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.
Coinegraion: The Engle and Granger approach Inroducion Generally one would find mos of he economic variables o be non-saionary I(1) variables. Hence, any equilibrium heories ha involve hese variables require
Chaper 8: Regression wih Lagged Explanaory Variables Time series daa: Y for =1,..,T End goal: Regression model relaing a dependen variable o explanaory variables. Wih ime series new issues arise: 1. One
Opion Pricing And Mone Carlo Simulaions George M. Jabbour, (Email: firstname.lastname@example.org), George Washingon Universiy Yi-Kang Liu, (email@example.com), George Washingon Universiy ABSTRACT The advanage of Mone Carlo
Profi Tes Modelling in Life Assurance Using Spreadshees PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Erik Alm Peer Millingon 2004 Profi Tes Modelling in Life Assurance Using Spreadshees
TEMPORAL PATTERN IDENTIFICATION OF TIME SERIES DATA USING PATTERN WAVELETS AND GENETIC ALGORITHMS RICHARD J. POVINELLI AND XIN FENG Deparmen of Elecrical and Compuer Engineering Marquee Universiy, P.O.
These noes largely concern auocorrelaion Issues Using OLS wih Time Series Daa Recall main poins from Chaper 10: Time series daa NOT randomly sampled in same way as cross secional each obs no i.i.d Why?
Deparmen of Economics Discussion Paper 00-07 Muliple Srucural Breaks in he Nominal Ineres Rae and Inflaion in Canada and he Unied Saes Frank J. Akins, Universiy of Calgary Preliminary Draf February, 00
A Brief Inroducion o he Consumpion Based Asse Pricing Model (CCAPM We have seen ha CAPM idenifies he risk of any securiy as he covariance beween he securiy's rae of reurn and he rae of reurn on he marke
Principal componens of sock marke dynamics Mehodology and applicaions in brief o be updaed Andrei Bouzaev, firstname.lastname@example.org Why principal componens are needed Objecives undersand he evidence of more han one
The Transpor Equaion Consider a fluid, flowing wih velociy, V, in a hin sraigh ube whose cross secion will be denoed by A. Suppose he fluid conains a conaminan whose concenraion a posiion a ime will be
Chaper 7. esponse of Firs-Order L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural
DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS Hong Mao, Shanghai Second Polyechnic Universiy Krzyszof M. Osaszewski, Illinois Sae Universiy Youyu Zhang, Fudan Universiy ABSTRACT Liigaion, exper
Term Srucure of Prices of Asian Opions Jirô Akahori, Tsuomu Mikami, Kenji Yasuomi and Teruo Yokoa Dep. of Mahemaical Sciences, Risumeikan Universiy 1-1-1 Nojihigashi, Kusasu, Shiga 525-8577, Japan E-mail:
Prof. Harris Dellas Advanced Macroeconomics Winer 2001/01 The Real Business Cycle paradigm The RBC model emphasizes supply (echnology) disurbances as he main source of macroeconomic flucuaions in a world
Coinegraion Analysis of Exchange Rae in Foreign Exchange Marke Wang Jian, Wang Shu-li School of Economics, Wuhan Universiy of Technology, P.R.China, 430074 Absrac: This paper educed ha he series of exchange
Why Did he Demand for Cash Decrease Recenly in Korea? Byoung Hark Yoo Bank of Korea 26. 5 Absrac We explores why cash demand have decreased recenly in Korea. The raio of cash o consumpion fell o 4.7% in
Random Walk in -D Random walks appear in many cones: diffusion is a random walk process undersanding buffering, waiing imes, queuing more generally he heory of sochasic processes gambling choosing he bes
INVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS Ilona Tregub, Olga Filina, Irina Kondakova Financial Universiy under he Governmen of he Russian Federaion 1. Phillips curve In economics,
Journal of Applied Economics, Vol. IV, No. (Nov 001), 313-37 GOOD NEWS, BAD NEWS AND GARCH EFFECTS 313 GOOD NEWS, BAD NEWS AND GARCH EFFECTS IN STOCK RETURN DATA CRAIG A. DEPKEN II * The Universiy of Texas
MACROECONOMIC FORECASTS AT THE MOF A LOOK INTO THE REAR VIEW MIRROR The firs experimenal publicaion, which summarised pas and expeced fuure developmen of basic economic indicaors, was published by he Minisry
Usefulness of he Forward Curve in Forecasing Oil Prices Akira Yanagisawa Leader Energy Demand, Supply and Forecas Analysis Group The Energy Daa and Modelling Cener Summary When people analyse oil prices,
ON THUSTONE'S MODEL FO PAIED COMPAISONS AND ANKING DATA Alber Maydeu-Olivares Dep. of Psychology. Universiy of Barcelona. Paseo Valle de Hebrón, 171. 08035 Barcelona (Spain). Summary. We invesigae by means
Opimal Invesmen and Consumpion Decision of Family wih Life Insurance Minsuk Kwak 1 2 Yong Hyun Shin 3 U Jin Choi 4 6h World Congress of he Bachelier Finance Sociey Torono, Canada June 25, 2010 1 Speaker
Inroducion o Hypohesis Tesing Research Quesion Is he average body emperaure of healhy aduls 98.6 F? HT - 1 HT - 2 Scienific Mehod 1. Sae research hypoheses or quesions. µ = 98.6? 2. Gaher daa or evidence
Disribuing Human Resources among Sofware Developmen Proecs Macario Polo, María Dolores Maeos, Mario Piaini and rancisco Ruiz Summary This paper presens a mehod for esimaing he disribuion of human resources
Bid-ask Spread and Order Size in he Foreign Exchange Marke: An Empirical Invesigaion Liang Ding* Deparmen of Economics, Macaleser College, 1600 Grand Avenue, S. Paul, MN55105, U.S.A. Shor Tile: Bid-ask
An empirical analysis abou forecasing Tmall air-condiioning sales using ime series model Yan Xia Deparmen of Mahemaics, Ocean Universiy of China, China Absrac Time series model is a hospo in he research
A Noe on Using he Svensson procedure o esimae he risk free rae in corporae valuaion By Sven Arnold, Alexander Lahmann and Bernhard Schwezler Ocober 2011 1. The risk free ineres rae in corporae valuaion
Mahemaics in Pharmacokineics Wha and Why (A second aemp o make i clearer) We have used equaions for concenraion () as a funcion of ime (). We will coninue o use hese equaions since he plasma concenraions
Inroducion Chaper 14: Dynamic D-S dynamic model of aggregae and aggregae supply gives us more insigh ino how he economy works in he shor run. I is a simplified version of a DSGE model, used in cuing-edge
Page 1 of 5 NTNU Noregs eknisk-naurviskaplege universie Fakule for informasjonseknologi, maemaikk og elekroeknikk Insiu for maemaiske fag - English Conac during exam: John Tyssedal 73593534/41645376 Exam
A general decomposiion formula for derivaive prices in sochasic volailiy models Elisa Alòs Universia Pompeu Fabra C/ Ramón rias Fargas, 5-7 85 Barcelona Absrac We see ha he price of an european call opion
Price elasiciy of demand for crude oil: esimaes for 23 counries John C.B. Cooper Absrac This paper uses a muliple regression model derived from an adapaion of Nerlove s parial adjusmen model o esimae boh
Serrasqueiro and Nunes, Inernaional Journal of Applied Economics, 5(1), 14-29 14 Deerminans of Capial Srucure: Comparison of Empirical Evidence from he Use of Differen Esimaors Zélia Serrasqueiro * and
Chaper H Inducance and Transien Circuis Blinn College - Physics 2426 - Terry Honan As a consequence of Faraday's law a changing curren hrough one coil induces an EMF in anoher coil; his is known as muual
Graduae Macro Theory II: Noes on Neoclassical Growh Model Eric Sims Universiy of Nore Dame Spring 2011 1 Basic Neoclassical Growh Model The economy is populaed by a large number of infiniely lived agens.
Chaper 3: Forecasing From Time Series Models Par 1: Whie Noise and Moving Average Models Saionariy In his chaper, we sudy models for saionary ime series. A ime series is saionary if is underlying saisical
Small and Large Trades Around Earnings Announcemens: Does Trading Behavior Explain Pos-Earnings-Announcemen Drif? Devin Shanhikumar * Firs Draf: Ocober, 2002 This Version: Augus 19, 2004 Absrac This paper
ABSTRACT Time Series Analysis Using SAS R Par I The Augmened Dickey-Fuller (ADF) Tes By Ismail E. Mohamed The purpose of his series of aricles is o discuss SAS programming echniques specifically designed
Individual Healh Insurance April 30, 2008 Pages 167-170 We have received feedback ha his secion of he e is confusing because some of he defined noaion is inconsisen wih comparable life insurance reserve
Fourier Series Soluion of he Hea Equaion Physical Applicaion; he Hea Equaion In he early nineeenh cenury Joseph Fourier, a French scienis and mahemaician who had accompanied Napoleon on his Egypian campaign,
Quarerly Repor on he Euro Area 3/202 II.. Deb reducion and fiscal mulipliers The deerioraion of public finances in he firs years of he crisis has led mos Member Saes o adop sizeable consolidaion packages.
Chaper 4 Properies of he Leas Squares Esimaors Assumpions of he Simple Linear Regression Model SR1. SR. y = β 1 + β x + e E(e ) = 0 E[y ] = β 1 + β x SR3. var(e ) = σ = var(y ) SR4. cov(e i, e j ) = cov(y
Business Condiions & Forecasing Exponenial Smoohing LECTURE 2 MOVING AVERAGES AND EXPONENTIAL SMOOTHING OVERVIEW This lecure inroduces ime-series smoohing forecasing mehods. Various models are discussed,
Supplemenary Appendix for Depression Babies: Do Macroeconomic Experiences Affec Risk-Taking? Ulrike Malmendier UC Berkeley and NBER Sefan Nagel Sanford Universiy and NBER Sepember 2009 A. Deails on SCF
Pricing Fixed-Income Derivaives wih he Forward-Risk Adjused Measure Jesper Lund Deparmen of Finance he Aarhus School of Business DK-8 Aarhus V, Denmark E-mail: email@example.com Homepage: www.hha.dk/~jel/ Firs
The Relaionship beween Sock Reurn Volailiy and Trading Volume: The case of The Philippines* Manabu Asai Faculy of Economics Soka Universiy Angelo Unie Economics Deparmen De La Salle Universiy Manila May
econsor www.econsor.eu Der Open-Access-Publikaionsserver der ZBW Leibniz-Informaionszenrum Wirschaf The Open Access Publicaion Server of he ZBW Leibniz Informaion Cenre for Economics Gil-Alaña, Luis A.
. Two quesions for oday. A. Why do bonds wih he same ime o mauriy have differen YTM s? B. Why do bonds wih differen imes o mauriy have differen YTM s? 2. To answer he firs quesion les look a he risk srucure
1. he graph shows he variaion wih ime of he velociy v of an objec. v Which one of he following graphs bes represens he variaion wih ime of he acceleraion a of he objec? A. a B. a C. a D. a 2. A ball, iniially
ENGIN112: Inroducion o Elecrical and Compuer Engineering Fall 2003 Prof. Russell Tessier Undersanding Sequenial Circui Timing Perhaps he wo mos disinguishing characerisics of a compuer are is processor
REFERENCES RC Circuis: Elecrical Insrumens: Mos Inroducory Physics exs (e.g. A. Halliday and Resnick, Physics ; M. Sernheim and J. Kane, General Physics.) This Laboraory Manual: Commonly Used Insrumens:
Chaper Suden Lecure Noes - Chaper Goals QM: Business Saisics Chaper Analyzing and Forecasing -Series Daa Afer compleing his chaper, you should be able o: Idenify he componens presen in a ime series Develop
Dependen Ineres and ransiion Raes in Life Insurance Krisian Buchard Universiy of Copenhagen and PFA Pension January 28, 2013 Absrac In order o find marke consisen bes esimaes of life insurance liabiliies
Modeling VIX Fuures and Pricing VIX Opions in he Jump Diusion Modeling Faemeh Aramian Maseruppsas i maemaisk saisik Maser hesis in Mahemaical Saisics Maseruppsas 2014:2 Maemaisk saisik April 2014 www.mah.su.se
Does Opion Trading Have a Pervasive Impac on Underlying Sock Prices? * Neil D. Pearson Universiy of Illinois a Urbana-Champaign Allen M. Poeshman Universiy of Illinois a Urbana-Champaign Joshua Whie Universiy
VII. THE FIRM'S INVESTMENT DECISION UNDER CERTAINTY: CAPITAL BUDGETING AND RANKING OF NEW INVESTMENT PROJECTS The mos imporan decisions for a firm's managemen are is invesmen decisions. While i is surely
Even and Odd Funcions 23.3 Inroducion In his Secion we examine how o obain Fourier series of periodic funcions which are eiher even or odd. We show ha he Fourier series for such funcions is considerabl
Review of Economic Dynamics 3, 486522 Ž 2000. doi:10.1006redy.1999.0084, available online a hp:www.idealibrary.com on Hiring as Invesmen Behavior Eran Yashiv 1 The Eian Berglas School of Economics, Tel
Defaul Risk in Equiy Reurns MRI VSSLOU and YUHNG XING * BSTRCT This is he firs sudy ha uses Meron s (1974) opion pricing model o compue defaul measures for individual firms and assess he effec of defaul
Mah 344 May 4, Complex Fourier Series Par I: Inroducion The Fourier series represenaion for a funcion f of period P, f) = a + a k coskω) + b k sinkω), ω = π/p, ) can be expressed more simply using complex
Economics Honors Exam 2008 Soluions Quesion 5 (a) (2 poins) Oupu can be decomposed as Y = C + I + G. And we can solve for i by subsiuing in equaions given in he quesion, Y = C + I + G = c 0 + c Y D + I
Do Propery-Casualy Insurance Underwriing Margins Have Uni Roos? Sco E. Harringon* Moore School of Business Universiy of Souh Carolina Columbia, SC 98 firstname.lastname@example.org (83) 777-495 Tong Yu College
USE OF EDUCATION TECHNOLOGY IN ENGLISH CLASSES Mehme Nuri GÖMLEKSİZ Absrac Using educaion echnology in classes helps eachers realize a beer and more effecive learning. In his sudy 150 English eachers were
COMPUTATION OF CENTILES AND Z-SCORES FOR HEIGHT-FOR-AGE, WEIGHT-FOR-AGE AND BMI-FOR-AGE The mehod used o consruc he 2007 WHO references relied on GAMLSS wih he Box-Cox power exponenial disribuion (Rigby
10/19/004 A Mahemaical Descripion of MOSFET Behavior.doc 1/8 A Mahemaical Descripion of MOSFET Behavior Q: We ve learned an awful lo abou enhancemen MOSFETs, bu we sill have ye o esablished a mahemaical
Norh merican cuarial Journal Volume 6, Number 1, p.166-170 (2002) Re-eaminaion of he Join Morali Funcions bsrac. Heekung Youn, rkad Shemakin, Edwin Herman Universi of S. Thomas, Sain Paul, MN, US Morali
Proceedings of he Firs European Academic Research Conference on Global Business, Economics, Finance and Social Sciences (EAR5Ialy Conference) ISBN: 978--6345-028-6 Milan-Ialy, June 30-July -2, 205, Paper
Inverse Funcions Reference Angles Inverse Trig Problems Trig Indeniies HANDOUT 4 INVERSE FUNCTIONS KEY POINTS A.) Inroducion: Many acions in life are reversible. * Examples: Simple One: a closed door can
Iner-American Developmen Bank Banco Ineramericano de Desarrollo (BID) Research Deparmen Deparameno de Invesigación Working Paper #647 Do Credi Raing Agencies Add Value? Evidence from he Sovereign Raing
Absrac number: 05-0407 Single-machine Scheduling wih Periodic Mainenance and boh Preempive and Non-preempive jobs in Remanufacuring Sysem Liu Biyu hen Weida (School of Economics and Managemen Souheas Universiy
Chaper 2 Unsrucured Experimens 2. Compleely randomized designs If here is no reason o group he plos ino blocks hen we say ha Ω is unsrucured. Suppose ha reamen i is applied o plos, in oher words ha i is
House Price Index (HPI) The price index of second hand houses in Colombia (HPI), regisers annually and quarerly he evoluion of prices of his ype of dwelling. The calculaion is based on he repeaed sales