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International Journal of Forecasting 8 (1992) 3-13North-Holland

Forecasting stock market prices: Lessonsfor forecasters *

Clive W.J. G-angerUni versit y of Cali forni a, Sun Di ego, USA

Abstract: In recent years a variety of models which apparently forecast changes in stock market priceshave been introduced. Some of these are summarised and interpreted. Nonlinear models are particularlydiscussed, with a switching regime, from forecastable to non-forecastable, the switch depending onvolatility levels, relative earnings/price ratios, size of company, and calendar effects. There appear to bebenefits from disaggregation and for searching for new causal variables. The possible lessons forforecasters 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, stockmarket prices have been the most analysed eco-

nomic data during the past forty years or so. Thebasic question most asked is - are (real) pricechanges forecastable? A negative reply leads tothe 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 includesthe prices P, _ , j 2 0. In a sense this hypothesishas 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 iscreated and indefinite wealth is possible. How-

Correspondence to: C.W.J. Granger, Economics Dept., 0508,Univ. of California, San Diego, La Jolla, California, USA92093-0508.* Invited lecture, International Institute of Forecasters, New

York Meeting, July 1991, work partly supported by NSFGrant SES 89-02950. I would like to thank two anonymousreferees for very helpful remarks.

ever, a deeper theory - known as the EfficientMarket Hypothesis - suggests that mere fore-castability is not enough. There are various formsof 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 makeeconomic profits by trading on the basis ofthis information set.

By economic profits is meant the risk-adjustedreturns net of all costs. An obvious difficultywith this hypothesis is that is is unclear how tomeasure risk or to know what transaction costsare faced by investors, or if these quantities arethe 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 martingalehypothesis, and thus will mainly consider theforecastability 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 surveyof this hypothesis is LeRoy (1989).

By the beginning of the seventies I think that itwas generally accepted by forecasters and re-

0169.2070/92/$05.00 0 1992 - Elsevier Science Publishers B.V. All rights reserved


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4 C. W .J. Crunger / Forecast ing stock mark et pri ces

searchers in finance that the random walk hy-pothesis (or H,,,) was correct, or at least verydifficult 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,+, =~A P,+ I,-, +~t +l ,

where 0 is near zero, 1, contributes only to thevery low frequencies and E, is zero mean whitenoise. A survey by Fama (1970) reached a similarconclusion. The information sets used were:

I,,: lagged prices or lags of logged prices.

ZZt: Ilr plus a few sensible possible explanatoryvariables 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 possiblereporting bias - if a method of forecasting wasfound an academic might prefer to profit from itrather than publish. In fact, by this period Ithought that the only sure way of making moneyfrom the stock market was to write a book aboutit. I tried this with Granger and Morgenstern(1970), but this was not a financially successfulstrategy.

However, from the mid-seventies and particu-larly in the 1980s there has been a burst of new

activity looking for forecastability, using newmethods, data sets, longer series, different timeperiods and new explanatory variables. What isinteresting is that apparent forecastability is oftenfound. An important reference is Guimaraes,Kingsman and Taylor (1989). The objective of thispart is to survey some of this work and to suggestlessons 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 andapproximated by the change in logprices.]

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 capitalasset 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 andPoors 500. p is the non-diversifiable risk for thestock. This is a good, but not necessarily ideal,measure of risk and which can be time-varyingalthough this is not often considered in the stud-ies discussed below.

Section 2 reviews forecasting models which canbe classified as regime-switching. Section 3 looksat the advantages of disaggregation, Section 4considers the search for causal variables, Section5 looks at technical trading rules, Section 6 re-views cointegration and chaos, and Section 7 looksat 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

andX, = (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 alagged value of x, one has the switching thres-hold autoregressive model (STAR) discussed indetail in Tong (1990), but z, can be a separatevariable, 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 anexcess return) it is forecastable in at least oneregime if either y, or y2 is non-zero.

2.a. Forecastabil i t y w it h Low Vol ati l i ty

LeBaron (1990) used R,, the weekly returns ofthe Standard and Poor 500 index for the period194661985, giving about 2,000 observations. Heused as the indicator variable a measure of therecent volatility

1

,,= c Rf_,1=0


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and the regime of interest is the lowest one-fifthquantile of the observed C? values in the first halfof the sample. The regime switching model wasestimated using the first half of the sample andpost-sample true one-step forecasts were evalu-

ated over the second half. For the low volatilityregime he finds a 3.1 percent improvement inforecast mean squared error over a white noisewith non-zero mean (that is, an improvementover a model in which price is taken to be arandom walk with drift). No improvement wasfound for other volatility regimes. He first takescy (the constant) in the model to be constantacross regimes, relaxing this assumption did notresult in improved forecasts. Essentially the modelfound is

R, = cr + 0.18R,_1 + if have low volatility

R,=cu+~, otherwise,

where LY s a constant. This non-linear model wasinitially found to fit equally well in and out ofsample. However, more recent work by LeBarondid not find much forecasting ability for themodel.

2. b. Earnings and size port fol i os

Using the stocks of all companies quoted oneither the New York or American Stock Ex-changes for the period 1951 to 1986, Keim (1989)formed portfolios based on the market value ofequity (size) and the ratio of earnings to price(E/P) and then calculated monthly returns (inpercentages). Each March 31 all stocks wereranked on the total market value of the equity(price x number of shares) and ten percent withthe lowest ranks put into the first (or smallest)portfolio, the next 10% in the second portfolioand so forth up to the shares in the top 10%ranked giving the largest portfolio. The portfo-lios were changed annually and average monthlyreturns calculated. Similarly, the portfolios wereformed from the highest E/P values to the lowest(positive) values. [Shares of companies with nega-tive earnings went into a separate portfolio.] Thetable 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) 2d 1.59 (0.22)

Sh 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)

negativeearnings (1.39) (0.39)

Source: Keim (1989).

It is seen that the smallest (in size) portfolioshave a substantially higher average return thanthe 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 andeach of these was then sub-divided into five partson E/P values. A few of the results are given inthe following table as average monthly returnswith 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 andthe smallest size has both a high average returnand a beta value only slightly above that of arandomly selected portfolio (which should have abeta of 1.0). The result was found to hold forboth non-January months and for January, al-though returns in January were much higher, aswill be discussed in the next section. Somewhatsimilar results have been found for stocks onother, non-U.S. exchanges. It should be notedthat 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 bythe size and E/P variables at the start of theyear. However, as rankings are used, these vari-ables for a single stock are related to the actualvalues of the variables for all other stocks.


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2.c. Seasonal e f f e t s

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/Pvalues and the smallest size gave an averagereturn of 7.46 (standard error 1.41) over Januarysbut only 1.39 (0.27) in other months. A secondexample is the observation that the small capital-ization companies (bottom 20% of companiesranked by market value of equity) out-performedthe S&P index by 5.5 percent in January for theyears 1926 to 1986. These small firms earnedinferior returns in only seven out of the 61 years.Other examples are given in Ikenberry andLakonishok (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 andzero in other months.

2.d. Price reversals

A number of studies have found that sharesthat do relatively poorly over one period areinclined to perform well over a subsequent pe-

riod, thus giving price change reversals. A surveyis provided by DeBondt (1989). For example, Dyland Maxfield (1987) selected 200 trading days inrandom in the period January 1974 to January1984, each day the three NYSE or AMEX stockswith the greatest percentage price loss (on aver-age - 12%) were noted. Over the next ten trad-ing days, these losers earn a risk-adjusted returnof 3.6 percent. Similarly the three highest gainerslost an average 1.8% over the next ten days.Other studies find similar evidence for daily,weekly and even monthly returns. Transactioncosts will be fairly heavy and a strategy based onthese results will probably be risky.

However, Lehman (1990) considered a portfo-lio whose weights depended on the return of asecurity the previous week minus the overall re-turn, with positive weights on previous losers andnegative weights (going short) on previous win-ners. The portfolio was found to consistently pro-duce positive profits over the next week, with veryfew losing periods and so with small risk. Trans-action costs were substantial but worthwhile prof-

its were achieved for transaction costs at a levelappropriate for larger traders. Thus, after allow-ing for risk and costs, a portfolio based on pricereversal 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 years1964 to 1980, the three year returns for decileportfolios of extreme previous losers exceed thecomparable returns of extreme previous winnersby an average 70 percent.

In this case the indicator variable is the ex-treme relative loss value of the share. As beforethe apparent forecastability leads to a simpleinvestment strategy, but knowledge is required ofthe value taken by some variable based on allstocks in some market.

2.e. Remolsal of extreme values

It is well known that the stock markets occa-sionally experience extraordinary movements, asoccurred in October 1987, for example. Friedmanand Laibson (1989) point out that these largemovements are of overpowering importance andmay obscure simple patterns in the data. Theyconsider the Standard and Poor 500 quarterlyexcess 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 withsignificant lag coefficient of 0.207 resulting in anR2 value of 0.036. The two regimes are thus theordinary excess returns, which seem to be fore-castable, and the extra-ordinary returns which arenot, from the lagged data at least.

3. Benefits of disaggregation

A great deal of the early work on stock marketprices used aggregates, such as the Dow Jones orStandard and Poor indices, or portfolios of arandom selection of stocks or some small groupof individual stocks. The availability of fast com-puters with plenty of memory and tapes withdaily data for all securities on the New York andAmerican 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


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which can possibly detect regularities that werenot 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 eachmonth. 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 wasestimated over the following few years, but thischoice was dropped when forecasting propertieswere considered.] Many of the averaged aj weresignificantly different from zero, particularly atlags one and twelve, but other average coeffi-cients were also significant, including at lags 24and 36. A few examples are shown, with t-valuesin brackets.

a1 a,2_aI4 Rf

all months -0.09(18) 0.034(9) 0.019(6.5) 0.108January -0.23 (9) 0.08 (5) 0.034C2.6) 0.178Feb. 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 of10% or more. As noticed earlier, January hasmore forecastability than other months and it wasfound that a group of large firms had regressionswith higher Rf in February to December than allfirms using these regressions (without the Rterms), stocks were ranked each month on theirexpected forecastability and ten portfolios formedfrom the 10% most forecastable (P,), second10% and so forth up to the 10% least fore-castable (P,,,). The average abnormal monthlyreturns (i.e. after risk removal) on the best andworse portfolios for different periods were

All months January Feb.-Dec.

PI 0.011 0.024 0.009P 10 - 0.014 - 0.020 -0.017

Source: Jegadeesh (1990).

There is thus seen to be a substantial benefitfrom using the best portfolio rather than theworst one based on the regressions. Benefits werealso found, but less substantial ones, using twelve

month ahead forecasts. Once transaction costsare taken into account the potential abnormalreturns from using P, are halved, but are stillaround 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 previousprices but it is also obviously sensible to searchfor other variables that provide some forecastabil-ity. The typical regression is

Ap, = constant + pKl _ , + E_

where & is a vector of plausible explanatory, orcausal variables, with a variety of lags considered.For example Darrat (1990) considered a monthlyprice index from the Toronto Stock Exchange forthe period January 1972 to February 1987 andachieved 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 themodulus of t-values in brackets. Several othervariables were considered but not found to besignificant, including changes of short-term rates,inflation rate, base money and the US-Canadianexchange rate, all lagged once. An apparentlyhigh significance R2 value is obtained but noout-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 period1929 to December 1987 to form NYSE value-weighted real market returns, Rr+k, over thetime span (t + 1, t + k). The regression

log R,tk,t = (Ye + p,( dividend/price ratio at t )


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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 simplemodel. However, again no post-sample evaluationis attempted.

Pesaran and Timmerman (1990) also employsimple models that produce useful forecastabilityand they also conduct a careful evaluation of themodel. As an example of the kind of model theyproduce, the following equation has as its depen-dent variable (Y,> the quarterly excess return onthe Standard and Poor 500 portfolio:

Y = - 0.097 + t_T7: dividend yield ( t - 2)

- 1.59 inflation rate (t - 3)(2.8)

- ? )IT-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)

.rice of S&P index (t)

T-bill is the one month interest rate end meansit is measured at the end of the third month ofthe quarter, begin indicates that it is measuredat the end of the first month of the quarter. Thetwo T-bill terms in the equation are thus effec-tively the change in the T-bill interest rate fromone month to the next, plus one at the end of thequarter. As just lagged variables are involved anda reasonable R: value is found, the model canpotentially be used for forecasting. [It might benoted that Rz climbed to 0.6 or so for annualdata.] Some experimentation with non-linearlagged 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 wasconsidered:(i) Buy the S&P 500 index if the excess return

was predicted to be positive according toequation (4.2), with the equation being se-quentially re-estimated. Thus only data avail-

able at the time of the forecast was used inmaking the forecast.

(ii) If the predictor was negative, the invest inT-bills.

The following table shows the rate of returnsachieved by either using a buy-and-hold marketportfolio, or the switching portfolio obtained fromthe above trading rule or by just buying T-bills.As the switching rule involves occasional buyingand 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% 0Interest rate of

returns 9.5 1 13.30 12.39 6.34Standard deviation

of returns 8.23 5.43 5.41 0.70Wealth at end of

period 1394 3736 2961 595

The period considered from 1960.1 to 1988.IV and thewealth 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 isseen to produce the greatest returns and as alower risk-level than the market (S&P 500) port-folio. A variety of other evaluation methods andother regressions are also presented in the paper.

It would seem that dividend/price ratios andinterest rates have quite good long-run forecast-ing abilities for stock price index returns.

5. A new look at old techniques - Technicaltrading rules

A strategy that is popular with actual specula-tors, but is disparaged by academics, is to use anautomatic, or technical trading rule. An exampleis to use perceived patterns in the data, such asthe famous head and shoulders, and to devise arule based on them. Much technical analysis isdifficult to evaluate, as the rules are not preciseenough. The early literature did consider varioussimple rules but generally found little or no fore-casting value in them. However, the availability of


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fast computers has allowed a new, more intensiveevaluation to occur, with rather different results.

Brock, Lakonishok and LeBaron (1991) con-sider two technical rules, one comparing the mostrecent value to a recent moving average, and theother is a trading range breakout. Only the firstof these is discussed here.

The first trading rule is as follows:

Let M, = average of previous 50 prices, forma band B, = (1 f 0.01) M,, so that the bandis plus and minus 1% around M,. If P,, thecurrent price, is above the band, this is abuy signal, if it is below the band, this is asell signal.

Using 90 years of daily data for the Dow-JonesIndex (giving a sample of over twenty-three thou-sand values) for the period 1897 to 1986, the rulesuggested 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 ifhave buy signal and go short on a sell signal gavean average daily return of 0.00094 (t = 5.4). Thefirst two t-values are for the return minus thedaily unconditional average return, the buy-sellr-value is relative to zero. If this buy-sell strategy

was used 200 times a year, it gives a return of 20.7percent for the year. However, this figure ignorestransaction costs, which could be substantial. Thetrading rule was considered for four sub-periodsand performed similarly for the first three butless well for the most recent sub-period of 1962-1986, where the buy-sell strategy produced a dailyreturn of 0.00049. Other similar trading ruleswere considered and gave comparable results.Thus, this rule did beat a buy-and-hold strategyby a significant amount if transaction costs arenot considered. The authors also consider a muchmore conservative rule, with a fixed ten day hold-ing period after a buy or sell signal. The aboverule then averages only 3; buy and sell signals ayear, giving an annual expected return of 8.5%compared to an annual return for the Dow Indexof about 5%, again ignoring transaction costs.These, and the results for the other trading rulesconsidered suggest that there may be regular butsubtle patterns in stock price data, which wouldgive useful forecastability. However, very longseries 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 to1976. Let M, be an equi-weighted moving aver-age over the past five months. If P, is the price ofthe index in month t, define a dummy variable:

D,= 1 if P,>M, given P,+, CM ,_,

= -1 if 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 movingaverage of order 17, for each of the three sub-periods 1792-1851, 1852-1910 and 1910-1976. Ineach case the sum of the alphas is near one, assuggested by the efficient market theory and inthe more recent period the gammas were allsignificant, individually and jointly, suggestingsome nonlinearity in the prices. No forecastingexercise was considered using the models. Theuse of data with such early dates as 1897 or 1792is surely only of intellectual interest, because ofthe dramatic institutional changes there have oc-curred since then.

Neftci also proves, using the theory of optimalforecasts, that technical trading rules can only behelpful with forecasting if the price series areinherently nonlinear.

6. New techniques - Cointegration and chaos

Since the early statistical work on stock prices,up to 1975, say, a number of new and potentiallyimportant statistical models and techniques havebeen developed. Some arrive with a great flourishand 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 its first differ-ence is stationary. A pair, X,, Y,, of Z(1) seriesare called cointegrated if there is a linear combi-nation of them, Z, =X, - AY,, say, which is Z(0).


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The properties and implications of such series aredescribed in Granger (19861, Engle and Granger(19911, and many other publications in economet-rics, macroeconomics and finance. If the seriesare 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 oneof N.~, tiy,. being non-zero. It follows that either X,must help forecast Y, , or Y, must help forecastX , , or both. Thus, if dividends and stock pricesare found to be cointegrated, as theory suggests,then prices might help forecast dividends, whichwould not be surprising, but dividends not help

forecast prices, in agreement with the efficientmarket hypothesis. However, for the same reasonone would not expect a pair of stock prices to becointegrated, as this would contradict the effi-cient market hypothesis. In fact several papershave been produced that claim to find cointegra-tion between pairs of prices or of portfolios, butthe error-correction models are not presented orthe forecasting possibility explored, and so thiswork will not be surveyed. It should be noted thatcointegration would be inconsistent with the

well-respected capital asset pricing model(CAPM) which says that the price P,, of the ithasset 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 socointegration should not occur between log P,,and log P,,,. Similarly, there should be no cointe-gration between portfolios, Gonzalo (1991) hasfound no cointegration between three well knownaggregates, the Dow-Jones Index, the Standardand Poor 500 Index and the New York StockExchange Equal Value Index.

A class of processes generated by particulardeterministic maps, such as

y,=4y,-, I -Y,-,)

with

0 < Yo < 1>

have developed a great deal of interest and canbe called white chaos. These series have the

physical appearance of a stochastic independent(i.i.d.1 process and also the linear properties of awhite noise such as zero autocorrelations and aflat spectrum. The question naturally arises ofwhether the series we have been viewing asstochastic white noise are actually white chaosand are thus actually perfectly forecastable - atleast in the short run and provided the actualgenerating mechanism is known exactly. The lit-erature on chaos is now immense, involves excit-ing and deep mathematics and truly beautifuldiagrams, and also is generally optimistic, sug-gesting that these processes occur frequently. Infact, a clear case can be made that they do notoccur in the real world, as opposed to in labora-tory physics experiments. There is no statisticaltest that has chaos as the null hypothesis. Therealso appears to be no characterizing property of achaotic process, that is a property that is true forchaos but not for any completely stochastic pro-cess. These arguments are discussed in Liu,Granger and Heller (1991). It is true that somehigh-dimensional white chaotic processes are in-

distinguishable from iid series, but this does notmean that chaos occurs in practice. In the abovepaper, a number of estimates of a statistic knownas the correlation dimension are made for variousparameter values using over three thousandStandard and Poor 500 daily returns. The result-ing values are consistent with stochastic whitenoise (or high dimensional chaos) rather than lowdimensional - and thus potentially forecastable -white chaos. A little introspection also make itseem 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 toforecast the mean of price changes, and the stud-ies reviewed above all attempt to do this. Theefficient market theory says little about the fore-


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castability of functions of price changes or re-turns, such as higher moments. If R s a return ithas been found that Rf and is clearly fore-castable and I I ven more so, from laggedvalues. Taylor (1986) finds evidence for this usingU.S. share prices and Kariya, Tsukuda and Maru(1990) get similar results for Japanese stocks. Forexample, if R s the daily return from the U.S.Standard and Poor index the autocorrelations forR re generally very small, the autocorrelationsfor Rf are consistently above 0.1 up to lag 100and for I re about 0.35 up to lag 100. It isclear that these functions of returns are veryforecastable, but this is not easily converted intoprofits, although there are implications for theefficiency of options markets. The results areconsistent with certain integrated GARCH mod-els but this work is still being conducted and thefinal conclusions have yet to be reached.

8. Lessons for forecasters

Despite stock returns once having been thoughtto be unforecastable, there is now plenty of opti-mism that this is not so, as the examples givenabove show. Is this optimism justified, and if yes,what are the lessons for forecasters working with

other data sets? As there is an obvious possibleprofit motive driving research into the forecasta-bility of stock prices, or at least returns, one canexpect more intensive analysis here than else-where. Whereas too many forecasters seem to becontent with just using easily available data, withthe univariate or simple transfer function fore-casting techniques that are found on popularcomputer packages, stock market research is moreambitious 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 usingdisaggregated data, from carefully removing out-hers or exceptional events, and especially fromconsidering non-linear models. Many of the lattercan 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 avector of explanatory variables possibly includinglagged returns and w, is the switching variable,possibly a lagged component of & or some linearcombination of these components. It has alwaysbeen important to discover appropriate explana-tory variables &, and with this new class ofmodels it is especially important to find the ap-propriate switching variable, if it exists and isobservable. This class of models is discussed inGranger and Terasvirta (1992), where tests andestimation procedures are outlined.

The papers also suggest that some sub-periodsmay be more forecastable than others - such assummer months or January - and this is worthexploring. If many component series are avail-able, then ranks may produce further informationthat is helpful with forecasting. There seems tobe many opportunities for forecasters, many ofwhom need to break away from simple linearunivariate ARIMA or multivariate transfer func-tions. It is often not easy to beat convincinglythese simple methods, so they make excellentbase-line models, but they often can be beaten.

Before the results discussed in previous sec-tions are accepted the question of how they shouldbe evaluated has to be considered. Many of the

studies in this, and other forecasting areas, are ofthe if only I had know this at the beginning ofthe period I could have made some money classi-fication. For a forecasting model to be acceptedit has to show that it actually forecasts, it is notsufficient 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, asshown by Nelson and Kim (1990). The possibilityof data mining having occurred, with many mod-els having been considered, and just the best onepresented is also a worry. Only out-of-sampleevaluation is relevant and, to some extent, avoidsthese difficulties. It is surprising that more of thestudies surveyed do not provide results of fore-casting exercises.

Not only do the models that are proposed asproviding useful forecasts of price changes orreturns need to be evaluated, to provide prof-itable strategies the forecast returns need to becorrected for risk levels and also for transactioncosts. Many of the studies discussed earlier fail to


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12 C. W .J. Granger / Forecasti ng stock murket pri ces

do this, and so, at present, say nothing about thecorrectness of the efficient market hypothesis(EMH). However, this criticism does not alwaysapply, for example for the carefully conductedanalysis by Pesaran and Timmerman (1990). Doesthis mean that the EMH should be rejected? Onehas to say - not necessarily, yet. If a methodexists that consistently produces positive profitsafter allowing for risk correction and transactioncosts and if this method has been publicly an-nounced for some time, then this would possiblybe evidence against EMH. There are so manypossibly relevant trading rules that it is unrealisticto suppose that investors have tried them all,especially those that have only been discoveredby expensive computation and sophisticated sta-tistical techniques. Once knowledge of an appar-ently trading rule becomes wide enough, onewould expect behaviour of speculators to removeits profitability, unless there exists another trad-ing rule the speculators think is superior and thusconcentrate on it. Only if a profitable rule isfound to be widely known and remains profitablefor an extended period can the efficient markethypothesis be rejected. It will be worthwhilechecking 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 theoriesrather than to try to verify them. Clearly verifica-tion of EMH is impossible.

References

Brock, W., J. Lakonishok and B. LeBaron. 1991, Simpletechnical trading rules and the stochastic properties ofstock returns, Working paper 9022, Social Science Re-search Institute, Unrversity of Wisconsin, Madison.

Dark. F.H. and K. Kate, 1986, Stock market over-reaction inthe Japanese stock market. Working Paper. Iowa StateUniversity.

Darrat, A.F., 1990, Stock returns, money and fiscal deficits,Journal of Financial and Quant it ati w Anal ysis, 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 marketover-react? Working paper, University of Arizona.

Engle. R.F. and C.W.J. Granger, 1991, Long-rmr EcorlomicRelati onships; Reading.7 in Coint egruti on (Oxford UniversityPress. Oxford).

Fama, E.F.. 1970, Efficient capital markets: A review oftheory and empirical work. Journal of fi nance, 25. 3X3-417.

Friedman, B.M. and D.1. Laibson, 1989, Economic implica-tions of extraordinary movements n stock prices, Workingpaper, 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., MathematicalMethods in Investment and Finance (North-Holland. Am-sterdam).

Granger, C.W.J., 1986, Development in the study of cointe-grated economic variables. Oxford Bull eti n of Economicsand Stat ist i cs. 48, 213-228.

Granger, C.W.J. and 0. Morgenstern. 1970, Predictabil i ty c~fStock M arket Pri ces (Heath-Lexington).

Granger, C.W.J. and T. Terasvirta, 1992, M odell ing NonlinearEconomic Relutionships (Oxford University Press, Oxford).

Guimaraes, R.M.C., B.G. Kingsman and S.J. Taylor, 1989. ARerrppraisal of the Efli ciency of Financial M arkets (Sprin-ger-Verlag, Berlin).

Hodrick. R.J.. 1990. Dividend yields and expected stockreturns: Alternative procedures for inference and meas-

urement, Working paper, Kellogg Graduate School ofManagement, Northwestern University.

Ikenberry, D. and J. Lakonishok, 1989. Seasonal anomalies infinancial markets: A survey. in: Guimaraes et al. (1989).

Jegadeesh, N., 1990, Evidence of predictable behaviour ofsecurity returns. Joftr tz al of Finan ce, 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 randomwalk hypothesis for Japanese stock prices in S. Taylormodels, 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, Universityof Wisconsin.

Lehman, B.N., 1990. Fads, martingales, and market efficiency.Quart erl y journal of Economics, 105 (1) l-28.

LeRoy, SF., 1989. Efficient capital markets and martingales,Journal of Ecottomi c Li terature 27, 1583-1621.

Liu, T.. C.W.J. Granger and W. Heller, 1991, Using thecorrelation exponent to decide if an economic series ischaotic, Working paper, Economics Department, Univer-sity of California. San Diego.

Neftci. S.N.. 1991. Naive trading rules in fincancial marketsand Wiener-Kolmogorov prediction theory, Journal ofBusmess. s49-571.

Nelson, C.R. and M.J. Kim, 1990, Predictable stock returns:reality or statistical illusion? Working paper, EconomicsDepartment, University of Washington, Seattle.

Pesaran, M.H. and A.G. Timmerman, 1990, The statisticaland economic significance of the predictability of excessreturns on common stocks, Program in Applied Econo-metrics Discussion paper 26, University of California,Los Angeles.

Taylor. S., 1986. M odelli ng Financial Ti me Serre.s (Wiley,Chichester).

Tong, H.. 1990. Non& w ar Tim e Seri es (Oxford UniversityPress. Oxford).


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C. KJ. Granger / Forecast ing st ock mark et pri ces

Biography Clive GRANGER is Professor of Economics atthe University of California, San Diego having moved from aprofessorship 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 ninebooks and nearly 1.50 papers. He is a Fellow of the Economet-ric Society.

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