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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

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

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 +
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

Figure 6.1 Random Walk for Stock Price.

Returns (Yo)


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


Figure 6.3 Positive Feedback Traders: Stock Prices.



t Y


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
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')

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

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

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.

Daily Stock Returns
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
return over N periods is:
The variance ratio (VR) used by Poterba and Summers uses returns at differe
and is defined as:


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
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

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