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on stocks depend in a predictable way upon information readily available (i.e
of the week it is). This is a violation of the EMH under the assumption of a co
premium since returns are in part predictable. However, in the ˜real world™ it m
the case that this predictability implies that investors can earn supernormal p
transactions costs need to be taken into account.
It should be clear from the above discussion that in order to test the EMH
some view of how prices or rates of return are determined in the market.
require an economic model of the determination of equilibrium returns and a
Our test of whether agents use information efficiently is conditional on our hav
the correct model to explain either stock prices or the rate of return on such stoc
or rejection of the efficient markets hypothesis may be either because we have
equilibrium ˜pricing model™ or because agents genuinely do not use information
As noted above, another way of describing the EMH is to say that in an effic
it is impossible for investors to make supernormal profits. Under the EMH inve
a return on each security which covers the riskiness of that security (and they
sufficient profits just to cover their costs in order to stay in the industry, th
business of buying and selling shares). However, there must not be opportunitie
to make abnormal profits by dealing in stocks. The latter is often referred to
game™ property .
One particular equilibrium valuation model is the CAPM. The CAPM pred
equilibrium the expected excess return on asset i should depend only on its b
expected excess return on the market.


Since the excess return on the market portfolio is constant at a particular po
then two assets will have different expected excess returns only if their respe
its risk class and this would be a violation of the EMH. However, note tha
be a violation of the EMH under the assumption that the CAPM is the true
equilibrium asset pricing. This again emphasises the fact that tests of the EM
involve the joint hypotheses (i) that agents use information rationally, (ii) th
use the same equilibrium model for asset pricing which happens to be the ˜tru


5.2 IMPLICATIONS OF THE EMH
The view that the return on shares is determined by the actions of™rational a
competitive market and that equilibrium returns reflect all available public in
is probably quite a widely held view among financial economists. The slight
assertion, namely that stock prices also reflect their fundamental value (i.e. t
future dividends) is also widely held. What then are the implications of the EM
to the stock market?
As far as a risk averse investor is concerned the EMH means that he shou
˜buy and hold™ policy. He should spread his risks and hold the market portfo
20 or so shares that mimicethe market portfolio). Andrew Carnegie™s advice
your eggs in one basket and watch the basket™ should be avoided. The role for
analysts, if the EMH is correct, is very limited and would, for example, inclu

(i) advising on the choice of the 20 or so shares that mimic the market por
(ii) altering the proportion of wealth held in each asset to reflect the ma
portfolio weights (the xi* of the CAPM) which will alter over time. The
both as expected returns change and as the riskiness of each security rela
market changes (i.e. covariances of returns),
(iii) altering the portfolio as taxes change (e.g. if dividends are more highly
capital gains then for high rate income tax payers it is optimal, at the
move to shares which have low dividends and high expected capital gai
(iv) ˜shopping around™ in order to minimise transactions costs of buying and
Under EMH the current share price incorporates all relevant publicly availabl
tion, hence the investment analyst cannot pick winners by reanalysing publicl
information or by using trading rules (e.g. buy ˜low™, wait for a price rise and s
Thus the EMH implies that a major part of the current activities of investmen
is wasteful. We can go even further. The individual investor can buy a pro
an indexfund (e.g. mutual fund or unit trusts). The latter contains enough se
closely mimic the market portfolio and transactions costs for individual inv
extremely low (say less than 2 percent of the value of the portfolio). Practiti
as investment managers do not take kindly to the assertion that their skills
redundant. However, somewhat paradoxically they often support the view that
is ˜efficient™. But their use of ˜efficient™ is usually the assertion that the stock
low transactions costs and should be free of government intervention (e.g. z
duty, minimal regulations on trading positions and capital adequacy).
sell funds in particular sectors (e.g. chemicals or services) or specific geograp
(e.g. Japanese stocks). There is a marketing reason for this. If finance house
number of such funds, then they effectively hold the market portfolio, while the
can speculate on individual ˜packages™ of mutual funds. Also with this strategy
house will usually have at least one fund that it can boast has ˜beaten the ma

Takeovers, Conglomerates and Financial Institutions
Let us turn now to some public policy issues. The stock market is supposed to
˜correct™ signals for the allocation of real resources (i.e. fixed investment). O
proportion of corporate investment is financed from new issues (e.g. about 4 p
gross basis in the UK); nevertheless, the rate of return of a quoted company o
market provides a measure of the opportunity cost of funds corrected for risk
can be used in discounting future expected profits from a physical investm
(i.e. in investment appraisal). Other things equal, if profits from a firm™s new
project are expected to be high the existing share price will be ˜high™ and th
obtain its funds by issuing fewer shares. However, if the share price does
firndamentals but is influenced by whim or fads of ˜irrational™ investors the
is broken. An abnormally low share price which reflects ill-informed extrane
(e.g. irrational market prejudice) will then inhibit a firm from embarking on
rational calculation) is a viable investment project.
The above analysis also applies to takeovers. If the stock market is myo
only considers profits and dividends that accrue in the near future, then manag
of a takeover may distribute more in current dividends rather than using th
profits to undertake profitable real investment say on R&D expenditure. Th
will boost the share price if the market is myopic. This is generally known
termism™. A possible response by government to such short-termism might b
hostile takeovers (e.g. as in Japan). The impact of short-termism on share pr
also be exacerbated by an incentive system whereby part of a manager™s rem
is in the form of share options. See Chapter 8 for a discussion on the theo
empirical issues that are relevant to the debate on short-termism.
The opposite view to the above, namely that hostile takeovers are welfare
(i.e. in terms of the output and profits of the firm) requires the assumption th
are efficient and that takeovers enable ˜bad™ incumbent managers to be replac
scenario, the hostile bidder recognises that the incumbent ˜bad™ manageme
shareholders to mark down the firm™s share price. The hostile bidder pays
excess of the existing share price. After replacing the ˜bad™ managers and re
the firm, the ensuing higher future profits are just sufficient to compensate for
price he paid for the shares.
In the 1960s and 1970s there was a wave of conglomerate formation follo
1980s by leveraged buyouts and conglomerate breakups (i.e. ˜asset stripping™).
erate mergers were sometimes justified on the grounds that the acquisition o
firms by ˜firm A™ reduced risk to the shareholder who held A™s shares since the
erate™ constituted a diversified portfolio of firms. The latter is an analogous a
not diversify their share holdings.)
Note that if share prices do reflect fundamentals but ˜news™ occurs freque
expected to make a substantial impact on a firm™s future performance, then one
expect to observe highly volatile share prices, even if the market is efficient. H
on occasions such volatility had adverse implications for parts of the real ec
an ˜externality™) - for example, that a stock market crash led to insolvencies
institutions, a ˜credit crunch™ and less physical investment - this would at le
a prima facie argument for governments to try and limit share price moveme
closing markets for a ˜cooling-off period™). Also, where systemic risk is invo
˜run™ on one bank causes a run on other banks) one might be prepared to proh
institutions from holding ˜highly volatile™ assets such as shares (e.g. banks in
By definition, ˜news™ is random around zero. Hence ˜news™ will not influen
of share prices over a long horizon. Therefore except in exceptional circumsta
would be unlikely to cause panics leading to a ˜run™ on banks or financial ins
their depositors. However, if the market is inefficient and prices are subject to
˜irrational swings™ then stock price volatility may be greater than that predicte
efficient markets hypothesis. Here, a prima facie case for financial institutio
enough resources (reserves) to weather such storms seems stronger. This is on
for general capital adequacy rules applied to financial institutions. If there are
risks (i.e. a form of externality) then, in principle, government action is require
that the level of capital reflects the marginal social costs of the systematic
than the marginal private costs (for any individual financial institution).
What are the implications of market efficiency in stock and bond markets f
corporate finance? If the market is efficient then there is no point in delaying
investment project in the hope that ˜financing conditions will improve™ (i.e. th
price will be higher): under the EMH the current price is the correct price a
expected future earnings from the project. Also under the EMH the firm™s cos
cannot be lowered by a given mix of securities (e.g. by altering the proportio
and equity). The Modigliani-Miller theorem (in the absence of taxes and b
suggests that the cost of capital is independent of the capital mix (i.e. debt-e
in an efficient market. The issue of capital mix can also be applied to the matu
structure of debt. Since rates on long and short corporate bonds fully reflec
information the proportion of long debt to short-dated debt will also not alter
capital to the firm. For example, under the expectations hypothesis, low lon
high current short rates simply reflect an expectation of lower future short rates
no advantage ex ante to financing an investment project by issuing long bonds
˜rolling over™ a series of short bonds. (This is discussed in Chapter 9 on the term
It follows from the above arguments that the role of the corporate treas
˜active manager™ either as regards the choice over the appropriate ˜mix™ of
finance or in analysing the optimum time to float new stock or bond issues is f
the EMH. Of course, if the market is not efficient the corporate treasurer m
to ˜beat the market™ and he can also alter the stock market valuation of the
chosen dividend policy or by share repurchase schemes, etc.
case for government intervention. However, given uncertainty about the imp
government policies on the behaviour of economic agents, the government s
intervene if, on balance, it feels the expected return from its policies outweig
attached to such policies. Any model of market inefficiency needs to ascerta
from efficiency the market is on average and what implications this has for pu
decisions and economic welfare in general. This is a rather difficult task giv
knowledge, as subsequent chapters will show.


53 EXPECTATIONS, MARTINGALES AND FAIR GA
.
As previously mentioned, the EMH can be formally stated in a number of diff
We do not wish to get unduly embroiled in the finer points of these alterna
our main concern is to see how the hypothesis may be tested and used in un
the behaviour of asset prices and rates of return. However, some formal de
the EMH are required. To this end, let us begin with some properties of
mathematical expectations; we can then state the basic axioms of rational e
such as unbiasedness, orthogonality and the chain rule of forecasting. Next w
the concepts of a martingale and a fair game. We then have the basic tools
alternative representations and tests of the ˜efficient markets hypothesis™.

Mathematical Expectations
If X is a random variable (e.g. heights of males in the UK) which can ta
values X I , Xz, X3 . . . with probabilities Ttj then the expected value of X, deno
defined as


i=l

If X is a continuous random variable (-00 < X < 00) with a continuous
distribution f ( X ) (e.g. normal distribution) then




Conditional probability distributions or conditional density functions are used
in the RE literature. For example, a fair die has a probability of (1/6)th
on any number from 1 to 6. However, suppose a friend lets you know t
to be used is biased and lands on the number 6 for half the time and on
numbers equally for the remaining throws. Conditional on the information
friend you would then alter your probabilities to (1/2) for a 6 and (1/10) for the
five numbers. Your conditional expected value would therefore be differen
expected value from an unbiased die, since the associated probabilities (or
density function) are different. The conditional expectation based on the info
where f(X, 152,) is the conditional density function. A conditional expectati
viewed as an optimal forecast of the random variable X t , based on all relevant i
52,. The conditional forecast error is defined as where
= Xr+l - E(Xr+l 152,)
&r+l

This (mathematical) conditional forecast error can be shown (always) to b
average:
E(Er+llQ) = E(Xr+lIQ) - E(Xr+llQr) = 0

We can rearrange (5.14) as:


and hence reinterpret (5.16) as stating that the conditional expectations are a
forecast of the outturn value.
The second property of conditional mathematical expectations is that the fo
is uncorrelated with all information at time t or earlier which, stated mathema
=0
E(E:+lQrIQr)

This is known as the orthogonality property of conditional expectations. Th
reason why (5.17) holds is that if RI could be used to reduce the forecast erro
it could be used to improve the forecast: hence all relevant information coul
been used in forecasting X r + l . It also follows that an optimal conditional fore
where subsequent forecast errors are unpredictable.
Note that an optimal forecast need not necessarily predict X,+l accurately.
can be large and the conditional expectation E t X l + j may only explain a sm
the variation in actual X , + j . What is important is that the optimal forecast
improved upon (in the sense of using 5 2 r to reduce the forecast errors E t + j ) .
worth noting that it is only the behaviour of the mean of the forecast error th
restricted in (5.17). The variance of the conditional forecast error denoted E ( q
not be constant and indeed may in part be predictable. The latter is of import
discussing the implications for market volatility within the framework of the
Consider for a moment making a forecast in January (at time t ) as to what t
+
you will make in February (t 1) will be, about the outcome of the variable X
(i.e. X r + 2 ) . Mathematically this may be represented as
)I
Er [Er+1 ( X t + 2 I fir+ 1

If information 52, at time t is used efficiently then you cannot predict today ho
change your forecast in the future, hence


where E , (Xl+l) is equivalent to E(X,+1(Rr).This is the rule of iterated e
which may be succinctly represented as:
- - - = Er
ErEt+lEt+2
of outcomes. Economic agents are therefore assumed to behave as if they fo
subjective expectations as the mathematical expectations of the true model of th
This is generally referred to as ˜Muth-RE™ (Muth, 1961).
To get a feel for what this entails consider a simple supply and demand
say, wheat. The supply and demand curves are subject to random shocks (e
in the weather on the supply side and changes in ˜tastes™ on the demand side
based products such as cookies). The actual equilibrium price depends in p
actual value of such ˜shocks™ which will only be revealed after the market h
Conceptually the individual RE farmer has to determine his supply of wheat, at
and the expected supplies of wheat of all other farmers (based on known fact
technology, prices of inputs, etc.). He makes a similar calculation of the kno
influencing demand such as income, xf. He then solves for the expected e
price by setting the demand and supply shocks to their expected values of zero
farmers behave as if they use a competitive stochastic model of supply and de
difference between the equilibrium or expected price and the outturn price is
unforecastable ˜error™ due to the random shocks to the supply and demand fun
additional information available to the farmer can reduce such errors any furth
RE orthogonality property holds). The stochastic reduced form is

Pf+1 = p;+1 + Ef+1 = fcx;™ x ; ) + Ef+1
where P f ; , = f ( x f , x : ) is the equilibrium price based on the known factor
influence supply and demand. The forecast error is the random variable &
under Muth-RE the uncertainty or randomness in the economy (e.g. the weat
product innovations) gives rise to agents™ forecast errors for the actual equilib
To test whether agents™ actual subjective expectations obey the axioms of
ical conditional expectations we either need an accurate measure of individ
subjective expectations or we need to know the form of the true model of th
used by all agents. Survey data on expectations can provide a ˜noisy™ proxy v
each agent™s subjective expectations. If we are to test that actual forecast erro
properties of conditional mathematical expectations via the second method (i.e
true model of the economy) the researcher has to choose a particular model fr
the many available on the ˜economist™s shelf™ (e.g. Keynesian, monetarist, re
cycle, etc.). Clearly, a failure of the forecast errors from such a model to ob
axioms could be due to the researcher choosing the wrong model from the ˜s
is, agents in the real world actually use another model.) The latter can provid
nient alibi for a supporter of RE, since he can always claim that failure to con
axioms is not due to agents being non-rational but because the ˜wrong™ econo
was used.

Martingale and Fair Game Properties
Suppose we have a stochastic variable X , which has the property:
Thus a fair game has the property that the expected ˜return™ is zero given Q r .
trivially that if X, is a martingale y,+l = X,+l - X, is a fair game. A fair game
sometimes referred to as a martingale difference. A fair game is such that th
return is zero. For example, tossing an (unbiased) coin with a payout of $1
and minus $1 for a tail is a fair game. The fair game property implies that t
to the random variable yt is zero on average even though the agent uses a
information Ql in making his forecast.
One definition of the EMH is that it embodies the fair game property for
stock returns yr+l = R,+l - Rf,,, where R:+l is the equilibrium return give
economic model of the supply and demand for risky assets (e.g. CAPM). The
abnormal) return is the profit from holding the risky asset. The fair game prope
that on average the abnormal return is zero™. Thus an investor may experience
and losses (relative to the equilibrium return RT+1) in specific periods but the
out to zero over a series of ˜bets™.

Stock Prices and Martingales
Let us assume a simple model of returns, namely that the equilibrium or requ
is a constant = k . The fair game property implies that the conditional expec
return is zero:
E[(R,+l - k)lQ,] = 0
Given the definition of R,+l we have


where we have used the logarithmic approximation for the proportionate price c
+
ln(P,+l/P,) = ln[l (P,+1 - P,)/P,] % AP,+l/Pr). Since, in general, the div
ratio is non-zero and varies over time then, in general, the (log of the) price le
be a martingale in this class of model. In fact any increase in the expected c
must be exactly offset by a lower expected dividend yield. Hence in the efficie
literature when it is said that stock prices follow a martingale, it should be
that this refers to stock prices including dividends. In fact we could define a
price variable q, such that
ln(qr+l /qr 1 = Rr+l
Then the logarithm of qr is a martingale (under the assumption that expected
constant ).

Fair Game and the Rational Valuation Formula
If we let equilibrium or required returns by investors = k, then the fair gam
implies
E [(RI+l - k,+l)lQ,] = 0
for returns also implies that the price of a stock equals the DPV of future
(Note, however, that in the presence of rational bubbles - discussed in Chap
fair game property holds but the RVF does not). The apparent paradox tha
returns can be unforecastable (i.e. a fair game), yet prices are determined by
fundamentals, is resolved.
A straightforward test of whether returns violate the fair game property
assumption of constant equilibrium returns is to see if returns can be predicte
data, Q r . Assuming a linear regression:


#
then if # ™ 0 (or E f + l is serially correlated), the fair game property is violate
l
test of the fair game property is equivalent to the orthogonality test for RE.

Economic Models and the Fair Game Property
Samuelson (1965) points out that the fair game model with constant requir
that is E,(Rt+l - k) = 0, can be derived under (restrictive) assumptions abo
preferences. All investors would have to have a common and constant time
rate, have homogeneous expectations and be risk neutral. Investors then pre
whichever asset has the highest expected return, regardless of risk. All ret
therefore be equalised and the required (real) rate of return equals the real i
which in turn equals the constant rate of time preference.

Martingales and Random Walks
A stochastic variable X , is said to follow a random walk with drift parameter


where is an identically and independently distributed random variable w
Er+l




As the are independent random variables then the joint density function
El
f ( & , ) f ( ˜ ˜ for m # s and this rules out any dependence between E, and
)
linear or non-linear. The first way in which the martingale model is less rest
the random walk is that for a martingale and E t need only be uncorrelat
1inea r ly re 1ated) ,
A random walk without drift has 6 = 0. Clearly X f + l is a martingale an
Xr+l - Xr is a fair game (for 6 = 0). However, the random walk is more rest
a martingale since a martingale does not restrict the higher conditional mo
02) be statistically independent. For example, if the price of a stock (inc
to
dividend payments) is a martingale then successive price changes are unpred
it allows the conditional variance of the price changes E ( E : + ˜ ( Xto )be predi
˜
Suppose that at any point in time all relevant (current and past) information fo
the return on an asset is denoted Qf while market participants, p , have an i
set Q/ which is assumed to be available without cost. In an efficient market,
assumed to know all relevant information (i.e. Qf = Q f ) and they know th
(true) probability density function of the possible outcomes for returns



Hence in an efficient market, investors know the true (stochastic) economic
generates future returns and use all relevant information to form their ˜best™
the expected return. This is the rational expectations element of the EMH.
Ex post, agents will see that they have made forecast errors and this w
ex-post profits or losses
- W&+IIQf)
= &+I
7),+1
P


where the superscript ˜p™ indicates that the expectations and forecast errors are
on the equilibrium model of returns used by investors2. The expected or e
return will include an element to compensate for any (systematic) risk in the
to enable investors to earn normal profits. (Exactly what determines this ris
depends on the valuation model assumed.) The EMH assumes that excess
forecast errors) only change in response to news so that r&l are innovations w
to the information available (i.e. the orthogonality property of RE holds).
For empirical testing a definition is needed of what constitutes ˜relevant in
and three broad types have been distinguished.

Weak Form: the current price (return) is considered to incorporate all the i
0

in past prices (returns).
Semi-strong Form: the current price (return) incorporates all publicly avail
0

mation (including past prices or returns).
Strong Form: prices reflect all information that can possibly be known
0

˜insider information™ (e.g. such as an impending announcement of a t
merger).
In empirical testing and general usage, tests of the EMH are usually considered
of the semi-strong form. We can now sum up the basic ideas that constitute th

(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
+ 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
difficulties.
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
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 . ™ ˜ + ˜
˜,
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
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(a™
Estimated Equation SEE
R
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
Y™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 ˜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
(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 ˜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.


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 ˜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™.


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 ˜ 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




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)™ 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
in the Stock Market
In Chapter 4 it was noted that any specific model of expected returns implie
model for stock prices. The RVF provides a general expression for stock pr
depend on the DPV of expected future dividends and discount rates. The EM
that expected equilibrium returns corrected for risk are unforecastable and v
this implies that stock prices are unforecastable. Hence tests of the EMH ar
examining the behaviour of both returns and stock prices. In principle, both ty
(rational bubbles are ignored here) should give the same inferences but, in pr
is not always the case. This chapter aims to:

Provide a set of alternative tests to ascertain whether stock returns are fo
0

Tests are conducted for returns measured over different horizons and result
on an examination of the correlogram (autocorrelations) of returns, from
tests using a variety of alternative information sets and a variance ratio te
tion, the profitability of ˜active™ trading strategies based on forecasts from
equations will be examined.
Next stock prices and the empirical validity of the EMH as represented b
0

formula are examined. Shiller (1979, 1981) and LeRoy and Porter (1981
a set of variance bounds tests. They show that if stock prices are dete
fundamentals (i.e. RVF), this puts a limit on the variability of prices whic
be tested empirically, even though expected future dividends are unobserv

To make the above tests of the EMH operational we require an equilibrium mo
prices. We can think of the equilibrium or expected return on a risky asset as
of a risk-free rate r, (e.g. on Treasury bills) and a risk premium, r p ,



At present, there is no sharp distinction between nominal and real variables an
(6.1) could be expressed in either form. Equation (6.1) is a non-operational i
economic model of the risk premium is required. Many (early) empirical tests o
either assume rpr and r, are constant or sometimes they assume only rp, is c
examine data on excess returns, (&+I - rt). If we assume expected equilibri
+ y™Qr +
=k
Rr+l Er+l

where S 2 r = information available at time t . A test of y™ = 0 provides evide
˜informational efficiency™ element of the EMH. These regression tests vary,
on the information assumed which is usually of the following type:

(i) data on past returns l?t-j(j = 0, 1,2, . . .rn) - that is, weak form effic
(ii) data on scale variables such as the dividend price ratio, the earnings pr
interest rates at time t or earlier,
(iii) data on past forecast errors E r - j ( j = 0, 1, . . .m).
When (i) and (iii) are examined together this gives rise to ARMA models, f
the ARMA(1,l) model:
+
&+I = k + y1Rr Er+l - ˜ 2 ˜ r

If one is only concerned with weak form efficiency then the autocorrelation c
between Rr+l and R r - j ( j = 0, 1, . . . rn) can be examined to see if they are n
is also possible to test weak form efficiency using a variance ratio test on re
different horizons. All of these alternative tests of weak form efficiency should
give the same inferences (ignoring small sample problems). The EMH gives no
of the horizon over which the returns should be calculated. The above tests ca
be done for alternative holding periods of a day, week, month or even over m
We may find violations of the EMH at some horizons but not at others.
Suppose the above tests show that informational efficiency does not hold. H
mation at time t can be used to help predict future returns. Nevertheless,
highly risky for an investor to bet on the outcomes predicted by a regressio
which has a high standard error or low R2. It is therefore worth investigating wh
predictability really does allow one to make abnormal profits in actual trading, a
account of transactions costs and possible borrowing constraints, etc. Thus the
somewhat distinct aspects to the EMH as applied to data on returns: one is inf
efficiency and the other is the ability to make supernormal profits.
Volatility tests directly examine the RVF for stock prices. Under the assump
and that expected one-period returns are a known constant, the RVF gives


i= 1

If we had a reliable measure of expected dividends then we could calculate
of (6.4). A test of this model of stock prices would then be to see if var(C
equalled var(P,). Shiller (1982), in a seminal article, obviated the need for data o
dividends. He noted that under RE, actual and expected dividends only differ by
(forecast) error and therefore so do the actual price Pf and the perfect foresigh
defined as P = CG™D,+i. (Note that P uses actual dividends). Shiller dem
: :
two papers led to plethora of contributions using variants of this basic methodo
commentaries emphasised the small sample biases that might be present, wh
work examined the robustness of the volatility tests, under the assumption tha
are a non-stationary process. It is impossible to describe all the nuances in
which provides an excellent illustration of the incremental improvements obta
scientific approach, applied to a specific, yet important, economic issue. The
of this chapter concentrates on the difficulties in implementing and interpre
of these direct tests of the RVF based on variance bounds. Chapter 16 ret
question of excess volatility and the issue of non-stationary data is dealt
improved analytic framework.
In this chapter illustrative examples are provided of tests of the EMH for st
and there is a discussion of volatility tests based on stock prices. It is by n
straightforward matter to interpret the results from the wide variety of tests a
to whether investor behaviour in the stock market is consistent with the EM
as presenting illustrative empirical results there is an indication of how the va
of test are interrelated.

Smart Money and Noise Traders
Before discussing the details of the various empirical tests enumerated above
briefly discussing the implications for stock returns and prices of there being
rational or noise traders in the market. This enables us to introduce the concep
reversion and excess volatility in a fairly simple way. Assume that the mark
a particular type of noise trader, namely a positive feedback trader whose d
the stock increases after there has been a price rise. To simplify matters assu
rational traders or smart money believe that (one plus) the expected or equilib
is constant
+
1 E,R,+1 = k*




Hence the expected proportionate change in price (including any dividend p
a constant which we assume equals 8 percent. If only the smart money is pre
market then the price only responds to new information or news and therefore
prices and the return per period are unpredictable. An example where only a
of news hit the market is shown in Figures 6.1 and 6.2, respectively. For ex
price change from A to B could be due to ˜good news™ about dividends. Price
random. The price level follows a random walk (with drift), that is it tends to
from its starting point in a random fashion and rarely crosses its starting point.
on the stock is unpredictable and past returns cannot be used to predict fut
Indeed any information available is of no use in predicting returns.
Now consider introducing positive feedback traders into the market. Afte
news about dividends revealed at A, the positive feedback traders purchase
Time

Figure 6.1 Random Walk for Stock Price.



t
Returns (Yo)
D
Sbck




I C



Figure 6.2 Stock Returns.

initially increasing its price still further. The price reaches a peak at Y (Figure
if there is some ˜bad™ news at Y, the positive feedback traders sell (or short se
price moves back towards fundamental value. Prices are said to be mean reve
n o things are immediately obvious. First, prices have overreacted to fu
(i.e. news about dividends). Second, prices are more volatile than would be pr
the change in fundamentals. It follows that prices are excessivezy volatile co
what they would be under the EMH. Volatility tests based on the early work
and LeRoy and Porter attempt to measure this excess volatility in a precise w
The per period return on the stock is shown in Figure 6.4. Over short horiz
are positively serially correlated: positive returns are followed by further posit
(points P,Q,R) and negative returns by further negative returns (points R,S,T).
horizons, returns are negatively serially correlated. An increase in returns betwe
is followed by a fall in returns between R and T (or Y). Thus, in the presence o
˜˜
7


Time

Figure 6.3 Positive Feedback Traders: Stock Prices.




I
Returns
stock

R




t Y

O%
-
Time

Figure 6.4 Positive Feedback Traders: Stock Returns.

traders, returns are serially correlated and hence predictable. Also, returns are
correlated with changes in dividends. Hence regressions which look at whet
are predictable have been interpreted as evidence for the presence of noise tra
market.
From Figure (6.5) one can also see why feedback traders can cause chan
variance of prices over different return horizons. In an efficient market, su
prices will either rise or fall by 15 percent per annum. After two years the
in prices might be 30 percent higher or 30 percent lower. The variance in r
N = 2 years equals twice the variance over one year and in general
var(P') = N var(R')
Time

Figure 6.5 Mean Reversion and Volatility. Source: Engel and Morris (199

or
var(RN)
=1
N var(R*)
However, with mean reversion the variance of the returns over N ( = 2) ye
less than N times the variance over one year. This is because prices over
fundamental value in the short run but not in the long run. (Compare points
B,B' in Figure 6.5.) This is the basis of the Poterba-Summer tests of vola
different horizons discussed below. (For a useful summary of these tests see
Morris (1991).)
So far we have assumed that the smart money believes that equilibrium expec
are constant. Let us examine how the above regression (or serial correlation) te
interpreted when expected returns are not constant. Expected equilibrium ret
vary either because subjective attitudes to risk versus return (i.e. preference
or because of changes in the risk-free rate of interest or because shares are
inherently more risky at certain periods (i.e. the variance of the market portfol
over time). Let us take the simple example whereby the risk-free (real) inte
varies and that in equilibrium the (real) return is given by:



In determining stock prices, agents will forecast future interest rates in order t
the discount factor for future dividends. If there is an unexpected fall in the in
then the stock price will show an unexpected rise (A to B in Figure 6.6).
If the lower interest rate is expected to persist then the rate of growth
(i.e. stock returns) will be low in all future periods. Thus even though we are in
market, the response of the smart money may cause prices to follow the p
which is rather similar to the time path when noise traders are present (i.e. A
Figure 6.3). In short, if equilibrium real interest rates are mean reverting then e
and actual returns will also be mean reverting in an efficient market. We ta
Time

Figure 6.6 Stock Prices and Interest Rates. Source: Engel and Morris (199


issue in a more formal way later in this chapter when discussing tests of the co
CAPM based on both returns and stock price data.
The above intuitive argument demonstrates that it may be difficult to infer
given observed path for returns is consistent with market efficiency or with the
noise traders. This problem arises because tests of the EMH are based on a spe
of equilibrium returns and if the latter is incorrect then this version of the EM
rejected by the data. However, another model of equilibrium returns might c
support the EMH. Nevertheless, an indication of a possible test is to note that in
where only the smart money operates, a large positive return is immediately f
smaller positive returns. When noise traders are present (Figure 6.3) large posi
are immediately followed by further large positive returns (i.e. in the short run
a slightly different autocorrelation pattern. However, it is clear that unless we
clear and well-defined models for the behaviour of both the smart money an
traders it may well be difficult to sort out exactly who is dominant in the
who exerts the predominant influence on prices. Formal models which incorp
trader behaviour and tests of these models are only just appearing in the lit
these will be discussed further in Chapters 8 and 17.


6.1 PREDICTABILITY IN STOCK RETURNS
Daily Stock Returns
6.1.1
Over short horizons such as a day, one would not expect equilibrium returns t
variable. Hence daily changes in stock prices probably provide a good app
to daily abnormal returns on stocks. Fortune (1991) provides an interesting
statistical analysis of the random walk hypothesis of stock prices using over
observations on the S&P 500 share index (closing prices, 2 January 1980-21
1990). Stock returns are measured as the proportionate daily change in (clo
+ 0.0006 HOL + 0.0006 JAN +
- 0.0017 WE El

(3.2) (0.2) (0.82)

E2= 0.0119, SEE = 0.0108, (.) = t statistic.
The variable WE = 1 if the trading day is a Monday and zero otherwise, H
the current trading day is preceded by a one-day holiday (zero otherwise) an
for trading days in January (zero otherwise). The only statistically significa
variable (for this data spanning the 1980s) is for the ˜weekend effect™ which i
price changes over the weekend are less on average than those on other trading
January effect is not statistically significant in the above regression for the
S&P 500 index (but it could still be important for stocks of small compa
error term in the equation is serially correlated and follows a moving averag
5, although only the MA(l), MA(4) and MA(5) terms are statistically signifi
previous periods™ forecast errors E are known (at time t) this is a violation of inf
efficiency, under the null of constant equilibrium returns. If there is a positi
error in one period then this will be followed by a further increase in the ret
period, a slight decline in the next three periods, followed finally by a rise in pe
The MA pattern might not be picked up by longer-term weekly or monthly
might therefore have white noise errors and hence be supportive of the EMH
However, the above data might not indicate a failure of the EMH where t
defined as the inability to persistently make supernormal profits. Only abou
(K2= 0.01) of the variability in daily stock returns is explained by the regress
potential profitable arbitrage possibilities are likely to involve substantial risk. I
unexpectedly on Tuesday then a strategy to beat the market based on (6.7) wo
buying the portfolio at close of trading on Wednesday and then selling the por
a further two days. Alternatively, because of the weekend effect, short selling
and purchasing the portfolio on a Monday yields a predictable return on avera
percent. In these two cases, as the portfolio in principle consists of all the
the S&P 500 index, this would involve very high transactions costs which
outweigh any profits from these strategies.
The difficulty in assessing regression tests of the above kind (particularly
index) are that they may reject the strict REEMH element of ˜unpredicta
not necessarily the view of the EMH which emphasises the impossibility
supernormal profits. Since the coefficients in equation (6.7) are in a sense ave
the sample data, one would have to be pretty confident that these ˜average ef
to persist in the future. To make money using (6.7) one would have to undertak
investments sequentially (e.g. each weekend based on the WE coefficient).
on Friday, the investor would find he had ˜won™ on some Mondays and could
the shares on Monday at a lower price. On other Mondays he would have
because the arrival of good news between Friday and Monday (i.e. the cur
error Er > 0.0017WE would increase prices). Of course if in the fiture the
on WE remains negative his repeated strategy will earn profits (ignoring t
costs). But at a minimum one would wish to test the temporal stability of c
such as that on WE before embarking on such a set of repeated gambles. I
the ˜supernormal profits™ view of the EMH one needs to examine ˜real wor
strategies in individual stocks within the portfolio, taking account of all t
costs, bid-ask spreads and managerial and dealers™ time and effort. (The la
be measured by the opportunity cost of the manpower involved compared
investment strategies which may also earn substantial profits, e.g. analysis of m
short, if the predictability indicated by regression tests cannot yield supernor
in the real world, one may legitimately treat the statistically significant ˜info
the regression equation as not economically relevant.

6.1.2 Long Horizon Returns
Are stock returns mean reverting, that is higher than average returns are fo
lower returns in the future? Fama and French (1988a) and Poterba and Summ
find evidence of mean reversion in stock returns over long horizons (i.e. in ex
months). Fama and French estimate an autoregression where the return over
t - N to t, call this Rr-N,r is correlated with Rr,f+N(LeRoy, 1989):



Fama and French consider return horizons N from one to ten years. They
or no autocorrelation, except for holding periods of between N = 2 and N =
which is less than zero. There was a peak at N = 5 years when % -0.5,
that a 10 percent negative return over five years is, on average, followed by
positive return over the next five years. The E2 in the regressions for the three
horizons are about 0.35. Such mean reversion (P < 0) is consistent with tha
˜anomalies literature™ where a ˜buy low, sell high™ trading rule earns persiste
profits (see Chapter 8). However, the Fama-French results appear to be ma
inclusion of the 1930s sample period (Fama and French, 1988a).
Poterba and Summers (1988) investigate mean reversion by looking at va
holding period returns over diferent horizons. If stock returns are random
ances of holding period returns should increase in proportion to the length of
period. To see this, assume the one-period expected return E,R,+I can be appro
Er In Pt+l - In P,.If the expected return is assumed to be constant (= p ) and
RE, this implies the random walk model of stock prices (including cumulated



The average return over N periods is approximately



Under RE the forecast errors are independent, with zero mean, hence th
E,
return over N periods is:
=NI
ErR;,
The variance ratio (VR) used by Poterba and Summers uses returns at differe
and is defined as:



where




and R, is the return over one month. Poterba and Summers show that the varia
closely related to tests based on the sequence of sample autocorrelation coef
(for various values of N). If the pN are statistically significant at various lags th
principle show up in a value for VR which is less than unity. Not surprisingly,
significant values for the regression coefficient /I in the Fama- French regre
imply VR # 1. However, the statistical properties of these three statistics V
in small samples are not equivalent and this is reported below.
Poterba and Summers (1988) find that the variance of returns increases at a ra
less than in proportion to N, which implies that returns are mean reverting (for
years). This conclusion is generally upheld when using a number of alternative
indexes, although the power of the tests is low when detecting persistent ye
returns. However, note that these results from the Poterba-Summers test m
due to restrictive ancillary assumptions being incorrect, for example constan
(real) returns or constant variance (a2). fact portfolio theory suggests tha
In
the latter assumptions is likely to be correct particularly over long horizons an
the Fama and French and Poterba and Summers results could equally well be
in terms of investors having time varying expected returns or time varying va
not as a violation of the EMH. Again it is the problem of inference when testin
hypothesis of market efficiency and an explicit (and maybe incorrect) returns
random walk).

Power of Tests
It is always possible that in a sample of data, a particular test statistic fai
the null hypothesis of randomness in returns, even when the true model
which really are non-random (i.e. predictable). The ˜power of a test™ is the
of rejecting a false null. To evaluate the power properties of the statistics p
/IN, Poterba and Summers set up a true model of stock prices which has a
(i.e. non-random) transitory component. The logarithm of actual prices p r
to comprise a permanent component p: and a transitory component U[. The
component follows a random walk.

pt = Pr* -+ ut
p: = p:-1 4- E t
(where and ut are independent) the degree of persistence is determined by ho
is to unity. Poterba and Summers set p1 to 0.98 implying that innovations in th
price component have a half-life of 2.9 years (if the basic time interval is co
be monthly). From the above three equations we obtain:

+ “Er - PlEr-1) + (vr - b 1 ) I
Apr = PlAPr-1
and therefore the model implies that the change in (the logarithm of) pric
an ARMA(1,l) process. Poterba and Summers then generate data on p, taki
drawings from independent distributions of er and v, with the relative share of t
of A p , determined by the relative size of 0 and 0 . They then calculate the st
: :
VR and from a sample of the generated data of 720 observations (i.e. the sam
as that for which they have historic data on returns). They repeat the calculati
times to obtain the frequency distributions of the statistics p ˜ VR and B. The
,
all three test statistics have little power to distinguish the random walk mode
above alternative of a highly persistent yet transitory price component.

A u toregressive Moving Average (ARMA) Representations
If weak form efficiency doesn™t hold then actual returns R,+l might not only de
past returns but could also depend on past forecast errors


The simplest representation is the ARMA( 1,l)model:


The autoregressive element is the independent variable R, and the ˜total™
(denoted U,) consists of a linear combination of white noise errors:


The error term uf is a moving average error (of order 1) and is serially cor
Ut
see the latter, note that
+ m ,+
= E ( E ˜ + ˜e = 8a2
COV(U,,
since E ( E r - s E f - m ) = 0 for m # s. Hence cov(u,, ur-l) # 0 and u, is serially co
order 1).Past forecast errors are in the agents™ information set and hence even
weak form efficiency is still violated if 8 # 0. By including p lags of Rt and
one can estimate a general ARMA(p, q) model for returns which may be rep


where y(L) is a polynomial in the lag operator such that
+ y1L + y2L2 + y3L3 + . . . + ypLP
y(L) = 1
provide fairly general and flexible representations of the behaviour of any
variable and tests are available to choose the ˜best™ values of p and q given th
data on R (although this usually involves judgement as well as formal statisti
Regressions based on ARMA models are often used to test the informationa
assumption of the EMH. In fact Poterba and Summers attempt to fit an A
model to their generated data on stock returns which, of course, should fit t
construction. However, in their estimated model (equation 6.18) they find y1
8 = 1 and because y1 and 8 are ˜close to™ each other, the estimation package
not ˜separate out™ (identify) and successfully estimate statistically distinct va
and 8. When Poterba and Summers do succeed in obtaining estimates of y1
less than 10 percent of the regressions have parameters that are close to the (k
values. This is another example of an estimated model failing to mimic the tru
a finite sample.
Poterba and Summers are aware that their results on mean reversion are su
problems of inference in small samples and that any element of predictability c
taken as evidence against the EMH, if the chosen equilibrium returns model
model. Cecchetti et a1 (1990) take up the last point and question whether the
Poterba and Summers (1988) and Fama and French (1988a) that stock prices
reverting should be interpreted in terms of the presence of noise traders. The
serial correlation of returns does not in itself imply a violation of efficiency. Fo
in the consumption CAPM if agents smooth their consumption, then stock
mean reverting. Cecchetti et a1 go on to demonstrate that empirical finding
reversion are consistent with data that could have been generated by an equilibri
They take a specific parameterisation of the consumption CAPM as their rep
equilibrium model and use Monte Carlo methods to generate artificial data
then subject the artificial data sets to the variance ratio tests of Poterba and Su
the long horizon return regressions of Fama and French. They find that measur
reversion in stock prices calculated from historic returns data nearly always l
60 percent confidence interval of the median of the Monte Carlo distributions
the equilibrium consumption CAPM.
Like all Monte Carlo studies the results are specific to the parameters
the equilibrium model. Cecchetti et a1 note that in the Lucas (1978) equilibri
consumption equals output which equals dividends, and their Monte Carlo st
tigates all three alternative ˜fundamental variables™. Taking dividends as an ex
Monte Carlo simulations assume
+ (a0+ a&1) +
h D , = lnD,-1 E,

The term S , is a Markov switching variable which has transition probabilities
Pr(S, = OJS,-l = 1) = 1 - p ,
Pr(S, = llS,-,) = p ,
Pr(S, = lIS,-l = 0) = 1 - q
Pr(S, = OIS,-1 = 0) = q,
Since a is restricted to be negative then S, = 0 is a ˜high growth™ state E , A ln
1
+
and S , = 1 is a low growth state E , A In D,+l = a 0 a (with a < 0). Ther
1
1
With preferences given by a constant coefficient of relative risk aversion (
+
of utility function U ( D ) = (1 y)-'D(l-Y) with --oo < y < 0, the solution to
equation is:


i=l

Simplifying somewhat the artificial series for dividends when used in the abov
(with representative values for 6 and y ) gives the generated series for pri
obey the parameterised consumption CAPM general equilibrium model. Gen
+ -
on returns Rt+l = [(Pt+1 Dt+l)/Dt] 1 are then used to calculate thc M
distributions for the Poterba-Summers variance ratio statistic and the Fama-F
horizon return regressions.
Essentially, the Cecchetti et a1 results demonstrate that with the available
observations of historic data on US stock returns, one cannot have great faith
ical results based on say returns over a 10-year horizon, since there are
10-non-overlapping observations. The historic data is therefore too short to ma
between an equilibrium model and a 'fads' model, based purely on an empiric
of historic returns data. Note, however, that the Cecchetti et a1 analysis does n
the possibility that variance bounds tests (see below) or other direct tests of th
Chapter 16) may provide greater discriminatory power between alternative
course, there are also a number of ancillary assumptions in the Cecchetti et
terisation of the consumption CAPM model which some may contest - an i
applies to all Monte Carlo studies. However, the Cecchetti et a1 results do m
more circumspect in interpreting weak-form tests of efficiency (which use on
lagged returns), as signalling the presence of noise traders.

6.1.3 Multivariate Tests
The Fama and French and Poterba and Summers results are univariate tests. H
number of variables other than past returns have also been found to help pred
returns. For example, Keim and Stambaugh (1986) using monthly excess retu
common stocks (over the Treasury bill rate) for the period from about 19
find that for a number of portfolios (based on size) the following (somewha
variables are usually statistically significant:

(i) the difference in the yield between low-grade corporate bonds and th
one-month Treasury bills,
(ii) the deviation of last periods (real) S&P index from its average over
years,
(iii) the level of the stock price index based only on 'small stocks'.
stocks, the regressions only explain about 0.6-2.0 percent of the actual exc
(These results are broadly similar to those found by Chen et a1 (1986), see sec
testing the APT.)
Fama and French (1988b) extend their earlier univariate study on the predi
expected returns over different horizons and examine the relationship betwee
and real) returns and the dividend yield DIP.


The equation is run for monthly and quarterly returns and for annual returns of
years on the NYSE index. They also test the robustness of the equation by runn
various subperiods. For monthly and quarterly data the dividend yield is often
significant (and /3 > 0) but only explains about 5 percent of the variability
and quarterly actual returns. For longer horizons the explanatory power inc
example, for nominal returns over the 1941-1986 period the explanatory po
2-, 3-, 4-year return horizons are 12, 17, 29 and 49 percent. The longer retu
regressions are also useful in forecasting ˜out-of-sample™.
The difficulty in assessing these results is that they are usually based on t
model possible for equilibrium returns, namely that expected real returns ar
The EMH implies that abnormal returns are unpredictable, not that actual
unpredictable. These studies reject the latter but as they do not incorporate a
sophisticated model of equilibrium returns we do not know if the EMH would
in a more general model. For example, the finding that /3 > 0 in (6.19) implies
current prices increase relative to dividends (i.e. ( D / P ) t falls) then returns
the future price tends to fall. This is an example of mean reversion in prices
be viewed as a mere statistical correlation where (DIP) may be a proxy for
equilibrium returns.
To take another example if, as in Keim and Stambaugh, an increase in
on low-grade bonds reflects an increase in investors™ general perception of
then we would under the CAPM or APT expect a change in equilibrium
returns. Here predictability could conceivably be consistent with the EMH sinc
no abnormal returns. Nevertheless the empirical regularities found by many
provide us with some useful stylised facts about variables which might influe
in a statistical sense.
When looking at regression equations that attempt to explain returns, an e
cian would be interested in general diagnostic tests (e.g. are the residuals norm
uncorrelated, non-heteroscedastic and the RHS variables weakly exogenous),
sample forecasting performance of the equations and the temporal stability of
eters. In many of the above studies this useful statistical information is not a
presented so it becomes difficult to ascertain whether the results are as ˜robu
seem. However, Pesaran and Timmermann (1992) provide a study of stock ret
attempts to meet the above criticisms of earlier work. First, as in previous st
run regressions of the excess return on variables known at time t or earlier.
however, very careful about the dating of the information set. For example, in
over one year, one quarter and one month for the period 1954-1971 and subp
annual excess returns a small set of independent variables including the divi
annual inflation, the change in the three-month interest rate and the term premiu
about 60 percent of the variability in the excess return. For quarterly and mo
broadly similar variables explain about 20 percent and 10 percent of exce
respectively. Interestingly, for monthly and quarterly regressions they find a
effect of previous excess returns on current returns. For example, squared prev
returns are often statistically significant while past positive returns have a diffe
than past negative returns, on future returns. The authors also provide diagnos
serial correlation, heteroscedasticity, normality and ˜correct™ functional form
test statistics indicate no misspecification in the equations.
To test the predictive power of these equations they use recursive estimation
+
predict the sign of next periods excess return (i.e. at t 1) based on estimated
which only use data up to period t. For annual returns, 70-80 percent of th
returns have the correct sign, while for quarterly excess returns the regression
a (healthy) 65 percent correct prediction of the sign of returns (see Figures 6.
Thus Pesaran and Timmermann (1994) reinforce the earlier results that excess
predictable and can be explained quite well by a relatively small number of i
variables.

Profitable Trading Strategies
Transactions costs in stock and bond trades arise from the bid-ask spread (i.e.
stock at a low price and sell to the investor at a high price) and the commissi
on a particular ˜buy™ or ˜sell™ order given to the broker. Pesaran and Timme
˜closing prices™ which may be either ˜bid™ or ˜ask™ prices. They therefore assu




Figure 6.7 Actual and Recursive Predictions of Annual Excess Returns (SP 500). Sou
and Timmermann (1994). Reproduced by permission of John Wiley and Sons Ltd
I
1,
-.26890[ 198302 199004
196001 196704 197503
..
- Predictions
Acutal a I 0.




Figure 6.8 Actual and Recursive Predictions of Quarterly Excess Returns (SP 5
Pesaran and Timmermann (1994). Reproduced by permission of John Wiley and Sons

trading costs are adequately represented by a fixed transactions cost per $ of
assume costs are higher for stocks c, than for bonds c b . They consider a sim
rule, namely:
If the predicted excess return (from the recursive regression) is positive then hold th
portfolio of stocks, otherwise hold government bonds with a maturity equal to the
the trading horizon (i.e. annual, quarterly, monthly).
The above ˜switching strategy™ has no problems of potential bankruptcy sinc
not sold short and there is no gearing (borrowing). The passive benchmark stra
of holding the market portfolio at all times. They assess the profitability of th
strategy over the passive strategy for transactions costs that are ˜low™, ˜medium
(The values of c, are 0, 0.5 and 1.0 percent for stocks and for bonds cb equ
0.1 percent.)
In general terms they find that the returns from the switching strategy are
those for the passive strategy for annual returns (i.e. switching once per year in Jan
when transactions costs are ˜high™ (see Table 6.1). However, it pays to trade at q
monthly intervals only if transactions costs are less than 1/2 percent for stocks.
they find that the standard deviation of returns for the switching portfolio for
Table 6.1) and quarterly but not monthly returns is below that for the passive por
under high transactions cost scenario). Hence the ˜annual™ switching portfolio
the passive portfolio on the mean-variance criterion (for the whole data period 1
The above results are found to be robust with respect to different sets of regre
excess return equations and over subperiods 1960- 1970, 1970- 1980 and 198
Table 6.1 are reported the Sharpe, Treynor and Jensen indices of mean-variance
for the switching and passive portfolios for the one-year horizon. For any po
these are given by:
s = (ERP - r)/cTp
T = (ERP - r ) / &
+ s(Rm - r),
(RP - r)t = J
Transaction Cost (%)
1 .o
Stocks 1.0
0.5 0.0
0.0 0.5
- - 0.0
-
Treasury Bills 0.1 0.1 0
Arithmetic Mean
Return (%) 10.78 10.72 10.67 12.70 12.43 12.21 6
SD of Return (%) 13.09 13.09 13.09 7.24 7.20 7.16 2
0.76
0.30 0.82
Sharpe™s Index 0.31 0.30 0.79
0.081

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