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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
Since

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 â€˜orthogonality testâ€™.
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
I1
Rational Bubbles
The idea of self-fulfilling â€˜bubblesâ€™ or â€˜sunspotsâ€™ in asset prices has been discus
since organised markets began. Famous documented â€˜firstâ€™ 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
â€˜crashâ€™ 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 â€˜fundamentalsâ€™ but by what â€˜average opin
average opinion to beâ€™. 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 judgesâ€™ 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 â€˜pin downâ€™ 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 â€˜herdingâ€™, or â€˜market psychologyâ€™, n
recent work emphasises that such sharp movements or â€˜bubblesâ€™ may be cons
the assumption of rational behaviour. Even if traders are perfectly rational, the a
price may contain a â€˜bubble elementâ€™ and therefore there can be a divergenc
the stock price and its fundamental value. This chapter investigates the phen
rational bubbles and demonstrates:
7.1 EULER EQUATION AND THE RATIONAL
VALUATION FORMULA
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
00

i=l

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:
00

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

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)
6D
+ + 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
n;mathematically:
= B,(an)-â€™ with probability n
with probability 1 - n
=O
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 â€˜
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 â€˜bubble premiumâ€™ 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:
f

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

7.2 TESTS OF RATIONAL BUBBLES
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
*,
=
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
can
*
Under the null of no bubbles, the indirect and direct estimates of should be
+
f
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-â€™ CD?)-â€™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 (â€˜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 Westâ€™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
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, â€˜growsâ€™
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 â€˜trackâ€™ 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 â€˜periodically collapsin
The type of rational bubble that Evans examines is one that is always positi
â€˜eruptâ€™ 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
bubblesâ€™ (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
5
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