THE SMALL SAMPLE PROPERTIES OF TESTS OF THE EXPECTATIONS HYPOTHESIS: A MONTE CARLO INVESTIGATION E. GARGANAS, S.G.HALL

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1 ISSN THE SMALL SAMPLE PROPERTIES OF TESTS OF THE EXPECTATIONS HYPOTHESIS: A MONTE CARLO INVESTIGATION E. GARGANAS, S.G.HALL Taaka Busiess School Discussio Papers: TBS/DP04/6 Lodo: Taaka Busiess School,

2 The small sample properies of ess of he expecaios hypohesis: A Moe Carlo ivesigaio By E. Gargaas (Naioal Bak of Greece), S.G. Hall (Imperial College) ABSTRACT I his paper, we exed resuls from he fiace lieraure ha explores small-sample bias, due o persise variables, i ess of prese value asse pricig models. Usig Moe Carlo experimes we aalyze he fiie sample behaviour of a variey of ess of he expecaios hypohesis of he erm srucure, whe ieres raes are icreasigly persise. Resuls overwhelmigly idicae ha asympoic iferece ca be misleadig. I he Campbel ad Shiller (1991) vecor auoregressio (VAR) framework, he Wald es overrejecs he correc ull ad he slope of he correspodig ordiary leas squares (OLS) es is srogly biased. The disorios ca be exreme for recursive esimaors, such as, he Kalma filer ime varyig parameer model. Ocober 2004 JEL classificaio: C15; E43; G12; Keywords: Ieres raes; Expecaios hypohesis; Small sample bias; Kalma filer 2

3 1. Iroducio The icoclusive empirical evidece agais he expecaios hypohesis of he erm srucure of ieres raes has ofe led researchers o sugges ha he heory remais popular primarily o a priori grouds, as i sill seems o provide he mos iuiive explaaio for he erm srucure. I is pure form he expecaios hypohesis (EH) saes ha he log rae is a forecas of expeced fuure shor raes, assumig zero or a ime-ivaria risk premium ad raioal expecaios. A he same ime, hese empirical puzzles have brough abou he developme of a small variey of ieresig explaaios for he heory s rejecio. A recurrig issue i he rece empirical lieraure, has bee he adopio of a formal mehodology for excludig he uforuae case i which he EH is merely rejeced as a resul of saisical or ecoomeric disorios i small samples (ad correcig for hem). The paricular case of srog small sample bias due o persise ieres raes is a ypical problem ha applies o may markes, icludig ha of he U.S. Geerally, here are hree broad explaaios for empirical failures of he EH. Firs, he heory simply is rue. The key behavioural assumpios do o hold e.g. ages are irraioal or risk averse. This has ured a relaively small par of research away from US ieres raes for which he EH is usually rejeced (e.g. Campbel ad Shiller, 1991; Bekaer e al., 1997) owards oher, maily Europea, marke raes for several of which he EH is o rejeced (e.g., Hardouvelis, 1994; Gerlach ad Smes, 1997; Bekaer ad Hodrick, 2000). Secod, he EH ca be saved by icorporaig ime-varyig risk premia o he model specificaio (GARCH models) (Egle, e al., 1987). Third, he saisical ess hemselves lead o false rejecios due o heir poor small sample properies. This leads o a family of explaaios icludig peso problems wih regime shifs (e.g. Bekaer e al., 2001), learig behaviour ad, fially, persise variables (Makiw ad Shapiro, 1986; Bekaer e al., 1997). The laer is he explaaio ha we se ou o explore i his paper, usig arificial raher ha real world ieres raes, i a Moe Carlo simulaio evirome. 3

4 I a iflueial paper, Bekaer e al. (1997) show ha he small sample properies of sadard ess of he EH ca be surprisigly wised whe ieres raes are highly persise. I heir resuls, he wo well kow Campbel ad Shiller (1991) spread regressio specificaios ad he wo VAR mehodology regressios, all provide srogly biased coefficies ha deped o he persisece parameer of a AR(1) oe-period rae, eve for samples of 2000 observaios. The high dispersio of he small sample disribuio of he slope coefficie, makes i more difficul o rejec he EH i small samples. They coclude ha small sample iferece usig carefully desiged Moe Carlo simulaio experimes has esseially become a sie qua o for esig he EH i moder fiace research (e.g. Logsaf, 2000). I his paper, we coduc a series of Moe Carlo experimes o examie he small sample properies of he perfec foresigh spread regressio es, he variace bouds iequaliy ad four VAR mehodology saisics (Campbel ad Shiller, 1991), all i respose o a icreasig persisece parameer. Give ha he lieraure is o always coclusive o he properies of hese small sample disribuios, depedig o he daa geeraig assumpios (or he heoreical approximaios i closed form esimaes), we cofirm ha for mos of hese ess high persisece leads o a bias owards overrejecio of he EH whe asympoic iferece is used. The surprisig excepio is he variace bouds raio for which he EH is uderrejeced. We also fid a paricularly srog bias i he case of he Wald es of he VAR coefficie resricios ad i he slope coefficie of he regressio of he (VAR-cosruced) heoreical agais he acual spread. Cosiderig he furher issue of ieres rae mauriy horizo we fid ha i maers sigificaly, for all ess. Overall, our Moe Carlo resuls idicae ha he saisical size disorios are larger for some ess ha for ohers. This would help explai why i is ofe he case ha i acual markes differe ess may be more favourable or less favourable o he EH. The paper is orgaised as follows. Secio 2 briefly oulies ess of he EH model. Secio 3 explais why we may expec small sample bias o be paricularly srog for mos of hese ess whe ieres raes are highly persise. We discuss he lieraure fidigs ad focus o he properies of he slope coefficie i he Campbel ad Shiller perfec foresigh spread regressio. Secio 4 explores Moe Carlo evidece regardig he small sample behaviour of 6 sadard EH ess ad a Kalma filer model. A sesiiviy aalysis is performed wih respec o a chagig persisece coefficie i a AR(1) daa geeraig process for he shor rae. The las secio he summarises he key fidigs wih some closig remarks. 4

5 2. Empirical ess of he EH 2.1 The perfec foresigh spread regressio ad volailiy ess The mai erm srucure relaioship implied by he EH usig coiuously compouded ieres raes ad a logarihmic approximaio for pure discou bods is: R = 1 1 Εr + c i = 0 (1) where he shor rae, wih oe period o mauriy, is deoed by r, he log rae wih (>1) periods o mauriy is represeed by R ad c is a cosa erm premium. Equaio (1) he saes ha he log rae, a cosa, hrough ime, erm premium, c. R, is a weighed average of expeced fuure shor ieres raes, r, plus I ca be show ha he fudameal EH equaio (1) may be rasformed hrough subracio of he shor rae, r, from boh sides, io he followig expressio (seig he erm premium c o zero, for simpliciy): i 1 (2) R r = 1 Ε r + i i= 1 where S = R r is he acual spread bewee he log ad he shor rae. Campbell ad Shiller * S = 1 i r + i = 1 (1991) defie he variable ( ) i 1 i expressio (2) as he spread prediced by he model if ages have perfec foresigh. The perfec foresigh spread (PFS) is hus cosruced as a weighed average of chages i fuure shor raes, usig ex-pos realisaios of ieres raes from he sample. Expressio (2) he saes ha, accordig o he EH, he acual spread, S, is he expeced PFS, Ε S *. This formulaio (2) is ormally adoped whe ieres raes are believed o be o-saioary I(1) processes. 5

6 Addig o eiher of hese expressios ((1),(2)) he raioal marke expecaios (RE) assumpio ha he shor rae is equal o is forecas plus a whie - oise error erm: r = Ε r + η (3) has led o a umber of ess of he EH proposed i he lieraure. Oe of he OLS regressio ess suggesed by Campbell ad Shiller (1991) ha we cosider here is: S * = α + βs + ε (4) We expec β = 1 ad α = 0 uder he ull of he joi pure expecaios hypohesis (EH wih zero erm premium c) wih he RE assumpio (hereby simply referred o as EH). The error erm is serially correlaed ad follows a MA(-2) process ad a GMM correcio o he variace covariace marix of he coefficies ca be used. We also cosider he correspodig variace boud iequaliy, reformulaed by Campbel ad Shiller (1991) i erms of S ad S * as: Var ( S ) Var( S ) (5) The variace (or sadard deviaio) of he ieres rae spread may o exceed ha of he PFS whe he EH is rue. Equivalely, he raio of he correspodig sadard deviaios (or variaces) mus be less ha 1. I is geerally expeced ha he wo ess (regressio (4) ad variace boud (5)) mus give similar ifereces. 2.2 The Campbell - Shiller VAR mehodology Campbell ad Shiller (1991) show ha he fudameal EH equaio (1) may sill be esed by replacig he RE assumpio wih a auoregressive scheme for predicig he perfec foresigh spread (i expressio (2)), as log as he acual spread S is icluded i he forecasig variables. 6

7 Assumig, firs, ha ad r S are saioary sochasic processes, vecor [ ] S r Z is also = a saioary sochasic process. I is he assumed ha a p-h order vecor auoregressio (VAR) represeaio of Z may be used o forecas he PFS. This bivariae VAR(p) may be wrie i compaio form (as a 1 s order VAR): z ω (6) + 1 = Az + +1 where, A, is a square (2p x 2p) marix of coefficies ad z is a (2p x 1) vecor of regressors, [ ] r r r S S... S z = -1 p+ 1 1 p The compaio form VAR may he be used o forecas (chages i) fuure shor raes as required by he EH equaio (2), by defiig wo (2p x 1) selecio vecors e1 ad e2 such ha S = e1 z ad r = e2 z. These vecors choose he wo key variables of ieres, S ad r, hrough oe eleme equal o 1, which is he 1 s eleme, for e1, ad he p+1 s eleme (row) for e2, wih all remaiig (2p-1) elemes beig zero. Usig he he VAR from equaio (6) o forecas oe period ahead: Ε Ε r 1 = e2 z 1 = e2 + + Az (7) More geerally, he chai rule of forecasig for muli period expecaios usig he VAR implies: E r e2 A z i + i = (8) The geeralised VAR forecas for equaio (2): Ε r + i from equaio (8) ca he be replaced i he EH 1 * i i S = Ε S = 1 e2 A z i = 1 (9) 7

8 The forecas of he PFS usig he VAR, i he righ had side of equaio (9), is he heoreical spread, S. Therefore, if he EH (2) is rue, he acual spread should be equal o he heoreical spread, e 1 z = S. From (9), Campbell ad Shiller (1991) derive he followig resricios o he VAR parameers whe he EH is rue: 1 f ( a) = e1 e2 A I 1 1 ( I A )( I A) ( I A) = 0 (10) The (2p x 1) o-liear resricios, f ( a), expressed i (10) deped oly o he esimaed parameers, Â, of he uresriced VAR model (6). A Wald es may he be used (as i Campbel ad Shiller, 1987) o es heir validiy: W = f 1 ( a) { [ f ( a) ]} f ( a) var (11) The variace of he resricios, var [ f ( a) ] VAR coefficies, Σ:, depeds o he variace-covariace marix of he var = (12) [ f ( a) ] f ( a) Σ f ( a) a a where f a (a) is he firs derivaive of he resricios evaluaed a he coefficie esimaes (from A). I addiio o he Wald es, he VAR mehodology employs a small lis of oher measures of proximiy bewee S ad S, ha ca be used o evaluae he EH, as i suggess hrough (9) ha he wo series mus be equal. Besides he iformal use of a graph of he wo series, saisical measures iclude a raio of sadard deviaios, he correlaio saisic, ad a OLS regressio es: ( S ) sd( S ) = 1 sd (13) ( S, S ) = 1 ρ = Corr (14) 8

9 S = α + βs + ν (15) where we expec α=0 ad β=1. 3. Small sample bias ad ieres rae persisece Cosiderig agai he previous es saisics, i ca be show ha early all of hem suffer from small sample bias ha becomes quie severe i some cases, depedig o he ime series properies of ieres raes ad of course he size of he sample. All OLS based ess icludig he VAR regressio ess poeially provide biased coefficie esimaes (Bekaer e al., 1997). Similarly, he variace bouds family of ess (Flavi, 1983) also eds o disor iferece i samples of ypical size. This secio focuses o he bias i he slope coefficie of he PFS regressio es i (4). Briefly, i his case, he OLS coefficie esimaes are biased due o he presece of a sochasic ad o ecessarily exogeous regressor (S ). Firs, we model he oe period ieres rae as a firs order auoregressive (AR(1)) process. Besides he desirable propery of greaer aalyical racabiliy (i he EH framework), he AR(1) model usually describes well ieres rae daa for acual ecoomies. Thus, he oe period ieres rae, r, is: r = µ + ρ 1 + ε (16) r where ρ 1 ad he error erm, ε, is a ormally ad idepedely disribued process wih mea zero ad a cosa variace, σ ε 2. Whe he EH (+RE) is rue, he log rae ad he spread are proporioal o r, hece, hey oo are persise. I greaer deail, whe µ = 0 (i 16) ad he AR(1) is subsiued ou o forecas fuure oe-period shor raes i he fudameal EH equaio (1) he log rae is: R 1 1 j = r 1 ρ = j = 0 1 ( 1 ρ ) r ( ρ ) (17) 9

10 Accordigly, he spread, S, is also aalogous o r : ( 1 ρ ) 1 S = R r = 1 r ( ) ( ) = η ρ, r 1 ρ (18) 1 where ( ) ( 1 ρ ) η ρ, = 1. ( 1 ρ ) Usig he AR(1) represeaio for r, expressio (18) implies ha he spread is persise ad he persisece parameer is, agai, ρ: S ( ρ )( ρr 1 + ε ) = ρη( ρ, ) r 1 + η( ρ, ) ε = ρs ν = 1 η, + (19) The PFS regressio (4) ad expressio (19) esseially form he followig sysem of equaios: S S * = α + βs = ξ + ρs 1 + u + v 2 where u ~ i.i.d(0, σ ) 2 where v ~ i.i.d(0, σ ) u v (20) (21) The secod equaio implies ha S is o fixed i repeaed samples, violaig a OLS requireme for he esimaio of he PFS equaio (20). Moreover, he error erms, u ad v, are coemporaeously correlaed wih covariace σ uv. Sadard ecoomeric heory ells us ha if S, he idepede variable i he PFS regressio, is sochasic he his is o ecessarily problemaic for he ypical properies of he OLS esimaor. The Gauss-Markov heorem may sill hold provided he error process of he regressor is idepede of he error process of he regressio iself. However, eve i ha case, he small sample properies of he OLS esimaor may be poor. 1 Neverheless, i his sysem, here is also a correlaio (σ uv ) bewee he disurbaces of he idepede variable (S ) ad he disurbaces of he PFS regressio (20) iself ad herefore he Gauss-Markov heorem o loger applies. I he paricular example, his is because he acual spread iheris o oly he persisece bu also he sochasic compoe of he shor rae, ε, 10

11 from he AR(1) model (16). The PFS o he lef had side of (20) also coais ha compoe as i icludes r +1 (from EH (2)). Therefore, he OLS coefficie esimaes, ˆ α ad ˆ β, are cosise bu biased, i samples of ypical size. Moreover he samplig disribuio of his small-sample esimaor is o sadard. This problem is well kow i he fiace lieraure, as i applies o forecasig equaios ofe used i ess of asse pricig heories, icludig he PFS regressio (e.g., Sambaugh 1986, 1999; Makiw ad Shapiro, 1986). Sambaugh (1986) has show ha if he sochasic regressor is also persise (as S is here) he he bias i esimaig he OLS slope coefficie depeds o he bias i esimaig he persisece coefficie, ρ, as follows: Ε ( ˆ σ uv β β ) = ( ˆ ρ ρ ) σ Ε 2 v (22) The bias i esimaig he auocorrelaio coefficie (ρ) of a auoregressive process is egaive (dowward) (Marrio ad Pope, 1954). The closer he rue coefficie, ρ, is o 1, he more severe his ype of bias becomes. Moreover, i he PFS regressio, he covariace, σ uv, is egaive so he slope bias (i (22)) is posiive (upward). Bekaer e al. (1997), illusrae ha i he case of he PFS regressio he slope bias expressio (22) will implicily ivolve esimaio of higher order auocorrelaio coefficies (due o he fuure shor raes ivolved i cosrucig he PFS) as well. They use Kedal s (1954) aalyical approximaios for he bias i higher order auocorrelaio coefficies o reach a more specific expressio (ha 22) of he bias i βˆ, i he PFS regressio. The bias is expressed as: 1 1 ˆ 1 (1 ) Ε( β ρ 1) = θ j = θ 0 j η (1 ρ) (1 ρ ) (23) ( ρ, ) j= 1 1 where ( ) ( 1 ρ ) j= 1 η ρ, = 1 ad θ ( 1 ρ) j is Kedall s approximaio o he esimaed auocorrelaio coefficie bias: 1 E.g. see pp i Cuhberso e al. (1992). 11

12 ( ρ ρ ) ( 1+ ρ ) 1 θ j = E ˆ j j = ( ) 2 (24) T 1 ( ) ( j ) j 1 ρ + jρ 1 ρ where j is he order of auocorrelaio. Hece, i his case: Ε ( ˆ β 1) = f (, T, ρ). The esimaed βˆ is posiively biased, above 1. The bias icreases wih higher ρ, smaller T ad smaller. The bias i expressio (23) is paricularly srog for very high auocorrelaio coefficies, ρ, akig values bewee 0.90 ad Bekaer e al. (1997) show ha he empirical disribuio is highly dispersed ad skewed o he righ. I akes almos as may as 2,000 observaios for symmery. Cosiderig, ow, he VAR ess ad saisics, Bekaer e al. (1997) show ha he VAR OLS coefficies are all biased, depedig o persisece. The bias passes hrough o early all he VAR-based ess ad merics. I summary, hey show ha: a) o a firs order approximaio, Corr spread ( S, S ) = 1, he coefficie of correlaio bewee he spread S ad he heoreical S is ubiased for small samples ad b) he sadard deviaio raio ( S ) sd( ) sd is S biased (upwards) as ρ rises, exceedig is heoreical value of 1 2. Bekaer ad Hodrick (2001) show ha he small sample disribuio of he Wald es of he EH resricios for he VAR coefficies is severely disored. The Wald saisic eds o overrejec he EH (biased upwards). 4. Moe Carlo experimes 4.1 Moe Carlo Srucure The resuls discussed i he previous secio show ha small sample iferece is esseial for esig he EH, especially whe ieres rae daa display a high degree of persisece. Give ha asympoic iferece is misleadig i fiie samples, well-desiged Moe Carlo simulaio (MCS) experimes ca be used o correc for small sample bias (e.g., Logsaf, 2000). Aalyical values from closed form esimaes ca eiher be edious o reach (i he case of he VAR) or of doubful validiy for high ρ. Moreover, esimaes of he bias i he coefficies are o sufficie for saisical iferece give also he skewess of he small sample disribuios. 2 Campbel ad Shiller (1991) also repor Moe Carlo evidece for he small sample disribuios of he VAR saisics. They fid a smaller bias, o large eough o overur he rejecio of he EH for heir daa. 12

13 We use Moe Carlo simulaio experimes o explore he sesiiviy of he small sample disribuios of he differe EH ess, described i secio 2, o chagig persisece levels. The AR(1) process for r i (16) is ceral o he Moe Carlo aalysis, as we are heoreically ieresed i he resposiveess of he shape of hese disribuios o persisece, ρ. However, researchers wishig o es acual daa migh boosrap heir esimaes ad correc he daa geeraig process for he bias i ρ, while also adopig a richer represeaio of realiy wih a VAR or VAR-GARCH model for he shor rae (e.g., Logsaf, 2000; Bekaer e al., 1997). We geerae daa uder he ull of EH (=EH+RE). This requires us o use he fudameal EH equaio (1) o geerae he log rae R (for =12), which i ur requires us o subsiue expeced fuure oe period raes. We, firs, geerae (T+200) observaios of he oe period rae, r, usig he AR(1) model i equaio (16) wih ormally disribued radom errors ad he recursively subsiue for fuure shor raes (RE) i he log rae model (EH). For he geeraed series, we he discard he firs 200 observaios o miimise he ifluece of iiial codiios. The acual spread we cosruc he PFS ( S 12 S is he he differece bewee he wo ieres rae series, R 12 ad r, ad *12 ) (from he EH equaio (2)) as he weighed average of chages i he forwarded shor raes, subsiuig recursively he AR(1) model. Havig geeraed series for he PFS ad he acual spread, complyig wih he EH, we esimae i each experime: 1) he PFS OLS regressio es (4) ad 2) i a ew roud of Moe Carlo experimes, he sadard deviaio raio of he wo series (5). I a hird roud of experimes, wih he same process for daa geeraio, ad afer calculaig he acual spread 12 S ad r we esimae a VAR(1) model ad use he coefficie esimaes o cosruc he heoreical spread, S. We he esimae i each replicaio he correspodig OLS regressio es (15) ad he VAR merics (correlaio ad sadard deviaio raio). For all hese experimes, we derive he small sample disribuios of he coefficies ad saisics, replicaig he exercise 5,000 imes. We repea he 5,000 Moe Carlo replicaios for arificial samples of (T=) 124, 524, 2,000 ad 20,000 observaios. Aiheic variaes are used i all experimes wih he excepio of he variace boud es for which we repea 10,000 imes (isead of 5,000) he Moe Carlo rials. For he AR(1) parameers i (16) we assume fixed values for µ = 1.24 ad σ 2 ε = 2.28, wih ρ chagig. We also validae he code by ryig he same se of parameer values for he AR(1) process as hose used i Bekaer e al. (1997) i heir similar experime. For he slope of he PFS regressio es, we fid bea=

14 agais heir 1.51 wih a sadard deviaio s.d.=0.62 agais 0.65 (furher similariies are documeed i figure 1 ad figure 2). 14

15 4.2 Simulaio resuls The perfec foresigh spread OLS es Table 1 repors Moe Carlo resuls (for 5,000 rials) ha oulie he small sample disribuio of he slope coefficie β i he PFS regressio es from experimes i which (AR(1)) ieres raes (from equaio (16)) were geeraed, as described i he previous secio, for differe ρ. We also explore larger sample properies of he es for samples of 524, 2,000 ad 20,000 observaios. Table 1. OLS slope coefficie β for PFS 12: ρ ad sample size sesiiviy Pael A, repors Moe Carlo resuls for differe samples ad for a broad rage of ρ bewee 0 ad 1. For each ρ, he mea slope coefficie β ad he sadard deviaio aroud his esimae are repored. The firs value repored is from OLS esimaio ha icludes a cosa erm, while he value udereah deoes OLS esimaio wihou a cosa erm. Pael B, repors resuls for very high values of ρ ad a sample of 124 observaios, for hree quarile levels of he small sample disribuio (besides he mea ad he sadard deviaio). Pael A Larger samples: Number of Observaios (T) = ,000 20,000 Mea βˆ Mea βˆ Mea βˆ Mea βˆ ρ (s.e.) (s.e.) (s.e.) (s.e.) ρ Mea βˆ (s.e.) Pael B High Persisece T=124 Quariles (0.044) (0.044) (0.015) (0.016) (0.007) (0.007) (0.002) (0.002) (0.226) (0.226) (0.053) (0.056) (0.023) (0.023) (0.011) (0.011) (0.004) (0.004) (0.416) (0.395) (0.078) (0.081) (0.038) (0.038) (0.019) (0.019) (0.006) (0.006) (0.689) (0.628) (0.146) (0.150) (0.077) (0.078) (0.041) (0.040) (0.013) (0.013) (2.305) (1.673) (0.159) (0.158) (0.079) (0.080) (0.041) (0.041) (0.013) (0.013) (0.159) (0.158) *Noe: 1s row for each value of he auocorrelaio coefficie, ρ, refers o sadard OLS value of he slope ad he secod row refers o he resriced-cosa OLS value of he slope. 15

16 Overall, he resuls are i accordace wih he aalyic approximaio 3 ad Moe Carlo evidece i Bekaer e al. (1997). For very high ρ, above 0.85, (i Pael B) we fid sigifica posiive bias ad disorios of he small sample disribuio of he esimaor. However, for lower degrees of persisece (Pael A), a egaive bias is also prese. For example, he aalyical approximaio esimae for ρ = 0.75 ad a sample of 124 observaios is β = 1.08, which implies a very small posiive bias (agais he heoreical value of 1). The mea esimae i our Moe Carlo resuls is 0.98, suggesig a very small egaive bias i he esimae of he slope coefficie, β. The bias is prese for every ρ i pael A of he able, for samples of 124 ad 524 observaios. I is due o small sample iflueces, oher ha ha from he persisece of ieres raes, perhaps also o be expeced. The resuls i Pael A of able 1 sugges wo paers as we icrease ρ ad he sample size, T. As ρ varies from 0 o 0.75 he mea slope, βˆ, (ad he sadard deviaio aroud i) rises from 0.92 o 0.98 for 124 observaios ad similarly for samples up o 2000 observaios 4. Thus as auocorrelaio rises, he implied posiive bias i esimaig β (described i Bekaer e al., 1997) rises, parly offseig he egaive bias foud i small samples. Cosiderig ow larger samples, he value of he slope β approximaes he heoreical value of 1, cofirmig he cosisecy propery of he OLS esimaor. I is srikig ha eve for a subsaially large sample of 2000 observaios he mea slope, which is very close o 1, sill slighly rises wih ρ, a he hird decimal poi. The slope disribuio collapses exacly o 1, wih a very small sadard deviaio, oly for a sample of 20,000 observaios. Pael B of able 1, repors resuls for high persisece, wih ρ bewee 0.90 ad 0.99 ad a sample of 124 observaios. The posiive bias ow becomes large (e.g. mea βˆ =1.63 for ρ = 0.96) ad he small sample disribuio is widely dispersed (sadard deviaio = 0.69). We fid ha 97.5% of he slope coefficie esimaes for ρ = 0.96 are below 3.11, implyig ha if a slope coefficie is esimaed o be as high as 3 i acual daa for a marke of he same degree of ieres rae persisece, he PEH should sill o be rejeced, usig his able. The resuls sugges ha, usig small sample, raher ha sadard asympoic, iferece, i is more difficul o rejec he ull 3 This is proposiio 3 i Bekaer e al. (1997), saed i equaio (23) here. 4 We also repor he ui roo case, for ρ = 1, which behaves differely. The aalyical approximaio of he bias refers oly o values smaller ha 1. 16

17 hypohesis whe ieres rae persisece is high due o he large dispersio of he small sample disribuio 5. Figure 1. PFS es slope coefficie: small sample agais asympoic disribuio Desiy curves (scaled) for small sample vs Asympoic Normal disribuio 0.04 Relaive frequecy Value of he slope coefficie (β ) small sample db N(1,0.0924) Noe: MCS resuls for sample size T=124 observaios ad ρ= NW-s.e. = Figure 2. Moe Carlo desiy curves for larger samples 0.25 ρ = Relaive Frequecy Value of slope coefficie (β) We repeaed all he above experimes excludig his ime he cosa erm from he OLS regressios. We fid ha for high ρ, he mea esimae of β, i he small sample disribuio (i able 1) ad he media (which is o repored i he able) are closer o 1 (see figure 3) ha hey 5 See Bekaer e al,

18 were i sadard OLS. The bias is subsaially reduced hough o elimiaed. However, he sadard deviaio of he small sample disribuio is agai high. Therefore, for high values of ρ, he slope coefficie bias is almos elimiaed whe he cosa erm is excluded from OLS regressios bu i is sill more difficul o rejec he ull hypohesis (EH), ha i is i large samples. The disicio bewee he wo esimaios is o impora for lower ρ bewee 0 ad 0.75 (pael A). Figure 3. Shif i he mea of he small sample disribuio: zero OLS cosa erm 0.03 Icludig vs excludig OLS cosa (124 obs) Relaive frequecy Value of he slope coefficie (β ) exclude c iclude c Table 2 repors Moe Carlo evidece for he sesiiviy of he small sample disribuio across differe spread horizos 6. The spread is calculaed bewee he (EH-cosruced) log rae for = 12, 9, 6, 3, 2 ad he AR(1) oe period rae. The sample cosiss of 124 observaios ad ρ is fixed o 0.93 (µ = 1.24). 6 For a sudy of marke efficiecy ess across differe horizos see Campbell (2001). 18

19 Table 2. OLS es for PFS across differe spreads: ρ=0.93, T=124. Spread mauriy, Mea β (s.e.) (aalyic al) - raio agais 1 Quariles: media (0.416) (1.415) (0.464) (1.429) (0.517) (1.443) (0.589) (1.459) (0.628) (1.465) Noe: Aalyical refers o he heoreical approximaio, usig proposiio 3 i Bekaer e al. (1997). Resuls i able 2 show ha as he mauriy () of he log rae i he spread S becomes shorer, he mea βˆ (ad he media) from he MCS experimes ad he sadard deviaio ed o rise. I he case of S 2, he mea Moe Carlo esimae of βˆ is 1.45 (compared o he aalyical esimae of 1.47), agais 1.27 foud for S 12. These resuls are broadly i agreeme wih he aalyical approximaios: as rises ( ˆ β 1 ) E (i equaio (23)) becomes smaller. Therefore, i all he previous Moe Carlo resuls i able 1, for which S 12 was used as a bechmark case, he bias i βˆ is udersaed whe oe refers o oher spreads. 19

20 4.2.2 Variace bouds I he ex roud of Moe Carlo experimes we explore he small sample disribuio of he * variace boud raio (i (5)), d. ( S ) s. d. ( S ). s, for he same se of values for chagig ρ ad T. We use agai he AR(1) oe period rae o geerae he log rae uder he ull hypohesis (EH), for 10,000 replicaios 7. Table 3: Volailiy raio for chagig ρ ad T * The raio VR= s d. ( S ) s. d. ( S ). is calculaed over 10,000 rials, for =12. Pael A, covers a wide rage of values for ρ, bewee 0 ad 1 for samples of 124, 524, 2,000 ad 20,000 observaios. Pael B cosiders oly high ρ for a small sample of 124 observaios. ρ Pael A Larger samples: Number of Observaios (T) = ,000 20,000 Mea VR (s.e.) % of raios >1 (ou of all rials) Mea VR (s.e.) Mea VR (s.e.) Mea VR (s.e.) Pael B High Persisece Number of Observaios (T)= 124 ρ Mea VR (s.e.) Quarile s (0.025) 45.8% (0.008) (0.004) (0.001) (0.077) (0.029) 15.7% (0.012) (0.006) (0.002) (0.085) (0.040) 3.4% (0.018) (0.009) (0.003) (0.078) (0.063) 0.6% (0.030) (0.016) (0.005) (0.077) (0.018) 0.0% (0.005) (0.001) (0.000) (0.018) We double he umber of replicaios o 10,000, i his case, as he aiheics echique does o make a differece for calculaig sadard deviaios. 20

21 We fid ha he mea volailiy raio esimaed over all rials, for a sample of 124 observaios ad for ρ = 0, is exacly equal o 1 ad he 97.5% quarile (which is o repored i he able) is a 1.06, which violaes he boud. This seems surprisig a firs, give ha he heoreical value of 1 is he exac upper boud ad ha daa were geeraed uder he ull. The violaio of he boud is observed for higher ρ as well. This is he ype of violaio of he boud explaied i J. Flavi s kow paper (1983), hough for ieres rae, raher ha spread, volailiy raios. I small samples he raio is biased upwards edig o violae he boud ad rejec he EH oo ofe, whe he ull hypohesis (EH) is rue. However, i our Moe Carlo resuls, as ρ rises, he mea raio eds o fall. For ρ = 0.75 ad a sample of 124 observaios, he mea raio is equal o 0.79 agais 0.91 for ρ = 0.5 ad correspodigly he 97.5% quarile level is foud a 0.94 agais 1.01 (o i he able). The sadard error aroud he mea Moe Carlo esimae also rises wih ρ. Therefore, i opposie direcio o he upward small sample bias here is also a dowward bias o he esimaed raio, which is sroger whe persisece rises. The persisece effec is more evide i pael B (of able 3) for ρ > The secod colum of he able (3) shows he opposig biases more clearly. Whe ρ is low here are freque violaios of he boud (e.g., a ρ = 0, 46% of he raios are above 1). As persisece rises, however, i is foud ha violaios fall owards 0, as a resul of he domiaig, dowward, persisece bias foud i hese experimes. For larger samples, biases ed o be elimiaed. However, he bias due o higher persisece sill exiss give ha he raio eds o fall whe persisece is higher. Eve for a very large sample of 20,000 observaios, a ρ = 0.50, he mea raio is 0.93, fallig o 0.77 a ρ = Overall, large sample resuls do o collapse o a specific value, ulike he case of OLS, where he esimaor was foud cosise. Ulike he OLS case, he variace bouds raio is o expeced heoreically o coverge o a specific value ad i mus oly saisfy he iequaliy. 8 The ui roo sceario, ρ = 1, is agai a special case ad he mea raio dramaically falls o

22 Figure 4. Moe Carlo hisogram: Volailiy raio ρ = 0.93, 124 observaios Table 4. Volailiy raios across differe spreads: ρ =0.93, T=124 Spread mauriy, 12 Mea VR (s.e.) (0.085) media (0.076) (0.063) (0.046) (0.035) Table 4, repors resuls for differe mauriy horizos,, of he spreads S. The level of persisece is fixed a ρ = 0.93 i all experimes, for samples of 124 observaios. Similarly o our previous resuls for he OLS regressio es, whe he log rae mauriy,, becomes shorer, he bias is larger ad he mea raio for S 2 falls o 0.17 agais 0.47 for S 12. Agai, he resuls repored i able 3 ad referrig o S 12, udersae he persisece bias for shorer mauriy spreads. 22

23 4.2.3 VAR ess ad merics I his roud of Moe Carlo experimes we explore he properies of he VAR model. Usig he same parameer values, we geerae r as a AR(1) process ad he log rae accordig o he EH, as before. I each rial, we geerae daa uder he ull, we esimae by OLS a bivariae VAR(1) model: S a S a r u (25) + 1 = , + 1 r a S a r u (26) + 1 = , + 1 We he calculae he heoreical spread ad he associaed saisics of proximiy o he acual spread: i) he sadard deviaio raio, sd(s )/sd(s ), ii) he correlaio saisic, Corr(S,S ), iii) he slope coefficie from he OLS regressio of S agais S ad iv) he Wald es of resricios for he VAR coefficies, A, implied by he EH (i equaio (10)) ad heir (Newey-Wes) robus variace covariace marix. Table (5) repors resuls describig he small-sample disribuio of sd(s )/sd(s ) ad is sesiiviy o chagig values of ρ ad sample size, T. I accordace wih he aalyic approximaio i Bekaer e al. (1997), he raio sd(s )/sd(s ) is biased upwards i a small sample, whe ρ is risig. For ρ up o 0.75, i pael A, he mea raio, sd(s )/sd(s ), is oly slighly above he heoreical value of 1 ad risig wih ρ. As he sample size, T, rises, for each ρ, he mea raio falls closer o 1. The bias is more proouced for ρ bewee 0.85 ad 0.96, i pael B, as he mea sadard deviaio raio rises from 1.01 o The small sample disribuio becomes more dispersed, wih sadard deviaios aroud he mea raios correspodigly risig from 0.18 o For ρ = 0.96, heoreical sadards of closeess o 1 would say ha sd(s )/sd(s ) = 1.6, is o close eough, however, uder small-sample iferece usig able 5, 1.6 is he mea raio, (wih a high sadard deviaio of 0.66) ad he EH is correcly suppored. 23

24 Table 5: VAR merics: sd(s )/sd(s ) for PFS 12: ρ ad T sesiiviy Pael A Larger samples: Number of Observaios (T) = ,000 Pael B High Persisece Number of Observaios (T)= 124 ρ Mea raio (s.e.) Mea raio (s.e.) Mea raio (s.e.) ρ Mea raio (s.e.).975 (1-righ ail) (0.011) (0.005) (0.028) (0.177) (0.018) (0.010) (0.005) (0.353) (0.035) (0.019) (0.098) (0.663) (0.094) (0.053) (0.017) (1.785) Table 6. VAR merics: sd(s )/sd(s) across spreads, ρ =0.93, T=124 Spread mauriy, 12 9 Mea raio (s.e.) (0.353) (0.447) media (0.528) (0.618) (0.6513)

25 Table 6 preses resuls for he sadard deviaio raio across differe mauriy horizos. The mea raio (ad sadard deviaio aroud i) rises as he log rae mauriy,, of he spread, S, shores. This resul is o i accordace wih he correspodig aalyic approximaio i Bekaer e al. (1997), accordig o which his saisic does o (o a firs approximaio) deped o he mauriy horizo,. However, a similar paer ca be observed i heir Moe Carlo resuls. I he same Moe Carlo exercise we esimae he correlaio coefficie, Corr(S,S ), ad fid ha i is ubiased wih respec o risig persisece. As expeced from heory, i collapses almos perfecly o is heoreical value (=1), for all ρ. Moreover, i coras wih heory, we fid ha he mea Corr(S,S ) varies o a small degree across mauriy horizos,. We do o repor hese resuls i a able, as ay variaios are oo small. Lookig ow a he mai saisical es of he VAR mehodology, he Wald es of he VAR cross-equaio resricios, able 7 shows resuls from he same experimes (for varyig ρ ad sample sizes, T). Give ha here are o racable aalyic approximaios for he heoreical bias i his saisic, wih respec o persisece, we oly rely o resuls from Moe Carlo evidece. Similarly o Bekaer ad Hodrick (2001), we fid ha he small sample properies of he Wald es of he EH resricios are very poor. 25

26 Table 7. VAR ess: Wald es for PFS 12: ρ ad T sesiiviy Pael A Larger samples: Number of Observaios (T) = ,000 ρ 0.95 quarile (1-righ ail) ρ Mea (s.e.) Pael B High Persisece T= (8.510) (13.434) (9.895) (10.935)

27 Table 8. VAR ess: Wald es across spreads, ρ =0.93, T=124 The firs row of he able quoes criical values from he χ 2 (2) disribuio, from sadard saisical ables. Spread S Mea (s.e.) quariles media χ 2 (2 d.f.) Na S 2 S (3.470) (3.992) S (5.091) S (6.924) S (11.521) Overall, we fid a proouced upward bias i he mea Wald es saisic, esimaed over 5,000 Moe Carlo rials, ad a more dispersed disribuio, as ρ rises, eve i larger samples. For fixed ρ, he bias falls, as T rises. More paricularly, whe persisece, ρ, is low ad samples are large he Wald es saisic is sill close o he criical values suggesed by he χ 2 disribuio, i pael A (able 7). For example, for large T=2,000, whe ρ =0.5, he 95% quarile level is a 6.21 agais he (χ 2 ) criical value of However, whe ρ is raised o 0.75, for he same T, he 95% quarile level rises o Pael B (i able 7) shows ha for very high values of ρ ad for a sample of 124 observaios, he small sample disribuio of he Wald saisic is cosiderably disored. Uder small sample iferece ad for ρ = 0.96, he EH should oly be rejeced for Wald es values higher ha (agais he sadard iferece value of 5.99), a he 95% quarile. Hece usig he Wald es uder asympoic iferece he EH eds o be over-rejeced i small samples. Nex, we explore i Moe Carlo resuls i able 8, differeces across chagig mauriy horizos,, for he spread, fixig ρ = 0.93, for a sample of 124 observaios. The bias i he mea Wald es saisic is higher for loger horizo spreads ad he 95% quarile level rises o for S 12 agais oly 9.67, for S 2. Therefore, ulike all previous ess, S 12 which was used i all he 27

28 experimes repored i able 7, oversaes he degree of persisece bias ha would apply o shorer mauriy spreads. We examie las he OLS regressio es of he heoreical spread S agais S i Moe Carlo resuls repored i able 9. I should be oed ha durig hese experimes, OLS esimaio failed o a grea exe i large samples. Therefore, our resuls focus o a small sample of 124 observaios, for differe ρ. Table 9. OLS βˆ from regressio of S agais S, T= 124, high ρ ρ Mea β (s.e.) - raio agais 1 quariles: media (1-righ ail) 0.05 (lef ail) (0.011) (0.036) (0.095) (0.178) (0.354) (0.665) (2.521) The resuls show ha a posiive bias i he slope coefficie is prese ad he small sample disribuio is highly dispersed. The mea (ad he media) βˆ rises above 1 as ρ, rises owards The sadard deviaio aroud βˆ also rises. Focusig o high ρ, able 9 shows a more proouced slope coefficie bias ha is eve larger ha ha recorded for βˆ i he PFS es, i able 1. Table 9 shows ha, for a real marke i which ρ is ear 0.92, if we fid βˆ = 1.90 he he EH mus sill o be rejeced, usig small sample 28

29 iferece, a he 97.5% quarile level. Moreover, i is ieresig o oe ha i larger samples he mea βˆ foud here coverges o 1 more quickly ha i did i he PFS regressios. Figure 5. Moe Carlo resuls: OLS βˆ (S agais S ) hisograms for high ad low ρ Table 10. OLS βˆ from regressio of S agais S, across spreads, T=124 Spread S Mea β (s.e.) -raio agais 1 quariles: media (1-righ ail) 0.05 (lef ail) S (0.642) S 3 S (0.614) (0.528) S 9 S (0.448) (0.354)

30 Table 10 preses resuls across mauriy horizos for differe S. Similarly o our resuls for he PFS regressio es i able 2, he small sample bias i βˆ eds o fall whe he log rae mauriy horizo,, is larger. For S 2, he mea βˆ is 1.48 ad i falls for larger (e.g., for S 12 βˆ = 1.26). Thus, he EH may be rejeced, a he 97.5% level, for βˆ = 2.9 for some spreads (e.g., for S 9 ) while i may o be rejeced, for he same value, for oher spreads (e.g. S 2 ) The Kalma filer-ime varyig parameer model Oe of he explaaios for possible empirical failures of he EH is ha especially whe here are regime chages, parameers may be usable raher ha cosa as i is assumed by OLS. I learig models, isead of raioally kowig a cosa parameer model, ages are updaig he model s parameers as hey gai more iformaio hrough ime. I his las roud of Moe Carlo experimes we cosider he properies of a ime varyig parameer model (TVP). We fid ha TVP models are a ieresig case of exreme small sample bias. Alhough hese models help elimiae he bias from regime shifs, hey are cosiderably more exposed o persisece bias ha he OLS esimaor. We geerae daa uder he ull (EH), exacly as i he previous experimes, for a AR(1) shor rae ad a log rae cosruced accordig o he EH. We he esimae, i each Moe Carlo rial, a ime varyig parameer model 9 of he same PFS equaio (4) ha was esimaed by OLS i secio (4.2.1). For a more horough descripio of he model, see Kim ad Nelso (1999). The wo mai equaios of he sae-space form are: S * = α + β S + ε (Measureme equaio) (3.23) α 1 = β 0 0 α 1 β 1 1 η1 + η 2 (Trasiio equaio) (3.24) where S * is agai he perfec foresigh spread (PFS). Give ha he EH is rue, we geerally expec ha ˆ 2 2 = 1ad ˆ α = 0, for mos. We esimae he wo hyper-parameers σ ad σ β ad he variace of he error erm i he measureme equaio, σ 2 ε. Table 11 repors resuls from 5,000 Moe Carlo replicaios for a sample of 124 observaios ad for ρ risig from 0 o η 1 η 2 9 We use Gauss code wrie by Kim ad Nelso (1999) for Kalma filer esimaio. 30

31 Table 11. Time varyig parameer PFS regressio, S 12 : Sesiiviy o ρ, T=124 The ime varyig cosa ad slope esimae are averaged over all replicaios ad all observaios (= T). ρ Mea σ 2 ε ( PFS equaio ) ( -raio) Mea σ 2 η1 ( α equaio ) (- raio) Mea σ 2 η2 ( β equaio ) (- raio) Mea α (s.d.) (- raio) Mea β (s.d.) (- raio) 0 1*10-8 (-0.004) *10-7 (0.001) *10-10 (-0.001) (9*10-5 ) (4*10-3 ) *10-9 (0.001) *10-8 (0.008) (14.581) (25.234) (19.160) (14.459) (13.000) (12.551) (11.220) 6*10-10 (0.009) 8*10-6 (0.824) 4*10-4 (3.698) (4.204) (1.341) (2.194) (4.071) (0.641) (0.002) (1.187) (-0.011) (2.103) (-0.003) (3.045) (0.007) (4.548) (-0.223) (6.500) (-0.154) (10.392) (-0.593) (0.020) (2.277) (0.032) (4.294) (0.065) (5.936) (0.131) (5.834) (0.232) (7.371) (0.549) (6.632) (3.094) (4.350) Cosiderig overall resuls from Kalma filer esimaio, for all ρ, he average σ 2 ε (over 5000 replicaios) is ear 0. The average hyper-parameer ha correspods o α, σ 2 η1, is relaively 2 large, implyig a very high sigal o oise raio ( σ σ 2 η1 ε ), compared o σ 2 η2, which correspods o β. Neverheless, he laer is also large eough o imply a high sigal o oise raio, for all ρ (excep a ρ = 0). Turig ow o he mea levels ad pahs for he wo TVPs, αˆ ad βˆ (i able 11), implied by he Kalma filer algorihm ad by he hyperparameer esimaes, we fid he same paer of sesiiviy o ρ as i he case of he OLS es of he EH (i secio 4.2.1). The ime varyig cosa erm αˆ is o average ear 0 bu he mea ime varyig slope coefficie βˆ rises from 1 owards much higher values, as ρ rises. The sadard deviaio aroud hese mea esimaes for αˆ ad βˆ rises as ρ also rises. More imporaly, he resuls sugges ha he upward bias i i he ime varyig parameer model, is impressively srog compared o ha foud for OLS esimaio i secio (e.g. for ρ = 0.96, βˆ = 4.64 agais 1.63 i OLS resuls). βˆ, 31

32 This is o surprisig give ha, i Kalma filerig he sample used for esimaio is esseially smaller ha 124 observaios. I he basic filer 10 algorihm, iformaio uil ime is used, which is before he ed of he sample, T(=124), ad i is gradually updaed a every ime icreme. As a resul, he sample size is smaller ha 124 a each. Moreover, he Kalma filer algorihm ca be ierpreed as a weighed leas squares esimaor, aachig smaller weighs o observaios ha are farher back i ime. Therefore, eve whe he las observaio T is reached, he ime varyig coefficie is esimaed usig a smaller (ha 124) widow of observaios. Kalma filer esimaes are more comparable o hose from a OLS esimaor ha uses a sample smaller ha 124 observaios. I a ime varyig parameer framework, he average values ad he able are oly iformaive whe complemeed by he correspodig plos of average coefficie esimaes for every. The properies of he esimaed ime varyig parameer model are bes summarised i he fourwidow plo i figure 6. The upper wo widows display he plos of he forecas error ad is associaed codiioal variace. The forecas error 11 is defied as: ν = ˆ ˆ ˆ β S ad he codiioal variace of he forecas error is 1 * y y 1 = S α ( ) f ν. 1 = Ε 1 10 Basic filer is used here o refer o simple filerig as opposed o smoohig which would use all iformaio up o T, he ed of he sample. 11 The firs few observaios display a higher forecas error, uil observaio 13. This is because a high uceraiy coefficie (=500,000), aached o he ages iiial guesses of he TVPs, is assumed i he code. 32

33 Figure 6. αˆ ad βˆ i deviaios from heoreical value (2 s.e. bads), for ρ =0.94 The lower wo widows i figure 6, display he plos of coefficies as deviaios from heir heoreical values ( αˆ =0 ad βˆ =1) (i.e. esimaes of he bias) 12. Figure 6 shows ha for ρ = 0. 94, βˆ - 1= 2.4, wihi a small bad of 2 sadard errors, while hypohesis would have bee rejeced based o sadard iferece. 13 αˆ = 0. The ull 12 The sadard error bads display he sadard deviaio of he disribuio obaied from he Moe Carlo rials. 13 We also fid ha ρ=0.99 is a limiig case. The resulig pah of he ime varyig coefficies is uusual ad he properies of he ime varyig model are o loger robus. The ime varyig slope coefficie, βˆ, rapidly rises over ime o 15. The TVP model is o applicable whe r is a ear- radom walk ad he EH is rue. 33

34 Figure 7. βˆ deviaio from 1, for 2000 observaios, ρ =0.93 TVP b for 2000 observaios bea: deviaio from observaio umber We also ivesigae larger samples of 524 ad of 2,000 observaios for his model ad fid ha he average Moe Carlo ˆ β 1 for samples of 524 ad 2000 observaios (figure 7) ad for a fixed ρ = is always equal o 2 as he sample size icreases. Ulike OLS, he TVP model is o a cosise esimaor, give ha, a each, he Kalma filer effecively uses a widow of observaios for he esimaio of βˆ, so ha raisig T will o elimiae he bias. 5. Coclusios We have used Moe Carlo Simulaio experimes o explore he small sample disribuio for a umber of ess of he EH. Our resuls cofirm ha he small sample disribuio for all hese saisics is more ad more disored i respose o risig persisece i ieres raes ad he spread. For all hese ess, we also fid ha mauriy horizo maers for small sample bias ad i mos cases, he laer is smaller for ess wih loger horizo spreads (S 12 ). I accordace wih resuls i Bekaer e al. (1997) we fid srog posiive bias i he OLS slope coefficie of he PFS es ad a small sample disribuio ha has greaer dispersio ad is skewed o he righ. The slope bias is almos elimiaed whe we ru he same regressios wihou a cosa erm. * Oe of he mos srikig resuls is ha he variace boud raio, s d. ( S ) s. d. ( S )., is biased dowwards, for high ρ, hece i eds o favour he EH oo ofe. This is i coras wih he 34

35 OLS es ha eds o rejec he EH oo ofe ad wih previous Moe Carlo evidece i he variace bouds lieraure, ha deals hough wih differe formulaios of he boud (Flavi, 1983). Furher Moe Carlo evidece shows ha all he VAR mehodology saisics are biased wih he excepio of Corr(S,S ). Small sample iferece would be preferable i he case of he sadard deviaio raio ha exhibis wide dispersio i is small sample disribuio. However, he case of he Wald es saisic sads ou, give ha asympoic iferece is pracically redered ieffecive whe persisece is high. The Wald es srogly eds o over-rejec he ull hypohesis i small samples, as ρ rises. We also look a he OLS slope coefficie i he regressio of S agais S, which, o our kowledge, is o cosidered elsewhere, ad fid a relaively high posiive bias. The Kalma filer ime varyig parameer model is o a ypical es of he EH, however, i belogs o a broader class of learig models ha may be more appropriae for esig he EH i samples which display impora regime shifs. A he same ime, i provides a appealig example of more exreme small sample bias (agais ha foud for OLS) as demosraed by our Moe Carlo evidece. Moreover, hese resuls are releva o a wider class of ecoomeric echiques ha may be applied o esig he EH or oher heoreical models. These would iclude weighed leas squares esimaio, recursive esimaio ad rollig widow esimaes, all of which effecively use a smaller widow of observaios ha OLS (for he same sample). The resuls preseed i his paper aim o provide a guide o praciioers wishig o es he EH i markes ha display high ieres rae auocorrelaio. Their coefficie esimaes are more likely o be comparable o he quarile levels documeed i our ables ha o he sadard criical values of asympoic iferece. Tha beig said, Moe Carlo simulaio would ehace esimaio ad iferece if ew criical values were used, derived from daa geeraed wih parameer calibraio o he specific marke. Fially, our resuls also serve as a heoreical ivesigaio of he properies of hese ess. I is hoped ha Moe Carlo evidece of differeces i he exe of small sample bias amog aleraive ess ca help explai he discrepacies foud i he empirical lieraure whe differe ess are employed. Towards he same direcio, a ivesigaio of he differeces i he power of hese ess, wih daa geeraio uder a specified aleraive o he ull hypohesis, would provide a possible exesio o our resuls. 35