ńňđ. 5
(âńĺăî 14)



Treynor’s Index 0.040 0.039 0.085
0.040 0.089
- 0.041
Jensen’s Index 0.045 0.043
(4.63) (4.25)
- - - (4.42)
Wealth at End of Period* 1855
1913 1884 3833 3559 3346 7
(a) The switching portfolio is based on recursive regressions of excess returns on the change in th
interest rate, the term premium, the inflation rate, and the dividend yield. The switching rule
portfolio selection takes place once per year on the last trading day of January.
(b) For a description and the rationale behind the various performance measures used in
Chapter 3. The ‘market’ portfolio denotes a buy and hold strategy in the S&P 500 ind
bills’ denotes a roll-over strategy in 12-month Treasury bills.
* Starting from $100 in January 1960.
Source: Pesaran and Timmermann (1994). Reproduced by permission of John Wiley and Sons L

One can calculate S and T for the switching and market portfolios. The Je
is the intercept J in the above regression. In general, except for the mont
strategy under the high cost scenario they find that these performance indices
the switching portfolio has the higher risk adjusted return.
Our final example of predictability based on regression equations is du
et a1 (1993) who provide a relatively simple model based on the gilt-equity
(GEYR). The GEYR is widely used by market analysts to predict capital gains
Govett 1991). The GEYR, = ( C / B , ) / ( D / P , ) where C = coupon on a consol
B, = bond price, D = dividends, P, = stock price (of the FT All Share index).
that UK pension funds, which are big players in the market, are concerned ab
flows rather than capital gains, in the short run. Hence, when ( D I P ) is low
( C / B ) they sell equity (and buy bonds). Hence a high GEYR implies a fa
prices, next period. Broadly speaking the rule of thumb used by market analy
if GEYR > 2.4, then sell some equity holdings, while for GEYR < 2, buy m
and if 2.0 6 GEYR < 2.4 then ‘hold’ an unchanged position.
A simple model which encapsulates the above is:

In static equilibrium the long-run GEYR is -a˜//? Clare et a1 (using qua
find this to be equal to 2.4. For every 0.1 that the GEYR exceeds 2.4, then the
equation predicts that the FT All Share index will fall 5 percent:
R' = 0.47
They use recursive estimates of the above equation to forecast over 1990(
and assume investors' trading rule is to hold equity if the forecast capital ga
the three-month Treasury bill rate (otherwise hold Treasury bills). This stra
the above estimated equation, gives a higher ex-post capital gain = 12.3 p
annum and lower standard deviation (a= 23.3) than the simpler analysts' 'rule
noted above (where the return is 0.4 percent per annum and (T = 39.9). The
provides some prima facie evidence against weak-form efficiency. Although
capital gains rather than returns are used in the analysis and transactions co
considered. However, it seems unlikely that these two factors would under

Returns and the Consumption CAPM
The C-CAPM model of equilibrium returns discussed in Chapter 4 can be te
using cross-section or time series data. A cross-section test of the C-CAPM
on the following rearrangement of 4.42 (see Chapter 4) which constitutes th
this model.

where we have assumed that the expected value of the marginal rate of substitut
cov(Rm,g") are constant (over time). Since a1 depends only on the market retu
consumption growth gc it should be the same for all stocks i. We can calculate
mean value & as a proxy for ERi and use the sample covariance for p c j , for ea
The above cross-section regression should then yield a statistically significan
a1 (and ao). In Chapter 3 we noted that the basic CAPM can be similarly te
the cross-section SML, where pci is 'replaced' by fimi = cov(Ri, Rm)/var(Rm
and Shapiro (1986) test these two versions of the CAPM using cross-section d
companies (listed on the NYSE) and sample values (e.g. for Ri, cov(Rm,gC)
from quarterly data over the period 1959-1982. They find that the basic CAP
outperforms the C-CAPM, since when Ei is regressed on both B m i and pci the
statistically significant while the latter is not.
A test of the C-CAPM based on time series data uses (4.45) (see Chapter 4

Since (6.21) holds for all assets (or portfolios i), then it implies a set of cros
restrictions, since the parameters (6, a)appear in all equations. Again this partic
national’ basket of assets, the coefficient restrictions in (4.46) are often found
these parameters are not constant over time. Hence the model or some of i
assumptions appear to be invalid.
We can also apply equation (6.21) to the aggregate stock market return and
(6.21) applies for any return horizon t j , we have, for j = 1 , 2 , . . .:

Equation (6.22) can be estimated on time series data and because it holds fo
j = 1 , 2 , . . ., etc. we again have a system of equations with ‘common’ parame
Equation (6.22) for j = 1, 2, . . . are similar to the Fama and French (1988a)
using returns over different horizons, except here we implicitly incorporate a ti
expected return. Flood et a1 (1986) find that the C-CAPM represented by (6.22
worse, in statistical terms, as the time horizon is extended. Hence they find a
version of) the C-CAPM and their results are consistent with those of Fama a
Overall, the C-CAPM does not appear to perform well in empirical tests.

A number of commentators often express the view that stock markets are
volatile: prices alter from day to day or week to week by large amounts wh
appear to reflect changes in fundamentals. If true, this constitutes a rejection o
Of course, to say that stock prices are excessively volatile requires one to ha
based on rational behaviour which provides a ‘yardstick’ against which one ca
Whether or not the stock market is excessively volatile is of course closely
with whether, if left to itself, the stock market is an efficient device for alloc
cial resources between alternative real investment projects (or firms). If stoc
not reflect economic fundamentals then resources will be misallocated. For e
unexpectedly low price for a share of a particular company may result in a t
another company. However, if the price is not giving a correct signal about th
efficiency of that company then the takeover may be inappropriate and wil
misallocation of resources.
Commonsense tells us, of course, that we expect stock prices to exhibit som
This is because of the arrival of ‘news’ or new information about companies. Fo
if the announced dividend payout of a company were unexpectedly high the
price is likely to rise sharply, to reflect this. Also if a company engaged
and development or mineral exploration suddenly discovers a profitable new i
mineral deposits, then this will effect future profits and hence future dividen
stock price. We would again expect a sudden jump in stock prices. However, t
we wish to address here is not whether stock prices are volatile but wheth
excessively volatile.
where k is the (known) required real rate of return and g, is the expected gro
dividends based on information up to time t. Barsky and De Long (1993) use
to calculate fundamental value V f = D , / ( k - g,) assuming that agents have to
update their estimate of the future growth in dividends. They then compare
actual S&P stock price in the US over the period 1880-1988. Even for the
of a constant value of (k - g)-’ = 20 and V, = 200, the broad movements
a long horizon of ten years are as high as 67 percent of the variability in P
Table 111, page 302) with the pre-Second World War movements in the two
being even closer. They then propose that agents estimate g, at any point in
long distributed lag of past dividend growth rates

For 8 = 1 the level of log dividends follows a random walk but for 8 = 0.97 pa
growth has some influence on g,. When g, is updated in this manner then th
of one-year changes in V , are as high as 76 percent of the volatility in P,
it should be noted that although long swings in P, are in part explained by
changes over shorter horizons such as one year or even five years are not wel
and movements in the (more stationary) price dividend ratio are also not well
The above evidence is broadly consistent with the view that real dividends and
move together in the long run (and hence are likely to be cointegrated) bu
fundamental value can diverge quite substantially for a number of years.
Bulkley and Tonks (1989) exploit the fact that PI # V, to generate profit
based on an aggregate UK stock price index (annual 1918- 1985). The predictio
for real dividends is In D,= 6 &, where a, and g, are estimated recursi
then assume that the growth rate of dividends is used in the RVF and the
fundamental value from recursive estimates using the regression b, = t, gs
is constrained to be the same as in the dividend equation (but Es varies period
They then investigate the profitability of a ‘switching’ trading rule whereby if a
PI exceeds the predicted price hf from the regression by more than K, perce
index and hold (risk-free) bonds. The investor then holds bonds until P, is
c, and then buys back the index. (At any date t, the value of K, is cho
would have maximised profits over the period (0,t - l).) The passive strateg
and hold the index. They find that (over 1930-1985) the switching strategy ea
annual excess return of 1.61 percent over the buy and hold strategy. Also,
of risk for the switching strategy is no higher than in the buy and hold strate
switching strategy only involves trades on seven separate occasions, transac
would have to be the order of 12.5 percent to outweigh net profits from the
Thus the above models, where an elementary learning process is introduced
broad movements in price and fundamentals (i.e. dividends) are linked over lon
However, they also indicate that for long periods prices may deviate from fun
free' tests and 'model-based' tests. In the former we do not have to assume a
statistical model for the fundamental variables('). However, as we shall see, t
that we merely obtain a point estimate of the relevant test statistic but we can
confidence limits on this measure. Formal hypothesis testing is therefore not po
one can do is to try and ensure that the estimator (based on sample data) of
statistic is an unbiased estimate of its population value(2). Critics of the earl
ratio tests highlighted the problem of bias in finite samples (Flavin, 1983).
A 'model-based test' assumes a particular stochastic process for dividends an
associated statistical distribution. This provides a test statistic with appropriate
limits and enables one to examine small sample properties using Monto Carl
However, with model-based tests, rejection of the null that the RVF is correct is
on having the correct statistical model for dividends. We therefore have the pr
joint null hypothesis.
A further key factor in interpreting the various tests on stock prices is w
dividend process is assumed to be stationary or non-stationary . Under non
the usual distributional assumptions do not apply and interpretation of the re
variance bounds tests is problematic. Much recent work has been directed to
procedures that take account of non-stationarity in the data. This issue is dis
variance bounds tests in this chapter and is taken up again in Chapter 16.

6.2.1 Shiller Volatility Tests
Shiller takes the rational valuation formula as his model of the determinatio
prices. Hence stock prices are determined by economic fundamentals, V t , n
discounted present value (DPV) of expected future dividends.

where P,+n is the expected 'terminal price' at time t n . It is assumed that a
take the same view of the future. Hence all investors form the same expectatio
dividends and it is also assumed for the moment that the discount factor 6(
is a constant in all future periods. 6 is defined as 1/(1 k) where k is the re
of return (see Chapter 4). Clearly the assumption of a constant nominal requ
return is rather unrealistic and hence tests of the model invariably assume a co
discount rate and therefore stock prices are also measured in real terms.
To set the ball rolling, it is instructive to note that if, for each time period
data on expected future dividends, the expected terminal price and the cons
we could work out the right-hand side of equation (6.23) and compare it with
stock price P,.Of course, at time t , we do not know what investors' forecasts o
future dividends would have been. However, Shiller (1981) proposed a simp
ingenious way of getting round this problem.
Data are available on actual dividends in the past, say from 1900 onwar
have the actual price PI+,, today, say in 1996. It is assumed 6 is a known
from 1900 onwards. As described above, the data series P: has been compute
following formula:
+ 6" Pr+n
P: = GiDl+i

When calculating P for 1900 the influence of the terminal price Pt+n is fair
since n is large and Gn is relatively small. As we approach the end-point o
term anPt+n carries more weight in our calculation of P*.One option is t
truncate our sample, say ten years prior to the present, in order to apply the DP
Alternatively, we can assume that the actual price at the terminal date is 'c
expected value E,P;+, and the latter is usually done in empirical work.
Comparing PI and P; we see that they differ by the sum of the forecas
dividends wt+i, weighted by the discount factors 6' where

If agents do not make systematic forecast errors then we would expect these fore
in a long sample of data to be positive about as many times as they are negati
average for them to be close to zero). This is the unbiasedness assumption of R
up again. Hence we might expect the (weighted) sum of U,+; to be relatively sm
broad movements in P; should then be correlated with those for P,. Shiller (198
a graph of (detrended) P, and P (in real terms) for the period 1871-1979 (F
One can immediately see that the correlation between Pt and P; is low thu
rejection of the view that stock prices are determined by fundamentals in a

. Year
I I 1

1870 1890 1910 1930 1950 1970

Figure 6.9 Source: Shiller (1981). Reproduced by permission of the American
where Z = sample mean and n = number of observations. In 1900 investors di
what future dividends were going to be and therefore the actual stock price
from the perfect foresight stock price. Hindsight has shown that investors ma
errors, qr, which may be represented as

(where q, is a weighted average of the forecast errors for dividends wr+i at t
etc.). If investors are rational then qr will be independent of all information
when investors made their forecast. In particular, q, will be independent of the
at time t. From (6.27) we obtain:

Informational efficiency (orthogonality) implies COv(P,, qr) is zero and (6.28)
+ var(qr)
var(P:) = var(Pr)
Since the variance of the forecast error is positive then:
var(P:) > var(Pr)
VR = var(P:)/var(P,) > 1

Hence if the market sets stock prices according to the rational valuation fo
(‘identical’) agents are rational in processing information and the discoun
constant, we would expect the variance inequality in equation (6.31) to ho
variance ratio (VR) (or standard deviation ratio (SDR)) to exceed unity.
For expositional reasons it is assumed that P is calculated as in (6.24) usin
formula. However, in much of the empirical work an equivalent method is
DPV formula (6.24) is consistent with the Euler equation:
+ D,+1) t = 1 , 2 , .. . n
P = 6(P:+,
Hence if we assume a terminal value for P + we can use (6.32) to calcula
by backward recursion. This, in fact, is the method used in Shiller (1981)
whatever method is used to calculate an observable version of PT, a termina
is required. One criticism of Shiller (1981) is that he uses the sample mean o
the terminal value, i.e.

6.2.2 First Generation Volatility Tests
Empirical tests of the RVF often use ‘real variables’, that is nominal variables
stock price and dividends are deflated by some general price index of goods (e.g
price index (CPI)). In this case the discount rate S must also be in real terms.
on volatility tests assumes a constant real discount rate and Shiller (1981) fou
stock prices are excessively volatile, that is to say, inequality (6.31) is grossly v
SDR = 5.59). However, LeRoy and Porter (1981) using a slightly different f
(see Appendix 6.1) found that although the variance bound is violated, the re
of borderline statistical significance.

Time Varying Real Interest Rates
So far, in our analysis, we have assumed that the real discount factor 6 i
However, we could rework the perfect foresight price, assuming the real requ
k, and hence 6, = (1 k,)-’ varies over time. For example, we could set k, t
actual real interest rate which existed in all future years r,, plus a constant ris
(k, = r, r p ) . Hence k, varies in each year and Pf is calculated as:

with a terminal value equal to the end of sample actual price. However, when t
bounds test is repeated using this new measure of PI* it is still violated (e.g. M
(1989) and Scott (1990)).
We can turn the above calculation on its head. Knowing what the variab
actual stock price was, we can calculate the variability in real returns k, tha
necessary to equate var(Pf ) with var(P,). Shiller (1981) performs this calcula
the assumption that P, and D,have deterministic trends. Using the detrended
finds that the standard deviation of real returns needs to be greater than 4
annum for the variance in the perfect foresight price to be brought into equali
variance of actual prices. However, the actual historic variability in real inter
much smaller than that required to ‘save’ the variance bounds test. Hence the e
for the violation of the excess volatility relationship does not appear to lie w
varying ex-post real interest rate.

Consumption CAPM
Another line of attack, in order to ‘rescue’ the violation of the variance bound, i
that the actual or ex-post real interest rate (as used above) may not be a particu
proxy for the ex-ante real interest rate. The consumption CAPM, where the
maximises the discounted present value of the utility from future consumpt
to a lifetime budget constraint, can give one a handle on what the ex-ante r
rate of return by investors (i.e. the discount rate) depends upon the rate of
consumption and the constant rate of time preference.
For simplicity, assume for the moment that dividends have a growth rate
D,= Do(gd)' and the perfect foresight price is:

Hence P:* varies over time (Grossman and Shiller, 1981) depending on the c
of consumption relative to a weighted harmonic average of future consump
Clearly, this introduces much greater variability in P: than does the consta
rate assumption (i.e. where the coefficient of relative risk aversion, a = 0).
the constant growth rate of dividends by actual ex-post dividends while re
C-CAPM formulation, Shiller (1987) recalculates the variance bounds tests
for the.period 1889-1985 using a = 4. The pictorial evidence (Figure 6.10) su
up to about 1950 the variance bounds test is not violated.
However, the relationship between the variability of actual prices and perfe
prices is certainly not close, in the years after 1950. Over the whole period S
that the variance ratio is not violated under the assumption that a = 4, which s
view as an implausibly high value of the risk aversion parameter. Thus, on bala
not appear as if the assumption of a time varying discount rate based on the co
CAPM can wholly explain movements in stock prices.

Small Sample Problems
Flavin (1983) and Kleidon (1986) point out that there are biases in small
measuring var(P,) and var(Pr) which might invalidate some of the 'first

120 1

0' I
1 I 1

1890 1910 1930 1950 1970
t (Year)

Figure 6.10 Consumption-based Time-Varying Interest Rates. Source: Grossman
(1981). Reproduced by permission of the American Economic Association
the degree of bias depending on the degree of serial correlation in
Since P: is more strongly autocorrelated than P, then var(P:) is esti
greater downward bias than var(P,). Hence it is possible that the sam
yield var(Pr) -= var(P,) in a finite sample, even when the null of the RV
(ii) Shiller’s use of the sample average of prices as a proxy for terminal val
t + n also induces a bias towards rejection.

There is a further issue surrounding the terminal price, noted by Gilles and LeR
The correct value of the perfect foresight price is

i= 1

which is unobservable. The observable series P:ln should be constructed using
price PI+, at the end of the sample, since this ensures P:in = E(PTI52,). Howev
still a problem since the sample variance of P& understates the true (but un
variance of PF. Intuitively this is because Pzn is ‘anchored’ on P,+,, and
take account of innovations in dividends which occur after the end of the s
implicitly sets these to zero but P; includes these, since the summation is
Clearly this problem is minimal if the sample is very large (infinite) but may b
in finite samples.
Flavin’s (1983) criticisms of these ‘first generation tests’ assumed, as did
of these tests, stationarity of the series being used. Later work tackled the i
validity of variance bounds tests when the price and dividend series are non
(i.e. have a stochastic trend). The problem posed by non-stationary series
population variances are functions of time and hence the sample variances
constant) are not correct measures of their population values. However, it is n
how to ‘remove’ these stochastic trends from the data, in order meaningfully t
variance bounds tests. It is to this issue that we now turn.

6.2.3 Volatility Tests and Stationarity
Shiller’s volatility inequality is a consequence purely of the assumption tha
stock price is an unbiased and optimal predictor of the perfect foresight price

P; = Pt ut

where u, is a random error term, with E(u,152,) = 0. Put another way, P,
cient statistic to forecast P ; . No information other than P, can improve on
of P:: in this sense P, is ‘optimal’. The latter implies that the conditional fo
E [ ( P : - P)t(52,]is independent of all information available at time t or earlie
is independent and therefore uncorrelated with 52,. Since P, c 52, then P, is i
of u, (i.e. cov(P,, u t ) = 0). The latter is the ‘informational efficiency’ or RE or
assumption. Using the definition of covariance for a stationary series, it follo
where we have used cov(P,, U,) = 0. The definition of the correlation coefficie
P, and P: is

Substituting for ‘cov’ from (6.36) in (6.35) we obtain a variance equality

= P(Pt p: )M:
M f ) 9

Since the maximum value of p = 1 then (6.37) implies the familiar variance

< w:)
Under the assumption of informational efficiency and that P, is etermined
fundamentals, then (6.38) must hold in the population for any stationary se
Stationary series have a time invariant and constant population mean varianc
deviation) and covariance. The difficulty in applying (6.38) to a sample of data
whether the sample is drawn from an underlying stationary series in the popu
It is also worth noting that the standard deviations in (6.38) are simple un
measures. If a time series plot is such that it changes direction often and he
its mean value frequently (i.e. is ‘jagged’) then in a short sample of data we
a ‘good’ estimate of the population value of a(P)from its sample value. How
time series wanders substantially from its (constant) mean value in long slow s
one will need a long sample of data to obtain a good estimate of the ‘true’
variance (i.e. a representative series of ‘cycles’ in the data set is required, n
one-quarter or one-half a cycle). In fact, stock prices appear to move in quite l
or cycles (see Figure 6.9) and hence a long data set is required to measure acc
true standard deviation (or variance).
If a series is non-stationary then it has a time varying population mean or v
hence (6.38) is ˜ndefined‘˜â€™. then need to devise an alternative variance in
terms of a transformation of the variables P , and P; into ‘new’ variables that are
The latter has led to alternative forms of the variance inequality condition. T
is that it is often difficult to ascertain whether a particular series is station
from statistical tests based on any finite data set. For example, the series g
+ +
Xt = xt-l E , is non-stationary while x, = O.98xf-1 E, is stationary. Howe
finite data set (on stock prices) it is often difficult statistically to discriminate b
two, since in a regression xt = a bx,-l + E,, the estimate of ‘b’ is subject t
error and often one could take it as being either 1 or 0.98. (Also the distribu
test statistic that b = 1 is ‘non-standard’.)
The fact that P , is an optimal forecast of PT does not necessarily imply
accurate forecast but only that one cannot improve upon the forecast based o
For example, if dividends are accurately described by Df = a wt where
noise, then using the DPV formula, P , will be constant. However, the varia
will depend on a weighted average of the variance of 0 and is certainly greate
p(P,, P : ) = 1. Then the equality in (6.37) holds and a ( P t ) = a(Pr). The in
position is most likely to occur in practice as agents make imperfect forecast
hence 0 < p ( P , , P:) < 1.
Let us return now to the issue of whether (real) dividends and (therefore fo
6 ) P and P , are non-stationary. How much difference does non-stationarit
practice when estimating the sample values of a(P,) and a(P:) from a finite da
can generate artificial data for these variables (i.e. the population) under the
of non-stationarity and see if in the generated sample of data the variance
(6.38) is met (i.e. Monte Carlo studies). For example, in his early work Sh
‘detrended’ the variables P, and PT by dividing by a simple deterministic tre
where b is estimated from the regression lnP, = a bt over the whole sam
If P, follows a stochastic trend then ‘detrending’ by assuming a determinis
statistically invalid. Use of A, will not ‘correctly’ detrend the series. The qu
arises as to whether the violation of the variance bounds found in Shiller’s (1
is due to this inappropriate detrending of the data.
Kleidon (1986)(5)and LeRoy and Parke (1992) examine this question us
Carlo methods. For example, Kleidon (1986) assumes that expected dividend
constant (= 0) so actual dividends are generated by a (geometric) random walk
lnD, =O+lnDr-l + E f
where Et is white noise. This yields a non-stationary stochastic trend for D
generated series for D, for rn observations using (6.39) one can use the DPV
generate a time series of length ‘rn’ for P; and for P,. One can then establi
var(P,) > var(PT) in the artificial sample of data of length rn. One can
‘experiment’ n times (each time generating m observations) and see how m
var(P,) > var(P:). Since the EMH/fundamentals model is ‘true’ by constru
would not expect the variance bound to be violated in a large number of cas
repeated experiments. (Some violations will be due to chance or ‘statistical o
fact Kleidon (1986) finds that when using the generated data and detrending b
the variance bound is frequently violated even though the EMH/fundamenta
true. The frequency of violations is 90 percent (when using Shiller’s method of d
while the frequency of ‘gross violations’ (i.e. VR > 5) varied considerably de
the rate of interest (discount rate) assumed in the simulations. (For example,
percent the frequency of gross violations is only about 5 percent, but for r = 5
the figure rises dramatically to about 40 percent.)
Shiller (1988) refined Kleidon’s procedure by noting that Kleidon’s combine
tions for the growth rate of dividends (= 8) and the level of interest rates
implausible value for the dividend price ratio. Shiller allows 8 to vary with r
the artificially generated data, the dividend price ratio equals its average his
Under the null of the RVF he finds that the gross violations of the varianc
substantially less than those found by Kleidon.
Further, Shiller (1989, page 85) notes that in none of the above Monte Ca
is the violation of the variance inequality as large as that actually found by Shi
past values of dividends (i.e. an AR(q) process where q is large) and may not
root (i.e. the sum of the coefficients on lagged dividends is less than unity).
This debate highlights the problem of trying to discredit results which use
by using ‘specific special cases’ (e.g. random walk) in a Monte Carlo anal
Monte Carlo studies provide a highly specific ‘sensitivity test’ of empirical
such experiments may not provide an accurate description of real world data.
problem is that in a finite ‘real world’ data set one often does not know what i
realistic’ statistical representation of the data. Often, data are equally well repr
a stationary or non-stationary univariate or multivariate series. However, one
with Shiller (1989) that on a priori economic grounds it is hard to accept tha
believe that when faced with an unexpected increase in current dividends, of
they expect that dividends will be higher by z percent, in allfitureperiods. Ho
latter is implied by the (geometric) random walk model of dividends (equat
used by Kleidon and others. Shiller (1989) prefers the view that firms attempt
nominal dividends (and hardly ever cut dividends). In this case real dividend
in the above studies) may appear to be ‘close to’ a unit root series but the ‘tru
The outcome of all of the above arguments is that not only may the sm
properties of the variance bounds tests be unreliable but if there is non-statio
even tests based on large samples may be suspect. Clearly, all one can do
(while awaiting new time series data as ‘time’ moves on!) is to assess the ro
the volatility results under different methods of detrending. For example, Shi
reworks some of his earlier variance inequality results using P,/E)’ and P:/
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.
The LeRoy and Porter (1981) variance bounds test is based on the mathematical prop
conditional expectation of any random variable is less volatile than the variable itself. Th
begins with a forecast of future dividends based on a limited information set A, = (D
The forecast of future dividends based on A, is defined as

The actual stock price P, is determined by forecasts based on the full information set

P, = E(P:lQt,)
Applying the law of iterated expectations to (2) gives

Using (1) and (3):
k = E(P,lAf)

F,is the conditional expectation of P I ,then from (4)the LeRoy-Porter variance in

Assuming stationarity, the sample variances of ifand P, provide consistent estima
population values given in (5). Given an ARMA model for D,and a known value of
p, can be constructed using (1).
As in Shiller (1981), the procedure adopted by LeRoy and Porter yields a variance
However, the LeRoy-Porter analysis also gives rise to a form of ‘orthogonality test’.
define the one-period forecast error of the $ return as
where P = C6iDr+i.If er+i is stationary:

where the er+, are mutually uncorrelated under RE. Also, under RE the covariance t
that is P, and er+, are orthogonal, hence

Equation (10) is both a variance equality and an orthogonality test of the RVF.
It is perhaps worth noting at this juncture that equation (9) is consistent with Shiller's
If in the data we find a violation of Shiller's inequality, that is var(P,) > var(P:), then f
implies cov(P,, C6'e,+,) < 0. Hence a weighted average of one-period $ forecast errors
with information at time t, namely P,. Thus violation of Shiller's variance bound im
weighted average of) one-period $ returns are forecastable. A link has therefore be
between Shiller's variance bounds test and the predictability of one-period $ returns. H
returns here are $ returns not percentage returns and therefore violation of Shiller's var
cannot be directly compared with those studies that find one-period returns are forec
Fama and French 1988b). In Chapter 19 we pursue this analysis further and a log-line
the RVF allows us to directly link a failure of the RVF with the predictability of one
multiperiod percentage returns.
The West (1988) test is important because it is valid even if dividends are non-sta
cointegrated with the stock price) and it does not require a proxy for the unobservable
LeRoy-Porter variance inequality, the West test is based on a property of mathematical e
Specifically, it is that the variance of the forecast error with a limited information set
greater than that based on the full information set Q r .
The West inequality can be shown to be a direct implication of the LeRoy-Porter
We begin with equation (1) and note that

Applying the law of iterated expectations E [ E ( . J h , + ˜ ) ) A , ]E ( - ) & ) to (11)

Substituting for the LHS of (12) from (1)

Now define the forecast error of the $ return, based on information Ar, as

Substituting from (13) in (14):
Equation (16) is the West inequality and using US data West (1988) finds (16) is violated
level of dividends is non-stationary the population variances in (16) are constant (station
sample variances provide consistent estimators. However, unfortunately the latter is on
level of dividends is non-stationary (e.g. D,+1 = D,+l wf+l).LeRoy and Parke (1992
the properties of the West test if dividends follow a geometric random walk (i.e. lnD,+
&,+I)and the reader should consult the original paper for further information.
The LeRoy-Porter and West inequalities are both derived by considering varian
limited information set and under the complete information set. Hence one might gue
inequalities provide similar inferences on the validity of the RVF. This is indeed the
now be demonstrated. The LeRoy-Porter equality (10) holds for any information set a
it holds for At which implies

The LeRoy-Porter inequality (5) is
var@,> < var(P,)

Substituting for var(b,) from (17) and for var(P,) from (10) we obtain the West inequ
c 7
Rational Bubbles
The idea of self-fulfilling ‘bubbles’ or ‘sunspots’ in asset prices has been discus
since organised markets began. Famous documented ‘first’ bubbles (Garber, 19
the South Sea share price bubble of the 1720s and the Tulipmania bubble. I
case, the price of tulip bulbs rocketted between November 1636 and January
to collapse suddenly in February 1637 and by 1739 the price had fallen to arou
of 1 percent of its peak value. The increase in stock prices in the 1920s and
‘crash’ in 1929, the stock market crash of 1987 and the rise of the dollar bet
and 1985 and its subsequent fall, have also been interpreted in terms of a se
bubble. Keynes (1936), of course, is noted for his observation that stock pric
be governed by an objective view of ‘fundamentals’ but by what ‘average opin
average opinion to be’. His analogy for the forecasting of stock prices was that
forecast the winner of a beauty contest. Objective beauty is not necessarily the
is important is how one thinks the other judges’ perceptions of beauty will b
in their voting patterns.
Rational bubbles arise because of the indeterminate aspect of solutions to rati
tations models, which for stocks is implicitly reflected in the Euler equation
prices. The price you are prepared to pay today for a stock depends on the pric
you can obtain at some point in the future. But the latter depends on the exp
even further in the future. The Euler equation determines a sequence of price
not ‘pin down’ a unique price level unless somewhat arbitrarily we impose
condition (i.e. transversality condition) to obtain the unique solution that p
fundamental value (see Chapter 4). However, in general the Euler equation do
out the possibility that the price may contain an explosive bubble. (There are s
qualifications to the last statement and in particular in the representative agen
Tirole (1985) he demonstrates uniqueness for an economy with a finite number
infinitely lived traders and he also demonstrates that bubbles are only possibl
rate of growth of the economy is higher than the steady state return on capital
While one can certainly try and explain prolonged rises or falls in stock price
some kind of irrational behaviour such as ‘herding’, or ‘market psychology’, n
recent work emphasises that such sharp movements or ‘bubbles’ may be cons
the assumption of rational behaviour. Even if traders are perfectly rational, the a
price may contain a ‘bubble element’ and therefore there can be a divergenc
the stock price and its fundamental value. This chapter investigates the phen
rational bubbles and demonstrates:
We wish to investigate how the market price of stocks may deviate, possibly su
from their fundamental value even when agents are homogeneous, rational and
is informationally efficient. To do so it is shown that the market price ma
fundamental value plus a 'bubble term' and yet the stock will still be wil
by rational agents and no supernormal profits can be made. The exposition is
by assuming (i) agents are risk neutral and have rational expectations and (i
require a constant (real) rate of return on the asset E,R, = k. The Euler equat

where 6 = 1/(1 k). We saw in Chapter 4 that this may be solved under RE b
forward substitution to yield the rational valuation formula for the stock price


if we assume the transversality condition holds (i.e. lim(G"E,D,+,) = 0, as n +
transversality condition ensures a unique price given by (7.2). The RHS of (7
the fundamental value Pf. The basic idea behind a rational bubble is that there
mathematical expression for Pt that satisfies the Euler equation, namely:

C SiEfDr+i+ Bf = Pf + B,
P, =

and the term B, is described as a 'rational bubble'. Thus the actual market price
from its fundamental value Ptf by the amount of the rational bubble B,. So f
no indication of any properties of B,: clearly if B, is large relative to fundam
then actual prices can deviate substantially from their fundamental value.
In order that (7.3) should satisfy (7.1) some restrictions have to be plac
dynamic behaviour of Bt and these restrictions are determined by establishing
contradiction. This is done by assuming (7.3) is a valid solution to (7.1) and
restricts the dynamics of Bt. Start by leading (7.3) by one period and taking ex
at time t:

where use has been made of the law of iterated expectations Ef(Ef+lDf+,) E
RHS of the Euler equation (7.1) contains the term 6(E,P,+1 E f D , + l ) and u
Substituting from (7.6) into (7.1)

But we now seem to have a contradiction since (7.3) and (7.7) cannot in ge
be solutions to (7.1). Put another way, if (7.3) is assumed to be a solution t
equation (7.1) then we also obtain the relationship (7.7) as a valid solution t
can make these two solutions (7.3) and (7.7) equivalent i f

Then (7.3) and (7.7) collapse to the same expression and satisfy (7.1). More
(7.8) implies
ErBt+m = Br/Jm
Hence (apart from the (known) discount factor) Bt must behave as a martinga
forecast of all future values of the bubble depend only on its current value.
bubble solution satisfies the Euler equation, it violates the transversality con
Bt # 0) and because Br is arbitrary, the stock price in (7.3) is non-unique.
What kind of bubble is this mathematical entity? Note that the bubble
solution provided the bubble is expected to grow at the rate of return required f
willingly to hold the stock (from (7.8) we have E(B,+I/B,) - 1 = k). Inves
care if they are paying for the bubble (rather than fundamental value) because
element of the actual market price pays the required rate of return, k. Market p
however, do not know how much the bubble contributes to the actual price:
is unobservable and is a self-fulfilling expectation.
Consider a simple case where expected dividends are constant and the v
bubble at time t , Bt = b(> 0), a constant. The bubble is deterministic and g
rate k, so that EtB,+m = (1 k)mb.Thus once the bubble exists, the actual sto
t m even if dividends are constant is from (7.3)
+ + k)m
Pr+m = - b(l
(1 - 8 )
Even though fundamentals (i.e. dividends) indicate that the actual price should
the presence of the bubble means that the actual price can rise continuo
(1 k) > 1.
In the above example, the bubble becomes an increasing proportion of the
since the bubble grows but the fundamental value is constant. In fact even whe
are not constant the stock price always grows at a rate which is less than the rat
of the bubble (= k) because of the payment of dividends:
make (supernormal) profits since all information on the future course of div
the bubble is incorporated in the current price: the bubble satisfies the fair gam
The above model of rational bubbles can be extended (Blanchard, 1979) to
case where the bubble collapses with probability (1 - n)and continues with
= B,(an)-’ with probability n
with probability 1 - n
This structure also satisfies the martingale property. These models of rational
should be noted, tell us nothing about how bubbles start or end, they merely te
the time series properties of the bubble once it is underway. The bubble is ‘
to the fundamentals model of expected returns.
As noted above, investors cannot distinguish between a price rise that is du
fundamentals from one that is due to fundamentals plus the bubble. Individu
mind paying a price over the fundamental price as long as the bubble element y
the required rate of return next period and is expected topersist. One implication
bubbles is that they cannot be negative (i.e. Bf < 0). This is because the bubb
falls at a faster rate than the stock price. Hence a negative rational bubble ultim
in a zero price (say at time t N). Rational agents realise this and they there
that the bubble will eventually burst. But by backward induction the bubble
immediately since no one will pay the ‘bubble premium’ in the earlier perio
actual price P, is below fundamental value P f , it cannot be because of a ratio
If negative bubbles are not possible, then if a bubble is ever zero it cannot r
arises because the innovation (B,+1 - E,B,+I) in a rational bubble must have a
If the bubble started again, the innovation could not be mean zero since the bu
have to go in one direction only, that is increase, in order to start up again.
In principle, a positive bubble is possible since there is no upper limit on st
However, in this case, we have the rather implausible state of affairs where
element B, becomes an increasing proportion of the actual price and the funda
of the price become relatively small. One might conjecture that this implies that
will feel that at some time in the future the bubble must burst. Again, if inve
that the bubble must burst at some time in the future (for whatever reason),
burst. To see this suppose individuals think the bubble will burst in the year 2
must realise that the market price in the year 2019 will reflect only the fundam
because the bubble is expected to burst over the coming year. But if the pri
reflects only the fundamental value then by backward induction this must be
price in all earlier years. Therefore the price now will reflect only fundamen
it seems that in the real world, rational bubbles can really only exist if th
horizon is shorter than the time period when the bubble is expected to burst
here is that one would pay a price above the fundamental value because one be
someone else will pay an even greater price in the future. Here investors are m
the price at some future time f N depends on what they think other investor
price will be.
that in calculating the perfect foresight price an approximation to the infin
discounting in the RVF is used and the calculated perfect foresight price is P

and Pt+N is the actual market price at the end of the data set. The variance b
the null of constant (real) required returns is var(P,) < var(PT). However, a
incorporated in this null hypothesis. To see this, note that with a rational bub

+ Bf
P, = P f
and E,(B,+N) = (1 k)NBf= 6 - N B f . If we now replace Pf+N in (7.13)
containing the bubble Pr+N = Pr+N Br+N then:

+ SNEfBf+N= P f + Bf
EfP: = Pf
and hence even in the presence of a bubble we have from (7.14) and (7.15)

An early test for bubbles (Flood and Garber, 1980) assumed a non-stochastic b
+ where Bo is the value of the bubble at the beginning of
is P f = P f
period. Hence in a regression context there is an additional term of the for
Knowing 6, a test for the presence of a bubble is then H o : Bo # 0. Unfortunate
(1/6) > 1, the regressor (1/6)' is exploding and this implies that tests on Bo
non-standard distributions and correct inferences are therefore problematic. (
details see Flood, Garber and Scott (1984)).
An ingenious test for bubbles is provided by West (1987). The test involves
a particular parameter by two alternative methods. Under the assumption of n
the two parameter estimates should be equal within the limits of statistica
while in the presence of rational bubbles the two estimates should differ. A
this approach (in contrast to Flood and Garber (1980) is that it does not requir
parameterisation of the bubble process: any bubble that is correlated with div
in principle be detected.
To illustrate the approach, note first that 6 can be estimated from (instrum
ables) estimation of the 'observable' Euler equation:

+ Uf+l
p , = Wf+1 Q+1)
where invoking RE, ut+l = -6[(P,+1 D f + l )- E,(P,+1 - D,+1)].Now assum
process for dividends
+ I4 <1
Df = aDt-1 vf
the true information set (but Er(E,+lJAr) 0). An indirect estimate of deno
be obtained from the regression estimates of 6 from (7.16) and a! from (7.17). H
direct estimate of \I/ denoted 6* be obtained from the regression of P, on D
Under the null of no bubbles, the indirect and direct estimates of should be
Consider the case where bubbles are present and hence P I = P, B, =
The regression of P, on D, now contains an omitted variable, namely the bubb
estimate of Q denoted &twill be inconsistent:
+ plim(T-’ CD?)-’l i m ( V c D f B t )
plim St = \I/ p
If the bubble B, is correlated with dividends then $t will be biased (u
cov(D,, B,) > 0) and inconsistent. But the Euler equation and the dividend
equations still provide consistent estimators of the parameters and hence of $
in the presence of bubbles, & # &t(and a Hausman (1978) test can be use
any possible change in the coefficients).(I)
The above test procedure is used by West (1987) whose data consists of
(1981) S&P index 1871-1980 (and the Dow Jones index 1928-1978). W
substantive difference between the two sets of estimates thus rejecting the
bubbles. However, this result could be due to an incorrect model of equilibri
or dividend behaviour. Indeed West recognises this and finds the results are
robust to alternative ARMA processes for dividends but in contrast, under tim
discount rates, there is no evidence against the null of no bubbles. Flood et a1 (1
out that if one iterates the Euler equation for a second period, the estimated (‘tw
Euler equation is not well specified and estimates of 6 may therefore be bia
the derivation of RVF requires an infinite number of iterations of the Euler eq
casts some doubt on the estimate of 6 and hence on West’s (1987) results.
West (1988a) develops a further test for bubbles which again involves com
difference between two estimators, based on two different information sets. O
information set A, consists of current and past dividends and the other infor
is the optimal predictor of future dividends, namely the market price P,. Und
of no bubbles, forecasting with the limited information set A, ought to yie
forecast error (strictly, innovation variance) but West finds the opposite. Thi
refutes the no-bubbles hypothesis but of course it is also not necessarily incons
the presence of fads.
Some tests for the presence of rational bubbles are based on investigating t
arity properties of the price and dividend data series in the RVF. An exogen
introduces an explosive element into prices which is not (necessarily) present in
mentals (i.e. dividends or discount rates). Hence if the stock price and divide
at the same rate, this is indicative that bubbles are not present. If P, ‘grows’
D, then this could be due to the presence of a bubble term B,. These intuiti
can be expressed in terms of the literature on unit roots and cointegration. Usin
(under the assumption of a constant discount rate) it can be shown that if the
dividends AD, is a stationary (ARMA) process and there are no bubbles, th
also a stationary series and P , and D,are cointegrated.
dividends and the RVF (without bubbles) gives P , = [6/(1 - 6)]D,. Since D
random walk then P, must also follow a random walk and therefore AP, is sta
addition, the (stochastic) trend in P,must ‘track’ the stochastic trend in D, so tha

is not explosive. In other words the RVF plus the random walk assumption fo
implies that zr must be a stationary I(0) variable. If z, is stationary then P, and
to be cointegrated with a cointegrating parameter equal to 6/(1 - 6). Testing f
(Diba and Grossman, 1988b) then involves the following:

(i) Demonstrate that PI and D, contain a unit root and are non-stationary
Next, demonstrate that AP, and AD, are both stationary I(0) series. T
adduced as evidence against the presence of an explosive bubble in P,,
(ii) The next step is to test for cointegration between P, and D,. Heuristical
involves a regression of P, = 2.0 &D, and then testing to see if the c
series z, = (P, - 20 - ?IDr) is stationary. If there are no bubbles, we e
be stationary I(O), but zr is non-stationary if explosive bubbles are prese
Using aggregate stock price and dividend indexes Diba and Grossman (1988
the above tests and on balance they find that AP, and AD, are stationary and
are cointegrated, thus rejecting the presence of explosive bubbles of the type r
by equation (7.8).
Unfortunately, the interpretation of the above tests has been shown to be
misleading in the presence of what Evans (1991) calls ‘periodically collapsin
The type of rational bubble that Evans examines is one that is always positi
‘erupt’ and grow at a fast rate before collapsing to a positive mean value,
process begins again. The path of the periodically collapsing bubble (see Figu
be seen to be different from a bubble that grows continuously.
Intuitively one can see why testing to see if P , is a non-stationarity 1(1) serie
detect a bubble component like that in Figure 7.1. The (Dickey-Fuller) test for
essentially tries to measure whether a series has a strong upward trend or an un
variance that is non-constant. Clearly, there is no strong upward trend in Figu
although the variance alters over time, this may be difficult to detect particu
bubbles have a high probability of collapsing (within any given time period). If
have a very low probability of collapsing, then we are close to the case of
bubbles’ (i.e. E,B,+I = &/a) examined by Diba and Grossman and here one m
standard tests for stationarity to be more conclusive.
Heuristically (and simplifying somewhat), Evans proceeds by artificially g
series for a periodically collapsing bubble. Adding the bubble to the fundament
under the assumption that D, is a random walk with drift) gives the generated
series. The generated stock price series containing the bubble is then subject
tests for the presence of unit roots. The experiment is then repeated a numbe
Evans finds that the results of his unit root tests depend crucially on n,the
i 75
; 50

” 70
20 80 90
10 40 60 100
Figure 7.1 Bubble Component. Source: Evans (1991). Reproduced by permission of t
Economic Association

(per period) that the bubble does not collapse. For values of n < 0.75, mo
percent of the simulations erroneously indicate that AP, is stationary. Also,
are erroneously found to be cointegrated. Hence using Monte Carlo simulat
demonstrates that a particular class of rational bubbles, namely ‘periodically
bubbles’, are often not detectable using standard unit root tests. (The reason
that ‘standard tests’ assume a linear autoregressive process whereas Evan’s
involve a complex non-linear bubble process). Thus the failure of Diba and G
detect continuously explosive bubbles in stock prices does not necessarily rul
types of rational bubble. Clearly, more sophisticated statistical tests of non-stat
required to detect periodically collapsing bubbles (see, for example, Hamilton

One of the problems with the type of bubble discussed so far is that the
a deus ex-machina and is exogenous to fundamentals such as dividends. T
term arises as an alternative solution (strictly the homogeneous part of th
to the Euler equation for stock prices. Froot and Obstfeld suggest a differe
bubble phenomenon which they term an intrinsic bubble. ‘Intrinsic’ is used b
bubble depends (in a non-linear deterministic way) on fundamentals, namely t
(real) dividends. The bubble element therefore remains constant if ‘fundament
constant but increases (decreases) along with the level of dividends. For th
intrinsic bubble, if dividends are persistent then so is the bubble term and s
will exhibit persistent deviations from fundamental value. In addition, the intrin
can cause stock prices to overreact to changes in dividends (fundamentals
consistent with empirical evidence.
To analyse this form of intrinsic bubble assume a constant real required rat
r (in continuous time). The Euler equation is
However, P, = P f B, is also a solution to the Euler equation, if BI is a mart
is B, = e-'[E,Br+l]. The 'intrinsic bubble' is constructed by finding a non-line
of dividends such that Br is a martingale and hence satisfies the Euler equation
Obstfeld show that a non-linear function, denoted B(D,), of the form:

B(D,) = cD; c > 0, h > 1

satisfies these conditions. If log dividends follow a random walk with drift p
+ +
and conditional variance 0 2 ,that is ln(Dt+l) = p ln(D,) &,+I, then the bubb
Pr is:
+ +
P, = P f B(D,) = aD, cD:

where(2),a = (er - ep+a2/2)-1. fundamentals solution Prf = CrD, is a stoc
sion of Gordon's (1962) growth model which gives Pf = (er - ep)-'Dr unde
It is clear from (7.23) that stock prices overreact to current dividends co
the 'fundamentals only' solution (i.e. a P f / a D , = a ) because of the bubble
dP,/dLI, = a CAD:-', c > 0). Froot and Obstfeld simulate the intrinsic bubb
assuming reasonable values for ( r , p , a2), estimated values of a, c and h (
and with Et+i drawn from an independent normal distribution. They compa
fundamentals path Pf, the intrinsic stochastic bubble path bt given by (7.2
addition, an intrinsic bubble that depends on time as well as dividends, which
to a path for prices denoted p,:

The intrinsic bubble which depends on time (7.24) allows a comparison with
bubble tests, which often invoke a deterministic exponential time trend (Flood a
1980, Blanchard and Watson, 1982 and Flood and Garber, 1994, page 1192)
lated values of these three price series are shown in Figure 7.2 and it is cle
intrinsic bubble can produce a plausible looking path for stock prices pt and
persistently above the fundamentals path Pf . (Although in other simulations t
bubble can be above the fundamentals path Pf and then 'collapse' towards
Figure 7.2, we see that in a finite sample the intrinsic bubble may not look exp
hence it would be difficult to detect in statistical tests that use a finite data set
dependent intrinsic bubble p , on the other hand yields a path which looks exp
this is more likely to be revealed by statistical tests.) In other simulations, th
bubble series f i r may end up (after 200 periods of the simulation) substantially
fundamentals price PI/.
Froot and Obstfeld test for the presence of intrinsic bubbles using a simp
mation of (7.23).
+ +
PtlD, = CO c@-' 77,
.d 11
6 10
1990 1920 1940 1960 1980 2010 2030 2050 2070 2080 2090

Figure 7.2 Simulated Stock Price Paths. Source: Froot and Obstfeld (1991). Rep
permission of the American Economic Association

where the null of no bubble implies H o : CO = a and c = 0 (where a = er
Using representative values of er = 1.09 per annum for the real S&P index
real dividend process, , = 0.011, o = 0.122 and therefore the sample averag
a equals 14. Hence under the null of no bubbles P, and D, should be cointeg
cointegration parameter CO = a, of about 14. In a simple OLS cointegrating
of P, on Df, Froot and Obstfeld find that P, = \Ir 370, and hence P, ov
dividends. In addition, P , - 140, is not stationary and therefore P , and D, ar


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