CI™, 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
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(a™
Estimated Equation SEE
0.06 1.131

0.20 1.891

0.07 11.519

0.21 24.17

(a) R™ 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 ˜real world™ 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 ˜true™ 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:

+ y™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

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 ˜informational efficiency™ 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˜ation˜˜):

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 ˜overidentified™).
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
+ (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 ˜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
informed™ 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.

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
The EMH assumes investors process information efficiently so that persisten
profits cannot be made by trading in financial assets. The return on
comprises a ˜payment™ 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 ˜real world™.

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 ˜ p ™ 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


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)™ is the correct
However, derivation of (5.47) in the text is less complex and provides th
required at this point.
Empirical Evidence on Efficienc

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