ñòð. 17 |

earnings E)â€™. 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

T

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 Shillerâ€™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)â€™) exceeds

Hence when dividends are non-stationary there is a tendency for spurious viola

variance bounds when the series are detrended by E)â€™. However, some comf

Deterministic

Using D,-1

2. 0.133 4.703 0.54 6.03

(6.03)

(0.23)

(0.06) (7.779)

Using 0.296 1.611 0.47 6.706

3. E˜O

(6.706)

(4.65) (0.22)

(0.048)

(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 â€˜trueâ€™ value of 1.00, is substantially less than the 4.1

in the real world data. (And in only one of the 1000 â€˜runsâ€™ did the â€˜generated

ratio exceed 4.16.)

Clearly since E:â€™ 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,

p

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 Mertonâ€™s criticism is of less importance if actual dividends paid out â€˜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 aâ€™(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

is

and LeRoy conclude that the Shiller-type variance bounds test â€˜is indecisiveâ€™

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 testâ€™ 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

T

+

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

T

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;â€™ 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 investorâ€™s expectation of dividends over this sample o

weighted average of the higher level of dividends and the â€˜normalâ€™ constant

But the outturn for dividends is constant and is lower than investorsâ€™ 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 â€˜fullâ€™ sample, forecast errors average zero. The Peso problem ari

we only â€˜observeâ€™ 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 â€˜t 1â€™ 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 â€˜the rumourâ€™. The key to the Peso problem is that the research

data for the periods over which â€˜the rumourâ€™ does not materialise. Although

investorâ€™s â€˜trueâ€™ expectations and hence the stock price, â€˜the takeoverâ€™ doe

occur and dividends actually remain at their constant value D.

Since â€˜the rumourâ€™ 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

2

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

il

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.

T

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

T

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 â€˜omitted variableâ€™. 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

/3)

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) â€˜collapsesâ€™ to the unbiase

2

The Peso problem arises because of one-off â€˜special eventsâ€™ 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

we

b=l

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,

R-

!

+ (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

Y

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

T

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

T

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.

6.3 SUMMARY

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.

Shillerâ€™s (1981) original seminal work using variance bounds inequalitie

0

decisively to reject the RVF. Subsequent work in the 1980s pointed out d

in Shillerâ€™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

0

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 Shillerâ€™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

ENDNOTES

Note, however, that the term â€˜model freeâ€™ is used here in a statistical sense

1.

and LeRoy (1991)). A model of stock prices always requires some assump

behaviour and to derive the RVF we need a â€˜modelâ€™ of expected retur

assumption of RE (see Chapter 4). Hence the RVF is not â€˜model freeâ€™ 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 â€˜aâ€™. test is biased towards rejection if the pro

A

rejection exceeds â€˜aâ€™. Hence in the variance bounds literature â€˜biasâ€™ 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:

n

i= 1

where X = sample mean, then 8â€™ 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 Kleidonâ€™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).

APPENDIX 6.1

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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