1 Journal of Empirical Finance 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 address: Ž K. Daniel rr$ - see fron maer q Elsevier Science B.V. All righs reserved. PII: S X
2 494 K. DanielrJournal of Empirical Finance 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 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.
3 K. DanielrJournal of Empirical Finance 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
4 496 K. DanielrJournal of Empirical Finance 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 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
5 K. DanielrJournal of Empirical Finance 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
6 498 K. DanielrJournal of Empirical Finance 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.
7 K. DanielrJournal of Empirical Finance 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
8 5 K. DanielrJournal of Empirical Finance 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.
9 K. DanielrJournal of Empirical Finance 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
10 5 K. DanielrJournal of Empirical Finance 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 (
11 K. DanielrJournal of Empirical Finance 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 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.
12 54 K. DanielrJournal of Empirical Finance 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.
13 K. DanielrJournal of Empirical Finance 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./
14 56 K. DanielrJournal of Empirical Finance 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.
15 K. DanielrJournal of Empirical Finance 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
16 58 K. DanielrJournal of Empirical Finance 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.
17 K. DanielrJournal of Empirical Finance 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.
18 5 K. DanielrJournal of Empirical Finance 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.
19 K. DanielrJournal of Empirical Finance 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 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.
20 5 K. DanielrJournal of Empirical Finance 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 < 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.