; 50
a
25
” 70
50
20 80 90
10 40 60 100
30
Period
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 nonlinear 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 nonstat
required to detect periodically collapsing bubbles (see, for example, Hamilton
7.3 INTRINSICBUBBLES
One of the problems with the type of bubble discussed so far is that the
a deus exmachina 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 nonlinear 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 nonline
of dividends such that Br is a martingale and hence satisfies the Euler equation
Obstfeld show that a nonlinear 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
The
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
P,
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,
13
12
.d 11
h
6 10
$9
I
a
7
6
5
4
3
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
x
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
tegrated (with a cointegration parameter equal to 14). The ˜fundamentals onl
Pf = aD, also implies that In P, and In Df are cointegrated with a cointegrat
eter of unity. However, estimates reveal this parameter to be in the range 1
that (In P,  In 0,) not be stationary. Hence taken at face value, these te
may
reject the (nobubble) fundamentals model. However, Froot and Obstfeld no
OLS cointegrating parameter could be heavily biased (Banerjee et al, 1993), a
power and size of these tests are problematic.
Froot and Obstfeld then consider a direct test for the presence of intrins
based on estimation of (7.25). A representative result is:
+
( P / D ) , = 14.6 0.04D˜™6“˜”
(2.28) (0.12)
Annual: 19001988, R2 = 0.57, (  ) = NewyWest standard errors.
Although there are some subtle econometric issues involved in testing for
A  1 = 0 in (7.25) in a finite sample, the evidence above is in part suppor
intrinsic bubble. The joint null, that c and h  1 equal zero is strongly rejected
the empirical evidence is not decisive since we do not reject the null that c =
5.4
5.2
5.0
4.0
4.6
4.4
4.2
4.0
3.8
3.6
3.4
1900 1910 1920 1930 1940 1950 1970 19
1960
Figure 7.3 Actual and Predicted Stock Prices. Source: Froot and Obstfeld (1991).
by permission of the American Economic Review
simulate values for the fundamentals price P f = 14.60t, and the price with
bubble P , given by (7.23) and compare these two series with the actual pri
path of the intrinsic bubble (Figure 7.3) is much closer to the actual path of s
than is Pf.The size of the bubble can also be very large as in the postSec
War period. Indeed, at the end of the period, the bubble element of the S&P
index appears to be large.
Finally, Froot and Obstfeld assess the sensitivity of their results to differen
models (using Monte Carlo methods) and to the addition of various additiona
of D or other deterministic time trends in the regression (7.22). The estim
t
basic intrinsic bubble formulation in (7.23) are quite robust.
Driffill and Sola (1994) repeat the FrootObstfeld model assuming divide
undergoes regime shifts, in particular that the (conditional) variance of divide
varies over the sample. A graph of (real) dividend growth for the US show
low variance between 1900 and 1920, followed by periods of fairly rapid ˜s
variance over 1920 1950 and then relatively low and constant variance post19
and Sola use the twostate Markov switching model of Hamilton (1989) (see
to model dividend growth and this confirms the results given by ˜eyeballing™
They then have two equations of the form (7.23) corresponding to each of the
of ˜high™ and ˜low™ variance. However, their graph of the price with an intrin
is very similar to that of Froot and Obstfeld (see l i t , Figure 7.3) so this particu
does not appear to make a major difference.
There are a number of statistical assumptions required for valid infere
approach of Froot and Obstfeld (some of which we have mentioned). For
Econometric Issues
There are severe econometric problems in testing for rational bubbles and th
tation of the results is also problematic. Econometric problems that arise
analysis of potentially nonstationary series using finite data sets, the behav
statistics in the presence of explosive regressors as well as the standard p
obtaining precise estimates of nonlinear parameters (as in the case of intrins
and of corrections for heteroscedasticity and moving average errors. Some of
are examined further in Chapter 20. Tests for rational bubbles are often co
having the correct equilibrium model of expected returns: we are therefore tes
hypothesis. Rejection of the nobubbles hypothesis may simply be a manifest
incorrect model based on fundamentals.
Another difficulty in interpreting results from tests of rational bubbles aris
Peso problem which is really a form of omitted variables problem. Suppos
in the market had information, within the sample period studied by the rese
dividends would increase in the future, but the researcher did not (or could n
this ˜variable™ in his model of fundamentals used to forecast dividends. In the
data, the stock price would rise but there would be no increase in dividends for
researcher. Stock prices would look as if they have overreacted to (current) div
more importantly such a rise in price might be erroneously interpreted as a b
problem of interpretation is probably most acute when there are severe price ch
one cannot rule out that the econometrician has omitted some factor (in say
model for dividends or discount rates) which might have a large impact o
dividends (or discount rates) but is expected to occur with a small probabili
to the issue of periodically collapsing bubbles, it appears unlikely that standar
detect such phenomena.
SUMMARY
7.4
Mathematically, rational bubbles arise because, in the absence of an (arbitrar
sality condition, the Euler equation yields a solution for stock prices that eq
mental value plus a ˜bubble term™, where the latter follows as a martingale p
key results in this chapter are as follows:
In the presence of a bubble, stock returns are unpredictable and therefore or
0
tests cannot be used to detect rational bubbles.
In the early literature bubbles were exogenous to fundamentals (e.g. divid
0
˜origin™ of the bubble cannot be explained and only the time path of th
given by these models.
Standard unit root and cointegration tests may be able to detect continuously
0
bubbles but are unlikely to detect periodically collapsing bubbles.
Intrinsic bubbles depend, in a nonlinear deterministic way, on economic fu
0
(e.g. dividends) yet still satisfy the Euler equation. The evidence for intrin
bubble which is subject to ˜eruptions™ and subsequent collapse. However,
involves information being passed between a heterogeneous set of agents wi
beliefs, so it does not fit neatly into the type of intrinsic bubble in this cha
assumed a homogeneous set of rational agents.
ENDNOTES
1. Strictly speaking, for the Hausman test to be valid we do not require di
discount rates) and the bubble to be correlated. This is because bias may
constant term of the regression of P, on D,(and lagged values of D)or b
excluded bubble has an exploding variance.
2. This solution arises because D,+l= D,exp(s,+l) and hence E(D
D,E[exp(E,+l )I. The lognormal distribution has the property that E[ex
+
exp(p a 2 / 2 )where &,+I is distributed as independent normal with m
variance 02.
h Anomalies, Noise Traders
I
and Chaos
8.1 THE EMH AND ANOMALIES
In an ˜efficient market™ all ˜players™ have access to the same information, th
the information in the same ˜rational way™ and all have equal opportunities for
and lending. In the real world these conditions are unlikely to be met. Fo
different investors may form different probability assessments about future o
use different economic models in determining expected returns. They may also
ences in transactions costs (e.g. insurance companies versus individuals when
shares), or face different tax rates, and of course they will each devote a differ
of resources (i.e. time and money) in collecting and processing information. O
these heterogeneous elements play a rather minor role then asset prices and rate
will be determined mainly by economic fundamentals and rational behaviour.
prices may deviate substantially and persistently from their fundamental values.
below it is often the assumption of heterogeneity in behaviour which allows us
why markets may not be efficient. In this chapter we examine the EMH from
different angle to the technical statistical research outlined in Chapter 6, in pa
By observing actual behaviour in the stock market one can seek to isolate
trading opportunities which persist for some time. If these ˜profitable trad
reflect a payment for risk and are persistent, then this refutes the EMH. Th
is referred to as stock market anomalies.
Theoretical models are examined in which irrational ˜noise traders™ in
interact with rational ˜smart money™ traders. This helps to explain why s
might deviate from fundamental value for substantial periods and why st
might be excessively volatile.
Continuing the above theme of the interaction of noise traders and smart m
shown how a nonlinear deterministic system can yield seemly random be
the time domain: this provides an applied example of chaos theory.
8.1.1 Weekend and January Effects
The weekend effect refers to the fact that there appears to be a systematic
daily rate of return on (some) stocks between the Friday closing and Monda
Monday (price is ˜low™) assuming that the expected profit more than covers t
costs and a payment for risk. This should then lead to a ˜removal™ of the ano
this should result in prices falling on Friday and rising on Monday.
The socalled January effect is a similar phenomenon to the weekend effec
rate of return on common stocks appears to be unusually high during the ea
the month of January. For the USA one explanation is due to yearend selli
in order to generate some capital losses which can be set against capital gai
to reduce tax liability. (This is known as ˜bed and breakfasting™ in the UK.)
investors wish to return to their equilibrium portfolios and therefore move into
to purchase stock. Again if the EMH holds, this predictable pattern of pri
should lead to purchases by nontax payers (e.g. pension funds) in Decembe
price is low and selling in January when the price is high, thus eliminating th
arbitrage opportunity. The January effect seems to take place in the first five t
of January (Keane, 1983) and also appears to be concentrated in the stocks of
(Reinganum, 1983).
8.1.2 The Small Firm Effect
Between 1960 and the middle of the 1980s all smallcapitalized companies
average a higher rate of return than the overall stock market index. Of course
to the CAPM this could be due to the higher risks attached to these small f
should be reflected in their higher beta values. However, Reinganum (1983) su
the rate of return, even after adjustment for risk, is higher on stocks of small
firms. Hence Reinganum has found that stocks of small firms do not lie on t
market line.
8.1.3 Closed End Funds
Closed end funds issue a fixed number of shares at the outset and trading in t
then takes place between investors. Shares which comprise the ˜basket™ in the
mutual fund are generally also traded openly on the stock market. The value
ought therefore to equal the market value of the individual shares in the fun
often the case that closed end mutual funds trade at a discount on their market
violates the EMH, for investors could buy the closed end fund™s shares at t
price and at the same time sell short a portfolio of stocks which are ident
held by the fund. The investor would thereby ensure he earned a riskless pro
the discount. Figure 8.1 shows that the discount on such funds can often be
(see Fortune (1991)) and appears to vary inversely with the return on the st
itself. For example, the bull markets of 19681970 and 19821986 are asso
declining discount where the bear markets of the 1970s and 1987 are asso
large discounts.
Several reasons have been offered for such closed end fund discounts. First,
fund members face a tax liability (in the form of capital gains tax), if the fund
securities after they have appreciated. This potential tax liability justifies pay
0

10

20
1 30 1
1982 1986
1970 1974 1978 1990
Figure 8.1 Average Premium (+) or Discount () on Seven ClosedEnd Funds. Sou
(1991), Fig. 3, p. 23. Reproduced by permission of The Federal Reserve Bank of Bos
price than the market value of the underlying securities. Second, some of th
the closed end funds are less marketable (i.e. have ˜thin™ markets). Third, agen
the form of management fees might also explain the discounts. However, Mal
found that the discounts were substantially in excess of what could be expla
above reasons, while Lee et a1 (1990) find that the discounts on closed end
primarily determined by the behaviour of stocks of small firms.
There is a further anomaly. This occurs because at the initial public offe
closed end fund shares, they incur underwriting costs and the shares in th
therefore priced at a premium over their true market value. The value of the
fund then generally moves to a discount within six months. The anomaly is
any investors purchase the initial public offering and thereby pay the underw
via the future capital loss. Why don™t investors just wait six months before
the mutual fund at the lower price?
8.1.4 The Value Line Enigma
The Value Line Investment Survey (VLIS) produces reports on public traded
ranks these stocks in terms of their ˜timeliness™, by which it means the des
purchasing them. In Figure 8.2 is shown the excess return of ˜rank 1™ stocks ov
stocks and of ˜rank 5™ stocks over ˜rank 3™ stocks. ˜Rank 3™ stocks are designa
that are expected to increase in line with the market, while rank 1 stocks are a
and rank 5 stocks a ˜bad buy™, in terms of their expected future returns. Cle
stocks earn a higher return than rank 3 stocks. This could be due to the fact
stocks have higher risk (reflected in higher betas) than do rank 3 stocks. Also
a trading strategy to be profitable account must be taken of transactions costs
(1981) found that even after adjustments for risk and transactions costs, a pass
of purchasing rank 1 stocks at the beginning of the year and selling them at
the year outperformed the passive strategy using rank 3 stocks. The Value Li
System therefore does provide profitable information for a buy and hold strate
is inconsistent with the EMH.
\
20 Less \I
c
Rank 3
1 1 I
1
I
Figure 8.2 Annual Excess Return on Stocks, Classified by Value Link Rank, 19
Source: Fortune (1991)™ Fig. 4, p. 24. Reproduced by permission of The Federal Rese
Boston
8.1.5 Winner™s Curse
There exists a strong negative serial correlation for stock returns for those
have experienced extreme price movements (particularly those which experie
fall followed by ˜a price rise). Thus for some stocks (or portfolios of stocks) the
reversion in stock price behaviour. Put another way there is some predictabili
returns. The issue for the EMH is then whether such predictability can lead to su
profits net of transactions costs and risk.
De Bondt and Thaler (1985) take 35 of the most extreme ˜winners™ and
extreme ˜losers™ over the five years from January 1928 to December 1932
monthly return data from the NYSE) and form two distinct portfolios of these c
shares. They follow these companies for the next five years (= ˜test period™). T
the exercise 46 times by advancing the start date by one year each time. Fi
calculate the average ˜test period™ performance (in excess of the return on the wh
index) giving equal weight (rather than value weights) to each of the 35 compa
find (Figure 8.3):
(i) The fiveyear price reversals for the ˜loser portfolio™ (at about plus 30 p
more pronounced than for the ˜winner portfolio™ (at minus 10 percent).
(ii) The excess returns on the ˜loser portfolio™ occur in January (i.e. ˜Janua
(iii) The returns on the portfolios are mean reverting (i.e. a price fall is fol
price rise and vice versa).
It is worth emphasising that the socalled ˜loser portfolio™ (i.e. one where p
fallen dramatically in the past) is in fact the one that makes high returns in
a somewhat paradoxical definition of ˜loser™. An arbitrage strategy of selling t