Journal of Empirical Finance 8 493 535 www.elsevier.comrlocaereconbase he power and size of mean reversion ess Ken Daniel ) Kellogg School of Managemen, Norhwesern UniÕersiy, Sheridan Road, EÕanson, IL 68-6, USA Keywords: Mean reversion ess; Power; Sock prices. Inroducion he power of mean reversion ess has long been a aci issue of he marke efficiency lieraure. Early ess of marke efficiency, as summarized in Fama Ž 97., found no economically significan evidence of serial correlaion in sock reurns. However, Summers Ž 986. laer suggesed ha his was because hese ess lacked power: Summers suggesed a model of AfadsB in which sock prices ake long swings away from heir fundamenal values, and showed ha even if a fads componen such as his accouned for a large fracion of he variance of reurns, he fads behavior migh be difficul o deec by looking a shor horizon auocorrelaions of reurns as hese early ess had done. he inuiion behind Summers reasoning was ha if sock prices ook large jumps away from heir AfundamenalB or full-informaion values, and hen only revered back owards he fundamenal price over a period of years, he auocorrelaions of monhly or daily reurns would capure only a small fracion of his mean reversion. Several aemps were made o develop ess ha would have greaer power agains AfadsB hypoheses such as Summers. Fama and French Ž 988a. used a long horizon regression of muli-year reurns on pas muli-year reurns, and Poerba and Summers Ž 988. used a variance raio es o look for fads-ype behavior in sock-index reurns. In addiion, variance raio es are used by Cochrane Ž 988. and Lo and MacKinlay Ž 988. o invesigae he ime series properies of producion and shor horizon reurns. ) el.: q-847-49-43. E-mail address: kend@nwu.edu Ž K. Daniel.. 97-5398rr$ - see fron maer q Elsevier Science B.V. All righs reserved. PII: S97-5398 38-X
494 K. DanielrJournal of Empirical Finance 8 493 535 Boh Fama and French and Poerba and Summers develop inuiion for why hese long horizon ess should have more power o deec fads ype behavior, and some effor has since been made o boh verify and formalize his inuiion. Lo and MacKinlay Ž 989. use Mone-Carlo mehods o compare he power of he variance raio, Box Pierce Q, and he Dickey and Fuller Ž 979. -ess. Jegadeesh Ž 99. used he approximae slope mehod Ž Badahur, 98; Geweke, 98. o evaluae he power of a generalized long horizon regression, and Richardson and Smih Ž 99. use his mehod o evaluae he power of he variance raio es and long horizon regression agains specific alernaives. Hodrick Ž 99. and Campbell Ž 99. propose similar analyses for mulivariae ess. However, hese papers are all comparisons of power across a discree se of ess and for a specific mean revering alernaive; none presens a mehod for deermining he mos powerful es or suggess how far heir ess migh be from opimal for he specified alernaive. Moreover, lile or no inuiion is provided as o he robusness of hese resuls wih respec o changes in he alernaive hypohesis. In his paper, we develop a mehodology for deermining asympoic es power. his mehod allows us Ž. o deermine he mos powerful es agains a specified alernaive; Ž. o deermine he disance of a es from he opimal es using an analyical measure of es power and Ž. 3 o deermine he implici alernaive o any es. Moreover, he sraighforward geomeric inerpreaion of es power we presen faciliaes consideraion of es robusness issues. I is imporan o noe here ha wha we presen a mehod for consrucing an opimal es once he alernaiõe hypohesis has been deermined. We do no rea he problem of acually specifying he alernaive hypohesis, which is very difficul problem, and is probably he reason ha so many ad hoc ess have been used in he finance field. Noneheless, in he debaes over wha ype of es is appropriae, es power is ofen an issue ha is ignored, or is addressed using Mone-Carlo mehods ha are no robus o small changes in he alernaive hypohesis. he mehod we presen here does allow us o address hose quesions. he mehodology we develop is applicable o all momen resricion ess where he insrumen is a linear combinaion of pas reurns. his class encompasses he long horizon regression es, he variance raio es, weighed specral ess Ž Durlauf., and insrumenal variable and generalized mehod of momens Ž GMM. ess involving pas reurns. Since our analyical es-power resuls are valid only asympoically and under local alernaives, we validae hese resuls for small sample and nonlocal alernaives using Mone-Carlo experimens. We find ha he asympoic resuls exend he implici alernaive of a es is ha alernaive agains which he es is he mos powerful, which we shall discuss laer.
K. DanielrJournal of Empirical Finance 8 493 535 495 well o small samples bu also show ha wo asympoically equivalen ess may have differen small sample properies. his es-power deerminaion mehod exends naurally o he consideraion of join ess of momen resricions. his issue is of imporance in he finance lieraure: in aemping o characerize a ime series of reurns, a common approach in he finance lieraure is o run a se of ess in order o deermine he ime series properies of he reurns series. For example, Fama and French run a se of eigh long horizon regressions a reurn horizons of,, 3, 4, 5, 6, 8 and years. Poerba and Summers Ž 988. perform variance raio ess for similar horizons. Boh find evidence of mean reversion a some horizons. However, as Richardson Ž 993. poins ou, he significance of hese resuls mus be based on he join significance of all ess. Richardson and Smih Ž 99. sugges calculaing he join significance by forming a x saisic where he variance covariance marix of he sample regression coefficiens is calculaed under he null hypohesis. A similar approach is adoped by Jegadeesh Ž 99. and ohers. However, we show ha a x join es of his form will have very low power, even if he individual ess are all powerful agains he alernaive. One way of inerpreing he Fama and French and Poerba and Summers ess is ha a number of horizons were used because he researchers had he alernaive ha reurns were mean revering, bu were unsure of he degree of persisence of he mean revering componen. hey, herefore, sudied a se regressions Žor variance raios. ha bounded he range of mean reversion raes hey expeced o see, and esimaed he rae of mean reversion by deermining he reurn horizon a which he regression coefficien was mos significan. his would have been a saisically correc procedure had hey correced for he fac ha hey had searched over a large number of regression coefficiens. his would have been similar o a procedure in which he mean reversion coefficien was esimaed from he daa, and hen, using his parameer esimae Ž for example in a GMM seing., a es was conduced of wheher he variance of he mean revering componen was significanly differen from zero. While a es such as his would have power ha is independen of he number of regressions run, he power of he x es of he join significance of he regression coefficien decreases as he number of regressions Ž variance raios. increases. hus, his mehod of esing is inherenly saisically weak. his resul is verified using Mone-Carlo Sudies. Finally, since our analyical es-power resuls are valid only asympoically and under local alernaives, we also conduc Mone-Carlo experimens o invesigae he robusness of hese resuls for small sample sizes and for nonlocal alernaives. We find ha he resuls are generally robus, bu we also explore siuaions where he asympoic heory will lead o incorrec conclusions. We exend he resuls of his secion o show how small differences in he small sample properies of a es can lead o srikingly differen saisical inferences. We show ha he long horizon regression, which uses analyical sandard errors as proposed by Richardson and Smih Ž 99., suffers from low power agains simple mean revering
496 K. DanielrJournal of Empirical Finance 8 493 535 alernaives, and ha his, no poor small properies, is he reason Richardson and Sock Ž 989. find no evidence of mean reversion using his es. We empirically calculae he small-sample correced disribuion for he Fama and French -saisics, which are based on Hansen and Hodrick Ž 98. calculaed sandard errors, and show ha here is sill a good deal of evidence in favor of a mean reversion hypohesis. We show why a es based on he HansenrHodrick based -saisic is more powerful even hough he wo es are asympoically equivalen. We proceed by showing ha all of hese are asympoically equivalen o weighed auocorrelaion ess, and develop he resul ha, for univariae ess, he mos powerful es saisic is ha which is a weighed sum of sample auocorrelaions a differen lags, for which he weighs are proporional o he expeced auocorrelaion under he alernaive hypohesis. he inuiion behind his mehod is sraighforward, and is based on he fac ha under he null hypohesis, he vecor of sample auocorrelaions a differen lags is asympoically mean zero, and is mulivariae-normally disribued wih a variance covariance marix V sž r. P I. In oher words, sample auocorrelaions a differen lags have he same variance and are uncorrelaed. If one changes he hypohesis from he null o he local alernaive hypohesis ŽDavidson and MacKinnon Ž 987.. Ž i.e., if he serial correlaion is small., he mean of he sample auocorrelaion vecor will shif in he direcion of he alernaive bu he variance covariance marix of sample auocorrelaions will remain he same. Given hese null and alernaive auocorrelaion disribuions, we show ha he mos powerful es saisic is a linear combinaion of sample auocorrelaions where he weighing vecor is proporional o he vecor of expeced sample auocorrelaions. One of he virues of wriing hese ess as weighed auocorrelaion ess is ha i leads o simple geomeric inerpreaion of es power, which we provide in Secion.3. We show in Secion.5. ha he weighed auocorrelaion es can jus as easily be wrien in he specral domain as a weighed periodigram es, wih an analogous resul ha he opimal es will have weighs proporional o he expeced periodigram under he alernaive. his es has same opimaliy properies as he weighed auocorrelaion es. We also show anoher version of he opimal es is one ess he orhogonaliy of he curren reurn o he opimal predicor of he curren reurn, based on he alernaive hypohesis. wo oher papers explore he opic of deermining an opimal es. Faus Ž 99. presens a mehod for deermining he opimal filered variance raion es based on maximum likelihood mehods. Perhaps mos closely relaed o his paper is Richardson and Smih Ž 994., which develops a general mehod for deermining he opimal es given a mean-revering alernaive. Using he approximae slope mehod as a measure of es power, Richardson and Smih reach conclusions on he opimal es saisics, which are similar o hose we presen in Secions. and.. In addiion, Richardson and Smih compare he power of heir opimal es for he Summers fads alernaive o he long horizon regression es, o he variance
K. DanielrJournal of Empirical Finance 8 493 535 497 raion es, and o he Jegadeesh Ž 99. regression boh asympoically Žusing he approximae slope measure. and in small-samples using Mone Carlo mehods. he paper is organized as follows. Secion develops he weighed auocorrelaion es and proves is opimaliy, and exends his developmen o he specral domain and o calculaion of he opimal insrumen. Secion 3 demonsraes he equivalence of commonly used mean reversion ess o weighed auocorrelaion ess, and invesigaes heir opimaliy and he implici alernaives of hese ess. Secion 4 exends he analysis o join es of resricions, and Secion 3.4 presens Mone-Carlo resuls on he small sample power of he ess. Secion 5 reexamines he Fama and French Ž 988b. long horizon es for mean reversion in ligh of his evidence. Secion 6 concludes he paper.. he opimal univariae ess he weighed auocorrelaion es In his secion, we derive he asympoic properies of he weighed auocorrelogram es and show ha his es is asympoically a uniformly mos powerful es agains a local alernaive for which he reurn generaing process can be described by an ARMA model. By uniformly mos powerful, we mean ha for any significance level Ž or probabiliy of ype I error. seleced by he economerician, he probabiliy of ype II error is minimized. We also provide a simple geomeric illusraion of he power of he es... he local alernaiõe hypohesis We begin wih a Piman sequence of local daa, or reurn, generaing processes Ž DGPs.: y 4 r smqa a qu. where u and a are given by u ;IID,s Eu 4 shs 4 -`. u u u u Ž L. a sf Ž L. e 3. e ;IIDŽ,se. Ee 4 shs e e 4 -` 4. EŽ u ey. s ; 5. fž L. and už L. are finie-order lag polynomials, and fž z. rwž y z. už z.x has roos ouside he uni circle. his reurn is seen o be composed of wo componens, he u componen, which is a differenced maringale, and he a or AalernaiveB componen, which has as ARMA represenaion. We assume ha he correlaion of e and u is zero. way. his assumpion is no criical in ha any ARMA process for reurns can be decomposed in his
498 K. DanielrJournal of Empirical Finance 8 493 535 a is he parameer which deermines how close he local alernaive is o he null hypohesis of whie noise reurns. Noice ha null hypohesis is nesed wihin he alernaive in he sense ha when as he null is rue, and when a is any value oher han zero, he alernaive is rue. Here, he reurn generaing process under he null hypohesis is allowed o be nonnormal, bu mus have a finie fourh momen. 3 Eq.. represens a sequence local DGPs: as he sample size of increases, he variance of mean-reversion componen grows smaller. he facor of yr4 in he reurn generaing process is chosen so ha, given a fixed size, he power of he 4 es will converge o some value in, as `. In he ineres of racabiliy, we mus deal wih asympoic power, ha is he power of he es `. However, if we were o increase wihou changing he imporance of he mean-revering componen, he power would always go o one as `. o allow asympoic power analysis, i is necessary o modify he alernaive hypohesis as grows, o move i AcloserB o he null so ha he asympoic probabiliy of rejecion under his local DGP is in Ž,.. As we show laer, his ype of convergence will occur only wih an exponen of yr4. Given he definiions in Eqs.. 5., he covariogram of he reurns series r is given by: f Ž z. f Ž z y. a c se a aq sse H z d z 6 y. G u Ž z. u Ž z. where G is he uni circle in he complex plane. We wrie he auocorrelaion esimaor for he r series as: ĉ Ýrr y rˆ s s 7. ĉ Ýr Expanding he numeraor yields: y y 4 4 Ý y Ý y y s s cˆ s rr s a a qu a a qu 3 Richardson and Smih Ž 99. also show ha heir opimal es is robus o limied kinds of heeroskedasiciy. 4 In he work of Davidson and MacKinnon Ž 987. and ohers concerning local alernaive hypoheses, his is usually a facor of y r. However, his is in a regression framework where only he DGP for he dependen variables varies wih. In our framework, where reurns are boh he dependen and independen variables, we wan he produc of hese wo o move owards he null a a rae of y r, so each par individually mus move a he rae of y r4.
K. DanielrJournal of Empirical Finance 8 493 535 499 ž / ac ž ' w y x/ a cˆ s aa q uu ' Ý Ý y y ^ ` _ ^ ` _ a asy 4 y ; N Ž,su. asy ; N,O ž / y 4 qa Ýauy q Ýay u 8. ^ ` _ y asy ; N,O 3 a c a By Eq. Ž A.3., he firs erm has an expeced value of and a variance of ' Ž 4 a r. Õ s O Ž y.. By Eq. Ž A.4. p, he second erm has a mean ha is asympoically zero and a variance of su 4 y. he expecaion of he las erm is zero since u and a are mean zero and independen. he asympoic variance of his erm is herefore: 5 y Ý y y q y sa E a u qa u q a u a 5 3 pž 3 y y y a a a u u u a sa s s qs c sa s s qc so /.9 he plim ` of he denominaor is su while, based on he cenral limi heorem, he numeraor ends o a sum of normally disribued random variables. Given his, he disribuion of he sample auocorrelaion is given by: ž / u asy a c a r ˆ ;N,. s ' he covariance of he auocorrelaion esimaor a differen lags is obained by performing erm by erm muliplicaion of he hree in he expansion of he
5 K. DanielrJournal of Empirical Finance 8 493 535 expansion of he covariance esimaor c A, B, and C. We hen have ha Cov c ˆs,cˆ se cˆs cˆ ye cˆs E cˆ ˆ se A PA qb PA qc PA s s s qa PB qb PB qc PB s s s qa PC qb PC qc PC s s s in Eq. 8.. Denoe hese hree erms by Independence of u and a plus assumpion. guaranees ha he expecaions of he APC, BPC and BPBPerms are zero. Expansion of he C PC s in a manner similar so ha in Eq. 9. above yields: E C PC s a s E a a u qa a u qa a u qa a u 3r asy a su a a s Ž c qc. 3r qs q ys q qs y ys y sy sq Ž 4. a a and from Eq. A.3, he expecaion of he remaining erm, As PAy a r ccs is asympoically: a a E A PA y cc s Õ 4 4 asy a a s s s and summing he las wo erms gives: asy a su a a Cov c ˆ,cˆ se cˆ cˆ ye cˆ E cˆ s Ž c qc. s s s 3 sy sq which is O Ž y3r4. p. Combining his resul wih he fac ha plim ` of he denominaor of Eq..7 is s and wih Eq. 8. yields: ž / a asy ˆ s u u a c ' r ;N,I... he weighed auocorrelogram es saisic We now proceed o find he mos powerful es. We proceed by firs deriving he opimal es among he class of ess ha are linear funcions of auocorrelaions, and hen showing in Secion.4 ha his linear es is opimal among all funcions of he auocorrelaions. Since he auocorrelaions Ž plus he variance.
K. DanielrJournal of Empirical Finance 8 493 535 5 summarize he properies of any series ha has an ARMA represenaion, his es will be globally opimal. We define he weighed auocorrelogram es saisic as: ž Ý / Âs w ˆ r. where, wihou loss of generaliy, he lengh of he vecor of weighs is normalized o one: Ýw s 3. From Eqs..7 and. we have ha: a Ý Ý asy a Â;N wc, w s u asy a ' ˆ A;N wc, a ž Ý /.4 s u Noice ha boh he mean and he variance of he disribuion of ' Aˆ are independen of. his means ha he probabiliy of rejecion as ` is in Ž,. yr4. Had we wrien he DGP in Eq.. wih an exponen of, his would no have been he case. If our alernaive hypohesis does no sugges a sign for a, we will use he es saisic A ˆ, which based on Eq. 4., has a noncenral x disribuion wih one degree of freedom and wih noncenraliy parameer NCP. a 4 a Ý NCP s wc 5. s u 4 ž / Since under he null hypohesis a s his saisic has a cenral x disribuion, o maximize he power of he es under he local alernaive represened by he ã DGP in Eq.., he weigh v mus be chosen o maximize he noncenraliy parameer Ž NCP., subjec o normalizaion consrain ha he sum of he squares of he weighs equals. he inuiion behind his resul is illusraed in Fig., where x densiy funcions wih NCPs of,, and 4 are ploed. Since he es saisic Aˆ for any se of weighs saisfying Eq..3 has a cenral x disribuion under he null, a single criical value will give all ess he same size. For example, a criical value of x ) s3.84 gives all ess a size of 5%. Maximizing he power of he es is hen equivalen o choosing he es for which i is mos likely ha he es saisic will
5 K. DanielrJournal of Empirical Finance 8 493 535 Fig.. es power as a funcion of he noncenraliy parameer. exceed he criical value of 3.84 given he alernaive is rue. In oher words, we need o find he value of he NCP which maximizes he inegral ` H x ) x Ž NCP.Ž x. d x Because a x disribuion wih a larger NCP firs-order sochasically dominaes a x wih a lower NCP, he es which has he highes NCP will always maximize his inegral, regardless of he size or criical value we choose. o deermine he se of weighs which maximizes he NCP, we solve he Lagrangian: Ý ž Ý / Ls wc a yl w y aking he firs-order condiions gives he opimal weighs: E L c a ) s w s Ew l E L ) s Ýw s El or, simplifying: ) w s c a a Ýc (
K. DanielrJournal of Empirical Finance 8 493 535 53 ha is, he opimal weighs are proporional o he auocorrelaion expeced under he alernaive hypohesis. Noe also ha, given a se of weighs, we can recover he implici alernaive of a es, which is he alernaive agains which he es has he greaes possible power. his can be useful in providing some inuiion as o wha sor of alernaives a given es will have power agains. In Secions 3. and 3.3, we will examine he implici alernaives of variance raio and long horizon regression es saisics. Finally, he power of a weighed auocorrelogram es agains a specified alernaive can be summarized by he parameer ž Ý wc a / cos Cs 6. a w c ž Ý /ž Ý / Using his parameer, he NCP as given in Eq..5 can be wrien as: NCPs a ž 4 Ýc a / 4 s u cos C he geomeric inerpreaion of his es saisic is explored in he nex secion Secion.3. From his equaion i is clear ha when he value of cos C is, he es will be an opimal es, and when he value is zero, here will be no power agains he alernaive 5, as will be explained in more deail in Secion.3..3. A geomeric inerpreaion of he weighed auocorrelogram es Before we prove he general opimaliy of he auocorrelaion es, i is useful o consider a simple geomeric inerpreaion of he es power resuls from he previous secion. Firs, noe ha he se of r s ˆ a differen lags can be expressed as a vecor in a p-dimensional space Žwhere p is he number of nonzero weighs in he es saisics.. In his coordinae sysem, he componen of he sample X auocorrelaion Ž ˆ r. vecor would be ˆ r sž ˆ r, r..., r. ˆ ˆp. Under boh he null and alernaive hypoheses, he p-vecor ˆ r is disribued spherically, ha is X E Ž rye ˆ w ˆ rx.ž rye ˆ w ˆ r x. s I where I is he p=p ideniy marix. However, under he null hypohesis, i is disribued abou he origin and under he local alernaive hypohesis r, ˆ iis 5 Where by Ano powerb, we mean ha he es has no power o discriminae beween he null and alernaive, or alernaively ha he disribuion of he es saisic is he same under he null as under he alernaive hypohesis.
54 K. DanielrJournal of Empirical Finance 8 493 535 cenered a he poin yr ac a, where c a is he p-vecor of alernaive auocorrelaax ion, i.e., c sžc a, c a..., c a. p. he es saisic Aˆ sž Ý. w ˆ r is herefore he square of he lengh of he projecion of ono he vecor of weighs w, where wsž w, w..., w. X p, based on our resricion ha he lengh of w is. he lengh of his projecion will be normally disribued as in Eq..4, wih mean a u ' a Ýwc. Rewriing his in s erms of he vecor we have defined. We have: < a asy a c < ' A;N ˆ cosc, 7. ž s u / a where C is he angle beween w and c, as is illusraed in Fig. and < c a < denoes a he lengh of he vecor c. Again, he es saisic Aˆ will be noncenral x ŽŽ 4 < a <. 4. disribued wih NCP s a c rsu cos C. hus, o maximize he NCP, we wan he vecor of weighs o poin in he same direcion as he vecor of expeced auocorrelaions, as his resuls in a C of zero and he maximum achievable value of cos C. On he oher hand, if CsŽ pr., hen we are looking in a direcion perpendicular o ha in which we expec o see deviaions, and he es will have no power. Fig.. A geomeric inerpreaion of he weighed auocorrelogram es.
K. DanielrJournal of Empirical Finance 8 493 535 55.4. Proof of opimaliy for a general class of funcions of auocorrelaions So far, we have only shown ha his es is opimal among he class of es which are linear funcions of he se of sample auocorrelaions. We now show ha his resul holds for all funcions of he p-vecor of sample auocorrelaions. p Firs noe any es ˆ r : R accep H, rejec H 4 is a mapping from he vecor of auocorrelaion o a binary choice variable. herefore, we can describe he es by he rejecion region V;R n, which is he se of auocorrelaion vecors rgr ˆ n, which are mapped ino rejec. Specifying he globally opimal es is equivalen o specifying he rejecion n,a4 region V such ha he probabiliy of ype I error is minimized. Leing f Ž P.: R n R denoe he probabiliy densiy funcions under he null and he alernaive and Vdenoe he complemen of V or he accepance region, his opimizaion problem can be wrien as: H max f a Ž ˆ r. d r such ha f n Ž ˆ r. d rsa H V V V Differeniaing he Lagrangian yields a firs-order condiion for maximizaion: ha on he boundary of he region, which we denoe by z;r ny, he raio of he densiy funcions under he alernaive is a consan: a f n f Ž ˆ r. Ž ˆ r. ˆ rgz sl 8. o prove he opimaliy of he linear weighed auocorrelaion es, we need o show ha he manifold z is defined by w X ˆ rsl;rgz ˆ for some w. o show his, we noe ha under he assumpions given in Eqs.. 5., he auocorrelaion vecor ˆ r is asympoically disribued mulivariae normal wih a variance covariance marix equal o s I, and ha, herefore, he disribuions under he null and alernaive are given by: n X f Ž ˆ r. s exp y ˆ r I ˆ r n ž ( Ž / p. s s and / a a X a f Ž ˆ r. s exp y Ž ryr. IŽ ryr ˆ ˆ. n ( Ž ž p. s s where r a is he vecor of auocorrelaions under he alernaive hypohesis. Now define he idempoen marix M as r a r ax Ms Iy X ž a a Ž r r./
56 K. DanielrJournal of Empirical Finance 8 493 535 Wih his we can wrie he log of he raio of he alernaive and null probabiliy densiy funcions as: f a Ž ˆ r. a X X a ys log s Ž ryr. MMŽ ryr n ˆ ˆ. f Ž ˆ r. ž / which afer some simplificaion, becomes: f a Ž ˆ r. ˆ r X r a ys log sy n X f Ž r. r r ž / r a r ž ax a a Ž r r./ r a r ax a X a qž ˆ ryr. X Ž ryr a a ˆ. Ž r r. X X X ˆ ˆ ˆ ˆ qr MMrqr X r ž ˆ / ž / a a We wan o find he value ˆ r of which makes his equal o l. he value of ˆ r ha saisfies his resricion is: yl rs ˆ ž / ra Since his resricion is equivalen o he linear resricion derived earlier, his means ha he linear resricion is opimal..5. Oher forms of he opimal es.5.. he specral domain: an opimal weighed periodigram es Durlauf Ž 99. proposes a specral based mehod of assessing wheher a ime series is a maringale. Basically, his mehod involves looking a he periodigram of he firs differences of he series: under he null hypohesis ha he series is a random walk, he expecaion of he specral densiy should be everywhere equal o Žs Ž.. x rp. hus, asympoically, he periodigram should be iid wih mean Žs Ž.. rp, and based on his he expecaion of he funcion x ž / l sxž. G Ž l. sh I Ž v. y dv p is zero for all l under he null hypohesis. GŽ l. is he Acumulaed periodigram.b By definiion, i will be equal o zero a ls and a lsp, and asympoically, i obeys a Brownian bridge process on w, x under he null hypohesis. Durlauf also suggess ha if A... a researcher believes ha he alernaive o he maringale model is a long-run mean reversion, maximizing es power migh dicae an examinaion of he low frequencies.b In his secion, we show how Durlauf s inuiion can be formalized, and how an opimal es in he specral domain can be consruced.
K. DanielrJournal of Empirical Finance 8 493 535 57 We show ha since he periodigram can be hough of as jus a represenaion of he auocorrelogram in anoher basis, he same inuiion will apply here: he researcher should apply weighs o he periodigram esimaes which are proporional o he expeced periodigram under he alernaive. he periodigram esimae of he specral densiy is given by: y I Ž v. s sˆ Ž j. e yijv Ý p jsyž y. x where sˆ Ž j. denoes he sample auocovariogram a lag j. Since s Ž j. s Ž yj. x ˆx x, his can be rewrien as: y yi jv ijv Ý ˆx ˆx p ž js / I v s s j e qe qs Consider he following modified specrum: I Ž v. y X yijv ijv IŽ v. s y s Ý ˆ rxž j.ž e qe. s Ž. p p ˆx js If we define he quaniy: fž j,v. s Ž e ijv qe yijv. scosž jv. we see ha he modified specrum is given by: y X I Ž v. s fž j,v. ˆ r Ž j. Ý p js x For v sž kp. r, kg, y 4, f Ž P. has he following properies: k y E fž j,v. fž j,v. s k/l Ý k l ½ ksl js Using his propery and he fac ha, asympoically, asy ' r;n ˆ Ž,I. we have ha: X E I Ž v. s;kg k k/l X X E I Ž v. I Ž v. s~ k l ksl p In oher words, he modified periodigram a frequencies v sž kp. r, kg k, y 4 is equivalen o he auocorrelogram in he sense ha i is asympoically mean zero and serially uncorrelaed. As an inuiive way of seeing his resul, recall ha, asympoically, he vecor of p auocorrelaions is spherically disribued in p-dimensional space. Fourier
58 K. DanielrJournal of Empirical Finance 8 493 535 ransforming he sample auocorrelaions o generae he specrum is geomerically jus ransforming he vecor of auocorrelaions ino anoher orhonormal basis; in his new basis, he vecor mus sill be spherically disribued. hus, we see ha he basis of periodigram esimaes has he same aracive properies as he auocorrelaion basis and ha, in fac, we can consruc a weighed periodigram es which will have he same opimaliy properies as he weighed auocorrelogram es. Jus as for he weighed auocorrelogram es, he weighs of he opimal es should be proporional o he expeced periodigram value under he alernaive hypohesis..5.. An opimal insrumenal Õariables es We show in his subsecion ha anoher expression of he opimal es is a regression in which he dependen variable is a one-period reurn and he independen variable is he linear combinaion of pas reurns which is he opimal predicor of he dependen variable, given ha he alernaive hypohesis is rue. 6 Since he orhogonaliy condiion is based on he characerisic ha under he null hypohesis reurns are no predicable using pas reurns, inuiively i seems ha he mos powerful insrumenal variables es for a given alernaive would be ha for which he insrumen was chosen o give he greaes possible predicive power under he alernaive. ha is, he opimal dependen variable should be ErNV w x y, where V y is he se of all pas reurns. We now demonsrae ha his inuiion is correc. We do his by showing ha an insrumenal variables es using he ErN w V x y as he insrumen is equivalen o he opimal weighed auocorrelaion es. he bes forecas of r given he se of pas reurns V y will be given by he projecion of V y ono r, which can be deermined in a regression framework, ha is rsb x qe where r y x s r y. 6 I has been noed by Hodrick Ž 99. ha we can wrie any linear orhogonaliy condiion involving reurns in his way. he es of he above orhogonaliy condiion is equivalen o eiher: Ž. a es of wheher a weighed average of fuure reurns given by Ý S sswr s qs is predicable using he reurns r ; Ž. S X or o a es of wheher a weighed average of fuure reurns Ýsswr s qs is predicable using he insrumen Ý R w Y r, where he weighs obey Ý` w X w Y rs r yr sy` s yr sw,and where he weighs are defined in his equaion so ha w X s for s- and s)s, and w Y s r s for r - and r ) R. hese ess are all precisely equivalen o he weighed auocorrelogram es if he sample momen variance is calculaed under he null hypohesis and using only he single period variance. If he sample momen variance is calculaed in some oher way, hen he ess will sill be asympoically equivalen.
K. DanielrJournal of Empirical Finance 8 493 535 59 We can use he OSL esimaor of b here since under he local alernaive he residuals will be uncorrelaed. herefore, ryr X y X X y ˆbs Ž xx. Ž xr. sý Ž xx. ry r. Ž X. Given he local-alernaive assumpion, we have ha xx f se I and herefore ha he projec coefficiens are c a a bs c s u. he regression of he single period reurn on he opimal predicor of his reurn under he alernaive is herefore jus a es of wheher: EŽ r Pb X x. s c a EŽ r Pr. s Ý y is zero. his is of course he same as he opional weighed auocorrelaion es. 3. he power of sandard es for mean reversion We now apply he mehod developed in he las secion o analyze hree sandard mean reversion ess: he long horizon regression, he modified long horizon regression, and he variance raio es. We show ha hese are asympoically equivalen o weighed auocorrelaion ess, calculaed he vecor of weigh implici in each es, and discuss he implici alernaive of each of he ess. In Secion 3.4, we evaluae he power of hese relaive o an opimal es using Mone-Carlo mehods. 3.. he long horizon regression Long horizon reurn regressions were used by Hansen and Hodrick Ž 98. o sudy forward rae predicions of exchange rae movemens and laer by Fama and French Ž 988a. o invesigae auocorrecion in sock reurns. he inuiion behind using a long horizon regression was ha such a es could capure behavior such as he long swings proposed by Summers Ž 986. because, in aggregaing reurns, he price movemens due o he ApredicableB long swings would be aggregaed, while he whie noise componens would be averaged ou. Consider he OLS regression coefficien Ž bˆ. for he regression rž,q. sa qb rž y,. qe Ž,q.
5 K. DanielrJournal of Empirical Finance 8 493 535 where r Ž, q. represens he sock s reurn from o q. he consisen OLS esimaor of b is given by cov ˆ Ž rž,q., rž y,.. ˆbs Ž 3.. sˆ Ž rž y,.. We can use he lineariy of he covariance operaor o wrie he OLS regression in Eq. 3. as: r Ý min s,ys cov ˆ r,rqs ss ˆbs sˆ rž y,. Because overlapping observaions are used, he residuals of he regressions will be correlaed and he OLS sandard error canno be used. o compue he sandard error of b, ˆ Hansen and Hodrick Ž 98. propose esimaing residual auocorrelaions a all lags up o he reurn horizon Ž i.e., up o y monhs., and hen calculaing he sandard error using a weighed sum of hese auocorrelaions. Richardson and Smih Ž 99. propose calculaing he variance covariance marix for he residuals assuming he null-hypohesis is rue. Boh mehods resul in consisen esimaion of he residual variance covariance marix V under he 7 local alernaive. Given his, a consisen esimaor of he variance of Ž bˆ y b. will be: X E byb ˆ byb ˆ s XX XVˆ X X X X y X X y where Vˆ is he variance covariance marix of he residuals which, under he null hypohesis, is a block diagonal marix where V s for < iyj< i,j G. Under he local alernaive we hen have ha: asy V s maxž y< iyj <,. i, j / asy X y X X y ˆ ž Ý ss Ž XX. XV XŽ X X. s minž s,ys. asy ˆ Ž. s r y, s asy minž s, y s. Ý ss Ý )ss minž s,ys. ˆ r 7 However, Richardson and Sock 99 have poined ou ha his esimaor will have poor small sample properies when he sample size is no considerably larger han he aggregaion inerval.
K. DanielrJournal of Empirical Finance 8 493 535 5 where s is he single period reurn variance. hus we see ha he -saisic is asympoically equivalen o a weighed auocorrelaion es, which has power agains an MA process wih lag polynomial weighs as show in Fig. 3. 3.. he modified long horizon regression Jegadeesh, 99 Jegadeesh addresses he quesion of he power of he Fama and French regression agains an ARŽ. fads alernaive such as ha discussed in Secion 3.4. of his paper. He looks a a generalized long horizon regression of he form: R,qjsa Ž J, K. qb Ž J, K. RyK,qe and assesses he power of he es as a funcion of he parameers J and K, using he Geweke Ž 98. approximae slope coefficien as a measure of he es power. He finds ha es power is maximized wih Js However, he also finds ha he opimal value of K is dependen on he parameerizaion of he fads alernaive chosen in he process given in Eq. Ž 3.3.: he closer f is o, he greaer he opimal value of K. he inuiion for his resul can be seen by referring o Fig. 6, which gives he auocorrelogram of reurns generaed by he ARŽ. fads model. Under he fads alernaive reurn, auocorrelaion is negaive a all lags, and is proporional o f, where is he lag lengh. o maximize he power of he modified long horizon regression, we need o choose J and K such ha he paern of effecive weighs Fig. 3. Equivalen lag polynomial weighs of long horizon regressions.
5 K. DanielrJournal of Empirical Finance 8 493 535 Fig. 4. Equivalen lag polynomial weighs of modified long horizon regression. will mos closely resemble hose in Fig. 6, ha is choose w o maximize Ýwc a. he effecive weighs of he modified long horizon regression are given by: 8 w smaxž,minž, JqKy, J, K.. Or imposing he requiremen ha he sine over he squares of he weighs be, we have: maxž,minž, JqKy, J, K.. w s Ž 3.. 3 < JyK < minž J, K. q minž J, K. qminž J, K. 3 and a plo of he normalized weighs for values of Js and for J) are given in Fig. 4. We deermine opimal weighs for a se of f s ranging from.95 o.99 and abulae he resuls in able. his is done by maximizing Ý w f over J and K, a where w is aken from Eq. 3.. Noe ha under he AR fads alernaive, c is proporional o f, so his maximizaion will yield an asympoically opimal es agains he local ARŽ. fads alernaive. In addiion o calculaing hese weighs, we also calculae he opimal reurn horizon for a Fama and French Ž 988a. like regressions Ž where J is consrained o equal K., and calculae he value of Ý w f for hese wo ess and for he opimal weighed auocorrelaion es, where, for his alernaive hypohesis, he opimal weighs are given by ( w s yf f y 8 Noe ha his is jus a more general version of he equaion for he long horizon regression weighs.
K. DanielrJournal of Empirical Finance 8 493 535 53 able Opimal aggregaion inervals and analyically calculaed es power agains a local alernaive for opimal weighed auocorrelaion es Ž WAC., modified long horizon regression Ž MLH regression. Ž Jegadeesh, 99., and long horizon regression Ž LH regression. f WAC es MLH regression LH regression Power Opimal J, K Power Opimal J Power.99 7.7 Ž, 5. or Ž 5,. 6.334 55 5.36.98 4.95 Ž, 6. or Ž 6,. 4.445 7 3.779.97 3.99 Ž, 4. or Ž 4,. 3.6 8 3.77.96 3.48 Ž, 3. or Ž 3,. 3.95 4.655.95 3.4 Ž, 5. or Ž 5,..746.367 here are several apparen differences beween Jegadeesh s resuls and ours, which are explained by his use of he Geweke Ž 98. approximae slope coefficien as opposed o our use of a measure of local power. Noe ha while Jegadeesh finds ha i is opimal o aggregae he independen variable and o use a single period reurn for he dependen variable, we find ha eiher he dependen or independen variable may be aggregaed, and he oher variable should be a single period reurn. he reason for he differences in he resuls is ha our es is opimal under a local alernaiõe, while Jegadeesh deermines opimaliy asympoically, bu using a nonlocal alernaive. o make he Geweke approximae slope coefficien equivalen o a es of a local alernaive, we need o calculae i in he limi as he variance of he emporary componen of prices relaive o he variance of he permanen componen Ž rf in Jegadeesh s noaion. goes o zero. When we recalculae he approximae slope coefficien given on page 5 of Jegadeesh Ž 99., we find boh ha he slope coefficien is he same wheher he dependen or independen variable is aggregaed, and ha he opimal aggregaion inervals are in agreemen wih hose given in able. o see he reason why he Geweke approximae slope coefficien would dicae ha he independen variable be aggregaed, while under he local hypohesis here would be no difference, we can look a he regression coefficien for wo possibiliies: if he dependen variable is an n period reurn and he independen variable is a single period reurn, hen he regression coefficien will be: n X y X ˆbs Ž xx. Ž yx. s Ý ˆ r s while he independen variable is an s period reurn and he dependen variable is a single period reurn hen he regression coefficien will be: Ýr Ž,q. n X y X ˆbs Ž xx. Ž yx. s Ý ˆ r r Ž,qn. s Ý
54 K. DanielrJournal of Empirical Finance 8 493 535 Under he null hypohesis or a local alernaive, he raion of variances should be rs, bu under a nonlocal alernaive hypohesis, he raio should be greaer because of he negaive auocorrelaion of reurns Žjus as he variance raio should be less han under he fads alernaive.. hus, he -saisic, which is jus b divided by he sandard error, is more likely o be significanly greaer han zero if he independen variable is aggregaed raher han he dependen. 9 3.3. he Õariance raio es he variance raio es has been used by Cochrane Ž 988. in esing for he presence of a permanen componen in producion daa, and by Poerba and Summers Ž 988. and Lo and MacKinlay Ž 988. in esing for predicabiliy in long and shor horizon sock reurns, respecively. Addiionally, Lo and MacKinlay Ž 989. have invesigaed he size of he variance raio es for boh homoskedasic and heeroskedasic null hypohesis, and have calculaed is power relaive o he Dickey Fuller -es and he Box Pierce Q saisic for various alernaives involving simple fads processes. he variance raio saisic for a reurn horizon J is he raio of he variance of J-period reurns o J imes he variance of one-period reurns: VRŽ J. s J J Ý Ýr qj yjr J s ž js / Ý s Ž r yr. he inuiion behind he use of he variance raio es is ha if reurns are uncorrelaed, he variance of a reurn of a given horizon will be proporional o he horizon and his raio will be. If, however, ransiory movemens in prices due o fads resul in posiive reurns regularly being followed by negaive reurns, shor horizon reurns will exhibi a proporionally higher variance. Using a muli-layer variance raio es, Poerba and Summers also find evidence of mean reversion a long horizons for real reurns on common socks over he 96 987 ime period. 9 Jegadeesh 99 and Hodrick 99 boh poin ou he compuaional advanages of aggregaing over he independen variable in ha under he null hypohesis, he regression residuals will hen be uncorrelaed. However, as we have shown, i is sraighforward o calculaed sandard error under he null hypohesis when a regression wih an aggregaed dependen variable is correced ino a weighed auocorrelaion es.
K. DanielrJournal of Empirical Finance 8 493 535 55 Fig. 5. Auocorrelaion weighs for variance-raio equivalen ess. As demonsraed by Cochrane 988, he variance raio saisic is equivalen o a es wheher a weighed average of auocorrelaions is equal o zero: Jy Ý Ž Jyj. rrqj Jy js JyNjN Ý ž J / jsyj Ý r s VRŽ J. s s ˆ r hus, we see ha he variance raio es is precisely equivalen o a weighed auocorrelaion es in which he paern of weighs forms an invered riangle as in Fig. 5. Noe also ha using he characerisics of he auocorrelaion esimaor given in Eqs..7 and., one can easily show ha asy Ž Jy.Ž Jy. ' PVR J ;N, 3 ž / which is in agreemen wih Lo and MacKinlay s Ž 988. resuls in heir Eqs. Ž 4a. and Ž 4b.. jž jq.ž jq. Using he relaion ha Ýisi s. 3 j
56 K. DanielrJournal of Empirical Finance 8 493 535 3.4. Small sample properies of he ess In his secion, we perform a se of Mone-Carlo sudies o sudy he small sample size and power properies of he weighed auocorrelaion es. We use as an alernaive hypohesis he ARŽ. fads model of Summers Ž 986., which has been used exensively in he lieraure of mean-reversion es power. In Secion 3.4.3, we presen he power comparisons among he differen es. 3.4.. he AR() fads alernaiõe hypohesis he ARŽ. fads model, suggesed by Summers Ž 986., is imporan from a hisorical perspecive in ha has been widely cied and used as a basis for power comparisons Žsee, e.g., Fama and French, 988a; Poerba and Summers, 988; Lo and MacKinlay, 989; Jegadeesh, 99; Hodrick, 99.. Summers poined ou ha if sock prices were equal o he fundamenal value plus a AfadsB componen, his fads componen migh no be deeced in low-order sample auocorrelaions. o some exen, his observaion promped some of he long horizon regression and variance raio ess, which were laer carried ou. However, he AR fads alernaive is unsaisfying in ha i is a model of overreacion raher han of raional variaion in expeced reurns. Moreover, i implies ha sock reurns will be negaively auocorrelaions a all lags, and he empirical evidence suggess ha shor horizon reurns are posiively auocorrelaed. he ARŽ. fads model posis ha observed sock prices Ž p. embody boh a permanen componen Ž p )., assumed o follow a random walk a drif, and a saionary componen Ž u., assumed o follow an auoregressive process of order one: psp ) qu p ) sp ) qmqe, e ;iid,s y e usfuy qn, -f-,n ;iid,sn Ž 3.3. he persisence of he emporary componen in deermined by f, while he share of he oal variance due o he emporary componen, g, is defined by sn g' Ž 3.4. s Ž qf. qs e n If e and n are independen, hen he ARŽ. fads model implies ha demeaned reurns r 'D p ym, follow and ARMAŽ,. process Ž yfl. rsž qu Lw. where: sn y Ž qf. s ys q Ž yf. 4 q Ž qf. s e n s e e us sn qfse 4 Ž. and w is an uncorrelaed sequence of errors wih s s y f q s ru. w Õ
K. DanielrJournal of Empirical Finance 8 493 535 57 Under his hypohesis, he auocavariogram and auocorrelogram for he reurn generaing process are given by c s~ s qs n e fors qf ž / yf n ž qf / y ys f forg y r s sy f forg Ž 3.5. m c sn yf m m c se Ž qf. qsn Plos of he auocorrelaion for several differen values of f are provided as Fig. 6. From his figure, i is seen ha he value of f conrols he degree of persisence of he emporary shocks: a value of f closer o makes he shocks more persisen. Noe ha a value of would make he shocks permanen. 3.4.. Analyical calculaion of es power able presens he analyically calculaed es power values agains he ARŽ. fads alernaive for he weighed auocorrelaion es, he modified long horizon regression Ž Jegadeesh, 99., and he sandard long horizon regression. his analyical comparison shows ha, for all values of f, he opimal es is mos powerful, and ha he modified long horizon regression is superior o he sandard Ž. Fig. 6. Auocorrelogram of simulaed AR fads model.
58 K. DanielrJournal of Empirical Finance 8 493 535 long horizon regression. We nex confirm hese analyical resuls for small samples using a Mone-Carlo experimen. 3.4.3. Mone-Carlo resuls We invesigae he power of four ess agains his alernaive: he long horizon regression wih Hansen and Hodrick Ž 98. calculaed sandard errors; he long horizon regression wih sandard errors calculaed under he null hypohesis Ž Richardson and Sock, 989; Richardson and Smih, 99.; he modified long horizon regression Ž Jegadeesh, 99.; and a weighed auocorrelaion es wih weighs given by: (yf iy wis f for is,..., 8 Ž 3.6. (yf 36 Ž. which is opimal agains he AR fads alernaive. All of he es saisics are correced for small sample bias using analyical correcions. We use Mone-Carlo mehods o calculae he size and power of he four ess. Firs, we calculae he empirical size of he ess. We simulae 6, reurns series and compile he resuling es saisics o deermine an empirical probabiliy disribuion of he es saisic under he null, and from his disribuion deermine he cuoff level for a size of 5%. We hen simulae daa under various parameerizaions of he alernaive hypohesis, and again compile he es saisics ino an Fig. 7. Power comparison of weighed auocorrelaion es, Jegadeesh regression, and FamarFrench regression as a funcion of g, for f s.95.
K. DanielrJournal of Empirical Finance 8 493 535 59 empirical disribuion. he repored empirical power is he fracion of he es saisics which fall ouside of he empirical cuoff value deermined in he size analysis. We repor Mone-Carlo deermined power levels for f s of.95 and.98, and for g s of.,.4,.6 and.8. g, which is given in Eq. Ž 3.4., is he proporion of he variance due o he emporary componen. Figs. 7 and 8 give he power levels for he four ess, for he se of g s, for a significance level of.5, for a persisence parameers f of.95 and.98, respecively. All power levels presened here are calculaed using, simulaed reurns series. In his and in Fig. 8, he reurn horizon used is ha which gives he highes power agains he paricular alernaive being evaluaed. In his figures, noe ha he power level for gs. is approximaely.5, which is o be expeced since he alernaive wih gs. is equivalen o he null, and we have se he criical value so ha he null will be falsely rejeced 5% of he ime. As g increases, he power increases for all ess, bu more quickly for he weighed auocorrelaion es. For hese parameers, he weighed auocorrelaion es is mos powerful, followed by he modified long horizon regression Žlabeled AJegadeeshB., followed by he long horizon regression using Hansen and Hodrick sandard errors Ž labeled AFF-HHB., followed by he long horizon regression using analyical sandard errors Ž labeled FF-R.. Excep for he relaion beween he FF-HH es and he FF-R es, abou which our asympoic heory makes no predicion, his is in agreemen wih he predicions as given in able. Fig. 8. Power comparison of weighed auocorrelaion es, Jegadeesh regression, and FamarFrench regression as a funcion of g, for f s.98.
5 K. DanielrJournal of Empirical Finance 8 493 535 hus, he asympoic predicions appear o be generally verified, alhough he differences in he wo long horizon regressions show ha he asympoic comparison is no a perfec predicor of es power in small samples. his issue is explored furher in Secion 5. 4. Join ess of weighed sums of auocorrelaions A number of sudies including Fama and French Ž 988b. and Poerba and Summers Ž 988. have looked a a se of mean-reversion es saisics, and calculaed significance levels based on he mos significan saisic. As Richardson Ž 993. poins ou, he saisical significance of he overall es should be evaluaed by joinly esing wheher all coefficiens are equal o zero. Richardson and Smih 99 propose a x join es embedded in he GMM framework of Hansen Ž 98. and use his join es o evaluae he significance of Fama and French s and Poerba and Summer s resuls. Jegadeesh Ž 99. uses he es o evaluae he join significance of his modified long horizon regressions. he inuiion behind he resuls in his secion is bes expressed in erms of he geomeric inerpreaion given in Secion.3. here, we showed ha a weighed auocorrelaion es saisic can be inerpreed as he lengh of he projecion of he vecor of auocorrelaions ono a vecor of weighs. Here we show ha a x es of wheher a se of n weighed auocorrelaion es saisics is zero is equivalen o a es of wheher he projecion of he auocorrelaion vecor ono he n-dimensional subspace spanned by he n weigh vecors has a lengh zero. An imporan implicaion of hese resuls is ha he x join es may lack power agains he very alernaives he economerician is ineresed in. For example, if he wishes o look for mean reversion in sock price daa, wihou having precise knowledge of he persisence of he mean reversion, he migh elec o run a se of n long horizon regressions and hen es heir significance using a x es. Even if he individual regressions have considerable power agains he alernaive, he power of he join es may be quie low. he reason for his is ha join es looks for deviaions from he null in he enire n dimensional subspace, even if he alernaive suggess deviaions only in a paricular direcion wihin he subspace. We perform he analysis in his secion wihin a GMM framework. All of he ess we are concerned wih are ess of wheher reurns are orhogonal o pas reurns, and GMM framework is a general way of analyzing his ype of Ž. resricion. We esablish he equivalence among he hree ess: a x or Wald join es of M regression coefficiens or variance raios, as in Richardson and Smih Ž 99.Ž. ; a GMM-es of a se of M overidenifying resricions; and Ž. 3 a x join es of a se of M weighed auocorrelogram ess. For exposiional reasons, we show his equivalence in erms of long horizon regressions, hough he mehod is applicable o any join es of any regressions, momen resricions, or variance raio ess.
K. DanielrJournal of Empirical Finance 8 493 535 5 4.. he long horizon regression in a GMM framework A single long horizon regression can be wrien in a generalized insrumenal variables framework by noing ha he OLS esimaes of a and b are based on he momen resricions ha EweŽ, q.xs and EweŽ, q. r, Ž q.x s : ha is, on he resricion ha he regression residual ež, q. s rž, q. ya yb ry Ž,. be orhogonal o he insrumens of and r, Ž q.. hese wo orhogonaliy condiions are used o esimae he sysem of wo unknowns. In he GMM framework, we can represen hese resricions in he following way: g Ž u. s Ý gž u. s ž s Ý rž,q. ya yb rž y,. / s rž,q. ya yb rž y,. rž y,. Ž 4.. he GMM esimaor of u sža b. X hen minimizes he disance of he sample momen vecor g Ž u. from zero. his is done by minimizing g Ž u. X W g Ž u., where W is some weighing marix. For his jus-idenified sysem, he choice of weighing marix is unimporan since for some choice of u, every elemen of he g Ž u. vecor will be equal o zero. In general, here are more momen resricions han variables o be esimaed. Hansen Ž 98. shows ha in his case, he opimal weighing marix is he inverse of he variance covariance marix of g Ž u. evaluaed a he rue value of u, u : ž / ` y ) y X Ý yj jsy` W ss s E g Ž u. g Ž u. Ž 4.. and ha, given his weighing marix, uˆ is consisen and asympoically normally disribued: asy X y y ' uyu ˆ ;N, DS D Ž 4.3. Ž. ž / where Eg Ž u. D se X Ž 4.4. Eu his represenaion can be exended o joinly esimae a se of regression coefficiens. his resuls in he following se of jus-idenified momen equaions: s ŽrŽ,qj. yb Ž j. rž yj,. rž yj,.. g Ž u. s Ýg Ž u. s Ý. s rž,qk. yb Ž k. rž yk,. rž yk,. Ž 4.5.
5 K. DanielrJournal of Empirical Finance 8 493 535 where now he reurns are demeaned and he inercep erms and corresponding momen resricions have been removed from he sysem. Noe ha, again, he esimaes of he b s will be idenical o he OLS regression esimaes since he sysem is jus idenified. he D and S marices can be consruced under he null hypohesis, following Richardson and Smih Ž 99.: ž / js D s Ž 4.6. ks j q j j qsž j,k. 4 3 S ss Ž 4.7. kž qk. j qsž j,k. 3 Ž w x. where s is he variance of a single period reurn Er and where jy Ý sž j,k. s Ž jyl. minž j,kyl. ls he variance covariance marix of he vecor of b esimaors hen obained by marix manipulaion from Eqs. 4.6 and 4.7. j q j qsž j,k. y 3 j jk X ˆ y V Ž b. s Ž DS D. s Ž 4.8. j qsž j,k. k q and he Wald es ha he se of b s are equal o zero is given by X y ˆ ˆ ˆ jk Jsb V b b Ž 4.9. Wih his mehod, GMM is used o esimae he se of regression b s, and hen a separae Wald es is performed o deermine wheher he b s are joinly significanly differen from zero, using VŽ bˆ. as calculaed analyically in he GMM 3k he es saisic Richardson and Smih Ž 99. propose is no equivalen o his es in small samples because he esimaed inercep erms Ž a s. obained from esimaing his overidenified sysem are slighly differen. However, since r is a consisen esimaor of až. under our assumpions, he ess will be idenical asympoically. We could se up an overidenified sysem in which he a s were esimaed using he momen resricion ha EŽ e P. s, bu his would complicae he analysis wihou adding addiional insigh.
K. DanielrJournal of Empirical Finance 8 493 535 53 framework. Alernaively, one could direcly impose he momen resricions which encompass his resricion Žha b s ;) :. rž,qj. rž yj,. X g s g u s Ý Ý. Ž 4.. rž,qk. rž yk,. s s Again, from Hansen Ž 98., a es saisic ha indicaes he AdisanceB of his model from he daa is given by X X ˆ y J s g u S g uˆ 4. which is asympoically x disribued wih n degrees of freedom. Jus as is done above, we can consruc S y under he null hypohesis, and from Eq. Ž 4.., we see ha when calculaed under he null, S will be idenical o he S given in Eq. Ž 4.7. because under he null hypohesis, b is zero for all reurn ) 4 ) horizons. Now, we define S ' Srs, where, from Eq. 4.7, S is now a funcion for j and k only. his means ha we can consruc an alernaive S, which we denoe as S, in he following way: s Ž j. s Ž j. j j ) Ss S s Ž k. s Ž k. k k where s j s r yj, Ý s Since under he null hypohesis, Žs Ž j.. rj is a consisen esimaor of s, he one-period reurn variance, S is a consisen esimaor of s 4 S ) ss. When Ž S. y is subsiued ino he definiion of J X in Eq. Ž 4.., we obain he following expression for he es saisic: X X y X j y j ˆ ) J s g Ž S. g sb Ž S. bˆ k k ^ ` _ ˆ w xy V b Comparing his equaion wih Eqs. Ž 4.6. Ž 4.8. confirms ha he Wald saisic in Eq. Ž 4.9. and he es saisic for he overidenified GMM sysem given here are idenical.
54 K. DanielrJournal of Empirical Finance 8 493 535 4.. Asympoic equiõalence o a se of weighed auocorrelaion ess We show now ha he es of he momen resricions in Eq. Ž 4.. is equivalen o he es of wheher a se of weighed sums of auocorrelaions are zero. Using hese resuls, we presen in Secion 4.3 an inuiive geomeric inerpreaion of he join es. Firs, noe ha he se of momen resricions in Eq. Ž 4.. are equivalen o j Ý minž s, jys. rrqs ss. Ý Ý. s s k Ý minž s, kys. rrqs ss g s g Ž u. s Ž 4.. Addiionally, S will again be of he same from here as in Eq. Ž 4.7.. However, since we can choose any consisen esimaor of he single period variance in Eq. Ž 4.7., we now choose sˆ ss s r, Ý s which is he variance calculaed using one-period reurns. he es of he overidenifying resricions can now be wrien Y X y J sg Ž S. g sˆ sˆ and ^`_ ^`_ ) X ) g g ž / ž / j rr qs Ý minž s, jys. ss r ) ) Ý Ý. s s k rr qs Ý minž s, kys. ss r g s g u s j Ý ss minž s, jys. ˆ r s. k Ý ss minž s, kys. ˆ r s s
K. DanielrJournal of Empirical Finance 8 493 535 55 so we see ha he momen condiions are equivalen o resricions ha weighed sums of auocorrelaions be equal o zero. Using vecor noaion, we can rewrie he momen condiions in Eq. Ž 4.3. as X wž j. ˆ r ) g s. Ž 4.3. X wž k. ˆ r where wž. j is a k-vecor whose ih elemen is maxž, minž s, kys.., and ˆ r is a k-vecor whose ih elemen is he auocorrelaion a lag i. Noe ha when g ) is expressed in his way, we can apply he asympoic relaionship X Es ˆˆ rr si o show ha ha S ), he variance covariance marix of g ), will be given by: X X X X w j rrw ˆˆ j PPP w j rrw ˆˆ k X. ) ) ) S se g g s.... X X X X wž k. rrw ˆˆ Ž j. PPP wž k. rrw ˆˆ Ž k. X wž j. wž j. PPP wž j. wž k.. s.... X X wž k. wž j. PPP wž k. wž k. which exensive algebraic manipulaion reveals o be idenical o he S ) s Žrs 4. S given in Eq. Ž 4.7.. hus, when we express he regressions as weighed sums of correlaions, we have a more sraighforward way of deriving he Richardson and Smih Ž 99. variance covariance marix as given in Eq. Ž 4.8.. Moreover, wriing he es in his way leads o a simple geomeric inerpreaion of he es power, which we provide in he nex secion. 4.3. Geomeric inerpreaion of he join es We can gain considerable inuiion ino he workings of he join es by giving he join es power issue a geomeric inerpreaion analogous o wha was done for he single weighed auocorrelaion es in Secion.3. Again, we consider a p-dimensional space in which he se of sample auocorrelaions is expressed as a vecor wih elemens Ž ˆ r, r..., r. ˆ ˆp. We showed in Secion.3 ha he es saisic A s Ý w ˆ r, was he square of he lengh of he projecion ono he vecor of weighs w, where wsž w. X, w..., w p, and ha his saisic was x ŽŽ 4 < a <. 4. disribued wih NCP a c rsu cos C under he alernaive hypohesis, where C is he angle beween he vecor c a and he vecor of weighs w. Since he alernaive hypohesis is noncenral x disribued and he null hypohesis is X
56 K. DanielrJournal of Empirical Finance 8 493 535 cenral x disribued, we showed ha he mos powerful es will maximize he NCP, which is done by packing weighs so ha C is zero or p. We can provide a similar inerpreaion of he join es. Consider again he join es of a se of M long horizon regressions, as expressed by he se of momen resricions in Eq. Ž 4.3.. If he es saisic ) X ) y ) Y J s g S g ;xm Ž 4.4. is rewrien as where J srˆ W Y X X X wž j.. wž k. Ws. X ž / I Wrˆ We see ha J is he square of he lengh of he projecion of ˆ r ono he M-dimensional manifold Ž or subspace., which is spanned by he M eigenvecors wž j...wž k.. Addiionally, jus as in Secion.3, he disribuion of he join es saisic J ŽŽ 4 < a <. 4. is xm disribued wih NCP a c rsu cos C where C is now he angle beween he vecor of auocovariances c a and he M-dimensional manifold conaining he weigh vecors. For his seing, he power of he es will depend boh on he angle C and on he number of resricions M. he mos powerful es will boh maximize cosž C. and minimize M. However, when he alernaive hypohesis is known precisely, here will be some radeoff: increasing he dimensionaliy of he W marix may decrease he expeced value of cosž C., hus increasing he expeced NCP of he es saisic, bu i will also increase he number of degrees of freedom of he es saisics x disribuion. An exreme example of his is Box Pierce Q es, which is he join es of wheher each of he p auocorrelaions is zero: for his join es, any process is an implici alernaive, bu of course i will have a very lile power agains any specific alernaive. Fig. 9 illusraes he problem wih he mehod of using a x join es of he significance of a se of ess. Suppose he economerician was using he firs hree auocorrelaions o invesigae mean reversion, and he believes ha each of he hree auocorrelaions are likely o have roughly equal posiive values. He migh hen run a join es of hree weighed auocorrelaion ess using he weigh vecors illusraed in Fig. 9. Each of hese hree ess would, individually, be powerful agains he alernaive. Bu, as our analysis shows, he join es will have considerably lower power: while he economerician wishes o place large weigh on sample auocorrelaions, vecors in a narrow region of Rq 3 he is in fac puing equal weigh on deviaions in any direcion in R 3.
K. DanielrJournal of Empirical Finance 8 493 535 57 Fig. 9. A geomeric illusraion of he join es problem. Anoher way of saing his inuiion is o say ha when he number of resricions being esed is increased, he AlargerB he manifold of implici alernaives becomes. he greaer he dimensionaliy of his manifold, he greaer he number of unreasonable alernaives he es is likely o have power agains, and he lower he ess power agains reasonable alernaives. We show in Secion 5 ha oher ypes of join ess can have greaer power. 5. Small sample size and power of long horizon regression mehods Fama and French Ž 988a. perform he following long horizon regression on he CRSP, EW, VW and size decile porfolio real reurns for 96 986. R,q sa qb R y,qe he regressions are done for reurn horizons of,, 3, 4, 5, 6, 8 and years, using monhly daa, and he Ž consisen. OLS coefficiens are calculaed. A -saisics is used as a es of saisical significance, ˆb Ž. s Ž 5.. sê b
58 K. DanielrJournal of Empirical Finance 8 493 535 where, because of he overlapping observaions and he resuling correlaed residuals, seb ˆ is calculaed using he Hansen and Hodrick Ž 98. mehod. Fama and he French find ha for he EW index and for size deciles 7, he slope coefficien for a reurn horizon of 4 years Ž 48 monhs. is more han sandard errors from. Addiionally, he slope coefficiens for reurn horizons of 3 and 5 years are significanly differen from for a number of he porfolios. Fama and French conclude from his ha here is evidence of mean reversion in he sock prices of small firms over he 96 986 period. However, he Fama and French evidence is no unambiguous. Firs, Richardson Ž 993. challenges he saisical reliabiliy of he Fama and French resuls on he ground ha he es does no properly accoun for implici muliple comparisons. ha is, one canno conclude from he saisical significance of he regression coefficien a a single reurn horizon ha here is evidence of mean reversion; a join es of significance of he coefficiens a all eigh reurn horizons mus be conduced. Using his es, Richardson finds ha he saionary random walk hypohesis canno be rejeced over he 96 986 sample period. Second, Richardson and Sock Ž 989. sugges ha he asympoic sandard errors used by Fama and French and by Richardson and Smih are flawed because of bad small sample properies. hey sugges anoher asympoic mehod of calculaing he bˆ sandard errors, which is based on holding he reurn horizon a a consan fracion of he sample size as he lengh of he daa series goes o infiniy. hey show ha he Jr limiing disribuion calculaed under hese assumpions has much beer small sample properies han he convenional asympoic disribuion, and finally, hey show ha even he indiõidual Fama and French regression coefficiens Žb s. are saisically insignifican when he significance is deermined using he Jr asympoics. hey, herefore, concluded ha saisical significance of he individual slope esimaes ha Fama and French finds is due o he poor small sample properies of he Hansen and Hodrick esimaor. We poin ou in his secion ha, while boh of hese criiques are well founded, heir conclusions ha he long horizon es saisics presened by Fama and French do no allow rejecion of he null hypohesis are due o he use of a differen es: Fama and French use HansenrHodrick calculaed -saisics while Richardson and Smih Ž 99. and Richardson and Sock use saisics calculaed under he null hypohesis of no serial correlaion. While hese wo saisics are asympoically equivalen, heir power differs in small samples, as was demonsraed in he Mone-Carlo resuls in Secion 3.4.3. We show in his secion ha if he small-sample correced p-values of he HansenrHodrick Ž. saisic are used insead of he p-values for he b ˆ saisic, here is sill evidence of mean reversion like. herefore, he reversal of Fama and French s conclusion is no due o he small sample properies of he esimaor, as claimed, bu raher o he difference in power of he wo ess. In addiion, we show ha he poor small sample properies of he b ˆ saisic can largely be correced by adjusing he OLS regression coefficien, he weighing
K. DanielrJournal of Empirical Finance 8 493 535 59 marix and sample auocorrelaions in he Hansen and Hodrick sandard error calculaion for small sample biases: Once hese correcions are made, he small sample properies of he asympoic saisics are grealy improved. 5.. Calculaion of he small sample disribuion We begin by deriving analyically he small-sample bias of he b ˆ OLS, which is available on reques from he auhor. hough edious o derive, he inuiion for he small sample bias is simply ha a demeaning series induces negaive serial correlaion. As an exreme example, consider calculaing he firs-order serial correlaion based on wo observaions: he calculaed value will always be negaive because once he observaions are demeaned, one of hem will be posiive and he oher negaive. Nex, using Mone-Carlo mehods, we calculae he empirical disribuion of he bols ˆ wih and wihou he bias adjusmen, for a sample of 7 poins Ž he lengh of Fama and French s sample.. his was done for reurn horizons of, 4, 36, 48, 6, 7, 96 and periods, corresponding o he -, -, 3-, 4-, 5-, 6-, 8-, and -year horizons used by Fama and French and Richardson and Sock. he cumulaive empirical disribuion wihou he bias correcion is ploed in Fig.. Fig.. FamarFrench regression cumulaive disribuion of HansenrHodrick -saisic Mone-Carlo resuls no bias correcion, 7 poins, ieraions.
53 K. DanielrJournal of Empirical Finance 8 493 535 Fig.. FamarFrench regression cumulaive disribuion of HansenrHodrick -saisic Mone-Carlo resuls bias correcion, 7 poins, 6 ieraions. Fig. shows ha, as demonsraed by Richardson and Sock, for long reurn horizons, he OLS b esimaor has a small sample disribuion ha is clearly no well represened by a mean-zero normal. Also, as hey poin ou, he likelihood of negaive values is quie high under he null hypohesis. However, Fig. shows ha he disribuions of he bias-adjused saisics for reurn horizons of 6 years are almos idenical o he asympoic disribuion, and ha for he reurn of 8 and years, he disribuion is acually narrower: he probabiliy of exreme negaive values is lower han wha is prediced by asympoic heory. 5.. Empirical resuls We performed he long horizon regression ess for real reurns using he ˆ bias-adjused HansenrHodrick saisics. We presen he regression b s in he bias correcion appears o be slighly oo-small here, in ha for all reurn horizons excep 8 and years, he mean of he disribuion is somewha negaive.
able Fama and French regressions real reurns: 96 985 wih significance levels based on small sample empirical disribuion Size decile Bias adjused b ˆ s and p-values porfolio Reurn horizon 3 4 5 6 8 ) EW y.5 Ž.34. y.3 Ž.77. y.33 Ž.7. ) y.37 Ž.. ) y.35 Ž.3. y.3 Ž.54. y. Ž.69..3 Ž.85.. Ž.58. y.4 Ž.97. y.6 Ž.88. ) y.39 Ž.4. y.34 Ž.46. y.6 Ž.397..39 Ž.95..64 Ž 964. ). Ž.558. y. Ž.9. y.6 Ž.75. )) y.4 Ž.6. )) y.45 Ž.3. y.7 Ž.64.. Ž.493..7 Ž.696. ) 3 y.4 Ž.394. y.7 Ž.4. y.8 Ž.5. ) y.37 Ž.. ) y.36 Ž.7. y.8 Ž.58. y.4 Ž.386..6 Ž.539. ) 4 y. Ž.45. y. Ž.5. y. Ž.6. ) y.35 Ž.. ) y.38 Ž.6. y. Ž... Ž.473..4 Ž.696. 5 y.5 Ž.356. y. Ž.84. y.9 Ž.44. ) y.3 Ž.5. y.3 Ž.3. y.6 Ž.5..5 Ž.57..4 Ž.79. 6 y.5 Ž.37. y. Ž.99. y.3 Ž.3. ) y.33 Ž.8. y.9 Ž.46. y.9 Ž.3..8 Ž.6..3 Ž.763. 7 y.6 Ž.3. y.6 Ž.48. y.33 Ž.5. y.6 Ž.58. y. Ž.43. y. Ž.48..4 Ž.74.. Ž.7. 8 y.6 Ž.37. y.3 Ž.7. y.3 Ž.9. y.4 Ž.49. y.8 Ž.35..3 Ž.58..7 Ž.763..4 Ž.776. 9 y.3 Ž.43. y. Ž.85. y.8 Ž.53. y.5 Ž.9. y. Ž.488.. Ž.84..34 Ž.878..33 Ž.799. y.6 Ž.33. y.5 Ž.5. y.9 Ž.4. y.4 Ž.3.. Ž.53.. Ž.848..34 Ž.876..3 Ž.79. VW y.3 Ž.399. y. Ž.96. y.6 Ž.7. y. Ž.3..6 Ž.64..6 Ž.89..36 Ž.884..3 Ž.78. Hansenr Hodrick -saisics EW Ž y.46. Ž y.64. Ž y.9. Ž y.85. Ž y.35. Ž y.7. Ž.46. Ž.8. Ž.5. Ž y.98. Ž y.57. Ž y.39. Ž y.96. Ž y.3. Ž.46. Ž.45. Ž.8. Ž y.84. Ž y.68. Ž y3.4. Ž y3.4. Ž y.65. Ž.4. Ž.45. 3 Ž y.3. Ž y.. Ž y.9. Ž y.84. Ž y.5. Ž y.8. Ž y.7. Ž.8. 4 Ž y.8. Ž y.76. Ž y.4. Ž y.89. Ž y3.. Ž y.3. Ž.. Ž.44. 5 Ž y.43. Ž y.59. Ž y.. Ž y.37. Ž y.. Ž y.88. Ž.. Ž.64. 6 Ž y.5. Ž y.47. Ž y.. Ž y.56. Ž y.96. Ž y.53. Ž.3. Ž.58. 7 Ž y.59. Ž y.93. Ž y.33. Ž y.86. Ž y.. Ž y.8. Ž.49. Ž.49. 8 Ž y.53. Ž y.7. Ž y.6. Ž y.97. Ž y.5. Ž.7. Ž.64. Ž.6. 9 Ž y.3. Ž y.58. Ž y.88. Ž y.95. Ž y.8. Ž.93. Ž.. Ž.66. Ž y.57. Ž y.9. Ž y.5. Ž y.88. Ž.. Ž.94. Ž.. Ž.63. VW Ž y.3. Ž y.49. Ž y.69. Ž y.6. Ž.3. Ž.4. Ž.5. Ž.6. denoes a p-value-.5. ) denoes a p-value-.5. )) denoes a p-value-.. K. DanielrJournal of Empirical Finance 8 ( ) 493 535 53
53 K. DanielrJournal of Empirical Finance 8 493 535 he upper pars of able. he HansenrHodrick -saisics for each of hese coefficiens are given a he boom of each able, and empirical p-values for hese -saisics are given under each coefficien. hese p-values are calculaed from he Mone-Carlo resuls ploed in Fig. and are, herefore, correc in small samples. he resuls here should be compared o hose in Richardson and Sock s Ž 989. able 4. hey find ha hree slope coefficiens are significan a he Ž wo-sided. 5% level. In conras, we find ha slope coefficiens are saisically significan a he Ž wo-sided. 5% level. Based on he problems wih he x join es discussed above, we use a differen saisic: we look a he mos significan of he eigh Fama French regression coefficiens. Supposedly, he reason for performing a number of regressions in he firs place is ha since he opimal es reurn horizon is dependen on he parameers of he alernae hypohesis, regressions should be run for a range of reurn horizons corresponding o he range of parameer values in he prior disribuion. he accepance or rejecion of he null hypohesis should hen be based on a join es of regression coefficiens. However, we have shown ha while he regression es may be powerful agains one of he range of alernaive hypoheses for a single regression, he power of he x join es agains he enire range of alernaives may be only peripherally relaed o he power of he individual regressions. herefore, as a join es, we use he mos significan saisic as a measure of he overall significance bu saisically correc for having seleced his saisic from he se of regression coefficiens. In order o deermine significance levels, we empirically calculae he disribuion of he mos negaive of he saisics as described before. Based on his join es, only he decile reurns exhibi evidence of mean-reversion a he 5% wo-ailed level. However, based on he saed alernaive of he ARŽ. fads model, he one-ailed es is appropriae, and he EW and decile, 3 and 4 porfolios exhibi mean reversion a a % one-ailed level. However, he economic significance of hese resuls is sill suspec based Jegadeesh s Ž 99. finding ha all significan mean reversion appears o be due o high reurns of small firms in January, and o he severe heeroskedasiciy in he sample period. 6. Conclusions We have developed a mehod ha allows analyical calculaion of he power of ess of mean reversion. his mehod allows us o calculae he power of any weighed auocorrelaion ess. We have shown he equivalence of his es wih he long horizon regression es, he variance raio es, weighed specral ess, and any insrumenal variable or generalized mehod of momen Ž GMM. ess, which use linear funcions of pas reurns as insrumens.
K. DanielrJournal of Empirical Finance 8 493 535 533 his mehod has allowed us o make power comparisons ess of his class and o deermine he implici alernaive he ess. In addiion, we have shown how o deermine he opimal es given a null hypohesis ha reurns are he sum of a differenced maringale and an alernaive process wih an ARMAŽ p, q. represenaion: in his seing, he opimal es will be a weighed sum of he sample auocorrelaions a differen lags, where he weighs are proporional o he expeced reurn auocorrelaions under he alernaive hypohesis. We have also provided a simple geomeric analogy ha gives he inuiion for his resul. In he specral domain, we have shown ha he weighed auocorrelaion es can jus as easily be wrien as a weighed periodigram es, wih an analogous resul ha he opimal weighed periodigram es will have weighs proporional o he expeced periodigram under he alernaive. his es shares he opimaliy propery of he weighed auocorrelogram es. We have also addressed he issue of join ess. We show ha he resuls exend easily o he case of muliple auocorrelaion or insrumen resricions as ha in he simple geomeric inuiion developed in he firs par of he paper. An imporan resul of his secion is ha he power of a join es of momen Ž resricions or x join es. of his sor may be only peripherally relaed o he power of he individual ess. Since our analyical es-power resuls are valid only asympoically and under local alernaives, we conduced Mone-Carlo experimens o invesigae he robusness of hese resuls for small sample sizes and for nonlocal alernaives found ha he resuls were robus, a leas for he limied se of alernaives we consider. Finally, we have shown how small differences in he small sample properies of a es can lead o srikingly differen saisical inferences. We show ha he long horizon regression, which uses analyical sandard errors, may have low power agains simple mean revering alernaives, and ha his, no poor small sample properies, is he reason Richardson and Sock Ž 989. find no evidence of mean reversion using his es. We empirically calculae he small-sample correced disribuion for he Fama and French -saisics, and find more evidence in favor of a mean reversion hypohesis. Acknowledgemens I am indebed o orben Anderson, John Cochrane, Buron Hollifield, Narasimhan Jegadeesh, Nicholas Polson, Sheridan iman, Waler orous and o he paricipans of he UBC, Universiy of Chicago, USC, and Wharon finance seminars and he annual meeings of he Canadian Economics Associaion for heir helpful discussions, commens and suggesions.
534 K. DanielrJournal of Empirical Finance 8 493 535 Appendix A. he saisical properies of he sample esimaors under he local alernaive If a process is described by: u Ž L. x sf Ž L. U u;iid,s Ž A.. u Eu 4 shs 4 -` Ž A.. where fž L. and už L. are finie-order lag polynomials, and where fž z. ruž z. has roos ouside he uni circle Ž for saionariy., hen Brockwell and Davis Ž 99. show ha if we define he vecors of covariances and correlaions in he following way: x x cˆ ˆ r x x c ˆ ' ˆ r '.. x x cˆ ˆ r where h y x ĉ x x Ý q ˆ x s ĉ c ˆ ' x x r ' h hen cˆ x and ˆ r x will have he following asympoic disribuions: cˆ x ;NŽ c x, y V. Ž A.3. ˆ r x ;N Ž r x, y W. Ž A.4. where he elemens of he variance covariance marices V and W are given by: ` Õ ij s Ž hy3. ccq i j Ý cc k kyiqj qc kqj c kyi 4 ksy` ` ij Ý kqi kqj kyi kqj i j k i j kqj j k kqi4 ksy` w s r r qr r q rrry rrr y rr r Under he assumpion ha x is generaed by a saionary, finie-order ARMA process, each of hese elemens is finie. References Badahur, R.R., 98. Raes of convergence of esimaes and es saisics. Annals of Mahemaical Saisics 38, 33 34. Brockwell, P.J., Davis, R.A., 99. ime Series: heory and Mehods. nd edn. Springer Verlag, New York. Campbell, J.Y., 99. Why Long Horizons? Asympoic Power Agains Persisen Alernaives 8, 459 49.
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