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the real price series are ˜detrended™ using a (backward) 30-year moving aver
earnings E)™. He also uses P,/D,-I and PT/D,-Iwhere D,-1 is real divide
previous year. To counter the criticism that detrending using a deterministic tre
1981) estimated over the whole sample period uses information not known
Shiller (1989) in his later work detrends P , and P using a time trend estimated
data up to time t (that is A, = exp[b,]t where the estimated b, changes as m
included - that is, recursive least squares).
The results using these various transformations of P, and Pf to try an
stationary series are given in Table 6.2. The variance inequality (6.38) is alwa
but the violation is not as great as in Shiller™s original (1981) study using
deterministic trend. However, the variance equality (6.37) is strongly violate
the variantd6).
To ascertain the robustness of the above results Shiller (1989) repeats Kleid
Carlo study using the geometric random walk model for dividends. He de
artificially generated data on F, and p; by a generated real earnings series E,
earnings are assumed to be proportional to generated dividends) and assumes
real discount rate of 8.32 percent (equal to the sample average annual real
stocks). In 1000 runs he finds that in 75.8 percent of cases a(P,/E)™) exceeds
Hence when dividends are non-stationary there is a tendency for spurious viola
variance bounds when the series are detrended by E)™. However, some comf
Using D,-1
2. 0.133 4.703 0.54 6.03
(0.06) (7.779)
Using 0.296 1.611 0.47 6.706
3. E˜O
(4.65) (0.22)
(a) The figures are for a constant real discount rate while those in parentheses are for a time
discount rate.
Source: Shiller 1989.

gained from the fact that for the generated data the mean value of V R = 1.4
although in excess of the ˜true™ value of 1.00, is substantially less than the 4.1
in the real world data. (And in only one of the 1000 ˜runs™ did the ˜generated
ratio exceed 4.16.)
Clearly since E:™ and P: are long moving averages, they both make lon
swings and one may only pick up part of the potential variability in P;/Ej
samples (i.e. we observe only a part of its full cycle). The sample variance may
be biased downwards. Since P , is not smoothed (P,/E:O) may well show more
than P;/E;O even though in a much longer sample the converse could apply.
boils down to the fact that statistical tests on such data can only be definitive if
long data set. The Monte Carlo evidence and the results in Table 6.2 do, howe
the balance of evidence against the EMH when applied to stock prices.
Mankiw et a1 (1991), in an update of their earlier 1985 paper, tackle the non-s
problem by considering the variability in P and P* relative to a naive foreca
the naive forecast they assume dividends follow a random walk and hence Er
for all j . Hence using the rational valuation formula the naive forecast is
P = [6/(1 - 6)]D,
where 6 = 1/(1+ k ) and k is the equilibrium required return on the stock. Now
the identity
P; - P; = (P; - P , ) ( P , - Pp)
The RE forecast error is PT - Pi and hence is independent of information at
hence of P , - Py. Dividing (6.40) by P , and squaring gives

and the inequalities are therefore
error. Equation (6.43) states that the ex-post rational price P: is more vola
the naive forecast Py than is the market price and is analogous to Shiller
inequality. An alternative test of the EMH is that Q = 0 in:
qt = E,

and the benefit of using this formulation is that we can construct a (asym
valid standard error for \z, (after using a GMM correction for any serial co
heteroscedasticity, see Part 7). Using annual data 1871-1988 in an aggregate
index Msnkiw et a1 find that equation (6.41) is rejected at only about the 5 perce
constant required real returns of k = 6 or 7 (although the model is strongly rej
the required return is assumed to be 5 percent). When they allow the required
nominal return to equal the (nominal) risk-free rate plus a constant risk pre
k, = r, r p ) then the EMH using (6.41) is rejected more strongly than for t
real returns case.
The paper by Mankiw et a1 (1991) also tackles another problem that has c
culties in the interpretation of variance bounds tests, namely the importance of t
price P r + N in calculating the perfect foresight price. Merton (1987) points out
of sample price P r + N picks up the effect of out-of-sample events on the (with
stock price, since it reflects all future (unobserved) dividends. Hence volatilit
use a fixed end point value for actual price may be subject to a form of m
error if P r + N is a very poor proxy for E,P:,,. Mankiw et a1 (and Shiller (1989
that Merton™s criticism is of less importance if actual dividends paid out ˜in
sufficiently high so that the importance of out-of-sample events (measured b
circumscribed. Empirically, the latter case applies to the data used by Mankiw
they have a long representative sample of data. However, Mankiw et a1 prov
ingenious yet simple counterweight to this argument (see also Shea (1989)).
foresight stock price (6.24) can be calculated for different horizons (n = 1, 2
so can qr used in (6.44). Hence, several values of q: (for n = 1 , 2 , . . . N) ca
lated in which many end-of-holding-period prices are observed in a sample.
they do not have to worry about a single end-of-sample price dominating thei
general they find that the EMH has greater support at short horizons (i.e. n =
rather than long horizons (i.e. n > 10 years).
In a recent paper, Gilles and LeRoy (1991) present some further evide
difficulties of correct inference when using variance bounds tests in the prese
stationarity data. They derive a variance bound test that is valid if divide
a geometric random walk and stock prices are non-stationary (but cointe
Chapter 20). They therefore assume the dividend price ratio is stationary and th
inequality is a2(P,lD,) < a2(P:ID,). The sample estimates of the variances (
US aggregate index as used in Shiller (1981)) indicate excess volatility since a
26.4 and a™(P:JD,) 19.4. However, they note that the sample variance o
is biased downwards for two reasons. First, because ( P f l D , ) is positively ser
lated (Flavin, 1983) and second because at the terminal date the unobservabl
assumed to equal the actual (terminal) price Pr+,. (Hence dividend innovatio
compared with 19.4 using actual sample data. On the other hand, the samp
a 2 ( P ,ID,) found to be a fairly accurate measure of the population variance. H
and LeRoy conclude that the Shiller-type variance bounds test ˜is indecisive™
LeRoy (1991), page 986). However, all is not lost. Gilles and LeRoy develop
on the orthogonality of P, and PT (West, 1988) which is more robust. Thi
nality test™ uses the geometric random walk assumption for dividends and
test statistic with much less bias and less sample variability than the Shille
The orthogonality test rejects the present value model quite decisively (alt
that there are some nuances involved in this procedure which we do not docu
Thus a reasonable summary of the Gilles-LeRoy study would be that the RVF
provided one accepts the geometric random walk model of dividends.
Scott (1990) follows a slightly different procedure and compares the beha
and P using a simple regression rather than a variance bounds test. If the DP
holds for stocks then P: = P, Er and in the regression
+ bP, -k
P: = a Er

the EMH implies a = 0, b = 1. Scott deflates P: and P, by dividends in the pr
so that the variables are stationary and he adjusts for serial correlation in the
residuals. He finds that the above restrictions are violated for US stock price d
P, is not an unbiased predictor of P:. The R2 of the regression is very low, so t
little (positive) correlation between P, and P and P, provides a very poor fore
ex-post perfect foresight price, PT. (Note, however, that it is the unbiasedness
that is important for the refutation of the efficient markets model, not the low
in itself is not inconsistent with the EMH.)
The EMH, however, does imply that any information SZ, included in (6.45)
be statistically significant. Scott (1990) regresses (PT - P r ) on the dividend
(i.e. dividend yield) and finds it is statistically significant, thus rejecting inf
efficiency. Note that since P: - P, may be viewed as a long horizon return Sc
is not inconsistent with those studies that find long horizon returns are predi
Fama and French (1988b)).
Shiller (1989, page 91) deflated P, and P: using a 30-year backward lookin
series E;™ and in the regression (corrected for serial correlation):

finds that 6 < 1. Although Shiller finds that 6 based on Monte Carlo evidenc
ward biased, such bias is not sufficient to account for the strong rejection (
found in the above regression on the real world data.

6.2.4 Peso Problems and Variance Bounds Tests
It should now be obvious that there are some complex statistical issues i
assessing the EMH. We now return to a theoretical issue which also can cau
which investors attach a small probability to the possibility of a large rise in
in the future. However, suppose this rise in dividends never occurs and actua
remain constant. An investor™s expectation of dividends over this sample o
weighted average of the higher level of dividends and the ˜normal™ constant
But the outturn for dividends is constant and is lower than investors™ true e
(i.e. D < E,D,+I). Investors have therefore made a systematic forecast error
sampleperiod. If the sample period is extended then we would also observe pe
investors expect lower dividends (which never occur), hence D > E,D,+1, and
the extended ˜full™ sample, forecast errors average zero. The Peso problem ari
we only ˜observe™ the first sample of data. To illustrate the Peso problem mo
maths will be simplified, and this issue examined by considering an asset
out a stream of expected dividend payments E,D,+1 all of which are discou
constant rate 6 . We can think of period ˜t 1™ as constituting rn data points
price is set equal to fundamental value:
pt = 6E,Df+1
where 6 = constant discount factor. Suppose there is a small probability 7r2
regime 2 so that the true expectation of investors is:

To simplify even further suppose that in regime 1, future dividends are exp
constant and ex-post are equal to D so that Dlyl = D.Regime 2 can be thou
rumour of a takeover bid which, if it occurs, will increase dividends so that
Call regime 2 ˜the rumour™. The key to the Peso problem is that the research
data for the periods over which ˜the rumour™ does not materialise. Although
investor™s ˜true™ expectations and hence the stock price, ˜the takeover™ doe
occur and dividends actually remain at their constant value D.
Since ˜the rumour™ exists then rational investors set the price of the share
fundamental value:

where D:yl = D,a constant. Variability in the actual price given by (6.48) wil
either because of changing views about ˜ 7 (the probability of a takeover) or
changing views about future dividends, should the takeover actually take pla
the takeover never takes place, then the constant level of dividends D will be p
the researcher will measure the ex-post perfect foresight price over our rn dat
the constant value P: = 6D and hence var(P:) = 0. However, the actual pric
as 712 and D: change and hence var(P,) > 0. Thus we have a violation of t
bound, that is var(P,) > var(P:), even though prices always equal fundamen
given by (6.48). This is a consequence of a sample of data which may not be rep
of the (whole) population of data. If we had a longer data set then the exp
might actually happen and hence P would vary along with the actual price.
when the true model for P, is based on fundamental value. Assume that d
regime 1, D:ill, vary over time. Since the takeover never actually occurs,
foresight price measured by the researcher is P = SDli)l. However, the act
determined by fundamentals and is given by:

where we have substituted P = SDli)l. Rearranging (6.50) gives

Comparing (6.51) and (6.49) we expect a # 0 and for a to be time varying
(Df;)l - Dli)l) are time varying. If the first term in (6.51) is time varying the
misspecified since it has an ˜omitted variable™. The OLS estimate of /3 from
of data will be biased because of the correlation between P, and dividends i
weak test of the absence of a Peso problem is to check on the temporal sta
and /3. Only if (a, are stable can one proceed to test H o : a = 0, /3 = 1
empirical studies often do not test the constancy of a and /3 before proceedin
usual test of Ho. Of course, if a and are temporally unstable this could b
host of other factors as well as Peso problems (e.g. use of a constant disco
forming P when the true discount rate is time varying). But non-constancy
would still invalidate any tests based on the assumption that these regression
are constant. Of course, for n = 0 equation (6.51) ˜collapses™ to the unbiase
The Peso problem arises because of one-off ˜special events™ which could
within the sample period but in actual fact do not. It considerably complica
hypotheses which are based on rational expectations such as the EMH whi
outturn data differ from expectations by a (zero mean) random error.

6.2.5 Volatility Tests and Regression Tests: A Comparison
Volatility tests are a joint test of informational efficiency and that price equ
mental value. Regression tests on the relationship between actual price P, and
foresight price P also test these two elements of the EMH. Regression tests su
of Fama and French (1988b) on returns are tests of informational efficiency
assumption that expected (real) returns are constant. But as we have seen in Ch
joint hypothesis that ER,+1 = k and that rational expectations holds, yields t
valuation formula and so in principle results from both types of test should yi
inferences. However, results from such tests might differ because of statistical
stationarity of the data, power of the tests).
The easiest way of seeing the relationship between the volatility tests and
tests is to note that in the regression

P; = a + bP, + E,
Substituting for p from the variance equality (6.37)in (6.53) obtain
Hence if the variance equality holds then we expect b = 1 in the regression (
Consider the regression tests involving P, and P; of the form

Under the orthogonality assumption of the EMH we expect
HO:a=c=O, b=l
reduces to
If this proves to be the case then (6.54)

+ qr
P; = P,
and hence the variance bounds test must also hold. The two tests are therefore
under the null hypothesis.
As a slight variant consider the case where c = 0 but b < 1 (as is found in m
empirical work described above). Then (6.54) reduces to
+ var(r],)
var(P:) = b2 var(P,)
+ var(q,)
var(P:) - var(P,) = (b2 - l)var(P,)
where informational efficiency implies cov(P,, q , ) = 0. (An OLS regression
impose this restriction in the sample of data.) Since b2 < 1 then the first te
RHS of (6.58)is negative and it is possible that the whole of the RHS of (6
negative. Hence if b < 1 this may also imply a violation of the variance boun
Next, consider the long-horizon regressions of Fama and French:
+ qt
RY = a 4- b,
+ (b + l)lnP,-N + qr
- blnP,
lnPr+N = a
where we have used R = In P r + N - In P,. Under the null hypothesis that expec
are constant (a # 0) and independent of information at time t or earlier then
Ho:b = 0. If H o is true then from (6.60)
lnP,+N = a + l n P , + q ,
Hence under the null, H o : b = 0, the Fama-French regressions are broadly
with the random walk model of stock prices. Finally note that the Scott (1990)
under the null is
+ +
InPT = a InP, qt
and hence it is similar to the Fama-French regression (6.60) under the null, b =
the perfect foresight price In P replacing the ex-post future price In P,+N.
T How
period of fixed length (see also Shiller (1989), page 91). In fact one can calc
a fixed distance from t and then the correspondence between the two regress
and (6.62) is even closer (e.g. Joerding (1988) and Mankiw et a1 (1989)).
In the wide-ranging study of Mankiw et a1 (1991) referred to above they al
the type of regression tests used by Fama and French. More specifically co
following autoregression of (pseudo) returns:

where PT" is the perfect foresight price calculated using a specific horizon (n =
Mankiw et a1 use a Monte Carlo study to demonstrate that under plausible con
hold in real world data, estimates of p and its standard error can be subject to v
small sample biases. These biases increase as the horizon n is increased. How
using their annual data set, under the constant real returns case (of 5, 6 or
per annum) it is still the case that H o : /3 = 0 is rejected at the 1-5 percen
most horizons between one and ten years (see Mankiw et a1 Table 5, page 4
P is constructed under the assumption that equilibrium returns depend on th
interest rate plus a constant risk premium then Ho: p = 0 is only rejected at
5-10 percent significance levels for horizons greater than five years (see the
page 471). Overall, these results suggest that the evidence that long-horizon r
n > 5 years) are forecastable as found by Fama and French are not necessaril
when the small sample properties of the test statistics are carefully examined.
It has been shown that under the null of market efficiency, regression tests u
P should be consistent with variance bounds inequalities and with regression
autoregressive models for stock returns. However, the small sample properti
tests need careful consideration and although the balance of the evidence i
against the EMH, this evidence is far from conclusive.

There have been innumerable tests of the EMH applied to stock prices and r
have discussed a number of these tests and the interrelationships between them
of the EMH are conditional on a particular equilibrium model for returns (or p
main conclusions are:
In principle, autocorrelation coefficients of returns data, regression-b
(including ARMA models) of stock returns and variance bounds tests for s
should all provide similar inferences about the validity of the EMH. Usua
not. This is in part because tests on returns are sometimes based on a slightl
equilibrium model to those based on stock prices. Also the small sample
of the test statistics differ.
Tests based on stock returns indicate that (ex-post) real returns and exc
are predictable but this is a violation of the EMH only if one accepts
erable evidence that actual trading strategies based on the predictions
equations can result in profits, net of dealing costs. The key question for
or otherwise of the EMH is whether these profits when corrected for ex-a
positive. There is certainly evidence that this might well be the case altho
always be argued that methods used to correct for the risk of the portfol
of sample variance of returns) are inadequate.
Shiller™s (1981) original seminal work using variance bounds inequalitie

decisively to reject the RVF. Subsequent work in the 1980s pointed out d
in Shiller™s original approach (e.g. Kleidon (1986) and Flavin (1983)) b
(1987) later work rather successfully answered his critics. However, very r
(e.g. Mankiw et a1 (1991) and Gilles and LeRoy (1991)) has certainly de
that violations of the RVF are statistically far from clear cut and considerabl
is required in reaching a balanced view on this matter.
Thus where the balance of the evidence for the EMH lies is very difficult

given the plethora of somewhat conflicting results and the acute problems o
inference involved. To this author it appears that the evidence cited in th
particularly that based on stock prices, is on balance marginally against th
The intuitive appeal of Shiller™s volatility inequality and the simple elegance o
insight behind this approach have become somewhat overshadowed by the pract
tical) issues surrounding the actual test procedures used. As we shall see in
6 some recent advances in econometric methodology have allowed a more
treatment of problems of non-stationarity and the modelling of time varying r

Note, however, that the term ˜model free™ is used here in a statistical sense
and LeRoy (1991)). A model of stock prices always requires some assump
behaviour and to derive the RVF we need a ˜model™ of expected retur
assumption of RE (see Chapter 4). Hence the RVF is not ˜model free™ if
interpreted to mean free of a specific economic hypothesis.
2. A formal hypothesis testing procedure requires one to specify a test stat
rejection region such that if the null (e.g. RVF) is true, then it will be re
a pre-assigned probability ˜a™. test is biased towards rejection if the pro
rejection exceeds ˜a™. Hence in the variance bounds literature ˜bias™ has
restrictive definition given in the text.
3. If a series ( X I , x2, . , . x , ) is drawn from a common distribution and the
estimated using:

i= 1

where X = sample mean, then 8™ is an unbiased estimator of o2 only if
mutually uncorrelated.
cross-section of data and not to a time series. He argues that evidence fro
time series of data is uninformative about the violation of the correctly
variance bound. This aspect of Kleidon™s work is not discussed here and the
reader should consult the original article and the clear exposition of this a
Gilles and LeRoy (1991).
6. It is worth nothing that LeRoy and Porter (1981) were aware of the p
non-stationarity in P , and they also adjusted the raw data series to try a
stationary variables. However, it appears as if they were not wholly su
removing these trends (see Gilles and LeRoy (1991), footnotes 3 and 4).

The LeRoy-Porter and West Tests
The above tests do not fit neatly into the main body of this chapter but are important l
the literature in this area. We therefore discuss these tests and their relationship to eac
to other material in the text.

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