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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

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

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

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, 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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