Spurious Regressions in Financial Economics?

21 Pages·2003·0.184 MB·English

by  Ferson W.E., Sarkissian S., Simin T.T.

THEJOURNALOFFINANCE(cid:1)VOL.LVIII,NO.4(cid:1)AUGUST2003
Spurious Regressions in Financial Economics?
WAYNEE.FERSON,SERGEISARKISSIAN,andTIMOTHYT.SIMINn
ABSTRACT
Eventhoughstockreturnsarenothighlyautocorrelated,thereisaspurious
regressionbiasinpredictiveregressionsforstockreturnsrelatedtotheclas-
sicstudiesofYule(1926)andGrangerandNewbold(1974).Dataminingforpre-
dictor variables interacts with spurious regression bias. The two e¡ects
reinforceeachother,becausemorehighlypersistentseriesaremorelikelyto
befoundsigni¢cantinthesearchforpredictorvariables.Oursimulationssug-
gestthatmanyoftheregressionsintheliterature,basedonindividualpredic-
torvariables,maybespurious.
PREDICTIVEMODELSFORCOMMONSTOCKRETURNShavelongbeenastaple of¢nancial
economics.Earlystudies,reviewedbyFama(1970),usedsuchmodelstoexamine
markete⁄ciency.Stockreturnsareassumedtobepredictable,basedonlagged
instrumentalvariables,inthecurrentconditionalassetpricingliterature.Stan-
dard lagged variables include the levels of short-term interest rates, payout-to-
priceratios for stock marketindexes,andyieldspreadsbetween low-grade and
high-gradebondsorbetweenlong-andshort-termbonds.Manyofthesevariables
behaveaspersistent,orhighlyautocorrelated,timeseries.
This paper studies the ¢nite sample properties of stock return regressions
with persistent lagged regressors.We focus on two issues.The ¢rst is spurious
regression,analogoustoYule(1926)andGrangerandNewbold(1974).Thesestu-
dieswarnedthatspuriousrelationsmaybefoundbetweenthelevelsoftrending
time series that are actually independent. For example, giventwo independent
random walks, it is likely that a regression of one on the other will produce a
‘‘signi¢cant’’slopecoe⁄cient,evaluatedbytheusualt-statistics.
In this paper, the dependent variables are asset rates of return, which are
not highly persistent.Thus, one may think that spurious regression problems
nFersonisattheCarrollSchoolofManagement,BostonCollegeandisaResearchAssoci-
ate, National Bureau of Economic Research; Sarkissian is on the Faculty of Management,
McGillUniversity;SiminisattheSmealCollegeofBusiness,PennsylvaniaStateUniversity.
WearegratefultoEugeneFamaforsuggestingthequestionthatmotivatesthisresearchand
toJohnCochrane,FrankDiebold,RichardC.Green,GordonHanka,RaymondKan,Donald
Keim, Je¡rey Ponti¡, Bill Schwert, RossenValkanov, and an anonymous referee forhelpful
comments. Ferson acknowledges ¢nancial support from the Pigott-Paccar professorship at
theUniversityofWashingtonandtheCollinsChairinFinanceatBostonCollege.Sarkissian
acknowledges ¢nancial support from FCAR and IFM2.Thispaperhasbene¢ted from work-
shopsatMcGillUniversity,attheJuly2000NBERAssetPricingGroup,the2000Northern
FinanceAssociationMeetings,andthe2001AmericanFinanceAssociationMeetings.
1393
1394 TheJournalofFinance
areunlikely.However,thereturnsmaybeconsideredtobethesumofanunob-
served expected return, plus unpredictable noise. If the underlying expected
returns are persistent time series, there is still a risk of spurious regression.
Becausetheunpredictablenoiserepresentsasubstantialportionofthevariance
of stock returns, the spurious regression results will di¡er from those in the
classicalsetting.
Thesecondissueis‘‘datamining’’asstudiedforstockreturnsbyLoandMac-
Kinlay (1990), Foster, Smith, andWhaley (1997), and others. If the standard in-
struments employed in the literature arise as the result of a collective search
through the data, they may have no predictive power in the future. Stylized
‘‘facts’’about the dynamic behavior of stock returns using these instruments
(e.g.,Cochrane(1999))couldbeartifactsofthesample.Suchconcernsarenatur-
al,giventhewidespreadinterestinpredictingstockreturns.
Wefocusonspuriousregressionandtheinteractionbetweendataminingand
spurious regression bias. If the underlying expected return is not predictable
over time, there is no spurious regression bias, even if the chosen regressor is
highlyautocorrelated.Inthiscase,ouranalysisreducestopuredataminingas
studiedbyFosteretal.(1997).
Whenexpectedreturnsarepersistent,spuriousregressionbiascallssomeof
the evidence of previous studies into question.We examine univariate regres-
sions for theStandard and Poors500 (S&P 500)excess returnusing13 popular
laggedinstrumentsoverthesampleperiodsoftheoriginalstudies.We¢ndthat7
of26t-ratiosorregressionR-squares,signi¢cantbytheusual¢vepercentcriter-
ia,areconsistentwiththenullhypothesisofaspuriousregression.
The spurious regression and data mining e¡ects reinforce each other. If
researchers have mined the data for regressors that produce high‘‘R-squares’’
in predictive regressions, the mining is more likely to uncover the spurious,
persistentregressors.Thestandardregressorsintheliteraturetendtobehighly
autocorrelated, as expected if the regressors result from a spurious mining
process.Forreasonableparametervalues,alltheregressionsthatwereviewfrom
theliterature areconsistent with a spurious mining process, evenwhenonlya
smallnumberofinstrumentsareconsideredinthemining.
This paper contributes to a substantial literature that studies the sampling
properties of predictive regressions for stock returns. Goetzmann and
Jorion (1993), Nelson and Kim (1993), Bekaert, Hodrick, and Marshall (1997),
and Stambaugh (1999) study biases due to dependent stochastic regressors.
Kim, Nelson, and Startz (1991) study structural-change-induced misspeci¢ca-
tion.CampbellandShiller(1988)considerdependentregressorswithunitroots.
Fama and French (1988a), Kandel and Stambaugh (1990), and Hodrick (1992)
focus on autocorrelation for long-horizon stock returns. Lanne (2001) and
Valkanov (2001) develop statistical inference methods in the presence of near
unitroots.PesaranandTimmermann(1995),BossaertsandHillion(1999),Goyal
andWelch (2002), and Simin (2002) examine model selection and out-of-sample
validity. Boudoukh and Richardson (1994) provide an overview of econometric
issues.Schwert(2002)reviewsanomaliesandtradingstrategiesbasedonpredict-
ability.
SpuriousRegressionsinFinancialEconomics? 1395
Therestofthepaperisorganizedasfollows.SectionIdescribesthedata.Sec-
tionIIpresentsthemodelsusedinthesimulationexperiments.SectionIIIpre-
sentsthesimulationresults.First,westudythepurespuriousregressionissuein
isolation.Thenweconsidertheinteractionbetweenspuriousregressionanddata
miningbiases.SectionIVo¡ersconcludingremarks.
I. TheData
TableIsurveysnineofthemajorstudiesthatproposeinstrumentsforpredict-
ingstockreturns.Thetablereportssummarystatisticsformonthlydata,cover-
ingvarioussubperiodsof1926through1998.Thesamplesizeandperioddepends
onthe studyandthevariable, and thetableprovides the details.We attempt to
replicate the data series that were used in these studies as closelyas possible.
Thesummarystatisticsarefromourdata.Notethatthe¢rst-orderautocorrela-
tionsfrequentlysuggestahighdegreeofpersistence.Forexample,theshort-term
Treasurybillyields,monthlybook-to-marketratios,thedividendyieldoftheS&P
500,andsomeoftheyieldspreadshavesample¢rstorderautocorrelationsof0.97
orhigher.
Table I also summarizes regressions for the monthly return of the S&P 500
stock index, measured in excess of the one-month Treasury bill return from
Ibbotson Associates, on the lagged instruments. These are OLS regressions
using one instrument at a time.We report the slope coe⁄cients, their t-ratios,
and the adjusted R-squares.The R-squares range from less than one percent
to more than seven percent, and 8 of the 13 t-ratios are larger than 2.0.
The t-ratios are based on the OLS slopes and Newey^West (1987) standard
errors,wherethenumberoflags is chosenbasedonthenumberofstatistically
signi¢cantresidualautocorrelations.1
The small R-squares inTable I suggest that predictability represents a tiny
fractionof thevariancein stock returns. However, even a small R-squaredcan
signal economically signi¢cant predictability. For example, Kandel and Stam-
baugh(1996)andFleming,Kirby,andOstdiek(2001)¢ndthatoptimalportfolios
respondbyasubstantialamounttosmallR-squaresinstandardmodels.Studies
combiningseveralinstrumentsinmultipleregressionsreporthigherR-squares.
Forexample,Harvey(1989),using¢veinstruments,reportsadjustedR-squaresas
highas17.9percentforsizeportfolios.FersonandHarvey(1991)reportR-squares
of 5.8 percent to 13.7 percent for monthly size and industry portfolio returns.
These values suggest that the‘‘true’’ R-squared, if we could regress the stock
returnonitstime-varyingconditionalmean,mightbesubstantiallyhigherthan
weseeinTableI.Toaccommodatethispossibility,weallowthetrueR-squaresin
our simulations to varyover the range from 0 to 15 percent. For expositionwe
focusonanintermediatevalueof10percent.
1Speci¢cally,wecompute12sampleautocorrelationsandcomparethevalueswithacuto¡
p
ffiffiffiffi
at two approximate standard errors: 2/ T, whereT is the sample size.The number of lags
chosen is the minimum lag length at which no higher order autocorrelation is larger than
twostandarderrors.Thenumberoflagschosenisindicatedinthefarrightcolumn.
1
3
9
6
TableI
CommonInstrumentalVariables:Sources,SummaryStatistics,andOLSRegressionResults
Thistablesummarizesvariablesusedintheliteraturetopredictstockreturns.The¢rstcolumnindicatesthepublishedstudy.Thesecondcolumn
denotesthelaggedinstrument.Thenexttwocolumnsgivethesample(Period)andthenumberofobservations(Obs)onthestockreturns.Columns
5and6reporttheautocorrelation(r )andthestandarddeviationoftheinstrument(s ),respectively.Thenextthreecolumnsreportregression
Z Z
resultsforS&P500excessreturnonalaggedinstrument.Theslopecoe⁄cientisb,thet-statisticist,andthecoe⁄cientofdeterminationisR2.The
lastcolumn(HAC)reportsthemethodusedincomputingthestandarderrorsoftheslopes.ThemethodofNewey^West(1987)isusedwiththe
numberoflagsgiveninparentheses.MA((cid:2))referstothenumberofmovingaveragetermsusedinthecovariancematrix.Theabbreviationsin
thetableareasfollows.TB1yistheyieldontheone-monthTreasurybill.Two-one,Six-one,andLag(two)-onearecomputedasthespreadsonthe
returnsofthetwo-andone-monthbills,six-andone-monthbills,andthelaggedvalueofthetwo-monthandcurrentone-monthbill.Theyieldonall T
h
corporatebondsisdenotedasALLy.TheyieldonAAAratedcorporatebondsisAAAy,andUBAAyistheyieldoncorporatebondswithabelow e
BAArating.Thevariable‘‘Cay’’isthelinearfunctionofconsumption,assetwealth,andlaborincome.Thebook-to-marketratiosfortheDowJones J
o
IndustrialAverageandtheS&P500arerespectivelyDJBMandSPBM. u
r
n
a
(1)Reference (2)Predictor (3)Period (4)Obs (5)rz (6)sz (7)b (8)t (9)R2 (10)HAC lo
f
F
Breen,Glosten,&Jagannathan(1989) TB1y 5404^8612 393 0.97 0.0026 (cid:3)2.49 (cid:3)3.58 0.023 NW(5) in
Campbell(1987) Two^one 5906^7908 264 0.32 0.0006 11.87 2.38 0.025 NW(0) a
n
Six^one 5906^7908 264 0.15 0.0020 2.88 2.13 0.025 NW(0) c
e
Lag(two)(cid:3)one 5906^7908 264 0.08 0.0010 9.88 2.67 0.063 NW(6)
Fama(1990) ALLy^AAAy 5301^8712 420 0.97 0.0040 0.88 1.46 0.005 MA(0)
Fama&French(1988a) Dividendyield 2701^8612 720 0.97 0.0013 0.40 1.36 0.007 MA(9)
Fama&French(1989) AAAy^TB1y 2601^8612 732 0.92 0.0011 0.51 2.16 0.007 MA(9)
Keim&Stambaugh(1986) UBAAy 2802^7812 611 0.95 0.0230 1.50 0.75 0.002 MA(9)
UBAAy^TB1y 2802^7812 611 0.97 0.0320 1.57 1.48 0.007 MA(9)
Kothari&Shanken(1997) DJBM 1927^1992 66 0.66 0.2270 0.28 2.63 0.078 MA(0)
Lettau&Ludvigson(2001) ‘‘Cay’’ 52Q4^98Q4 184 0.79 0.0110 1.57 2.58 0.057 MA(7)
Ponti¡&Schall(1998) DJBM 2602^9409 824 0.97 0.2300 2.96 2.16 0.012 MA(9)
SPBM 5104^9409 552 0.98 0.0230 9.32 1.03 0.001 MA(5)
SpuriousRegressionsinFinancialEconomics? 1397
To incorporate data mining, we compile a randomlyselected sample of 500
potentialinstruments,throughwhichoursimulatedanalystsiftstominethedata
forpredictorvariables.Weselectthe500seriesrandomlyfromamuchlargersam-
ple of 10,866 potential variables. The speci¢cs are described in the Appendix.
Essentially,theprocedureistogenerateuniformlydistributedrandomnumbers,
order the series from1to10,866 and randomlyextract 500 series.The 500 series
are randomly ordered, and permanently assigned numbers between 1 and
500.When a data miner in our simulations searches through, say 50 series, we
use the sampling properties of the 50 series to calibrate the parameters in the
simulations.
We alsouse our sample of potential instruments to calibratetheparameters
thatgoverntheamountofpersistenceinthe‘‘true’’expectedreturnsinthemodel.
On the one hand, if the instruments we see in the literature, summarized in
Table I,arise froma spurious mining process, theyarelikely tobe morehighly
autocorrelated than the underlying‘‘true’’expected stock return. On the other
hand, if the instruments in the literature are a realistic representation of
expected stock returns,theautocorrelationsinTableI maybeagoodproxyfor
thepersistence of the true expected returns.2 The mean autocorrelationofour
500seriesis15percentandthemedianis2percent.Elevenofthe13sampleauto-
correlationsinTableIarehigherthan15percent,andthemedianvalueis95per-
cent.We considera range of values for the true autocorrelationbasedonthese
¢gures,asdescribedbelow.
II. TheModels
Considerasituationinwhichananalystrunsatime-seriesregressionforthe
futurestockreturn,r ,onalaggedpredictorvariable:
tþ1
r ¼aþdZ þv : ð1Þ
tþ1 t tþ1
Thedataareactuallygeneratedbyanunobservedlatentvariable,Zn,as
t
r ¼mþZnþu ; ð2Þ
tþ1 t tþ1
whereu iswhitenoisewithvariance,s2.Weinterpretthelatentvariable,Znas
tþ1 u t
the deviations of the conditional mean return fromtheunconditional mean, m,
wheretheexpectationsareconditionedonanunobserved‘‘market’’information
setattimet.Thepredictorvariablesfollowanautoregressiveprocess:
ðZn;ZÞ0 ¼(cid:2)rn 0(cid:3)(cid:4)Zn ;Z (cid:5)0þ(cid:4)en;e(cid:5)0 ð3Þ
t t 0 r t(cid:3)1 t(cid:3)1 t t
2Therearegoodreasonstothinkthatexpectedstockreturnsmaybepersistent.Assetpri-
cing models like the consumption modelof Lucas (1978) describe expected stock returns as
functions of expected economic growth rates. Merton (1973) and Cox, Ingersoll, and Ross
(1985) propose real interest rates as candidate state variables, driving expected returns in
intertemporalmodels.Suchvariablesarelikelytobehighlypersistent.Empiricalmodelsfor
stock return dynamics frequently involve persistent, autoregressive expected returns (e.g.,
Conrad and Kaul (1988), Fama and French (1988b), Lo and MacKinlay (1988), or Huberman
andKandel(1990)).
1398 TheJournalofFinance
Theassumptionthatthetrueexpectedreturnisautoregressivefollowsprevious
studiessuchasConradandKaul(1988),FamaandFrench(1988b),LoandMac-
Kinlay(1988),andHubermanandKandel(1990).
Togeneratethearti¢cialdata,theerrorsðen;eÞaredrawnrandomlyasanor-
t t
malvectorwithmeanzeroandcovariancematrix,S.Webuildupthetime-series
oftheZandZnthroughthevectorautoregressionequation(3),wheretheinitial
valuesaredrawnfromanormalwithmeanzeroandvariances,Var(Z)andVar(Zn).
Theotherparametersthatcalibratethesimulationsare{m,s2,r,rn,andS}.
u
Wehaveasituationinwhichthe‘‘true’’returnsmaybepredictable,ifZncould
t
beobserved.ThisiscapturedbythetrueR-squared,Var(Zn)/[Var(Zn)þs2].Weset
u
Var(Zn)toequalthesamplevariance of theS&P 500return,inexcessofaone-
monthTreasurybill return, multiplied by 0.10.Whenthe true R-squared of the
simulationis10percent,theunconditionalvarianceofther thatwegenerate
tþ1
isequaltothesamplevariance oftheS&P500return,andthe¢rst-orderauto-
correlationissimilartothatoftheactualdata.Whenwechooseothervaluesfor
thetrue R-squared,these determinethevalues for theparameters2.We set to
u
equalthesamplemeanexcessreturnoftheS&P500overthe1926through1998
period,or0.71percentpermonth.
The extent of the spurious regression bias depends on the parameters r
and rn, which control the persistence of the measured and the true regressor.
ThesevaluesaredeterminedbyreferencetoTableIandfromoursampleof500
potentialinstruments.Thespeci¢csdi¡eracrossthespecialcases,asdescribed
below.
WhilethestockreturncouldbepredictedifZncouldbeobserved,theanalyst
t
usesthemeasuredinstrumentZ.IfthecovariancematrixSisdiagonal,Z and
t t
Znareindependent,andthetruevalueofdintheregression(1)iszero.
t
A. PureSpuriousRegression
Tofocusonspuriousregressioninisolation,wespecializeequation(3)asfol-
lows.ThecovariancematrixSisa2(cid:4)2diagonalmatrixwithvariances(s2,s2).
n
Foragivenvalue ofrn thevalue ofs2 is determinedas s2¼(1(cid:3)r2)Var(Zn).The
n n n
measuredregressorhasVar(Z)¼Var(Zn).Theautocorrelationparameters,rn¼r
areallowedtovaryoverarangeofvalues.(Wealsoallowrandrntodi¡erfrom
oneanother,asdescribedbelow.)
Following Granger and Newbold (1974), we interpret a spurious regression
asoneinwhichthe‘‘t-ratios’’inregression(1)arelikelytoindicateasigni¢cant
relation when the variables are really independent. The problem may come
from the numerator or the denominator of the t-ratio: The coe⁄cient or its
standard error may be biased. As in Granger and Newbold, the problem lies
with the standard errors.3 The reason is simple to understand.When the null
3WhileGrangerandNewbold(1974)donotstudytheslopesandstandarderrorstoidentify
the separate e¡ects, our simulations, designed to mimic their setting (not reported in the
tables), con¢rm that their slopes are well behaved, while the standard errors are biased.
Granger and Newbold use OLS standard errors, while we focus on the heteroskedasticity
andautocorrelation-consistentstandarderrorsthataremorecommoninrecentstudies.
SpuriousRegressionsinFinancialEconomics? 1399
hypothesisthattheregressionsloped¼0istrue,theerrortermu ofregression
tþ1
Equation(1)inheritsautocorrelationfromthedependentvariable.Assumingsta-
tionarity,the slope coe⁄cientis consistent,but standarderrors that donotac-
countfortheserialdependencecorrectlyarebiased.
Because the spurious regression problem is driven by biased estimates
of the standard error, the choice of standard error estimator is crucial. In our
simulationexercises, it ispossibleto¢nd ane⁄cientunbiasedestimator, since
weknowthe‘‘true’’modelthatdescribestheregressionerror.Ofcourse,thiswill
notbeknowninpractice.Tomimicthepracticalreality,theanalystinoursimu-
lations uses the popular autocorrelation-heteroskedasticity-consistent (HAC)
standard errors from Newey and West (1987), with an automatic lag selection
procedure. The number of lags is chosen by computing the autocorrelations
oftheestimatedresidualsandtruncatingthelaglengthwhenthesampleauto-
correlations become ‘‘insigni¢cant’’ at longer lags. (The exact procedure is
described in Footnote 1, and modi¢cations to this procedure are discussed
below.)
ThissettingisrelatedtoPhillips(1986)andStambaugh(1999).Phillipsderives
asymptoticdistributionsfortheOLSestimatorsoftheregression(1),inthecase
where r¼1, u (cid:5)0, and fen;eg are general independent mean zero processes.
tþ1 t t
Weallowanonzerovarianceofu toaccommodatethelargenoisecomponent
tþ1
of stock returns.We assume ro1 to focus on stationary, but possibly highly
autocorrelated,regressors.
Stambaugh (1999) studies a case where the errors fen;eg are perfectly cor-
t t
related,orequivalently,theanalystobservesandusesthecorrectlaggedstochas-
ticregressor.Abiasariseswhenthecorrelationbetweenu anden isnotzero,
tþ1 tþ1
related to the well-known small sample bias of the autocorrelation coe⁄cient
(e.g., Kendall (1954)). In the pure spurious regression case studied here, the
observedregressorZ isindependentofthetrueregressorZn,andu isinde-
t 7 tþ1
pendent of en .The Stambaughbias is zerointhis case.Thepoint isthatthere
tþ1
remainsaprobleminpredictiveregressions,intheabsenceofthebiasstudiedby
Stambaugh,becauseofspuriousregression.
B. SpuriousRegressionandDataMining
We consider the interaction between spurious regression and data mining,
where the instruments to be mined are independent as in Foster et al. (1997).
ThereareLmeasuredinstrumentsoverwhichtheanalystsearchesforthe‘‘best’’
predictor, based on the R-squares of univariate regressions. In Equation (3) Z
t
becomes a vector of length L, where L is the number of instruments through
which the analyst sifts.The error terms ðen;eÞ become an Lþ1vector with a
t t
diagonalcovariancematrix;thus,enisindependentofe.
t t
ThepersistenceparametersinEquation(3)becomean(Lþ1)-square,diagonal
matrix,withtheautocorrelationofthetruepredictorequaltorn.Thevalueofrn
iseithertheaveragefromoursampleof500potentialinstruments,15percent,or
the median value from Table I, 95 percent. The remaining autocorrelations,
denoted by the L-vector r, are set equal to the autocorrelations of the ¢rst L
1400 TheJournalofFinance
instrumentsinoursample of500potentialinstruments,whenrn¼15percent.4
Whenrn¼95percent,werescaletheautocorrelationstocenterthedistribution
at0.95whilepreservingtherangeinthe originaldata.5Thesimulations match
theunconditionalvariancesoftheinstruments,Var(Z),tothedata.The¢rstele-
mentofthecovariancematrixSisequaltos2.Foratypicalithdiagonalelement
n
ofS,denotedbys,theelementsofr(Z)andVar(Z)aregivenbythedata,andwe
i i i
sets2¼[1(cid:3)r(Z)2]Var(Z).
i i i
III. SimulationResults
We¢rstconsiderspuriousregressioninisolation.Thenwestudyspurious re-
gressionwithdatamining.
A. PureSpuriousRegression
Table II summarizes the results for the case of pure spurious regression.We
record the estimated slope coe⁄cient in regression (1), its Newey^West t-ratio,
andthecoe⁄cientofdeterminationateachtrialandsummarizetheirempirical
distributions. The experiments are run for two sample sizes, based on the
extremesinTableI.TheseareT¼66andT¼824inPanelsAandB,respectively.
InPanelC,wematchthesamplesizestothestudiesinTableI.Ineachcase,10,000
trialsofthesimulationarerun;50,000trialsproducessimilarresults.
TherowsofTableIIrefertodi¡erentvaluesforthetrueR-squares.Thesmal-
lestvalueis0.1percent,wherethestockreturnisessentiallyunpredictable,and
thelargest value is15 percent.The columns of Table IIcorrespond to di¡erent
values of rn, the autocorrelationof the true expected return, which runs from
0.00 to 0.99. Inthese experiments, we set r¼rn.The subpanels labeled Critical
t-statistic and Critical estimated R2 report empirical critical values from the
10,000 simulated trials, so that 2.5 percent of the t-statistics or ¢ve percent of
theR-squareslieabovethesevalues.
The subpanels labeled Mean d report the average slope coe⁄cients over the
10,000trials.The meanestimatedvaluesarealwayssmall,andveryclosetothe
truevalueofzeroatthelargersamplesize.Thiscon¢rmsthattheslopecoe⁄cient
estimatorsarewellbehaved,sothatbiasduetospuriousregressioncomesfrom
thestandarderrors.
4Wecalibratethetrueautocorrelationsinthesimulationstothesampleautocorrelations,
adjusted for¢rst-order ¢nite-samplebiasas: rr^ þ (1þ3rr^)/T, where rr^ is the OLS estimate of
theautocorrelationandTisthesamplesize.
5The transformation is as follows. In the 500 instruments, the minimum bias-adjusted
autocorrelation is (cid:3)0.571, the maximum is 0.999, and the median is 0.02. We center the
transformeddistributionaboutthemedianinTableI,whichis0.95.Iftheoriginalautocorre-
lationrislessthanthemedian,wetransformitto
:95þðr(cid:3)0:02Þfð0:95þ0:571Þ=ð0:02þ0:571Þg:
Ifthevalueisabovethemedian,wetransformitto
:95þðr(cid:3)0:02Þfð0:999(cid:3)0:95Þ=ð0:999(cid:3)0:02Þg:
SpuriousRegressionsinFinancialEconomics? 1401
Whenrn¼0,andthereisnopersistenceinthetrueexpectedreturn,thespur-
ious regression phenomenon is not a concern.This is true even when the mea-
sured regressor is highly persistent. (We con¢rm this with additional
simulations,notreportedinthetables,wherewesetrn¼0andvaryr.)Thelogic
is that when the slope in Equation (1) is zero and rn¼0, the regression error
has no persistence, so the standard errors are well behaved. This implies
that spurious regression is not a problem from the perspective of testing the
null hypothesis that expected stock returns areunpredictable, even if a highly
autocorrelatedregressorisused.
Table II shows that spurious regression bias does not arise to any serious
degree,providedrnis0.90orless,andthetrueR2isonepercentorless.Forthese
parameters,theempiricalcriticalvaluesforthet-ratiosare2.48(T¼66,PanelA),
and 2.07 (T¼824, Panel B).The empirical critical R-squares are close to their
theoreticalvalues.Forexample,fora¢vepercenttestwithT¼66(824)theFdis-
tributionimpliescriticalR-squaredvaluesof5.9percent(0.5percent).Thevalues
inTable II when rn¼0.90 and true R2¼1percent, are 6.2 percent (0.5 percent);
thus, the empirical distributions do not depart far from the standard rules of
thumb.
Variableslikeshort-terminterestratesanddividendyieldstypicallyhave¢rst-
ordersampleautocorrelationsinexcessof0.95,aswesawinTableI.We¢ndsub-
stantialbiaseswhentheregressorsarehighlypersistent.Considertheplausible
scenario with a sample ofT¼824 observations where r¼0.98 and true R2¼10
percent. In view of the spurious regression phenomenon, an analyst who was
notsurethatthecorrectinstrumentisbeingusedandwhowantedtoconducta
5percent,two-tailedt-testforthesigni¢canceofthemeasuredinstrumentwould
havetouseat-ratioof3.6.Thecoe⁄cientofdeterminationwouldhavetoexceed
2.2percenttobesigni¢cantatthe5percentlevel.Thesecuto¡saresubstantially
morestringentthantheusualrulesofthumb.
Panel C of Table II revisits the evidence from the literature in Table I.The
critical values for the t-ratios and R-squares are reported, along with the
theoretical critical values for the R-squares implied by the F distribution.
WesetthetrueR-squaredvalueequalto10percentandrn¼rineachcase.We
¢ndthat7ofthe17statisticsinTableIthatwouldbeconsideredsigni¢cantusing
thetraditionalstandards,arenolongersigni¢cantinviewofthespuriousregres-
sionbias.
WhilePanelsAandBofTableIIshowthatspuriousregressioncanbeapro-
blem in stock return regressions, Panel C ¢nds that accounting for spurious
regression changes the inferences about speci¢c regressors that were found to
besigni¢cantinpreviousstudies.Inparticular,we questionthesigni¢canceof
thetermspreadinFamaandFrench(1989),onthebasisofeitherthet-ratioorthe
R-squaredoftheregression.Similarly,thebook-to-marketratiooftheDowJones
index,studiedbyPonti¡andSchall(1998)failstobesigni¢cantwitheithersta-
tistic. Several other variables are marginal, failing on the basis of one but not
both statistics.These include the short-term interest rate (Fama and Schwert
(1977), using the more recent sample of Breen, Glosten, and Jagannathan
(1989)), the dividend yield (Fama and French (1988a)), and the quality-related
1402 TheJournalofFinance
yieldspread(KeimandStambaugh(1986)).Alloftheseregressorswouldbecon-
sideredsigni¢cantusingthestandardcuto¡s.
It is interesting tonotethat thebiases documented inTable IIdo not always
diminishwith larger sample sizes; in fact, the criticalt-ratios arelarger inthe
lowerrightcornerofthepanelswhenT¼824thanwhenT¼66.Themeanvalues
oftheslopecoe⁄cientsareclosertozeroatthelargersamplesize,sothelarger
criticalvaluesaredrivenbythestandarderrors.AsampleaslargeasT¼824is
notby itself acure for the spurious regression bias.This is typical of spurious
regression with a unit root, as discussed by Phillips (1986) for in¢nite sample
sizesandnonstationarydata.6Itisinterestingtoobservesimilarpatterns,even
withstationarydataand¢nitesamples.
Phillips(1986)showsthatthesampleautocorrelationintheregressionstudied
by Grangerand Newbold(1974) converges in limit to1.0. However, we ¢ndonly
mildly in£ated residual autocorrelations (not reported in the tables) for stock
returnsamplesaslargeasT¼2000,evenwhenweassumevaluesofthetrueR2
aslargeas40percent.Evenintheseextremecases,noneoftheempiricalcritical
valuesfortheresidualautocorrelationsarelargerthan0.5.Sinceu ¼0inthe
tþ1
casesstudiedbyPhillips,weexpecttoseeexplosiveautocorrelationsonlywhen
the true R2 is very large.When R2 is small, the white noise component of the
returns serves to dampen the residual autocorrelation.Thus, we are not likely
toseelargeresidualautocorrelationsinassetpricingmodels,evenwherespur-
iousregressionisaproblem.Theresiduals-baseddiagnosticsforspuriousregres-
sion, such as the Durbin^Watsontests suggestedby Grangerand Newbold, are
notlikelytobeverypowerfulinassetpricingregressions.Forthesamereason,
naive application of the Newey^West procedure, where the number of lags is
selectedbyexamining theresidualautocorrelations,is notlikely toresolvethe
spuriousregressionproblem.
NeweyandWest(1987)showthattheirprocedureisconsistentwhenthenum-
ber of lags used grows withoutbound as the sample sizeT increases, provided
thatthenumberoflagsgrowsnofasterthanT1/4.Thelagselectionprocedurein
TableIIexamines12lags.Eventhoughnomorethanninelagsareselectedforthe
actualdatainTableI,morelagswouldsometimesbeselectedinthesimulations,
andaninconsistencyresultsfromtruncatingthelaglength.7However,in¢nite
samples, an increase in the number of lags can make things worse.When‘‘too
many’’ lags are used, the standard error estimates become excessively noisy,
whichthickensthetailsofthesamplingdistributionofthet-ratios.Thisoccurs
6Phillips derives asymptotic distributions for the OLS estimators of equation (1), in the
p
ffiffiffiffi
casewhere r¼1, utþ1(cid:5)0. He shows that the t-ratiofor d diverges forlargeT, while t(d)/ T,
d, and the coe⁄cient of determination converge to well-de¢ned random variables. Marmol
(1998) extends theseresultstomultipleregressionswithpartially integratedprocesses,and
provides references to more recent theoretical literature. Phillips (1998) reviews analytical
toolsforasymptoticanalysiswhennonstationaryseriesareinvolved.
7Atverylargesample sizes,a hugenumberoflags cancontrolthebias.Weverify thisby
examiningsamplesaslargeasT¼5000,lettingthenumberoflagsgrowto240.With240lags,
the critical t-ratio when the true R2¼10 percent and r¼0.98 falls from 3.6 in Panel B of
TableIItoareasonablywell-behavedvalueof2.23.