ñòð. 15 
: :
VR and from a sample of the generated data of 720 observations (i.e. the sam
as that for which they have historic data on returns). They repeat the calculati
times to obtain the frequency distributions of the statistics p ˜ VR and B. The
,
all three test statistics have little power to distinguish the random walk mode
above alternative of a highly persistent yet transitory price component.
A u toregressive Moving Average (ARMA) Representations
If weak form efficiency doesnâ€™t hold then actual returns R,+l might not only de
past returns but could also depend on past forecast errors
The simplest representation is the ARMA( 1,l)model:
The autoregressive element is the independent variable R, and the â€˜totalâ€™
(denoted U,) consists of a linear combination of white noise errors:
The error term uf is a moving average error (of order 1) and is serially cor
Ut
see the latter, note that
+ m ,+
= E ( E ˜ + ˜e = 8a2
COV(U,,
since E ( E r  s E f  m ) = 0 for m # s. Hence cov(u,, url) # 0 and u, is serially co
order 1).Past forecast errors are in the agentsâ€™ information set and hence even
weak form efficiency is still violated if 8 # 0. By including p lags of Rt and
one can estimate a general ARMA(p, q) model for returns which may be rep
where y(L) is a polynomial in the lag operator such that
+ y1L + y2L2 + y3L3 + . . . + ypLP
y(L) = 1
provide fairly general and flexible representations of the behaviour of any
variable and tests are available to choose the â€˜bestâ€™ values of p and q given th
data on R (although this usually involves judgement as well as formal statisti
Regressions based on ARMA models are often used to test the informationa
assumption of the EMH. In fact Poterba and Summers attempt to fit an A
model to their generated data on stock returns which, of course, should fit t
construction. However, in their estimated model (equation 6.18) they find y1
8 = 1 and because y1 and 8 are â€˜close toâ€™ each other, the estimation package
not â€˜separate outâ€™ (identify) and successfully estimate statistically distinct va
and 8. When Poterba and Summers do succeed in obtaining estimates of y1
less than 10 percent of the regressions have parameters that are close to the (k
values. This is another example of an estimated model failing to mimic the tru
a finite sample.
Poterba and Summers are aware that their results on mean reversion are su
problems of inference in small samples and that any element of predictability c
taken as evidence against the EMH, if the chosen equilibrium returns model
model. Cecchetti et a1 (1990) take up the last point and question whether the
Poterba and Summers (1988) and Fama and French (1988a) that stock prices
reverting should be interpreted in terms of the presence of noise traders. The
serial correlation of returns does not in itself imply a violation of efficiency. Fo
in the consumption CAPM if agents smooth their consumption, then stock
mean reverting. Cecchetti et a1 go on to demonstrate that empirical finding
reversion are consistent with data that could have been generated by an equilibri
They take a specific parameterisation of the consumption CAPM as their rep
equilibrium model and use Monte Carlo methods to generate artificial data
then subject the artificial data sets to the variance ratio tests of Poterba and Su
the long horizon return regressions of Fama and French. They find that measur
reversion in stock prices calculated from historic returns data nearly always l
60 percent confidence interval of the median of the Monte Carlo distributions
the equilibrium consumption CAPM.
Like all Monte Carlo studies the results are specific to the parameters
the equilibrium model. Cecchetti et a1 note that in the Lucas (1978) equilibri
consumption equals output which equals dividends, and their Monte Carlo st
tigates all three alternative â€˜fundamental variablesâ€™. Taking dividends as an ex
Monte Carlo simulations assume
+ (a0+ a&1) +
h D , = lnD,1 E,
The term S , is a Markov switching variable which has transition probabilities
Pr(S, = OJS,l = 1) = 1  p ,
Pr(S, = llS,,) = p ,
Pr(S, = lIS,l = 0) = 1  q
Pr(S, = OIS,1 = 0) = q,
Since a is restricted to be negative then S, = 0 is a â€˜high growthâ€™ state E , A ln
1
+
and S , = 1 is a low growth state E , A In D,+l = a 0 a (with a < 0). Ther
1
1
With preferences given by a constant coefficient of relative risk aversion (
+
of utility function U ( D ) = (1 y)'D(lY) with oo < y < 0, the solution to
equation is:
i=l
Simplifying somewhat the artificial series for dividends when used in the abov
(with representative values for 6 and y ) gives the generated series for pri
obey the parameterised consumption CAPM general equilibrium model. Gen
+ 
on returns Rt+l = [(Pt+1 Dt+l)/Dt] 1 are then used to calculate thc M
distributions for the PoterbaSummers variance ratio statistic and the FamaF
horizon return regressions.
Essentially, the Cecchetti et a1 results demonstrate that with the available
observations of historic data on US stock returns, one cannot have great faith
ical results based on say returns over a 10year horizon, since there are
10nonoverlapping observations. The historic data is therefore too short to ma
between an equilibrium model and a 'fads' model, based purely on an empiric
of historic returns data. Note, however, that the Cecchetti et a1 analysis does n
the possibility that variance bounds tests (see below) or other direct tests of th
Chapter 16) may provide greater discriminatory power between alternative
course, there are also a number of ancillary assumptions in the Cecchetti et
terisation of the consumption CAPM model which some may contest  an i
applies to all Monte Carlo studies. However, the Cecchetti et a1 results do m
more circumspect in interpreting weakform tests of efficiency (which use on
lagged returns), as signalling the presence of noise traders.
6.1.3 Multivariate Tests
The Fama and French and Poterba and Summers results are univariate tests. H
number of variables other than past returns have also been found to help pred
returns. For example, Keim and Stambaugh (1986) using monthly excess retu
common stocks (over the Treasury bill rate) for the period from about 19
find that for a number of portfolios (based on size) the following (somewha
variables are usually statistically significant:
(i) the difference in the yield between lowgrade corporate bonds and th
onemonth Treasury bills,
(ii) the deviation of last periods (real) S&P index from its average over
years,
(iii) the level of the stock price index based only on 'small stocks'.
stocks, the regressions only explain about 0.62.0 percent of the actual exc
(These results are broadly similar to those found by Chen et a1 (1986), see sec
testing the APT.)
Fama and French (1988b) extend their earlier univariate study on the predi
expected returns over different horizons and examine the relationship betwee
and real) returns and the dividend yield DIP.
The equation is run for monthly and quarterly returns and for annual returns of
years on the NYSE index. They also test the robustness of the equation by runn
various subperiods. For monthly and quarterly data the dividend yield is often
significant (and /3 > 0) but only explains about 5 percent of the variability
and quarterly actual returns. For longer horizons the explanatory power inc
example, for nominal returns over the 19411986 period the explanatory po
2, 3, 4year return horizons are 12, 17, 29 and 49 percent. The longer retu
regressions are also useful in forecasting â€˜outofsampleâ€™.
The difficulty in assessing these results is that they are usually based on t
model possible for equilibrium returns, namely that expected real returns ar
The EMH implies that abnormal returns are unpredictable, not that actual
unpredictable. These studies reject the latter but as they do not incorporate a
sophisticated model of equilibrium returns we do not know if the EMH would
in a more general model. For example, the finding that /3 > 0 in (6.19) implies
current prices increase relative to dividends (i.e. ( D / P ) t falls) then returns
the future price tends to fall. This is an example of mean reversion in prices
be viewed as a mere statistical correlation where (DIP) may be a proxy for
equilibrium returns.
To take another example if, as in Keim and Stambaugh, an increase in
on lowgrade bonds reflects an increase in investorsâ€™ general perception of
then we would under the CAPM or APT expect a change in equilibrium
returns. Here predictability could conceivably be consistent with the EMH sinc
no abnormal returns. Nevertheless the empirical regularities found by many
provide us with some useful stylised facts about variables which might influe
in a statistical sense.
When looking at regression equations that attempt to explain returns, an e
cian would be interested in general diagnostic tests (e.g. are the residuals norm
uncorrelated, nonheteroscedastic and the RHS variables weakly exogenous),
sample forecasting performance of the equations and the temporal stability of
eters. In many of the above studies this useful statistical information is not a
presented so it becomes difficult to ascertain whether the results are as â€˜robu
seem. However, Pesaran and Timmermann (1992) provide a study of stock ret
attempts to meet the above criticisms of earlier work. First, as in previous st
run regressions of the excess return on variables known at time t or earlier.
however, very careful about the dating of the information set. For example, in
over one year, one quarter and one month for the period 19541971 and subp
annual excess returns a small set of independent variables including the divi
annual inflation, the change in the threemonth interest rate and the term premiu
about 60 percent of the variability in the excess return. For quarterly and mo
broadly similar variables explain about 20 percent and 10 percent of exce
respectively. Interestingly, for monthly and quarterly regressions they find a
effect of previous excess returns on current returns. For example, squared prev
returns are often statistically significant while past positive returns have a diffe
than past negative returns, on future returns. The authors also provide diagnos
serial correlation, heteroscedasticity, normality and â€˜correctâ€™ functional form
test statistics indicate no misspecification in the equations.
To test the predictive power of these equations they use recursive estimation
+
predict the sign of next periods excess return (i.e. at t 1) based on estimated
which only use data up to period t. For annual returns, 7080 percent of th
returns have the correct sign, while for quarterly excess returns the regression
a (healthy) 65 percent correct prediction of the sign of returns (see Figures 6.
Thus Pesaran and Timmermann (1994) reinforce the earlier results that excess
predictable and can be explained quite well by a relatively small number of i
variables.
Profitable Trading Strategies
Transactions costs in stock and bond trades arise from the bidask spread (i.e.
stock at a low price and sell to the investor at a high price) and the commissi
on a particular â€˜buyâ€™ or â€˜sellâ€™ order given to the broker. Pesaran and Timme
â€˜closing pricesâ€™ which may be either â€˜bidâ€™ or â€˜askâ€™ prices. They therefore assu
Figure 6.7 Actual and Recursive Predictions of Annual Excess Returns (SP 500). Sou
and Timmermann (1994). Reproduced by permission of John Wiley and Sons Ltd
I
1,
.26890[ 198302 199004
196001 196704 197503
..
 Predictions
Acutal a I 0.
Figure 6.8 Actual and Recursive Predictions of Quarterly Excess Returns (SP 5
Pesaran and Timmermann (1994). Reproduced by permission of John Wiley and Sons
trading costs are adequately represented by a fixed transactions cost per $ of
assume costs are higher for stocks c, than for bonds c b . They consider a sim
rule, namely:
If the predicted excess return (from the recursive regression) is positive then hold th
portfolio of stocks, otherwise hold government bonds with a maturity equal to the
the trading horizon (i.e. annual, quarterly, monthly).
The above â€˜switching strategyâ€™ has no problems of potential bankruptcy sinc
not sold short and there is no gearing (borrowing). The passive benchmark stra
of holding the market portfolio at all times. They assess the profitability of th
strategy over the passive strategy for transactions costs that are â€˜lowâ€™, â€˜medium
(The values of c, are 0, 0.5 and 1.0 percent for stocks and for bonds cb equ
0.1 percent.)
In general terms they find that the returns from the switching strategy are
those for the passive strategy for annual returns (i.e. switching once per year in Jan
when transactions costs are â€˜highâ€™ (see Table 6.1). However, it pays to trade at q
monthly intervals only if transactions costs are less than 1/2 percent for stocks.
they find that the standard deviation of returns for the switching portfolio for
Table 6.1) and quarterly but not monthly returns is below that for the passive por
under high transactions cost scenario). Hence the â€˜annualâ€™ switching portfolio
the passive portfolio on the meanvariance criterion (for the whole data period 1
The above results are found to be robust with respect to different sets of regre
excess return equations and over subperiods 1960 1970, 1970 1980 and 198
Table 6.1 are reported the Sharpe, Treynor and Jensen indices of meanvariance
for the switching and passive portfolios for the oneyear horizon. For any po
these are given by:
s = (ERP  r)/cTp
T = (ERP  r ) / &
+ s(Rm  r),
(RP  r)t = J
Transaction Cost (%)
1 .o
Stocks 1.0
0.5 0.0
0.0 0.5
  0.0

Treasury Bills 0.1 0.1 0
Arithmetic Mean
Return (%) 10.78 10.72 10.67 12.70 12.43 12.21 6
SD of Return (%) 13.09 13.09 13.09 7.24 7.20 7.16 2
0.76
0.30 0.82
Sharpeâ€™s Index 0.31 0.30 0.79
0.081
Treynorâ€™s Index 0.040 0.039 0.085
0.040 0.089
 0.041

Jensenâ€™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 â€˜marketâ€™ portfolio denotes a buy and hold strategy in the S&P 500 ind
billsâ€™ denotes a rollover strategy in 12month 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 giltequity
(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 â€˜holdâ€™ an unchanged position.
A simple model which encapsulates the above is:
In static equilibrium the longrun GEYR is a˜//? Clare et a1 (using qua
and
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
1969(1)1992(2)
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 threemonth Treasury bill rate (otherwise hold Treasury bills). This stra
the above estimated equation, gives a higher expost 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 weakform 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
conclusions.
Returns and the Consumption CAPM
The CCAPM model of equilibrium returns discussed in Chapter 4 can be te
using crosssection or time series data. A crosssection test of the CCAPM
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 crosssection regression should then yield a statistically significan
a1 (and ao). In Chapter 3 we noted that the basic CAPM can be similarly te
the crosssection SML, where pci is 'replaced' by fimi = cov(Ri, Rm)/var(Rm
and Shapiro (1986) test these two versions of the CAPM using crosssection d
companies (listed on the NYSE) and sample values (e.g. for Ri, cov(Rm,gC)
from quarterly data over the period 19591982. They find that the basic CAP
outperforms the CCAPM, since when Ei is regressed on both B m i and pci the
statistically significant while the latter is not.
A test of the CCAPM 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
nationalâ€™ 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 â€˜commonâ€™ 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 CCAPM represented by (6.22
worse, in statistical terms, as the time horizon is extended. Hence they find a
version of) the CCAPM and their results are consistent with those of Fama a
Overall, the CCAPM does not appear to perform well in empirical tests.
VOLATILITY TESTS
6.2
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 â€˜yardstickâ€™ against which one ca
volatilities.
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 â€˜newsâ€™ 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 18801988. Even for the
of a constant value of (k  g)â€™ = 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 preSecond 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 oneyear 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
ñòð. 15 