Description
In recent years a variety of models which apparently forecast changes in stock market prices have been introduced. Some of these are summarised and interpreted.
International Journal of Forecasting 8 (1992) 3-13
North-Holland
Forecasting stock market prices: Lessons
for forecasters *
Clive W.J. G-anger
University of California, Sun Diego, USA
Abstract: In recent years a variety of models which apparently forecast changes in stock market prices
have been introduced. Some of these are summarised and interpreted. Nonlinear models are particularly
discussed, with a switching regime, from forecastable to non-forecastable, the switch depending on
volatility levels, relative earnings/price ratios, size of company, and calendar effects. There appear to be
benefits from disaggregation and for searching for new causal variables. The possible lessons for
forecasters are emphasised and the relevance for the Efficient Market Hypothesis is discussed.
Keywords: Forecastability, Stock returns, Non-linear models, Efficient markets.
1. Introduction: Random walk theory
For reasons that are probably obvious, stock
market prices have been the most analysed eco-
nomic data during the past forty years or so. The
basic question most asked is - are (real) price
changes forecastable? A negative reply leads to
the random walk hypothesis for these prices,
which currently would be stated as:
H,,: Stock prices are a martingale.
i.e. E[ P,+, I I,] = P,,
where Z, is any information set which includes
the prices P, _ j, j 2 0. In a sense this hypothesis
has to be true. If it were not, and ignoring trans-
action costs then price changes would be consis-
tently forecastable and so a money machine is
created and indefinite wealth is possible. How-
Correspondence to: C.W.J. Granger, Economics Dept., 0508,
Univ. of California, San Diego, La Jolla, California, USA
92093-0508.
* Invited lecture, International Institute of Forecasters, New
York Meeting, July 1991, work partly supported by NSF
Grant SES 89-02950. I would like to thank two anonymous
referees for very helpful remarks.
ever, a deeper theory - known as the Efficient
Market Hypothesis - suggests that mere fore-
castability is not enough. There are various forms
of this hypothesis but the one I prefer is that
given by Jensen (1978):
HC,2: A market is efficient with respect to infor-
mation set 1, if it is impossible to make
economic profits by trading on the basis of
this information set.
By ‘economic profits’ is meant the risk-adjusted
returns ‘net of all costs’. An obvious difficulty
with this hypothesis is that is is unclear how to
measure risk or to know what transaction costs
are faced by investors, or if these quantities are
the same for all investors. Any publically avail-
able method of consistently making positive prof-
its is assumed to be in I,.
This paper will concentrate on the martingale
hypothesis, and thus will mainly consider the
forecastability of price changes, or returns (de-
fined as (P, - P,_ , +D,)/P, _ 1 where D, is divi-
dends), but at the end I will give some considera-
tion to the efficient market theory. A good survey
of this hypothesis is LeRoy (1989).
By the beginning of the seventies I think that it
was generally accepted by forecasters and re-
0169.2070/92/$05.00 0 1992 - Elsevier Science Publishers B.V. All rights reserved
4 C. W.J. Crunger / Forecasting stock market prices
searchers in finance that the random walk hy-
pothesis (or H,,,) was correct, or at least very
difficult to refute. In a survey in 1972 I wrote,
‘Almost without exception empirical studies.. . ’
support a model for p, = log f’, of the form
dP,+, =~AP,+ I ,-, +~t+l,
where 0 is near zero, 1, contributes only to the
very low frequencies and E, is zero mean white
noise. A survey by Fama (1970) reached a similar
conclusion. The information sets used were:
I,,: lagged prices or lags of logged prices.
ZZt: Ilr plus a few sensible possible explanatory
variables such as earnings and dividends.
The data periods were usually daily or monthly.
Further, no profitable trading rules were found,
or at least not reported. I suggested a possible
reporting bias - if a method of forecasting was
found an academic might prefer to profit from it
rather than publish. In fact, by this period I
thought that the only sure way of making money
from the stock market was to write a book about
it. I tried this with Granger and Morgenstern
(1970), but this was not a financially successful
strategy.
However, from the mid-seventies and particu-
larly in the 1980s there has been a burst of new
activity looking for forecastability, using new
methods, data sets, longer series, different time
periods and new explanatory variables. What is
interesting is that apparent forecastability is often
found. An important reference is Guimaraes,
Kingsman and Taylor (1989). The objective of this
part is to survey some of this work and to suggest
lessons for forecasters working on other series.
The notation used is:
p,
= a stock price,
PI
= log P,,
D,
= dividend for period t,
Rr
= return = (P, + D, - P,_,)/P,_,,
[In some studies the return is calcu-
lated without the dividend term and
approximated by the change in log
prices.]
rr
= return on a ‘risk free’ investment,
R, -rt
= excess return,
P
= risk level of the stock,
R, - r, - /3 X market return - cost of transac-
tion = risk-adjusted profits.
The risk is usually measured from the capital
asset pricing model (CAPM):
R, - r, = p (market excess return) + e,,
where the market return is for some measure of
the whole market, such as the Standard and
Poor’s 500. p is the non-diversifiable risk for the
stock. This is a good, but not necessarily ideal,
measure of risk and which can be time-varying
although this is not often considered in the stud-
ies discussed below.
Section 2 reviews forecasting models which can
be classified as ‘regime-switching’. Section 3 looks
at the advantages of disaggregation, Section 4
considers the search for causal variables, Section
5 looks at technical trading rules, Section 6 re-
views cointegration and chaos, and Section 7 looks
at higher moments. Section 8 concludes and re-
considers the Efficient Market Theory.
2. Regime-switching models
If a stationary series X, is generated by:
X, = (Y, + ylxI_, + E, if z, in A
and
X, = (Ye + y*x,_, + Ed if z, not in A,
then x, can be considered to be regime switching,
with z, being the indicator variable. If Z~ is a
lagged value of x, one has the switching thres-
hold autoregressive model (STAR) discussed in
detail in Tong (1990), but z, can be a separate
variable, as is the case in the following examples.
It is possible that the variance of the residual E,
also varies with regime. If x, is a return (or an
excess return) it is forecastable in at least one
regime if either y, or y2 is non-zero.
2.a. Forecastability with Low Volatility
LeBaron (1990) used R,, the weekly returns of
the Standard and Poor 500 index for the period
194661985, giving about 2,000 observations. He
used as the indicator variable a measure of the
recent volatility
10
&,,’ = c Rf_,
1=0
C. W.l. Granger / Forecasting stock market prices 5
and the regime of interest is the lowest one-fifth
quantile of the observed C? values in the first half
of the sample. The regime switching model was
estimated using the first half of the sample and
post-sample true one-step forecasts were evalu-
ated over the second half. For the low volatility
regime he finds a 3.1 percent improvement in
forecast mean squared error over a white noise
with non-zero mean (that is, an improvement
over a model in which price is taken to be a
random walk with drift). No improvement was
found for other volatility regimes. He first takes
cy (the constant) in the model to be constant
across regimes, relaxing this assumption did not
result in improved forecasts. Essentially the model
found is
R, =cr +0.18R,_1 + l t if have low volatility
R,=cu+~, otherwise,
where LY is a constant. This non-linear model was
initially found to fit equally well in and out of
sample. However, more recent work by LeBaron
did not find much forecasting ability for the
model.
2. b. Earnings and size portfolios
Using the stocks of all companies quoted on
either the New York or American Stock Ex-
changes for the period 1951 to 1986, Keim (1989)
formed portfolios based on the market value of
equity (size) and the ratio of earnings to price
(E/P) and then calculated monthly returns (in
percentages). Each March 31”’ all stocks were
ranked on the total market value of the equity
(price x number of shares) and ten percent with
the lowest ranks put into the first (or smallest)
portfolio, the next 10% in the second portfolio
and so forth up to the shares in the top 10%
ranked giving the ‘largest’ portfolio. The portfo-
lios were changed annually and average monthly
returns calculated. Similarly, the portfolios were
formed from the highest E/P values to the lowest
(positive) values. [Shares of companies with nega-
tive earnings went into a separate portfolio.] The
table shows the average monthly returns (mean)
for five of the portfolios in each case, together
with the corresponding standard errors:
Size
E/P
Mean (s.d.) Mean (s.d.)
smallest 1.79 (0.32) highest 1.59 (0.25)
2nd 1.53 (0.28) 2”d 1.59 (0.22)
S’h 1.25 (0.24) gLh 1.17 (0.22)
9th 1.03 (0.21) gth 1.11 (0.25)
largest 0.99 (0.20) lowest 1.19 (0.28)
negative
earnings (1.39) (0.39)
Source: Keim (1989).
It is seen that the smallest (in size) portfolios
have a substantially higher average return than
the largest and similarly the highest E/P portfo-
lios are better than the lowest.
The two effects were then combined to gener-
ate 25 portfolios, five were based on size and
each of these was then sub-divided into five parts
on E/P values. A few of the results are given in
the following table as average monthly returns
with beta risk values shown in brackets.
Size E/P ratio
Lowest
smallest 1.62
(1.27)
middle 1.12
(1.28)
largest 0.89
(1.11)
Source: Keim (1989).
Middle
1.52
(1.09)
1.09
(1.02)
0.97
(0.98)
Highest
1.90
(1.09)
1.52
(1.06)
1.43
(1.03)
The portfolio with the highest E/P ratio and
the smallest size has both a high average return
and a beta value only slightly above that of a
randomly selected portfolio (which should have a
beta of 1.0). The result was found to hold for
both non-January months and for January, al-
though returns in January were much higher, as
will be discussed in the next section. Somewhat
similar results have been found for stocks on
other, non-U.S. exchanges. It should be noted
that as portfolios are changed each year, transac-
tion costs will be moderately large.
The results are consistent with a regime-
switching model with the regime determined by
the size and E/P variables at the start of the
year. However, as rankings are used, these vari-
ables for a single stock are related to the actual
values of the variables for all other stocks.
6 C. W.J. Changer / Forecasting stock market prices
2.c. Seasonal effects
A number of seasonal effects have been sug-
gested but the strongest and most widely docu-
mented is the January effect. For example Keim
(1989) found that the portfolio using highest E/P
values and the smallest size gave an average
return of 7.46 (standard error 1.41) over Januarys
but only 1.39 (0.27) in other months. A second
example is the observation that the small capital-
ization companies (bottom 20% of companies
ranked by market value of equity) out-performed
the S&P index by 5.5 percent in January for the
years 1926 to 1986. These small firms earned
inferior returns in only seven out of the 61 years.
Other examples are given in Ikenberry and
Lakonishok (1989). Beta coefficients are also gen-
erally high in January.
The evidence suggests that the mean of re-
turns have regime changes with an indicator vari-
able which takes a value of unity in January and
zero in other months.
2.d. Price reversals
A number of studies have found that shares
that do relatively poorly over one period are
inclined to perform well over a subsequent pe-
riod, thus giving price change reversals. A survey
is provided by DeBondt (1989). For example, Dyl
and Maxfield (1987) selected 200 trading days in
random in the period January 1974 to January
1984, each day the three NYSE or AMEX stocks
with the greatest percentage price loss (on aver-
age - 12%) were noted. Over the next ten trad-
ing days, these losers earn a risk-adjusted return
of 3.6 percent. Similarly the three highest gainers
lost an average 1.8% over the next ten days.
Other studies find similar evidence for daily,
weekly and even monthly returns. Transaction
costs will be fairly heavy and a strategy based on
these results will probably be risky.
However, Lehman (1990) considered a portfo-
lio whose weights depended on the return of a
security the previous week minus the overall re-
turn, with positive weights on previous losers and
negative weights (going short) on previous win-
ners. The portfolio was found to consistently pro-
duce positive profits over the next week, with very
few losing periods and so with small risk. Trans-
action costs were substantial but worthwhile prof-
its were achieved for transaction costs at a level
appropriate for larger traders. Thus, after allow-
ing for risk and costs, a portfolio based on price
reversal was found to be clearly profitable.
Long term price reversals have also been docu-
mented. For example, Dark and Kato (1986)
found in the Japanese market that for the years
1964 to 1980, the three year returns for decile
portfolios of extreme previous losers exceed the
comparable returns of extreme previous winners
by an average 70 percent.
In this case the indicator variable is the ex-
treme relative loss value of the share. As before
the apparent forecastability leads to a simple
investment strategy, but knowledge is required of
the value taken by some variable based on all
stocks in some market.
2.e. Remolsal of extreme values
It is well known that the stock markets occa-
sionally experience extraordinary movements, as
occurred in October 1987, for example. Friedman
and Laibson (1989) point out that these large
movements are of overpowering importance and
may obscure simple patterns in the data. They
consider the Standard and Poor 500 quarterly
excess returns (over treasury bills) for the period
19541 to 1988IV. After removal of just four ex-
treme values, chosen by using a Poisson model,
the remaining data fits an AR(l) model with
significant lag coefficient of 0.207 resulting in an
R2 value of 0.036. The two regimes are thus the
‘ordinary’ excess returns, which seem to be fore-
castable, and the extra-ordinary returns which are
not, from the lagged data at least.
3. Benefits of disaggregation
A great deal of the early work on stock market
prices used aggregates, such as the Dow Jones or
Standard and Poor indices, or portfolios of a
random selection of stocks or some small group
of individual stocks. The availability of fast com-
puters with plenty of memory and tapes with
daily data for all securities on the New York and
American Exchanges, for example, allows exami-
nation of all the securities and this can on occa-
sion be beneficial. The situation allows cross-sec-
tion regressions with time-varying coefficients
C. W.J. Grunger / Forecasting stock market prices I
which can possibly detect regularities that were
not previously available. For example Jegadeesh
(1990) uses monthly data to fit cross-section mod-
els of the form
12
for each month. Thus, a lagged average relation-
ship is considered with coefficients changing each
month. Here R,, is the average return over a long
(four or six years) period which exclude the previ-
ous three years. [In the initial analysis, R was
estimated over the following few years, but this
choice was dropped when forecasting properties
were considered.] Many of the averaged aj were
significantly different from zero, particularly at
lags one and twelve, but other average coeffi-
cients were also significant, including at lags 24
and 36. A few examples are shown, with t-values
in brackets.
a1 a,2
_
aI4
Rf
all months -0.09(18) 0.034(9) 0.019(6.5) 0.108
January -0.23 (9) 0.08 (5) 0.034C2.6) 0.178
Feb. to Dec. -0.08(17) 0.03 (8) 0.017(6) 0.102
Source: Jegadeesh (1990).
There is apparently some average, time-vary-
ing structure in the data, as seen by R: values of
10% or more. As noticed earlier, January has
more forecastability than other months and it was
found that a group of large firms had regressions
with higher Rf in February to December than all
firms using these regressions (without the R
terms), stocks were ranked each month on their
expected forecastability and ten portfolios formed
from the 10% most forecastable (P,), second
10% and so forth up to the 10% least fore-
castable (P,,,). The average abnormal monthly
returns (i.e. after risk removal) on the ‘best’ and
‘worse’ portfolios for different periods were
All months January Feb.-Dec.
PI
0.011 0.024 0.009
P
10
- 0.014 - 0.020 -0.017
Source: Jegadeesh (1990).
There is thus seen to be a substantial benefit
from using the best portfolio rather than the
worst one based on the regressions. Benefits were
also found, but less substantial ones, using twelve
month ahead forecasts. Once transaction costs
are taken into account the potential abnormal
returns from using P, are halved, but are still
around 0.45% per month (from personal commu-
nication by author of the original study).
4. Searching for causal variables
Most of the studies discussed so far have con-
sidered forecasting of prices from just previous
prices but it is also obviously sensible to search
for other variables that provide some forecastabil-
ity. The typical regression is
Ap, = constant + p’Kl _ , + E, ,
_
where & is a vector of plausible explanatory, or
causal variables, with a variety of lags considered.
For example Darrat (1990) considered a monthly
price index from the Toronto Stock Exchange for
the period January 1972 to February 1987 and
achieved a relationship:
Ap, = tsTA volatility of interest rates (t - 1)
- :;::A production index (t - 1)
+ yiE3fA long-term interest rate (t - 10)
- 0.015 A cyclically-adjusted budget
(3.0)
deficit (t - 3))
R2 = 0.46,
Durbin-Watson = 2.01,
(4.1)
where only significant terms are shown and the
modulus of t-values in brackets. Several other
variables were considered but not found to be
significant, including changes of short-term rates,
inflation rate, base money and the US-Canadian
exchange rate, all lagged once. An apparently
high significance R2 value is obtained but no
out-of-sample forecastability is investigated.
This search may be more successful if a long-
run forecastability is attempted. For example Ho-
drick (1990) used monthly US data for the period
1929 to December 1987 to form NYSE value-
weighted real market returns, Rr+k, over the
time span (t + 1, t + k). The regression
log R,tk,t
= (Ye + p,( dividend/price ratio at t )
x C. W.J. Granger / Forecasting stock market prices
found R2 increasing as k increases, up to R2 =
0.354 at k =48. Thus, apparent long-run fore-
castability has been found from a very simple
model. However, again no post-sample evaluation
is attempted.
Pesaran and Timmerman (1990) also employ
simple models that produce useful forecastability
and they also conduct a careful evaluation of the
model. As an example of the kind of model they
produce, the following equation has as its depen-
dent variable (Y,> the quarterly excess return on
the Standard and Poor 500 portfolio:
Y = - 0.097 + t_T7: dividend yield ( t - 2)
- 1.59 inflation rate (t - 3)
(2.8)
- ?$)I T-bill (end, t - 1)
+ 0.025 T-bill (begin, t - 2)
(4.6)
+ 0.066A twelve month bond state (t - 1)
(5.5)
+ residual,
Rf =0.364, Durbin-Watson = 2.02.
(4.2)
Here dividend yield at time t is
dividend on S&P index ( t - 1)
. price of S&P index (t)
T-bill is the one month interest rate ‘end’ means
it is measured at the end of the third month of
the quarter, ‘begin’ indicates that it is measured
at the end of the first month of the quarter. The
two T-bill terms in the equation are thus effec-
tively the change in the T-bill interest rate from
one month to the next, plus one at the end of the
quarter. As just lagged variables are involved and
a reasonable R: value is found, the model can
potentially be used for forecasting. [It might be
noted that Rz climbed to 0.6 or so for annual
data.] Some experimentation with non-linear
lagged dependent variables produced some in-
creases in R:, to about 0.39, but this more com-
plicated model was not further evaluated.
A simple switching portfolio trading rule was
considered:
(i) Buy the S&P 500 index if the excess return
was predicted to be positive according to
equation (4.2), with the equation being se-
quentially re-estimated. Thus only data avail-
able at the time of the forecast was used in
making the forecast.
(ii) If the predictor was negative, the invest in
T-bills.
The following table shows the rate of returns
achieved by either using a ‘buy-and-hold’ market
portfolio, or the switching portfolio obtained from
the above trading rule or by just buying T-bills.
As the switching rule involves occasional buying
and selling, possibly quarterly, two levels of trans-
action costs are considered 4% and 1%.
Investment strategy
Market Switching T-bill
Transaction
costs 0 4% 1% 0
Interest rate of
returns 9.5 1 13.30 12.39 6.34
Standard deviation
of returns 8.23 5.43 5.41 0.70
Wealth at end of
period ” 1394 3736 2961 595
” The period considered from 1960.1 to 1988.IV and the
wealth accumulates from an investment of $100 in Decem-
ber 1959.
Source: Pesaran and Timmerman (I 990).
Although the results presented are slightly bi-
ased against the switching portfolio zero transac-
tion costs are assumed for the alternative invest-
ments, the trading rule based on the regression is
seen to produce the greatest returns and as a
lower risk-level than the market (S&P 500) port-
folio. A variety of other evaluation methods and
other regressions are also presented in the paper.
It would seem that dividend/price ratios and
interest rates have quite good long-run forecast-
ing abilities for stock price index returns.
5. A new look at old techniques - Technical
trading rules
A strategy that is popular with actual specula-
tors, but is disparaged by academics, is to use an
automatic, or technical trading rule. An example
is to use perceived patterns in the data, such as
the famous ‘head and shoulders’, and to devise a
rule based on them. Much technical analysis is
difficult to evaluate, as the rules are not precise
enough. The early literature did consider various
simple rules but generally found little or no fore-
casting value in them. However, the availability of
C. W.J. Granger / Forecasting stock market prices 9
fast computers has allowed a new, more intensive
evaluation to occur, with rather different results.
Brock, Lakonishok and LeBaron (1991) con-
sider two technical rules, one comparing the most
recent value to a recent moving average, and the
other is a ‘trading range breakout’. Only the first
of these is discussed here.
The first trading rule is as follows:
Let M, = average of previous 50 prices, form
a band B, = (1 f 0.01) M,, so that the band
is plus and minus 1% around M,. If P,, the
current price, is above the band, this is a
buy signal, if it is below the band, this is a
sell signal.
Using 90 years of daily data for the Dow-Jones
Index (giving a sample of over twenty-three thou-
sand values) for the period 1897 to 1986, the rule
suggested buying 50% of the time giving an aver-
age return next day of 0.00062 (t = 3.7) and sell-
ing 42% of the time, giving an average return of
- 0.00032 (t = 3.6). The return on the rule ‘buy if
have buy signal and go short on a sell signal’ gave
an average daily return of 0.00094 (t = 5.4). The
first two t-values are for the return minus the
daily unconditional average return, the ‘buy-sell’
r-value is relative to zero. If this buy-sell strategy
was used 200 times a year, it gives a return of 20.7
percent for the year. However, this figure ignores
transaction costs, which could be substantial. The
trading rule was considered for four sub-periods
and performed similarly for the first three but
less well for the most recent sub-period of 1962-
1986, where the buy-sell strategy produced a daily
return of 0.00049. Other similar trading rules
were considered and gave comparable results.
Thus, this rule did beat a buy-and-hold strategy
by a significant amount if transaction costs are
not considered. The authors also consider a much
more conservative rule, with a fixed ten day hold-
ing period after a buy or sell signal. The above
rule then averages only 3; buy and sell signals a
year, giving an annual expected return of 8.5%
compared to an annual return for the Dow Index
of about 5%, again ignoring transaction costs.
These, and the results for the other trading rules
considered suggest that there may be regular but
subtle patterns in stock price data, which would
give useful forecastability. However, very long
series are needed to investigate these rules.
Neftci (1991) investigates a similar moving av-
erage trading rule using different statistical meth-
ods and an even longer period - monthly Dow-
Jones Industrial Index starting in 1792, up to
1976. Let M, be an equi-weighted moving aver-
age over the past five months. If P, is the price of
the index in month t, define a dummy variable:
D,= 1 if P,>M, given P,+, CM,_,
= -1 if P, <M, given P,-, >&I _,
= 0 otherwise.
Regression results are presented for the equation
P *+,#= 5 CI ~P~_~ + 5 Y~D,_~+residual,
j = 0 j = 0
where the residual is allowed to be a moving
average of order 17, for each of the three sub-
periods 1792-1851, 1852-1910 and 1910-1976. In
each case the sum of the alphas is near one, as
suggested by the efficient market theory and in
the more recent period the gammas were all
significant, individually and jointly, suggesting
some nonlinearity in the prices. No forecasting
exercise was considered using the models. The
use of data with such early dates as 1897 or 1792
is surely only of intellectual interest, because of
the dramatic institutional changes there have oc-
curred since then.
Neftci also proves, using the theory of optimal
forecasts, that technical trading rules can only be
helpful with forecasting if the price series are
inherently nonlinear.
6. New techniques - Cointegration and chaos
Since the early statistical work on stock prices,
up to 1975, say, a number of new and potentially
important statistical models and techniques have
been developed. Some arrive with a great flourish
and then vanish, such as catastrophe theory,
whereas others seem to have longer staying power.
I will here briefly consider two fairly new ap-
proaches which have not been successful, so far,
in predicting stock prices.
An Z(1) series is one such that it’s first differ-
ence is stationary. A pair, X,, Y,, of Z(1) series
are called cointegrated if there is a linear combi-
nation of them, Z, =X, - AY,, say, which is Z(0).
10 C. W.J. Granger / Forecasting stock market prices
The properties and implications of such series are
described in Granger (19861, Engle and Granger
(19911, and many other publications in economet-
rics, macroeconomics and finance. If the series
are cointegrated there is necessarily an error-cor-
rection data generating model of the form (ignor-
ing constants):
,4X, = (Y,Z, _, + lagged AX,, AY, terms
+ white noise,
plus a similar equation for Ax, with at least one
of N.~, tiy,. being non-zero. It follows that either X,
must help forecast Y,+ , or Y, must help forecast
X ,+ , or both. Thus, if dividends and stock prices
are found to be cointegrated, as theory suggests,
then prices might help forecast dividends, which
would not be surprising, but dividends not help
forecast prices, in agreement with the efficient
market hypothesis. However, for the same reason
one would not expect a pair of stock prices to be
cointegrated, as this would contradict the effi-
cient market hypothesis. In fact several papers
have been produced that claim to find cointegra-
tion between pairs of prices or of portfolios, but
the error-correction models are not presented or
the forecasting possibility explored, and so this
work will not be surveyed. It should be noted that
cointegration would be inconsistent with the
well-respected capital asset pricing model
(CAPM) which says that the price P,, of the ith
asset is related to the price of the whole market
P,,,, bY
3 log P,, = b,d log P,, + E;, ,
where err is white noise. Summing over time gives
log Pi, = bi log P,,cr + i e,,t_j.
j=O
As the last term is the accumulation of a station-
ary series, it is 1(l) (ignoring trends) and so
cointegration should not occur between log P,,
and log P,,,. Similarly, there should be no cointe-
gration between portfolios, Gonzalo (1991) has
found no cointegration between three well known
aggregates, the Dow-Jones Index, the Standard
and Poor 500 Index and the New York Stock
Exchange Equal Value Index.
A class of processes generated by particular
deterministic maps, such as
y,=4y,-,(I -Y,-,)
with
0 < Yo < 1 >
have developed a great deal of interest and can
be called ‘white chaos’. These series have the
physical appearance of a stochastic independent
(i.i.d.1 process and also the linear properties of a
white noise such as zero autocorrelations and a
flat spectrum. The question naturally arises of
whether the series we have been viewing as
stochastic white noise are actually white chaos
and are thus actually perfectly forecastable - at
least in the short run and provided the actual
generating mechanism is known exactly. The lit-
erature on chaos is now immense, involves excit-
ing and deep mathematics and truly beautiful
diagrams, and also is generally optimistic, sug-
gesting that these processes occur frequently. In
fact, a clear case can be made that they do not
occur in the real world, as opposed to in labora-
tory physics experiments. There is no statistical
test that has chaos as the null hypothesis. There
also appears to be no characterizing property of a
chaotic process, that is a property that is true for
chaos but not for any completely stochastic pro-
cess. These arguments are discussed in Liu,
Granger and Heller (1991). It is true that some
high-dimensional white chaotic processes are in-
distinguishable from iid series, but this does not
mean that chaos occurs in practice. In the above
paper, a number of estimates of a statistic known
as the correlation dimension are made for various
parameter values using over three thousand
Standard and Poor 500 daily returns. The result-
ing values are consistent with stochastic white
noise (or high dimensional chaos) rather than low
dimensional - and thus potentially forecastable -
white chaos. A little introspection also make it
seem unlikely to most economists that a stock
market, which is complex, involving many thou-
sands of speculations, could obey a simple deter-
ministic model.
7. Higher moments
To make a profit, it is necessary to be able to
forecast the mean of price changes, and the stud-
ies reviewed above all attempt to do this. The
efficient market theory says little about the fore-
C. W.J. Granger / Forecasting stock market prices
11
castability of functions of price changes or re-
turns, such as higher moments. If R, is a return it
has been found that Rf and is clearly fore-
castable and I R, I even more so, from lagged
values. Taylor (1986) finds evidence for this using
U.S. share prices and Kariya, Tsukuda and Maru
(1990) get similar results for Japanese stocks. For
example, if R, is the daily return from the U.S.
Standard and Poor index the autocorrelations for
R, are generally very small, the autocorrelations
for Rf are consistently above 0.1 up to lag 100
and for I R, I are about 0.35 up to lag 100. It is
clear that these functions of returns are very
forecastable, but this is not easily converted into
profits, although there are implications for the
efficiency of options markets. The results are
consistent with certain integrated GARCH mod-
els but this work is still being conducted and the
final conclusions have yet to be reached.
8. Lessons for forecasters
Despite stock returns once having been thought
to be unforecastable, there is now plenty of opti-
mism that this is not so, as the examples given
above show. Is this optimism justified, and if yes,
what are the lessons for forecasters working with
other data sets? As there is an obvious possible
profit motive driving research into the forecasta-
bility of stock prices, or at least returns, one can
expect more intensive analysis here than else-
where. Whereas too many forecasters seem to be
content with just using easily available data, with
the univariate or simple transfer function fore-
casting techniques that are found on popular
computer packages, stock market research is more
ambitious and wide-ranging. It should be empha-
sized that the above is not a complete survey of
all of the available literature.
The sections above suggest that benefits can
arise from taking a longer horizon, from using
disaggregated data, from carefully removing out-
hers or exceptional events, and especially from
considering non-linear models. Many of the latter
can be classified as belonging to a regime switch-
ing model of the form
were Q(w) is a smooth monotonic function such
as a cumulative distribution function of a contin-
uous random variable so that 0 I @ I 1, _X, is a
vector of explanatory variables possibly including
lagged returns and w, is the ‘switching variable’,
possibly a lagged component of & or some linear
combination of these components. It has always
been important to discover appropriate explana-
tory variables &, and with this new class of
models it is especially important to find the ap-
propriate switching variable, if it exists and is
observable. This class of models is discussed in
Granger and Terasvirta (1992), where tests and
estimation procedures are outlined.
The papers also suggest that some sub-periods
may be more forecastable than others - such as
summer months or January - and this is worth
exploring. If many component series are avail-
able, then ranks may produce further information
that is helpful with forecasting. There seems to
be many opportunities for forecasters, many of
whom need to break away from simple linear
univariate ARIMA or multivariate transfer func-
tions. It is often not easy to beat convincingly
these simple methods, so they make excellent
base-line models, but they often can be beaten.
Before the results discussed in previous sec-
tions are accepted the question of how they should
be evaluated has to be considered. Many of the
studies in this, and other forecasting areas, are of
the ‘if only I had know this at the beginning of
the period I could have made some money’ classi-
fication. For a ‘forecasting model’ to be accepted
it has to show that it actually forecasts, it is not
sufficient to produce a regression model evalu-
ated only in sample. There is always the possibil-
ity of small-sample in-sample biases of coeffi-
cients which give overly encouraging results, as
shown by Nelson and Kim (1990). The possibility
of ‘data mining’ having occurred, with many mod-
els having been considered, and just the best one
presented is also a worry. Only out-of-sample
evaluation is relevant and, to some extent, avoids
these difficulties. It is surprising that more of the
studies surveyed do not provide results of fore-
casting exercises.
Not only do the models that are proposed as
providing useful forecasts of price changes or
returns need to be evaluated, to provide prof-
itable strategies the forecast returns need to be
corrected for risk levels and also for transaction
costs. Many of the studies discussed earlier fail to
12 C. W.J . Granger / Forecasting stock murket prices
do this, and so, at present, say nothing about the
correctness of the efficient market hypothesis
(EMH). However, this criticism does not always
apply, for example for the carefully conducted
analysis by Pesaran and Timmerman (1990). Does
this mean that the EMH should be rejected? One
has to say - not necessarily, yet. If a method
exists that consistently produces positive profits
after allowing for risk correction and transaction
costs and if this method has been publicly an-
nounced for some time, then this would possibly
be evidence against EMH. There are so many
possibly relevant trading rules that it is unrealistic
to suppose that investors have tried them all,
especially those that have only been discovered
by expensive computation and sophisticated sta-
tistical techniques. Once knowledge of an appar-
ently trading rule becomes wide enough, one
would expect behaviour of speculators to remove
its profitability, unless there exists another trad-
ing rule the speculators think is superior and thus
concentrate on it. Only if a profitable rule is
found to be widely known and remains profitable
for an extended period can the efficient market
hypothesis be rejected. It will be worthwhile
checking in a few years on the continued prof-
itability of the rules discussed earlier. This re-
search program agrees with the modern taste in
the philosophy of science to try to falsify theories
rather than to try to verify them. Clearly verifica-
tion of EMH is impossible.
References
Brock, W., J. Lakonishok and B. LeBaron. 1991, “Simple
technical trading rules and the stochastic properties of
stock returns, Working paper 9022, Social Science Re-
search Institute, Unrversity of Wisconsin, Madison.
Dark. F.H. and K. Kate, 1986, “Stock market over-reaction in
the Japanese stock market”. Working Paper. Iowa State
University.
Darrat, A.F., 1990, “Stock returns, money and fiscal deficits”,
Journal of Financial and Quantitatiw Analysis, 25, 387-398.
DeBondt, W.F.M.. 1984, Stock price reversals and over-reac-
tion to news events. A survey of theory and evidence, in:
Guimaraes et al. (19X9).
Dyl. E.A. and K. Maxfield. 1987, Does the stock market
over-react‘? Working paper, University of Arizona.
Engle. R.F. and C.W.J. Granger, 1991, Long-rmr Ecorlomic
Relationships; Reading.7 in Cointegrution (Oxford University
Press. Oxford).
Fama, E.F.. 1970, “Efficient capital markets: A review of
theory and empirical work”. Journal of finance, 25. 3X3-
417.
Friedman, B.M. and D.1. Laibson, 1989, “Economic implica-
tions of extraordinary movements in stock prices, Working
paper, Ecomomics Department, Harvard University.
Gonzalo, J., 1991, Private communication.
Granger, C.W.J.. 1972, “Empirical studies of captial markets:
A survey, in: G. Szego and K. Shell, eds., Mathematical
Methods in Investment and Finance (North-Holland. Am-
sterdam).
Granger, C.W.J., 1986, “Development in the study of cointe-
grated economic variables”. Oxford Bulletin of Economics
and Statistics. 48, 213-228.
Granger, C.W.J. and 0. Morgenstern. 1970, Predictability c~f
Stock Market Prices (Heath-Lexington).
Granger, C.W.J. and T. Terasvirta, 1992, Modelling Nonlinear
Economic Relutionships (Oxford University Press, Oxford).
Guimaraes, R.M.C., B.G. Kingsman and S.J. Taylor, 1989. A
Rerrppraisal of the Efliciency of Financial Markets (Sprin-
ger-Verlag, Berlin).
Hodrick. R.J.. 1990. “Dividend yields and expected stock
returns: Alternative procedures for inference and meas-
urement”, Working paper, Kellogg Graduate School of
Management, Northwestern University.
Ikenberry, D. and J. Lakonishok, 1989. Seasonal anomalies in
financial markets: A survey. in: Guimaraes et al. (1989).
Jegadeesh, N., 1990, “Evidence of predictable behaviour of
security returns”. J oftrtzal of Finance, 35, X81-898.
Jensen M.C., 1978. Some anomalous evidence regarding mar-
ket efficiency, Journal of Financiul Economics. 6, 95- 101.
Kariya, T., T. Tsukuda and J. Maru, 1990, Testing the random
walk hypothesis for Japanese stock prices in S. Taylor
models, Working paper 90-94. Graduate School of Busi-
ness, University of Chicago.
Keim, D.B., 19X9, Earnings yield and size effects: Uncondi-
tional and conditional estimates, in: Guimaraes et al.
(1989).
LeBaron, B., 1990. forecasting improvements using a volatility
index, Working paper, Economics Department, University
of Wisconsin.
Lehman, B.N., 1990. Fads, martingales, and market efficiency.
Quarterly journal of Economics, 105 (1) l-28.
LeRoy, SF., 1989. Efficient capital markets and martingales,
Journal of Ecottomic Literature 27, 1583-1621.
Liu, T.. C.W.J. Granger and W. Heller, 1991, Using the
correlation exponent to decide if an economic series is
chaotic, Working paper, Economics Department, Univer-
sity of California. San Diego.
Neftci. S.N.. 1991. Naive trading rules in fincancial markets
and Wiener-Kolmogorov prediction theory, Journal of
Busmess. s49-571.
Nelson, C.R. and M.J. Kim, 1990, Predictable stock returns:
reality or statistical illusion‘? Working paper, Economics
Department, University of Washington, Seattle.
Pesaran, M.H. and A.G. Timmerman, 1990, The statistical
and economic significance of the predictability of excess
returns on common stocks, Program in Applied Econo-
metrics Discussion paper #26, University of California,
Los Angeles.
Taylor. S., 1986. Modelling Financial Time Serre.s (Wiley,
Chichester).
Tong, H.. 1990. Non&war Time Series (Oxford University
Press. Oxford).
C. KJ . Granger / Forecasting stock market prices
Biography: Clive GRANGER is Professor of Economics at
the University of California, San Diego having moved from a
professorship in Applied Statistics and Econometrics of Not-
tingham University sixteen years ago. His work has empha-
sised forecasting, econometric modelling, speculative markets,
causality and cointegration. He has written and edited nine
books and nearly 1.50 papers. He is a Fellow of the Economet-
ric Society.
13
doc_241149294.pdf
In recent years a variety of models which apparently forecast changes in stock market prices have been introduced. Some of these are summarised and interpreted.
International Journal of Forecasting 8 (1992) 3-13
North-Holland
Forecasting stock market prices: Lessons
for forecasters *
Clive W.J. G-anger
University of California, Sun Diego, USA
Abstract: In recent years a variety of models which apparently forecast changes in stock market prices
have been introduced. Some of these are summarised and interpreted. Nonlinear models are particularly
discussed, with a switching regime, from forecastable to non-forecastable, the switch depending on
volatility levels, relative earnings/price ratios, size of company, and calendar effects. There appear to be
benefits from disaggregation and for searching for new causal variables. The possible lessons for
forecasters are emphasised and the relevance for the Efficient Market Hypothesis is discussed.
Keywords: Forecastability, Stock returns, Non-linear models, Efficient markets.
1. Introduction: Random walk theory
For reasons that are probably obvious, stock
market prices have been the most analysed eco-
nomic data during the past forty years or so. The
basic question most asked is - are (real) price
changes forecastable? A negative reply leads to
the random walk hypothesis for these prices,
which currently would be stated as:
H,,: Stock prices are a martingale.
i.e. E[ P,+, I I,] = P,,
where Z, is any information set which includes
the prices P, _ j, j 2 0. In a sense this hypothesis
has to be true. If it were not, and ignoring trans-
action costs then price changes would be consis-
tently forecastable and so a money machine is
created and indefinite wealth is possible. How-
Correspondence to: C.W.J. Granger, Economics Dept., 0508,
Univ. of California, San Diego, La Jolla, California, USA
92093-0508.
* Invited lecture, International Institute of Forecasters, New
York Meeting, July 1991, work partly supported by NSF
Grant SES 89-02950. I would like to thank two anonymous
referees for very helpful remarks.
ever, a deeper theory - known as the Efficient
Market Hypothesis - suggests that mere fore-
castability is not enough. There are various forms
of this hypothesis but the one I prefer is that
given by Jensen (1978):
HC,2: A market is efficient with respect to infor-
mation set 1, if it is impossible to make
economic profits by trading on the basis of
this information set.
By ‘economic profits’ is meant the risk-adjusted
returns ‘net of all costs’. An obvious difficulty
with this hypothesis is that is is unclear how to
measure risk or to know what transaction costs
are faced by investors, or if these quantities are
the same for all investors. Any publically avail-
able method of consistently making positive prof-
its is assumed to be in I,.
This paper will concentrate on the martingale
hypothesis, and thus will mainly consider the
forecastability of price changes, or returns (de-
fined as (P, - P,_ , +D,)/P, _ 1 where D, is divi-
dends), but at the end I will give some considera-
tion to the efficient market theory. A good survey
of this hypothesis is LeRoy (1989).
By the beginning of the seventies I think that it
was generally accepted by forecasters and re-
0169.2070/92/$05.00 0 1992 - Elsevier Science Publishers B.V. All rights reserved
4 C. W.J. Crunger / Forecasting stock market prices
searchers in finance that the random walk hy-
pothesis (or H,,,) was correct, or at least very
difficult to refute. In a survey in 1972 I wrote,
‘Almost without exception empirical studies.. . ’
support a model for p, = log f’, of the form
dP,+, =~AP,+ I ,-, +~t+l,
where 0 is near zero, 1, contributes only to the
very low frequencies and E, is zero mean white
noise. A survey by Fama (1970) reached a similar
conclusion. The information sets used were:
I,,: lagged prices or lags of logged prices.
ZZt: Ilr plus a few sensible possible explanatory
variables such as earnings and dividends.
The data periods were usually daily or monthly.
Further, no profitable trading rules were found,
or at least not reported. I suggested a possible
reporting bias - if a method of forecasting was
found an academic might prefer to profit from it
rather than publish. In fact, by this period I
thought that the only sure way of making money
from the stock market was to write a book about
it. I tried this with Granger and Morgenstern
(1970), but this was not a financially successful
strategy.
However, from the mid-seventies and particu-
larly in the 1980s there has been a burst of new
activity looking for forecastability, using new
methods, data sets, longer series, different time
periods and new explanatory variables. What is
interesting is that apparent forecastability is often
found. An important reference is Guimaraes,
Kingsman and Taylor (1989). The objective of this
part is to survey some of this work and to suggest
lessons for forecasters working on other series.
The notation used is:
p,
= a stock price,
PI
= log P,,
D,
= dividend for period t,
Rr
= return = (P, + D, - P,_,)/P,_,,
[In some studies the return is calcu-
lated without the dividend term and
approximated by the change in log
prices.]
rr
= return on a ‘risk free’ investment,
R, -rt
= excess return,
P
= risk level of the stock,
R, - r, - /3 X market return - cost of transac-
tion = risk-adjusted profits.
The risk is usually measured from the capital
asset pricing model (CAPM):
R, - r, = p (market excess return) + e,,
where the market return is for some measure of
the whole market, such as the Standard and
Poor’s 500. p is the non-diversifiable risk for the
stock. This is a good, but not necessarily ideal,
measure of risk and which can be time-varying
although this is not often considered in the stud-
ies discussed below.
Section 2 reviews forecasting models which can
be classified as ‘regime-switching’. Section 3 looks
at the advantages of disaggregation, Section 4
considers the search for causal variables, Section
5 looks at technical trading rules, Section 6 re-
views cointegration and chaos, and Section 7 looks
at higher moments. Section 8 concludes and re-
considers the Efficient Market Theory.
2. Regime-switching models
If a stationary series X, is generated by:
X, = (Y, + ylxI_, + E, if z, in A
and
X, = (Ye + y*x,_, + Ed if z, not in A,
then x, can be considered to be regime switching,
with z, being the indicator variable. If Z~ is a
lagged value of x, one has the switching thres-
hold autoregressive model (STAR) discussed in
detail in Tong (1990), but z, can be a separate
variable, as is the case in the following examples.
It is possible that the variance of the residual E,
also varies with regime. If x, is a return (or an
excess return) it is forecastable in at least one
regime if either y, or y2 is non-zero.
2.a. Forecastability with Low Volatility
LeBaron (1990) used R,, the weekly returns of
the Standard and Poor 500 index for the period
194661985, giving about 2,000 observations. He
used as the indicator variable a measure of the
recent volatility
10
&,,’ = c Rf_,
1=0
C. W.l. Granger / Forecasting stock market prices 5
and the regime of interest is the lowest one-fifth
quantile of the observed C? values in the first half
of the sample. The regime switching model was
estimated using the first half of the sample and
post-sample true one-step forecasts were evalu-
ated over the second half. For the low volatility
regime he finds a 3.1 percent improvement in
forecast mean squared error over a white noise
with non-zero mean (that is, an improvement
over a model in which price is taken to be a
random walk with drift). No improvement was
found for other volatility regimes. He first takes
cy (the constant) in the model to be constant
across regimes, relaxing this assumption did not
result in improved forecasts. Essentially the model
found is
R, =cr +0.18R,_1 + l t if have low volatility
R,=cu+~, otherwise,
where LY is a constant. This non-linear model was
initially found to fit equally well in and out of
sample. However, more recent work by LeBaron
did not find much forecasting ability for the
model.
2. b. Earnings and size portfolios
Using the stocks of all companies quoted on
either the New York or American Stock Ex-
changes for the period 1951 to 1986, Keim (1989)
formed portfolios based on the market value of
equity (size) and the ratio of earnings to price
(E/P) and then calculated monthly returns (in
percentages). Each March 31”’ all stocks were
ranked on the total market value of the equity
(price x number of shares) and ten percent with
the lowest ranks put into the first (or smallest)
portfolio, the next 10% in the second portfolio
and so forth up to the shares in the top 10%
ranked giving the ‘largest’ portfolio. The portfo-
lios were changed annually and average monthly
returns calculated. Similarly, the portfolios were
formed from the highest E/P values to the lowest
(positive) values. [Shares of companies with nega-
tive earnings went into a separate portfolio.] The
table shows the average monthly returns (mean)
for five of the portfolios in each case, together
with the corresponding standard errors:
Size
E/P
Mean (s.d.) Mean (s.d.)
smallest 1.79 (0.32) highest 1.59 (0.25)
2nd 1.53 (0.28) 2”d 1.59 (0.22)
S’h 1.25 (0.24) gLh 1.17 (0.22)
9th 1.03 (0.21) gth 1.11 (0.25)
largest 0.99 (0.20) lowest 1.19 (0.28)
negative
earnings (1.39) (0.39)
Source: Keim (1989).
It is seen that the smallest (in size) portfolios
have a substantially higher average return than
the largest and similarly the highest E/P portfo-
lios are better than the lowest.
The two effects were then combined to gener-
ate 25 portfolios, five were based on size and
each of these was then sub-divided into five parts
on E/P values. A few of the results are given in
the following table as average monthly returns
with beta risk values shown in brackets.
Size E/P ratio
Lowest
smallest 1.62
(1.27)
middle 1.12
(1.28)
largest 0.89
(1.11)
Source: Keim (1989).
Middle
1.52
(1.09)
1.09
(1.02)
0.97
(0.98)
Highest
1.90
(1.09)
1.52
(1.06)
1.43
(1.03)
The portfolio with the highest E/P ratio and
the smallest size has both a high average return
and a beta value only slightly above that of a
randomly selected portfolio (which should have a
beta of 1.0). The result was found to hold for
both non-January months and for January, al-
though returns in January were much higher, as
will be discussed in the next section. Somewhat
similar results have been found for stocks on
other, non-U.S. exchanges. It should be noted
that as portfolios are changed each year, transac-
tion costs will be moderately large.
The results are consistent with a regime-
switching model with the regime determined by
the size and E/P variables at the start of the
year. However, as rankings are used, these vari-
ables for a single stock are related to the actual
values of the variables for all other stocks.
6 C. W.J. Changer / Forecasting stock market prices
2.c. Seasonal effects
A number of seasonal effects have been sug-
gested but the strongest and most widely docu-
mented is the January effect. For example Keim
(1989) found that the portfolio using highest E/P
values and the smallest size gave an average
return of 7.46 (standard error 1.41) over Januarys
but only 1.39 (0.27) in other months. A second
example is the observation that the small capital-
ization companies (bottom 20% of companies
ranked by market value of equity) out-performed
the S&P index by 5.5 percent in January for the
years 1926 to 1986. These small firms earned
inferior returns in only seven out of the 61 years.
Other examples are given in Ikenberry and
Lakonishok (1989). Beta coefficients are also gen-
erally high in January.
The evidence suggests that the mean of re-
turns have regime changes with an indicator vari-
able which takes a value of unity in January and
zero in other months.
2.d. Price reversals
A number of studies have found that shares
that do relatively poorly over one period are
inclined to perform well over a subsequent pe-
riod, thus giving price change reversals. A survey
is provided by DeBondt (1989). For example, Dyl
and Maxfield (1987) selected 200 trading days in
random in the period January 1974 to January
1984, each day the three NYSE or AMEX stocks
with the greatest percentage price loss (on aver-
age - 12%) were noted. Over the next ten trad-
ing days, these losers earn a risk-adjusted return
of 3.6 percent. Similarly the three highest gainers
lost an average 1.8% over the next ten days.
Other studies find similar evidence for daily,
weekly and even monthly returns. Transaction
costs will be fairly heavy and a strategy based on
these results will probably be risky.
However, Lehman (1990) considered a portfo-
lio whose weights depended on the return of a
security the previous week minus the overall re-
turn, with positive weights on previous losers and
negative weights (going short) on previous win-
ners. The portfolio was found to consistently pro-
duce positive profits over the next week, with very
few losing periods and so with small risk. Trans-
action costs were substantial but worthwhile prof-
its were achieved for transaction costs at a level
appropriate for larger traders. Thus, after allow-
ing for risk and costs, a portfolio based on price
reversal was found to be clearly profitable.
Long term price reversals have also been docu-
mented. For example, Dark and Kato (1986)
found in the Japanese market that for the years
1964 to 1980, the three year returns for decile
portfolios of extreme previous losers exceed the
comparable returns of extreme previous winners
by an average 70 percent.
In this case the indicator variable is the ex-
treme relative loss value of the share. As before
the apparent forecastability leads to a simple
investment strategy, but knowledge is required of
the value taken by some variable based on all
stocks in some market.
2.e. Remolsal of extreme values
It is well known that the stock markets occa-
sionally experience extraordinary movements, as
occurred in October 1987, for example. Friedman
and Laibson (1989) point out that these large
movements are of overpowering importance and
may obscure simple patterns in the data. They
consider the Standard and Poor 500 quarterly
excess returns (over treasury bills) for the period
19541 to 1988IV. After removal of just four ex-
treme values, chosen by using a Poisson model,
the remaining data fits an AR(l) model with
significant lag coefficient of 0.207 resulting in an
R2 value of 0.036. The two regimes are thus the
‘ordinary’ excess returns, which seem to be fore-
castable, and the extra-ordinary returns which are
not, from the lagged data at least.
3. Benefits of disaggregation
A great deal of the early work on stock market
prices used aggregates, such as the Dow Jones or
Standard and Poor indices, or portfolios of a
random selection of stocks or some small group
of individual stocks. The availability of fast com-
puters with plenty of memory and tapes with
daily data for all securities on the New York and
American Exchanges, for example, allows exami-
nation of all the securities and this can on occa-
sion be beneficial. The situation allows cross-sec-
tion regressions with time-varying coefficients
C. W.J. Grunger / Forecasting stock market prices I
which can possibly detect regularities that were
not previously available. For example Jegadeesh
(1990) uses monthly data to fit cross-section mod-
els of the form
12
for each month. Thus, a lagged average relation-
ship is considered with coefficients changing each
month. Here R,, is the average return over a long
(four or six years) period which exclude the previ-
ous three years. [In the initial analysis, R was
estimated over the following few years, but this
choice was dropped when forecasting properties
were considered.] Many of the averaged aj were
significantly different from zero, particularly at
lags one and twelve, but other average coeffi-
cients were also significant, including at lags 24
and 36. A few examples are shown, with t-values
in brackets.
a1 a,2
_
aI4
Rf
all months -0.09(18) 0.034(9) 0.019(6.5) 0.108
January -0.23 (9) 0.08 (5) 0.034C2.6) 0.178
Feb. to Dec. -0.08(17) 0.03 (8) 0.017(6) 0.102
Source: Jegadeesh (1990).
There is apparently some average, time-vary-
ing structure in the data, as seen by R: values of
10% or more. As noticed earlier, January has
more forecastability than other months and it was
found that a group of large firms had regressions
with higher Rf in February to December than all
firms using these regressions (without the R
terms), stocks were ranked each month on their
expected forecastability and ten portfolios formed
from the 10% most forecastable (P,), second
10% and so forth up to the 10% least fore-
castable (P,,,). The average abnormal monthly
returns (i.e. after risk removal) on the ‘best’ and
‘worse’ portfolios for different periods were
All months January Feb.-Dec.
PI
0.011 0.024 0.009
P
10
- 0.014 - 0.020 -0.017
Source: Jegadeesh (1990).
There is thus seen to be a substantial benefit
from using the best portfolio rather than the
worst one based on the regressions. Benefits were
also found, but less substantial ones, using twelve
month ahead forecasts. Once transaction costs
are taken into account the potential abnormal
returns from using P, are halved, but are still
around 0.45% per month (from personal commu-
nication by author of the original study).
4. Searching for causal variables
Most of the studies discussed so far have con-
sidered forecasting of prices from just previous
prices but it is also obviously sensible to search
for other variables that provide some forecastabil-
ity. The typical regression is
Ap, = constant + p’Kl _ , + E, ,
_
where & is a vector of plausible explanatory, or
causal variables, with a variety of lags considered.
For example Darrat (1990) considered a monthly
price index from the Toronto Stock Exchange for
the period January 1972 to February 1987 and
achieved a relationship:
Ap, = tsTA volatility of interest rates (t - 1)
- :;::A production index (t - 1)
+ yiE3fA long-term interest rate (t - 10)
- 0.015 A cyclically-adjusted budget
(3.0)
deficit (t - 3))
R2 = 0.46,
Durbin-Watson = 2.01,
(4.1)
where only significant terms are shown and the
modulus of t-values in brackets. Several other
variables were considered but not found to be
significant, including changes of short-term rates,
inflation rate, base money and the US-Canadian
exchange rate, all lagged once. An apparently
high significance R2 value is obtained but no
out-of-sample forecastability is investigated.
This search may be more successful if a long-
run forecastability is attempted. For example Ho-
drick (1990) used monthly US data for the period
1929 to December 1987 to form NYSE value-
weighted real market returns, Rr+k, over the
time span (t + 1, t + k). The regression
log R,tk,t
= (Ye + p,( dividend/price ratio at t )
x C. W.J. Granger / Forecasting stock market prices
found R2 increasing as k increases, up to R2 =
0.354 at k =48. Thus, apparent long-run fore-
castability has been found from a very simple
model. However, again no post-sample evaluation
is attempted.
Pesaran and Timmerman (1990) also employ
simple models that produce useful forecastability
and they also conduct a careful evaluation of the
model. As an example of the kind of model they
produce, the following equation has as its depen-
dent variable (Y,> the quarterly excess return on
the Standard and Poor 500 portfolio:
Y = - 0.097 + t_T7: dividend yield ( t - 2)
- 1.59 inflation rate (t - 3)
(2.8)
- ?$)I T-bill (end, t - 1)
+ 0.025 T-bill (begin, t - 2)
(4.6)
+ 0.066A twelve month bond state (t - 1)
(5.5)
+ residual,
Rf =0.364, Durbin-Watson = 2.02.
(4.2)
Here dividend yield at time t is
dividend on S&P index ( t - 1)
. price of S&P index (t)
T-bill is the one month interest rate ‘end’ means
it is measured at the end of the third month of
the quarter, ‘begin’ indicates that it is measured
at the end of the first month of the quarter. The
two T-bill terms in the equation are thus effec-
tively the change in the T-bill interest rate from
one month to the next, plus one at the end of the
quarter. As just lagged variables are involved and
a reasonable R: value is found, the model can
potentially be used for forecasting. [It might be
noted that Rz climbed to 0.6 or so for annual
data.] Some experimentation with non-linear
lagged dependent variables produced some in-
creases in R:, to about 0.39, but this more com-
plicated model was not further evaluated.
A simple switching portfolio trading rule was
considered:
(i) Buy the S&P 500 index if the excess return
was predicted to be positive according to
equation (4.2), with the equation being se-
quentially re-estimated. Thus only data avail-
able at the time of the forecast was used in
making the forecast.
(ii) If the predictor was negative, the invest in
T-bills.
The following table shows the rate of returns
achieved by either using a ‘buy-and-hold’ market
portfolio, or the switching portfolio obtained from
the above trading rule or by just buying T-bills.
As the switching rule involves occasional buying
and selling, possibly quarterly, two levels of trans-
action costs are considered 4% and 1%.
Investment strategy
Market Switching T-bill
Transaction
costs 0 4% 1% 0
Interest rate of
returns 9.5 1 13.30 12.39 6.34
Standard deviation
of returns 8.23 5.43 5.41 0.70
Wealth at end of
period ” 1394 3736 2961 595
” The period considered from 1960.1 to 1988.IV and the
wealth accumulates from an investment of $100 in Decem-
ber 1959.
Source: Pesaran and Timmerman (I 990).
Although the results presented are slightly bi-
ased against the switching portfolio zero transac-
tion costs are assumed for the alternative invest-
ments, the trading rule based on the regression is
seen to produce the greatest returns and as a
lower risk-level than the market (S&P 500) port-
folio. A variety of other evaluation methods and
other regressions are also presented in the paper.
It would seem that dividend/price ratios and
interest rates have quite good long-run forecast-
ing abilities for stock price index returns.
5. A new look at old techniques - Technical
trading rules
A strategy that is popular with actual specula-
tors, but is disparaged by academics, is to use an
automatic, or technical trading rule. An example
is to use perceived patterns in the data, such as
the famous ‘head and shoulders’, and to devise a
rule based on them. Much technical analysis is
difficult to evaluate, as the rules are not precise
enough. The early literature did consider various
simple rules but generally found little or no fore-
casting value in them. However, the availability of
C. W.J. Granger / Forecasting stock market prices 9
fast computers has allowed a new, more intensive
evaluation to occur, with rather different results.
Brock, Lakonishok and LeBaron (1991) con-
sider two technical rules, one comparing the most
recent value to a recent moving average, and the
other is a ‘trading range breakout’. Only the first
of these is discussed here.
The first trading rule is as follows:
Let M, = average of previous 50 prices, form
a band B, = (1 f 0.01) M,, so that the band
is plus and minus 1% around M,. If P,, the
current price, is above the band, this is a
buy signal, if it is below the band, this is a
sell signal.
Using 90 years of daily data for the Dow-Jones
Index (giving a sample of over twenty-three thou-
sand values) for the period 1897 to 1986, the rule
suggested buying 50% of the time giving an aver-
age return next day of 0.00062 (t = 3.7) and sell-
ing 42% of the time, giving an average return of
- 0.00032 (t = 3.6). The return on the rule ‘buy if
have buy signal and go short on a sell signal’ gave
an average daily return of 0.00094 (t = 5.4). The
first two t-values are for the return minus the
daily unconditional average return, the ‘buy-sell’
r-value is relative to zero. If this buy-sell strategy
was used 200 times a year, it gives a return of 20.7
percent for the year. However, this figure ignores
transaction costs, which could be substantial. The
trading rule was considered for four sub-periods
and performed similarly for the first three but
less well for the most recent sub-period of 1962-
1986, where the buy-sell strategy produced a daily
return of 0.00049. Other similar trading rules
were considered and gave comparable results.
Thus, this rule did beat a buy-and-hold strategy
by a significant amount if transaction costs are
not considered. The authors also consider a much
more conservative rule, with a fixed ten day hold-
ing period after a buy or sell signal. The above
rule then averages only 3; buy and sell signals a
year, giving an annual expected return of 8.5%
compared to an annual return for the Dow Index
of about 5%, again ignoring transaction costs.
These, and the results for the other trading rules
considered suggest that there may be regular but
subtle patterns in stock price data, which would
give useful forecastability. However, very long
series are needed to investigate these rules.
Neftci (1991) investigates a similar moving av-
erage trading rule using different statistical meth-
ods and an even longer period - monthly Dow-
Jones Industrial Index starting in 1792, up to
1976. Let M, be an equi-weighted moving aver-
age over the past five months. If P, is the price of
the index in month t, define a dummy variable:
D,= 1 if P,>M, given P,+, CM,_,
= -1 if P, <M, given P,-, >&I _,
= 0 otherwise.
Regression results are presented for the equation
P *+,#= 5 CI ~P~_~ + 5 Y~D,_~+residual,
j = 0 j = 0
where the residual is allowed to be a moving
average of order 17, for each of the three sub-
periods 1792-1851, 1852-1910 and 1910-1976. In
each case the sum of the alphas is near one, as
suggested by the efficient market theory and in
the more recent period the gammas were all
significant, individually and jointly, suggesting
some nonlinearity in the prices. No forecasting
exercise was considered using the models. The
use of data with such early dates as 1897 or 1792
is surely only of intellectual interest, because of
the dramatic institutional changes there have oc-
curred since then.
Neftci also proves, using the theory of optimal
forecasts, that technical trading rules can only be
helpful with forecasting if the price series are
inherently nonlinear.
6. New techniques - Cointegration and chaos
Since the early statistical work on stock prices,
up to 1975, say, a number of new and potentially
important statistical models and techniques have
been developed. Some arrive with a great flourish
and then vanish, such as catastrophe theory,
whereas others seem to have longer staying power.
I will here briefly consider two fairly new ap-
proaches which have not been successful, so far,
in predicting stock prices.
An Z(1) series is one such that it’s first differ-
ence is stationary. A pair, X,, Y,, of Z(1) series
are called cointegrated if there is a linear combi-
nation of them, Z, =X, - AY,, say, which is Z(0).
10 C. W.J. Granger / Forecasting stock market prices
The properties and implications of such series are
described in Granger (19861, Engle and Granger
(19911, and many other publications in economet-
rics, macroeconomics and finance. If the series
are cointegrated there is necessarily an error-cor-
rection data generating model of the form (ignor-
ing constants):
,4X, = (Y,Z, _, + lagged AX,, AY, terms
+ white noise,
plus a similar equation for Ax, with at least one
of N.~, tiy,. being non-zero. It follows that either X,
must help forecast Y,+ , or Y, must help forecast
X ,+ , or both. Thus, if dividends and stock prices
are found to be cointegrated, as theory suggests,
then prices might help forecast dividends, which
would not be surprising, but dividends not help
forecast prices, in agreement with the efficient
market hypothesis. However, for the same reason
one would not expect a pair of stock prices to be
cointegrated, as this would contradict the effi-
cient market hypothesis. In fact several papers
have been produced that claim to find cointegra-
tion between pairs of prices or of portfolios, but
the error-correction models are not presented or
the forecasting possibility explored, and so this
work will not be surveyed. It should be noted that
cointegration would be inconsistent with the
well-respected capital asset pricing model
(CAPM) which says that the price P,, of the ith
asset is related to the price of the whole market
P,,,, bY
3 log P,, = b,d log P,, + E;, ,
where err is white noise. Summing over time gives
log Pi, = bi log P,,cr + i e,,t_j.
j=O
As the last term is the accumulation of a station-
ary series, it is 1(l) (ignoring trends) and so
cointegration should not occur between log P,,
and log P,,,. Similarly, there should be no cointe-
gration between portfolios, Gonzalo (1991) has
found no cointegration between three well known
aggregates, the Dow-Jones Index, the Standard
and Poor 500 Index and the New York Stock
Exchange Equal Value Index.
A class of processes generated by particular
deterministic maps, such as
y,=4y,-,(I -Y,-,)
with
0 < Yo < 1 >
have developed a great deal of interest and can
be called ‘white chaos’. These series have the
physical appearance of a stochastic independent
(i.i.d.1 process and also the linear properties of a
white noise such as zero autocorrelations and a
flat spectrum. The question naturally arises of
whether the series we have been viewing as
stochastic white noise are actually white chaos
and are thus actually perfectly forecastable - at
least in the short run and provided the actual
generating mechanism is known exactly. The lit-
erature on chaos is now immense, involves excit-
ing and deep mathematics and truly beautiful
diagrams, and also is generally optimistic, sug-
gesting that these processes occur frequently. In
fact, a clear case can be made that they do not
occur in the real world, as opposed to in labora-
tory physics experiments. There is no statistical
test that has chaos as the null hypothesis. There
also appears to be no characterizing property of a
chaotic process, that is a property that is true for
chaos but not for any completely stochastic pro-
cess. These arguments are discussed in Liu,
Granger and Heller (1991). It is true that some
high-dimensional white chaotic processes are in-
distinguishable from iid series, but this does not
mean that chaos occurs in practice. In the above
paper, a number of estimates of a statistic known
as the correlation dimension are made for various
parameter values using over three thousand
Standard and Poor 500 daily returns. The result-
ing values are consistent with stochastic white
noise (or high dimensional chaos) rather than low
dimensional - and thus potentially forecastable -
white chaos. A little introspection also make it
seem unlikely to most economists that a stock
market, which is complex, involving many thou-
sands of speculations, could obey a simple deter-
ministic model.
7. Higher moments
To make a profit, it is necessary to be able to
forecast the mean of price changes, and the stud-
ies reviewed above all attempt to do this. The
efficient market theory says little about the fore-
C. W.J. Granger / Forecasting stock market prices
11
castability of functions of price changes or re-
turns, such as higher moments. If R, is a return it
has been found that Rf and is clearly fore-
castable and I R, I even more so, from lagged
values. Taylor (1986) finds evidence for this using
U.S. share prices and Kariya, Tsukuda and Maru
(1990) get similar results for Japanese stocks. For
example, if R, is the daily return from the U.S.
Standard and Poor index the autocorrelations for
R, are generally very small, the autocorrelations
for Rf are consistently above 0.1 up to lag 100
and for I R, I are about 0.35 up to lag 100. It is
clear that these functions of returns are very
forecastable, but this is not easily converted into
profits, although there are implications for the
efficiency of options markets. The results are
consistent with certain integrated GARCH mod-
els but this work is still being conducted and the
final conclusions have yet to be reached.
8. Lessons for forecasters
Despite stock returns once having been thought
to be unforecastable, there is now plenty of opti-
mism that this is not so, as the examples given
above show. Is this optimism justified, and if yes,
what are the lessons for forecasters working with
other data sets? As there is an obvious possible
profit motive driving research into the forecasta-
bility of stock prices, or at least returns, one can
expect more intensive analysis here than else-
where. Whereas too many forecasters seem to be
content with just using easily available data, with
the univariate or simple transfer function fore-
casting techniques that are found on popular
computer packages, stock market research is more
ambitious and wide-ranging. It should be empha-
sized that the above is not a complete survey of
all of the available literature.
The sections above suggest that benefits can
arise from taking a longer horizon, from using
disaggregated data, from carefully removing out-
hers or exceptional events, and especially from
considering non-linear models. Many of the latter
can be classified as belonging to a regime switch-
ing model of the form
were Q(w) is a smooth monotonic function such
as a cumulative distribution function of a contin-
uous random variable so that 0 I @ I 1, _X, is a
vector of explanatory variables possibly including
lagged returns and w, is the ‘switching variable’,
possibly a lagged component of & or some linear
combination of these components. It has always
been important to discover appropriate explana-
tory variables &, and with this new class of
models it is especially important to find the ap-
propriate switching variable, if it exists and is
observable. This class of models is discussed in
Granger and Terasvirta (1992), where tests and
estimation procedures are outlined.
The papers also suggest that some sub-periods
may be more forecastable than others - such as
summer months or January - and this is worth
exploring. If many component series are avail-
able, then ranks may produce further information
that is helpful with forecasting. There seems to
be many opportunities for forecasters, many of
whom need to break away from simple linear
univariate ARIMA or multivariate transfer func-
tions. It is often not easy to beat convincingly
these simple methods, so they make excellent
base-line models, but they often can be beaten.
Before the results discussed in previous sec-
tions are accepted the question of how they should
be evaluated has to be considered. Many of the
studies in this, and other forecasting areas, are of
the ‘if only I had know this at the beginning of
the period I could have made some money’ classi-
fication. For a ‘forecasting model’ to be accepted
it has to show that it actually forecasts, it is not
sufficient to produce a regression model evalu-
ated only in sample. There is always the possibil-
ity of small-sample in-sample biases of coeffi-
cients which give overly encouraging results, as
shown by Nelson and Kim (1990). The possibility
of ‘data mining’ having occurred, with many mod-
els having been considered, and just the best one
presented is also a worry. Only out-of-sample
evaluation is relevant and, to some extent, avoids
these difficulties. It is surprising that more of the
studies surveyed do not provide results of fore-
casting exercises.
Not only do the models that are proposed as
providing useful forecasts of price changes or
returns need to be evaluated, to provide prof-
itable strategies the forecast returns need to be
corrected for risk levels and also for transaction
costs. Many of the studies discussed earlier fail to
12 C. W.J . Granger / Forecasting stock murket prices
do this, and so, at present, say nothing about the
correctness of the efficient market hypothesis
(EMH). However, this criticism does not always
apply, for example for the carefully conducted
analysis by Pesaran and Timmerman (1990). Does
this mean that the EMH should be rejected? One
has to say - not necessarily, yet. If a method
exists that consistently produces positive profits
after allowing for risk correction and transaction
costs and if this method has been publicly an-
nounced for some time, then this would possibly
be evidence against EMH. There are so many
possibly relevant trading rules that it is unrealistic
to suppose that investors have tried them all,
especially those that have only been discovered
by expensive computation and sophisticated sta-
tistical techniques. Once knowledge of an appar-
ently trading rule becomes wide enough, one
would expect behaviour of speculators to remove
its profitability, unless there exists another trad-
ing rule the speculators think is superior and thus
concentrate on it. Only if a profitable rule is
found to be widely known and remains profitable
for an extended period can the efficient market
hypothesis be rejected. It will be worthwhile
checking in a few years on the continued prof-
itability of the rules discussed earlier. This re-
search program agrees with the modern taste in
the philosophy of science to try to falsify theories
rather than to try to verify them. Clearly verifica-
tion of EMH is impossible.
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C. KJ . Granger / Forecasting stock market prices
Biography: Clive GRANGER is Professor of Economics at
the University of California, San Diego having moved from a
professorship in Applied Statistics and Econometrics of Not-
tingham University sixteen years ago. His work has empha-
sised forecasting, econometric modelling, speculative markets,
causality and cointegration. He has written and edited nine
books and nearly 1.50 papers. He is a Fellow of the Economet-
ric Society.
13
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