Forecasting Stock Market Prices: Lessons For Forecasters
InternationalNorth-HollandJournalof Forecasting8 (1992) 3-13Forecasting stock market prices: Lessonsfor forecasters *Clive W.J. G-angerUniversity of California, Sun Diego, USAIn 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/priceratios, 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.Abstract:Keywords: Forecastability,1. Introduction:Stock returns, Non-linear models, Efficient markets.Random walk theoryFor reasons that are probably obvious, stockmarket prices have been the most analysed economic 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, j 2 0. In a sense this hypothesishas to be true. If it were not, and ignoring transaction costs then price changes would be consistently forecastable and so a money machine iscreated and indefinite wealth is possible. How-Correspondenceto: C.W.J. Granger,EconomicsDept., 0508,Univ. of California,San Diego, La Jolla, California,USA92093-0508.* Invited lecture, InternationalInstitute of Forecasters,NewYork Meeting, July 1991, work partly supportedby NSFGrant SES 89-02950. I would like to thank two anonymousreferees for very helpful remarks.0169.2070/92/ 05.000 1992 - ElsevierSciencePublishersever, a deeper theory - known as the EfficientMarket Hypothesis - suggests that mere forecastability is not enough. There are various formsof this hypothesis but the one I prefer is thatgiven by Jensen (1978):HC,2:A market is efficient with respect to information 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 available method of consistently making positive profits is assumed to be in I,.This paper will concentrate on the martingalehypothesis, and thus will mainly consider theforecastability of price changes, or returns (defined as (P, - P, , D,)/P, 1 where D, is dividends), but at the end I will give some consideration 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-B.V. All rightsreserved
C. W.J. Crunger / Forecasting stock market P, ,in finance that the random walk hy(or H,,,) was correct, or at least veryto refute. In a survey in 1972 I wrote,without exceptionempiricalstudies. . ’a model for p, log f’, of the form AP, I,-, t l,where 0 is near zero, 1, contributesonly to thevery low frequenciesand E, is zero mean whitenoise. A survey by Fama (1970) reached a similarconclusion.The informationsets used were:I,,: lagged prices or lags of logged prices.ZZt: Ilr plusvariablesa few sensible possible explanatorysuch as earnings and dividends.The data periods were usually daily or monthly.Further, no profitabletrading rules were found,or at least not reported.I suggested a possiblereportingbias - if a method of forecastingwasfound 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 Grangerand Morgenstern(1970), but this was not a financiallysuccessfulstrategy.However, from the mid-seventiesand particularly in the 1980s there has been a burst of newactivity lookingfor forecastability,using newmethods, data sets, longer series, differenttimeperiods and new explanatoryvariables.What isinterestingis that apparent forecastabilityis oftenfound. An importantreferenceis 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: a stock price, log P,, dividend for period t,D, return (P, D, - P, ,)/P, ,,Rr[In some studies the return is calculated without the dividend term andapproximatedby the change in logprices.] return on a ‘risk free’ investment,rr excess return,R, -rt risk level of the stock,PR, - r, - /3 X marketreturn - cost of transaction risk-adjustedprofits.p,PIThe risk is usually measuredasset pricing model (CAPM):R, - r, p (marketexcess return)from the capital e,,where the market return is for some measure ofthe whole market,such as the StandardandPoor’s 500. p is the non-diversifiablerisk for thestock. This is a good, but not necessarilyideal,measure of risk and which can be time-varyingalthough this is not often consideredin the studies discussed below.Section 2 reviews forecasting models which canbe classified as ‘regime-switching’.Section 3 looksat the advantagesof disaggregation,Section 4considers the search for causal variables, Section5 looks at technicaltrading rules, Section 6 reviews cointegrationand chaos, and Section 7 looksat higher moments.Section 8 concludesand reconsiders the Efficient Market Theory.2. Regime-switchingIf a stationarymodelsseriesX, is generatedX, (Y, ylxI , E,ifz, in AandX, (Ye y*x, , Edifz, not in A,by:then x, can be considered to be regime switching,with z, being the indicatorvariable.If Z is alagged value of x, one has the switching threshold autoregressivemodel (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 forecastablein at least oneregime if either y, or y2 is non-zero.2.a. Forecastabilitywith Low VolatilityLeBaron (1990) used R,, the weekly returns ofthe Standardand Poor 500 index for the period194661985, giving about 2,000 observations.Heused as the indicator variable a measure of therecent volatility10&,,’ c1 0Rf ,
5C. W.l. Granger / Forecasting stock market pricesand the regime of interest is the lowest one-fifthquantile of the observed C? values in the first halfof the sample. The regime switching model wasestimatedusing the first half of the sample andpost-sampletrue one-step forecasts were evaluated over the second half. For the low volatilityregime he finds a 3.1 percentimprovementinforecast mean squared error over a white noisewith non-zeromean (that is, an improvementover a model in which price is taken to be arandom walk with drift). No improvementwasfound for other volatility regimes. He first takescy (the constant)in the model to be constantacross regimes, relaxing this assumptiondid notresult in improved forecasts. Essentially the modelfound isR, cr 0.18R, 1R, cu , ltif have low volatilityotherwise,where LYis a constant. This non-linearmodel wasinitially found to fit equally well in and out ofsample. However, more recent work by LeBarondid not find much forecastingability for themodel.2. b. Earnings and size portfoliosUsing the stocks of all companiesquoted oneither the New York or AmericanStock Exchanges for the period 1951 to 1986, Keim (1989)formed portfolios based on the market value ofequity (size) and the ratio of earningsto price(E/P)and then calculatedmonthly 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 portfolios 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 negative earnings went into a separate portfolio.] Thetable shows the average monthly returns (mean)for five of the portfoliosin each case, togetherwith the .25)(0.22)(0.22)(0.25)(0.28)(1.39)(0.39)Keim (1989).It is seen that the smallest(in size) portfolioshave a substantiallyhigher average return thanthe largest and similarly the highest E/P portfolios are better than the lowest.The two effects were then combined to generate 25 portfolios,five were based on size andeach of these was then sub-dividedinto five partson E/P values. A few of the results are given inthe following table as average monthly returnswith beta risk values shown in )Source:Keim (1989).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-Januarymonths and for January,although 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, transaction costs will be moderatelylarge.The resultsare consistentwith a regimeswitching model with the regime determinedbythe size and E/P variablesat the start of theyear. However, as rankings are used, these variables for a single stock are related to the actualvalues of the variables for all other stocks.
6C. W.J. Changer / Forecasting stock market prices2.c. Seasonal effectsA number of seasonal effects have been suggested but the strongestand most widely documented is the January effect. For example Keim(1989) found that the portfolio using highest E/Pvalues and the smallestsize gave an averagereturn of 7.46 (standard error 1.41) over Januarysbut only 1.39 (0.27) in other months. A secondexample is the observationthat the small capitalization companies(bottom20% 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.Otherexamplesare given in IkenberryandLakonishok (1989). Beta coefficients are also generally high in January.The evidencesuggests that the mean of returns have regime changes with an indicator variable which takes a value of unity in January andzero in other months.2.d. Price reversalsA numberof studies have found that sharesthat do relativelypoorly over one period areinclined to perform well over a subsequentperiod, thus giving price change reversals. A surveyis provided by DeBondt (1989). For example, Dyland Maxfield (1987) selected 200 trading days inrandom in the period January1974 to January1984, each day the three NYSE or AMEX stockswith the greatest percentageprice loss (on average - 12%) were noted. Over the next ten trading days, these losers earn a risk-adjustedreturnof 3.6 percent. Similarly the three highest gainerslost an average1.8% over the next ten days.Otherstudies find similar evidencefor daily,weekly and even monthlyreturns.Transactioncosts will be fairly heavy and a strategy based onthese results will probably be risky.However, Lehman (1990) considereda portfolio whose weights dependedon the return of asecurity the previous week minus the overall return, with positive weights on previous losers andnegative weights (going short) on previous winners. The portfolio was found to consistentlyproduce positive profits over the next week, with veryfew losing periods and so with small risk. Transaction costs were substantialbut worthwhile prof-its were achieved for transactioncosts at a levelappropriatefor larger traders. Thus, after allowing for risk and costs, a portfolio based on pricereversal was found to be clearly profitable.Long term price reversals have also been documented.For example,Dark and Kato (1986)found in the Japanesemarket that for the years1964 to 1980, the three year returns for decileportfolios of extreme previous losers exceed thecomparablereturns of extreme previous winnersby an average 70 percent.In this case the indicatorvariable is the extreme relative loss value of the share. As beforethe apparentforecastabilityleads to a simpleinvestmentstrategy, but knowledge is required ofthe value taken by some variable based on allstocks in some market.2.e. Remolsal of extreme valuesIt is well known that the stock markets occasionally experienceextraordinarymovements,asoccurred in October 1987, for example. Friedmanand Laibson (1989) point out that these largemovementsare of overpoweringimportanceandmay obscure simple patternsin the data. Theyconsiderthe Standardand Poor 500 quarterlyexcess returns (over treasury bills) for the period19541 to 1988IV. After removal of just four extreme values, chosen by using a Poisson model,the remainingdata fits an AR(l)model withsignificant lag coefficient of 0.207 resulting in anR2 value of 0.036. The two regimes are thus the‘ordinary’ excess returns, which seem to be forecastable, and the extra-ordinaryreturns which arenot, from the lagged data at least.3. Benefits of disaggregationA great deal of the early work on stock marketprices used aggregates, such as the Dow Jones orStandardand Poor indices, or portfoliosof arandom selection of stocks or some small groupof individualstocks. The availability of fast computers with plenty of memory and tapes withdaily data for all securities on the New York andAmericanExchanges, for example, allows examination of all the securities and this can on occasion be beneficial.The situation allows cross-section regressionswith time-varyingcoefficients
IC. W.J. Grunger / Forecasting stock market priceswhich can possibly detect regularitiesthat werenot previously available. For example Jegadeesh(1990) uses monthly data to fit cross-sectionmodels of the form12month ahead forecasts.Once transactioncostsare taken into account the potentialabnormalreturns from using P, are halved, but are stillaround 0.45% per month (from personal communication by author of the original study).4. Searchingfor each month. Thus, a lagged average relationship is considered with coefficients changing eachmonth. Here R,, is the average return over a long(four or six years) period which exclude the previous three years. [In the initial analysis, R wasestimatedover the following few years, but thischoice was dropped when forecastingpropertieswere considered.]Many of the averaged aj weresignificantlydifferentfrom zero, particularlyatlags one and twelve, but other average coefficients were also significant,includingat lags 24and 36. A few examples are shown, with t-valuesin brackets.Rfa1a,2aI4all monthsJanuaryFeb. to 4(9)0.08 (5)0.03 0).Source:All monthsJanuaryFeb.-Dec.0.011- 0.0140.024- 0.0200.009-0.017JegadeeshMost of the studies discussed so far have considered forecastingof prices from just previousprices but it is also obviously sensible to searchfor other variables that provide some forecastability. The typical regression isAp, constant p’Kl , (1990).There is thus seen to be a substantialbenefitfrom using the best portfoliorather than theworst one based on the regressions. Benefits werealso found, but less substantialones, using twelveE, ,where & is a vector of plausible explanatory,orcausal variables, with a variety of lags considered.For example Darrat (1990) considereda monthlyprice index from the Toronto Stock Exchange forthe period January1972 to February1987 andachieved a relationship:Ap, tsTA volatility- :;::Aof interestproduction yiE3fA long-termThere is apparentlysome average, time-varying structure in the data, as seen by R: values of10% or more. As noticed earlier, Januaryhasmore forecastabilitythan other months and it wasfound that a group of large firms had regressionswith higher Rf in February to December than allfirms using these regressions(withoutthe Rterms), stocks were ranked each month on theirexpected forecastabilityand ten portfolios formedfrom the 10% most forecastable(P,), second10% and so forth up to the 10% least forecastable(P,,,). The average abnormalmonthlyreturns (i.e. after risk removal) on the ‘best’ and‘worse’ portfolios for different periods werePIP 10for causal variablesrates (t - 1)index (t - 1)interest- 0.015 A cyclically-adjusted(3.0)deficitR2 0.46,Durbin-Watsonrate (t - 10)budget(t - 3))(4.1) 2.01,where only significantterms are shown and themodulusof t-values in brackets.Several othervariables were consideredbut not found to besignificant, including changes of short-termrates,inflation rate, base money and the US-Canadianexchangerate, all lagged once. An apparentlyhigh significanceR2 value is obtainedbut noout-of-sampleforecastabilityis investigated.This search may be more successful if a longrun forecastabilityis attempted.For example Hodrick (1990) used monthly US data for the period1929 to December1987 to form NYSE valueweightedreal market returns,Rr k, over thetime span (t 1, t k). The regressionlog R,tk,t (Ye p,( dividend/priceratio att)
C. W.J. Granger / Forecasting stock market pricesxfound R2 increasingas k increases, up to R2 0.354 at k 48. Thus, apparentlong-runforecastabilityhas been found from a very simplemodel. However, again no post-sampleevaluationis attempted.Pesaran and Timmerman(1990) also employsimple models that produce useful forecastabilityand they also conduct a careful evaluationof themodel. As an example of the kind of model theyproduce, the following equation has as its dependent variable (Y, the quarterly excess return onthe Standard and Poor 500 portfolio:Y - 0.097 t T7: dividendable at the time of the forecast was used inmaking the forecast.(ii) If the predictorwas 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 transaction costs are considered4% and 1%.Investmentyield ( t - 2)MarketstrategySwitchingT-billTransaction- 1.59 inflation(2.8)-? )Irate (t - 3)T-bill (end, 0.025costsInterest rate ofreturnsStandard deviationof returnsWealth at end ofperiod ”t - 1)T-bill (begin,t - 2)(4.6) 0.066A(5.5)twelve monthbond state (t - 1) residual,(4.2)Rf 0.364, Durbin-WatsonHeredividenddividend 2.02.yield at time t ison S&Pprice of S&Pindex ( t - 1)index (t).T-bill is the one month interest rate ‘end’ meansit is measuredat 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 equationare thus effectively 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 reasonableR: value is found, the model canpotentiallybe used for forecasting.[It might benoted that Rz climbed to 0.6 or so for annualdata.]Some experimentationwith non-linearlagged dependentvariablesproducedsome increases in R:, to about 0.39, but this more complicated model was not further evaluated.A simple switching portfolio trading rule wasconsidered:(i) Buy the S&P 500 index if the excess returnwas predictedto be positiveaccordingtoequation(4.2), with the equationbeing sequentiallyre-estimated.Thus only data avail-04%1%9.5 113.3012.396.348.235.435.410.701394373629610595” The period consideredfrom 1960.1 to 1988.IV and thewealth accumulatesfrom an investment of 100 in December 1959.Source: Pesaran and Timmerman(I 990).Although the results presentedare slightly biased against the switching portfolio zero transaction costs are assumed for the alternativeinvestments, the trading rule based on the regression isseen to produce the greatest returnsand as alower risk-level than the market (S&P 500) portfolio. A variety of other evaluationmethods andother regressions are also presentedin the paper.It would seem that dividend/priceratios andinterest rates have quite good long-run forecasting abilities for stock price index returns.5. A new looktrading rulesat old techniques-TechnicalA strategy that is popular with actual speculators, 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 technicalanalysis isdifficult to evaluate, as the rules are not preciseenough. The early literaturedid consider varioussimple rules but generally found little or no forecasting value in them. However, the availability of
C. W.J. Granger / Forecasting stock market pricesfast computers has allowed a new, more intensiveevaluationto occur, with rather different results.Brock, Lakonishokand LeBaron (1991) consider 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.Neftci (1991) investigatesa similar moving average trading rule using different statistical methods and an even longer period - monthly DowJones IndustrialIndex startingin 1792, up to1976. Let M, be an equi-weightedmoving average over the past five months. If P, is the price ofthe index in month t, define a dummy variable:if P, M,given P, , CM, , -1if P, M,given P,-, &I , otherwise.D, 10RegressionresultsP* ,# 5CI P 5j 0Using 90 years of daily data for the Dow-JonesIndex (giving a sample of over twenty-threethousand values) for the period 1897 to 1986, the rulesuggested buying 50% of the time giving an average return next day of 0.00062 (t 3.7) and selling 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 unconditionalaverage return, the ‘buy-sell’r-value is relative to zero. If this buy-sell strategywas used 200 times a year, it gives a return of 20.7percent for the year. However, this figure ignorestransactioncosts, which could be substantial.Thetrading rule was consideredfor four sub-periodsand performedsimilarly for the first three butless well for the most recent sub-periodof 19621986, where the buy-sell strategy produced a dailyreturnof 0.00049. Other similar tradingruleswere consideredand gave comparableresults.Thus, this rule did beat a buy-and-holdstrategyby a significantamount if transactioncosts arenot considered.The authors also consider a muchmore conservativerule, with a fixed ten day holding 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 ignoringtransactioncosts.These, and the results for the other trading rulesconsideredsuggest that there may be regular butsubtle patterns in stock price data, which wouldgive useful forecastability.However,very longseries are needed to investigate these rules.9are presentedfor the equationY D, residual,j 0where the residualis allowed to be a movingaverage of order 17, for each of the three subperiods 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,individuallyand jointly,suggestingsome nonlinearityin the prices. No forecastingexercise was consideredusing the models. Theuse of data with such early dates as 1897 or 1792is surely only of intellectualinterest, because ofthe dramatic institutionalchanges there have occurred since then.Neftci also proves, using the theory of optimalforecasts, that technical trading rules can only behelpful with forecastingif the price series areinherentlynonlinear.6. New techniques- Cointegrationand chaosSince the early statistical work on stock prices,up to 1975, say, a number of new and potentiallyimportantstatistical models and techniqueshavebeen developed. Some arrive with a great flourishand then vanish,such as catastrophetheory,whereas others seem to have longer staying power.I will here briefly considertwo fairly new approaches which have not been successful, so far,in predicting stock prices.An Z(1) series is one such that it’s first difference is stationary.A pair, X,, Y,, of Z(1) seriesare called cointegratedif there is a linear combination of them, Z, X, - AY,, say, which is Z(0).
10C. W.J. Granger/ ForecastingThe properties and implicationsof such series aredescribed in Granger (19861, Engle and Granger(19911, and many other publicationsin econometrics, macroeconomicsand finance. If the seriesare cointegratedthere is necessarily an error-correction data generatingmodel of the form (ignoring constants):,4X, (Y,Z, , laggedAX,,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 dividendsnot helpforecast prices, in agreementwith the efficientmarket hypothesis. However, for the same reasonone would not expect a pair of stock prices to becointegrated,as this would contradictthe efficient market hypothesis.In fact several papershave been produced that claim to find cointegration between pairs of prices or of portfolios, butthe error-correctionmodels are not presentedorthe forecastingpossibilityexplored,and so thiswork will not be surveyed. It should be noted thatcointegrationwould be inconsistentwith thewell-respectedcapitalassetpricingmodel(CAPM) which says that the price P,, of the ithasset is related to the price of the whole marketP,,,, bY3 log P,, b,d log P,, E;, ,whereerr is white noise. Summinglog Pi, bi log P,,cr iover time givese,,t j.j OAs the last term is the accumulationof a stationary series, it is 1(l) (ignoringtrends) and socointegrationshould not occur betweenlog P,,and log P,,,. Similarly, there should be no cointegration betweenportfolios,Gonzalo(1991) hasfound no cointegrationbetween three well knownaggregates,the Dow-JonesIndex, the Standardand Poor 500 Index and the New York StockExchange Equal Value Index.A class of processes generatedby particulardeterministicmaps, such asy, 4y,-,(I-Y,-,)stock market priceswith0 Yo 1 have developeda great deal of interest and canbe called ‘white chaos’. These series have thephysical appearanceof a stochastic independent(i.i.d.1 process and also the linear properties of awhite noise such as zero autocorrelationsand aflat spectrum.The questionnaturallyarises ofwhetherthe series we have been viewing asstochasticwhite noise are actually white chaosand are thus actually perfectly forecastable- atleast in the short run and providedthe actualgeneratingmechanismis known exactly. The literature on chaos is now immense, involves exciting and deep mathematicsand truly beautifuldiagrams,and also is generallyoptimistic,suggesting that these processes occur frequently.Infact, a clear case can be made that they do notoccur in the real world, as opposed to in laboratory physics experiments.There is no statisticaltest that has chaos as the null hypothesis. Therealso appears to be no characterizingproperty of achaotic process, that is a property that is true forchaos but not for any completelystochastic process. Theseargumentsare discussedin Liu,Granger and Heller (1991). It is true that somehigh-dimensionalwhite chaotic processes are indistinguishablefrom iid series, but this does notmean that chaos occurs in practice. In the abovepaper, a number of estimates of a statistic knownas the correlationdimensionare made for variousparametervaluesusing over threethousandStandard and Poor 500 daily returns. The resulting values are consistentwith stochasticwhitenoise (or high dimensionalchaos) rather than lowdimensional- and thus potentially forecastablewhite chaos. A little introspectionalso make itseem unlikely to most economiststhat a stockmarket, which is complex, involving many thousands of speculations,could obey a simple deterministic model.7. Higher momentsTo make a profit, it is necessary to be able toforecast the mean of price changes, and the studies reviewed above all attemptto do this. Theefficient market theory says little about the fore-
C. W.J. Granger/ Forecasting stock market pricescastabilityof functionsof price changes or returns, such as higher moments. If R, is a return ithas been found that Rf and is clearly forecastable and IR, I even more so, from laggedvalues. Taylor (1986)
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
Forecasting prices in stock markets is a matter of great interest both in the academic field and in business. The forecasting of stock prices and stock returns is possible using various techniques and methods. Many researchers study price trends in stock markets with the help of artificial neural networks [1-2] or fuzzy-trends [3, 4]. The
Introduction to Forecasting 1.1 Introduction What would happen if we could know more about the future? Forecasting is very important for: Business. Forecasting sales, prices, inventories, new entries. Finance. Forecasting financial risk, volatility forecasts. Stock prices? Economics. Unemplo
This research tries to see the influence of G7 and ASEAN-4 stock market on Indonesian stock market by using LASSO model. Stock market estimation method had been conducted such as Stock Market Forecasting Using LASSO Linear Regression Model (Roy et al., 2015) and Mali et al., (2017) on Open Price Prediction of Stock Market Using Regression Analysis.
Forecasting Stock Price Turning Points in the Tehran Stock Exchange Using Weighted Support Vector Machine. Journal of Entrepreneurship Education, 25(5), 1-12 . 2 1528-2651-25-5-797 Citation Information: Sayrani., M & Sharif, J.S. (2022). Forecasting Stock Price Turning Points in the Tehran Stock Exchange Using Weighted Support Vector Machine. .
between stock market and oil prices is still growing. Nevertheless, there are very few studies on the dynamic correlation between these two markets. A first approach on the dynamic co-movements between oil prices and stock markets was performed by Ewing and Thomson (2007), using the cyclical components of oil prices and stock prices.
stock prices up, but the discount rate effect prevails eventually, pushing the stock prices down. The resulting pattern in the new economy stock prices looks like a bubble but it obtains under rational expectations through a general equilibrium effect. The bubble-like pattern in stock prices arises in part due to an ex post selection bias.
Recently forecasting stock market return is gaining more attention, because of the fact that if the direction of the market is successfully predicted the investors may be better guided and also monetary rewards will be substantial. In recent years, many new methods for the modeling and forecasting the stock market has been developed [14].
measures used to proxy for stock market size and the size of real economy. Most of the existing studies use stock market index as a proxy for measuring the growth and development of stock market in a country. We argue that stock market index may not be a good measure of stock market size when looking at its association with economic growth.
