TESTING FOR SERIAL CORRELATION: GENERALIZED ANDREWS- PLOBERGER TESTS. John C. Nankervis

Transcription

1 TESTING FOR SERIAL CORRELATION: GENERALIZED ANDREWS- PLOBERGER TESTS John C. Nankervis Deparmen of Accouning, Finance and Managemen, Universiy of Essex, Colcheser, CO4 3SQ U.K. N. E. Savin Deparmen of Economics, Universiy of Iowa, Iowa Ciy, IA ABSTRACT This paper considers esing he null hypohesis ha a imes series is uncorrelaed when he ime series is uncorrelaed bu saisically dependen. This case is of ineres in economic and finance applicaions. The GARCH (1, 1) model is a leading example of a model ha ha generaes serially uncorrelaed bu saisically dependen daa. The ess of serial correlaion inroduced by Andrews and Ploberger (1996, hereafer AP) are generalized for he purpose of esing he null. The raionale for generalizing he AP ess is ha hey have aracive properies for case for which hey were originally designed: They are consisen agains all non-whie noise alernaives and have good all-round power agains nonseasonal alernaives compared o several ess in he lieraure, which includes he Box-Pierce (197) ess. These properies are inheried by he generalized AP ess. Our findings indicae ha he generalized AP ess provide subsanial improvemens in power compared o he generalized Box-Pierce ess. JEL classificaion: C12; C22 Keywords: Auoregressive moving average model; Lagrange muliplier es; Nonsandard esing; Saisically dependen ime series; Zero correlaion. Corresponding auhor: N. E. Savin, Deparmen of Economics, 18 Pappajohn Business Building, Iowa Ciy, IA Tel.: (319) ; fax: (319) address: gene-savin@uiowa.edu.

2 1. INTRODUCTION This paper considers esing he null hypohesis ha a imes series is uncorrelaed when he ime series is uncorrelaed bu saisically dependen. The case of uncorrelaed dependen ime series is of ineres in economic and finance applicaions. The GARCH (1, 1) model is a leading example of a model ha ha generaes serially uncorrelaed bu saisically dependen daa. The ess of serial correlaion inroduced by Andrews and Ploberger (1996, hereafer AP) are generalized for he purpose of esing he null. The raionale for generalizing he AP ess is ha hey have aracive properies for case for which hey were originally designed. They are consisen agains all non-whie noise alernaives and have good all-round power agains nonseasonal alernaives compared o several ess in he lieraure, which includes he Box-Pierce (197, hereafer BP) ess. The generalized AP ess inheri he properies of he AP ess in power comparisons. In simulaion experimens, he generalized AP ess ypically have subsanially beer power han he generalized BP (Lobao, Nankervis and Savin (22)) ess agains non-seasonal alernaives. AP inroduced ess of serial correlaion designed for he case where he ime series is generaed by ARMA (1, 1) models under he alernaive. As hey noed, i is naural o consider ess of his sor because ARMA (1, 1) models provide parsimonious represenaions of a broad class of saionary ime series. ARMA(1, 1) modcls for financial reurns series follow from he mean-reversion models of Poerba and Summers (1988) and he price-rend models of Taylor (1982, 25). However, esing for serial correlaion in an ARMA (1, 1) model is a nonsandard esing problem because he ARMA (1, 1) model reduces o a whie noise model whenever he AR and MA 1

3 coefficiens are equal. The esing problem is one in which a nuisance parameer is presen only under he alernaive hypohesis. Andrews and Ploberger (1994, 1995), Davies (1977, 1987) and Hansen (1996) have considered such esing problems. For he problem addressed by AP, he sandard likelihood raio (LR) saisic does no possess is usual asympoic chi-squared disribuion or is usual asympoic opimaliy properies. The LR es has he aracive feaure of being consisen agains all forms of serial correlaion (Poscher (199)). AP show ha his feaure also holds for ess inroduced ino he lieraure by Andrews and Ploberger (1994, 1995), namely, he sup Lagrange Muliplier (LM) and average exponenial LM and LR ess. By conras, his robusness feaure is no shared by some well-known ess of serial correlaion such as he BP ess and Durbin-Wason (195) ess. AP esablish he asympoic null disribuion for he LR, sup LM and average exponenial es saisics under he assumpion ha he generaing process is a condiionally homoskedasic maringale difference sequence (MDS). The asympoic criical values for hese ess were calculaed by simulaion. In addiion, he finie-sample criical values were calculaed by simulaion under he assumpion ha he innovaions are independenly and idenically normally disribued. In Mone Carlo power experimens, AP compared he finie-sample powers of he LR, sup LM, average exponenial, BP, Durbin Wason and several poin opimal invarian ess. The alernaives include Gaussian ARMA (1, 1) models. Agains his class of alernaives, he LR es was found o have very good all-around power properies for non-seasonal alernaive models, especially compared o BP ess. 2

4 For serially uncorrelaed bu saisically dependen ime series, he rue levels of he LR, sup LM and average exponenial ess can differ subsanially from he nominal levels when he ess use he asympoic criical values calculaed by AP. This paper generalizes a subse of he ess considered by AP so ha hey have he correc level asympoically when he ime series is serially uncorrelaed bu saisically dependen. The subse consiss of LM-based ess, namely he sup LM es and he average exponenial LM ess. The generalizaion is obained by using he rue asympoic covariance marix of he sample auocorrelaions, or a consisen esimaor. The same approach has been used by Lobao, Nankervis and Savin (22) o generalize he BP ess o seings where he imes series is serially uncorrelaed bu saisically dependen. The asympoic criical values repored by AP remain valid for he generalized LM-based ess. The finie-sample levels of he generalized LM-based ess wih asympoic criical values are assessed by simulaion. In Mone Carlo power experimens, hese ess are compared o generalized BP ess. The generalized AP ess ypically have subsanially beer power han he generalized BP agains non-seasonal alernaives. Hence, generalized AP ess can be recommended for use in economic and finance applicaions. The paper repors he resuls of an empirical applicaion o sock reurn indexes. 2. ARMA (1, 1) MODEL AND TEST STATISTICS This secion reviews he model, hypohesis and es saisics considered by AP. The model is he ARMA (1, 1) model, Y = φy 1+ ε πε 1 for = 2,3,..., (1) 3

5 where { Y : = 1,..., T} are observed random variables and { ε : = 1,2,...} are unobserved innovaions. AP reparameerize (1) as Y = ( β + π) Y + ε πε for = 2,3,..., 1 1 where β = φ π, he parameer space forπ is Π and ha for β is B. They assume ha Π and B are such ha he absolue value of he auoregressive coefficien π + β <1, Π is closed and B conains a neighborhood of zero. The former condiion rules ou uni roo and explosive behavior of { Y : = 1,..., T}. The null hypohesis is ha { Y : = 1,..., T} is whie noise, and he alernaive is ha { Y : = 1,..., T} is serially correlaed. These hypoheses are given by H : β = and H : β. (2) 1 When β =, he model (1) reduces o Y = ε, and he parameerπ is no longer presen. The esing problem is non-sandard becauseπ is no idenified when he null hypohesis is rue. Le LR T (π ) denoe he sandard LR saisic for esing H versus H 1 when π is known under he alernaive. Then he LR saisic for he unknown π is where π = T σ ˆ σ π 2 2 LR T( ) log( Y / Y( )), LR = sup LR ( π ), π Π T σ 1 T 2 = 2 Y Y T = 1 and T T i i ˆ σy( π) = σy Y π Y i 1 π Y i 1. T = 2 i= = 2 i= (3) Noe ha LR = LR ( ˆ π ), where ˆ π is he ARMA (1, 1) ML esimae of π. T, 4

6 AP proved ha an asympoically equivalen es saisic o he LR saisic (under he null and local alernaives) is he sup LM saisic where suplm ( π ), (4) π Π T T i 2 4 LM T( π ) = Y π Y i 1 (1 π ) / σy. T = 2 i= (5) The LR and sup LM ess are shown o saisfy an admissibiliy propery, and as a consequence, bea any given es in erms of weighed average power agains alernaives ha are local o, bu sufficienly disan from he null. Andrews and Ploberger (1994) inroduced average exponenial ess. These ess are asympoically opimal in he sense ha hey minimize weighed average power for a specific weigh funcion. The weigh funcions for he parameer β are mean zero normal densiies wih variances proporional o a scalar c >. The weigh funcion J for he parameer π is chosen by he invesigaor. For he simulaion resuls in AP, he funcion is aken o be uniform on Π, and similarly in his aricle. For each c (, ), he average exponenial LM es saisic is given by 1/2 1 c Exp-LM ct = (1 + c) exp LM T ( π ) dj( π ) 21+ c, (6) where LM T (π) is as defined in (5), and J( ) is probabiliy measure on Π, such as he uniform measure. The average exponenial LR es saisic, Exp-LR ct, is defined analogously wih LM T (π) are replaced by LR T (π). The limiing average exponenial LM es saisic as c and c are given by 5

7 Exp-LM = LM ( ) ( π ) T T π dj and 1 Exp-LM T = ln exp LM T( π ) dj ( π ) 2. (7) Andrews and Ploberger (1994) show ha he average exponenial ess have asympoic local power opimaliy properies. 3. ASYMPTOTIC AND FINITE-SAMPLE NULL DISTRIBUTIONS OF AP TEST STATISTICS AP esablished he asympoic null disribuion of he es saisics previously inroduced. This secion reviews he asympoic heory. AP use he following: (Assumpion 2 in AP) Assumpion 1. The random variables { Y : = 1,2,...} saisfy EY ( I 1) = a.s. for all 1, 4 δ EY ( I ) = σ a.s. for all 1 and sup 1 E Y + < for some δ >, Y where I denoes he field generaed by Y 1,, Y. The asympoic null disribuions of he es saisics are esablished by showing ha he sequences of sochasic processes{lr ( ) : T 1} and {LM T ( ):T 1}indexed by π Π converge weakly o a sochasic process G( ) and hen by applying he coninuous T mapping heorem. Le d denoe converge in disribuion of a sequence of random variables. Le { Zi : i } be a sequence of iid N(,1) random variables. Define 2 i G( π) = (1 π ) π Zi i= for π Π. (8) The following heorem is proved. Theorem 1. Under he Assumpion 1, 6

8 a.lm G(), T b. sup LM ( π) sup G( π), π Π c.exp-lm G( π) dj( π), T T d d π Π d 1 d. Exp-LM T ln exp( G( π)) dj( π), and 2 e. pars (a)-(d) hold wih LM replaced by LR. Theorem 1 holds for ime series where he asympoic covariance marix of he firs T-1 of he sample auocorrelaions is equal o he ideniy marix. As a consequence, Theorem 1 does no hold for many models used in economics and finance, for example, a GARCH (1, 1) one wih normal innovaions. The implicaions of he ideniy marix condiion for esing H are explored in he nex secion. AP commen ha he maringale difference condiion in he Assumpion 1 is no essenial for he resuls of Theorem 1 o hold. They sae ha wha is essenial is ha (a) EY ( ) = for all 1, (b) EY = > for all 1, (c) EYY ( ) = for all s <, (d) 2 2 ( ) σy s EYYYY ( s u v) = for all s u v unless s = and u = v, and (e) 1 T 2 i 2 π = i= T Y Y i 1 saisfies a cenral limi heorem for all π Π. Assumpion 1 implies condiions (a) - (e). Because he Gaussian GARCH model saisfies (a) - (e), i may be hough ha Theorem 1 also holds for he GARCH model. However, i worh poining ou ha assumpions (a) - (e) need o be augmened by EYY ( ) = σ when s < for Theorem 1 o hold. This las s Y condiion fails in he GARCH case. Hence, assumpions (a) (e) do no imply ha he asympoic disribuion AP s es saisics for he GARCH model are he same as for he 2 iid N(, σ ) case. 7

9 From Theorem 1, he LR, sup LM and average exponenial LR and LM ess are asympoically pivoal, ha is, he asympoic disribuions does no depend on any unknown parameers. Hence, he asympoic criical values for he ess can be simulaed by runcaing he series i π Z i= i a a large value TR. AP repor simulaed criical values of he ess in heir Table 1.The criical values are based on he parameer space Π = {, ±.1,, ±.79, ±.8}, TR = 5 and 4, repeiions. They also calculae finiesample criical values for he ess under Assumpion 1. The LR, sup LM and average exponenial LR and LM ess are shown by AP o be consisen agains all deviaions from he null hypohesis of whie noise wihin a class of weakly saionary srong mixing sequences of random variables. This consisency propery illusraes he robus power properies of he ess. The ess inroduced by AP can also be used o es wheher regression errors are serially correlaed. The pracically imporan exension includes he case of an ARMA (1, 1) model wih an inercep. The ess are consruced using he residuals { Yˆ : T} raher han he random variables{ Y : T} ; see AP for deails. Provided ha he regressors are exogenous, he resuling LR, sup LM and average exponenial LR and LM es saisics have he same asympoic disribuions as when he acual errors are used o consruc he saisics. Thus, he asympoic criical values previously calculaed by AP are applicable. Finally, AP invesigaed he accuracy of he asympoic criical values by simulaing he rue levels of he LR, sup LM, Exp-LR and Exp-LM, Ex-LM 1 and Exp- LM ess for nominal levels of.1,.5 and.1. The model used was he ARMA (1, 1) model wih inercep, normal innovaions and sample sizes from 25 o 5. The rue levels show ha he asympoic approximaions are sufficienly accurae for applicaions. 8

10 4. GENERALIZATION OF LM BASED TESTS In his secion, he LM-based ess considered by AP are generalized such ha he ess have he correc asympoic level under he null when he asympoic covariance marix of he sample auocorrelaions is no he ideniy marix. As noed previously, he Theorem 1 does no hold in his case. Our approach is based on Lobao, Nankervis and Savin (22), who have inroduced a generalizaion of he sandard Box-Pierce (BP) ess. We begin wih a review of he asympoic disribuion heory of he sample auocorrelaions when { Y : = 1,2,...} is a covariance saionary sequence of saisically dependen bu uncorrelaed random variables wih mean zero (or allow for mean μ. as we do below). Define he lag-j auocovariance by γ j = E(Y Y + j ) and he lag-j auocorrelaion by ρ j = γ j /γ. The sample mean, variance and lag-j auocovariance are given by, T- j = T and ˆ = ( YY )/ T. The usual esimaor of ρ j is r ˆ / ˆ j = γ j γ. T 2 ˆ γ ( Y ) / = 1 γ j = 1 + j Under general weak dependence condiions, Tr = T( r, r,..., r ) N(, V), (9) 1 2 K d where he ijh elemen of V is given in Hannan and Heyde (1972) and Romano and Thombs (1996). Consider esing he null hypohesis H (K): ρ = (ρ 1,, ρ K ) = agains he alernaive ρ. Suppose V is known. Then a es can be based on he es saisic 1 BPK( V) = Tr' V r, (1) where he saisic is asympoically chi-square disribued wih K degrees of freedom when H(K) is rue. In pracice, V is unknown. In he sandard BP saisic, V is replaced 9

11 by I, he ideniy marix. If V is no equal o he ideniy, he sandard BP es can produce misleading inferences. Thus, he esimaion of V is of ineres. Lobao, Nankervis and Savin (22) esimaed V afer imposing he resricions implied by he null H (K). Under he null, V simplifies o V as is ijh elemen = [( γ ) C ] where 2 C has d = c E Y ij Y i Y+ d Y+ d j d = = ( μ)( μ)( μ)( μ) for i,j =1,,K, (11) where he above formula covers he case EY ( ) = μ. The generalizaion of he BP es of H (K) proposed by Lobao, Nankervis and Savin (22) uses he es saisic ˆ ˆ 1 BP KV ( ) = Tr ( V ) r, (12) where ˆ V is a consisen esimaor of V under H (K). A consisen esimaor can be obained by using ˆ γ o esimae γ and a consisen parameric or nonparameric esimaor ofc. Our generalizaion of he LM-based ess explois he fac ha he LM es saisic is a funcion of sample auocorrelaions. Rewriing he LM T (π) saisic in (5) gives T T T 1 i i T π = T π YY i 1 σy π T π π ri+ 1 T = i= = i+ 2 i= LM ( ) / (1 ) (1 ), (13) where he ih sample auocorrelaion in (13) is T 2 i = ( i / ) Y =+ i 1 r YY T σ. i Suppose ha he series π ri + 1 is runcaed a a large value TR. Recall ha AP i= i runcaed he series π Zi for he purpose of simulaing he asympoic criical values i= 1

12 of he LR, sup LM and average exponenial LR and LM ess. Taking K = TR, we can wrie he saisic in (13) as 2 2 LM ( ) (1 )( ), T π = π p Tr where 2 1 = (1,,,..., TR ), 1 2 p ππ π r = ( r, r,..., r ). The vecor Tr is asympoically N (, TR V) where V is a TR TR marix. If V = I, he LM-based ess of H : β = can be carried ou using he asympoic criical values calculaed by AP. If V I, he rue levels of he ess wih AP asympoic criical values can differ subsanially from he nominal levels in finie-samples and asympoically. The level disorion of he LM-based ess when V I can be correced asympoically by using TLr in place of Tr in (13) wherev = LL. Our proposed 1 generalizaion is o replace V by V and consisenly esimae V by ˆ V. The generalized ess we consider are: supπ Π LM T ( π, V ) ˆ ˆ Exp-LM T( V ) = LM T( π, V ) dj( π), ˆ Exp-LM T( V ) ln exp LM T( π, V ) dj( π ), ˆ 1 ˆ = 2 (14) where ˆ 2 ˆ 2 LM T ( π, V ) = (1 π )( p TL r) wih ( Vˆ ) = Lˆ Lˆ. This generalizaion exends 1 o he case of an ARMA(1, 1) model wih an inercep. Lobao, Nankervis and Savin (22, page 733) presen a specific assumpion under which of V. The assumpion is: ˆ V is a consisen esimaor 11

13 Assumpion 2. Le Y be a covariance saionary process ha saisfies E Y s < for some s > 4. and all and is L 2 -NED of size -1/2 on a process U where U is an α- mixing sequence of size s/(s-4). See De Jong and Davidson (2). The asympoic criical values repored by AP remain valid for our generalizaion of he LM-based ess. We use criical values based on TR = 2, whereas AP use TR = 5. We have se, as AP did, max π =.8. Using TR = 2 raher han 5 has a negligible effec on he 1%, 5% and 1% asympoic criical values hrough erms 2 TR 1 (1 π ). This is seen in Table 1 where we repor he asympoic criical values for he hree es saisics for Π = {, ±.1,, ±.79, ±.8}, TR = 2 and TR = 5, using 15 million replicaions. The admissibiliy and opimaliy resuls will also hold under Assumpion 2, given a consisen esimaor V ˆ. The form of he V marix is simplified when he ime series is a maringale difference sequence (MDS). For a MDS process, he only possible nonzero elemens of C are he erms of he form EY Y Y. In (11) hese occur a d =. 2 ( μ) ( i μ)( j μ) Guo and Phillips (1998) have developed a version of he BP es for he MDS case. This special form of V marix can also be used in consrucing a generalizaion of he LM- based ess for general MDS processes. For cerain MDS processes, V is diagonal. The jh diagonal elemen is equal o ( γ ()) EY ( μ) ( Y μ). The BP es for he diagonal case has been repeaedly j developed in he lieraure; see, for example, Taylor (1984), Diebold (1986), Lo and MacKinlay (1989) and Lobao, Nankervis and Savin (21). The generalized BP es for 12

14 his case uses a consisen esimaor of he diagonal elemens. Likewise, generalizaions of he LM-based ess can be consruced for he diagonal case. The generalized versions of LM-based and BP ess require a consisen esimaor of V. We consider four differen esimaors. The firs is an esimaor of he diagonal V marix, denoed by V * *. A consisen esimaor of he jh diagonal elemen of V is vˆ = cˆ / and cˆ = ( Y Y) ( Y Y) / T, j = 1,..., TR. (15) * * 2 * T 2 2 jj jj ˆ γ jj = TR+ 1 j The second is an esimaor of V in he general MDS case, denoed byv GP. A consisen esimaor of he ijh elemen of GP V is T GP GP 2 GP 2 ˆ ij ij γ ij i j = TR+ 1 vˆ = cˆ / and cˆ = ( Y Y) ( Y Y)( Y Y) / T; i, j = 1,..., TR. (16) The hird and fourh esimaors are esimaors of V ha are also consisen in he non-mds case. We use he VARHAC esimaion procedure proposed by den Haan and Levin (1997). This procedure uses a vecor auoregressive (VAR) esimaor of he covariance marix where he order of equaion in he VAR is auomaically seleced. To presen he explici formula for he VARHAC esimaor of C, le wˆ i = ( Y Y)( Y i Y), wˆ ˆ ˆ = ( w1,..., wtr, )', and le S be he maximum lag order chosen for he VAR. The esimaed residuals from he VAR regressions are eˆ wˆ Aw ˆ ˆ s s s= 1 S =, where A ˆs are he marices of esimaed coefficiens from he VAR, and he esimaed innovaion covariance marix is 13

15 Then he VARHAC esimaor of T Σ= ee ˆˆ' T. = S+ 1 C is S S 1 1 s s s= 1 s= 1. (17) Cˆ = ( I Aˆ ) Σ ( I Aˆ ') We repor resuls for he VAR wih he AIC (Akaike (1973)) and he SBC (Schwarz (1978)) crieria. The resuling esimaors are denoed by V ˆ (AIC) and V ˆ (SBC). The maximum lag lengh is 3, 4, 5 and 8 for sample sizes T = 2, 5, 1 and 5, and, for any equaion in he VAR, he same lag lengh is used for each elemen of he vecor process. To miigae he effec of occasional exreme esimaes we used he procedure of Andrews and Monahan (1992), and se he minimum singular values of he S inverse of he recoloring marix, I ˆ A, o be MONTE CARLO EXPERIMENTS UNDER THE NULL s= 1 s This secion considers he rue levels of he LM-based ess and he BP ess when he ess use asympoic criical values. Special versions of he ess for he MDS case are included. The LM-based ess are he sup LM, Exp-LM and Exp-LM ess. The BP ess are included because hey are naural compeiors o he LM-based ess. The model we consider is he locaion model wih serially uncorrelaed bu dependen errors, Y = μ + ε for = 1,2,..., T, (18) where V I. The models used for he errors ε in he experimens include wo MDS processes and wo non-mds processes. 14

16 The wo MDS models for he errors are varians of he GARCH model of Bollerslev (1986). The models are Gaussian GARCH (1, 1) and he exponenial GARCH (1, 1) or EGARCH (1, 1). The Gaussian GARCH model has been exensively used in economics and finance. Nelson (1991) proposed he EGARCH model. Boh models are described in Campbell, Lo and MacKinlay (1997). The GARCH (1, 1) models are he following. GARCH (1, 1). ε = Z σ, where {Z }is an iid N(,1) sequence and σ = ω+ α ε + β σ. The consans α and β are such haα + β < 1. This condiion is needed so ha Y is covariance saionary. He and Teräsvira (1999) show ha he uncondiional fourh momen of Y for GARCH (1, 1) models of Y exiss if and only if β + 2αβυ + α υ < 1, where υ = E Z i. We se ω =.1, α =.9, and β = i Wih his parameer seing, he He and Teräsvira condiion for he exisence of he 2 fourh momen of Y is saisfied. For his process, γ = EY ( μ) =.5, EY ( μ) / γ =, 3 3/2 EY ( μ) / γ = 5.8, and V is diagonal. We noe ha our resuls 4 4 are invarian o he value of ω. Esimaes from sock reurn daa sugges ha α + βis close o one wih β also close o one; for example, see Bera and Higgins (1997). EGARCH (1, 1). ε = Z σ, where {Z }is an iid N(,1) sequence and where ln( σ ) = ω+ ψ Z + α Z + β ln( σ ). We se ω =.1, ψ =.5,α =.2, and β =.95. He, Teräsvira and Malmsen (22) show ha Y is saionary if β < 1 and ha wih Gaussian {Z } all momens of Y exis. We have ha (he skewness is an 2 esimae) γ = EY ( μ) = 1.8, EY ( μ) / γ =, and 3 3/2 EY ( μ) / γ = 23.4, and V is

17 no longer diagonal. Our resuls are invarian o he value of he inercep and he variance. The wo models for he non-mds errors are he nonlinear moving average model, and he bilinear model. Tong (199) considers he nonlinear moving average model, and Granger and Andersen (1978) he bilinear model. The moivaion for eneraining non- MDS processes is he growing evidence ha he MDS assumpion is oo resricive for financial daa; see, for example, El Babsiri and Zakoian (21). For boh models considered below, V is non-diagonal. Nonlinear Moving Average Model. Le ε = Z -1 Z -2 ( Z -2 + Z + c) where { Z } is a sequence of iid N(, 1) random variables and c = 1.. For his process, EY ( μ) = 5, EY ( μ) / γ =, EY ( μ) / γ = /2 4 4 Bilinear Model. Le ε = Z + b Z -1 ε -2, where { Z } is a sequence of iid N(, σ 2 ) random variables, b =.5 and σ 2 = 1.. The Y process is covariance saionary provided ha b 2 σ 2 < 1. The fourh momen of his process exiss if 3b 4 σ 4 < 1. For his process, ( μ) σ (1 bσ ) EY = = 1.333, EY ( μ) / γ =, 3 3/2 EY ( μ) / γ = 3(1 bσ ) /(1 3 bσ ) = Bera and Higgins (1997) have fied a bilinear model o sock reurn daa. Table 2 presens he finie-sample rejecion probabiliies (percen) of he nominal.5 LM, Exp-LM and Exp-LM ess of H : β = wih asympoic criical values for he GARCH error models. Table 3 does likewise for he nonlinear moving average and he bilinear models. The rejecion probabiliies for he sup LM, Exp-LM and Exp-LM ess are compued using Π= {.8,.79,...,.79,.8}, which is he same se as used by AP. 16

18 The rejecion probabiliies are based on TR = 2. Increasing TR does no produce a noiceable difference in he rejecion probabiliies when using Π as defined above. For comparison, he ables also include he BP6, BP12 and BP2 ess; he former wo were considered by AP. The MDS and non-mds models are simulaed for sample sizes T = 2, 5, 1 and 5 using 25, replicaions. Tables 2 and 3 show ha he differences beween he rue and nominal levels are subsanial when he ideniy marix is used. The differences are larges for he BP ess wih he EGARCH model and for he LM-based ess wih he nonlinear moving average model. The differences end o increase as he sample size increases. The increase is large for he EGARCH and nonlinear moving average models. Nex we consider he generalized ess in Table 2. We firs compare he LM based ess. The differences beween he rue and nominal levels end o be essenially eliminaed for GARCH when he ess use he consisen esimaorsv, ˆ GP V or Vˆ(SBC) and T 5. For EGARCH, he difference is essenially eliminaed when he ess use he consisen esimaors ˆ GP V or V ˆ(SBC) and T 5. In boh cases larger sample sizes are needed o eliminae he difference when Vˆ(AIC) is used, especially for he sup LM es. Overall, he Exp-LM and Exp-LM ess end o have beer conrol over he level han he sup LM ess. The esimaor of ˆ * ˆ * V is inconsisen in he EGARCH case: generalized LM-based ess using his esimaor end o over rejec. We nex compare he LM-based ess and he BP ess. The BP ess generally end o show less saisfacory conrol over he level han he LM-based ess, especially BP2. 17

19 We also calculaed he finie-sample rejecion probabiliies using he skewed (5) GARCH (1, 1) using he sandardized version given in Lamber and Lauren (21) and he mixures of normal GARCH (1,1) proposed by Haas, Minik and Paolella, (24). The above conclusions are no alered by he resuls for hese models. The comparison of he generalized ess in Table 3 is complicaed. We sar wih he nonlinear moving average model. The Exp-LM and Exp-LM ess based on he consisen esimaors V ˆ(AIC) and V ˆ (SBC) show essenially no level disorion when T 5. These ess end o perform beer han he sup LM es or he BP ess. Nex we consider he bilinear model. Large samples are needed o eliminae he level disorion of he LM-based ess. The Exp-LM and Exp-LM ess are generally beer han he sup LM ess. The disorion is essenially zero for he Exp-LM and Exp-LM ess based on V ˆ(AIC) when T = 5, bu he disorion persiss a T = 5 when V ˆ (SBC) is used. The esimaors ˆ * V and ˆ GP V are inconsisen for hese non-mds examples: generalized LM-based ess using hese esimaors have rue levels greaer han he nominal level a all sample sizes considered. Nex we compare he LM-based ess and he BP ess. The BP ess end o show beer conrol over he level han he LM-based ess. BP6 and BP12 perform beer han he LM-based ess when V ˆ(AIC) is used, and BP12 and BP2 perform beer han he LM-based ess when V ˆ (SBC) is used. Compuing. The random number generaor used in he experimens was he very long period generaor RANLUX wih luxury level p = 3; See Hamilon and James (1997). The program used for VARHAC was he version of he program by den Haan and Levin (hp://econ.ucsd.edu/~wdenhaan/varhac.hml) modified o run subsanially faser. 18

20 6. MONTE CARLO POWER COMPARISONS In his secion, he finie-sample level-correced powers of he sup LM, Exp-LM and Exp-LM ess are compared, and he powers of he LM-based ess are compared wih hose of he BP ess. The model considered is he locaion model (18), bu now wih serially correlaed errors. Following AP, he models used for he errors ε include AR(1) : ε = φε + u, 1 MA(1): ε = u + θu, j AR(6): ε = φ ε + u, j j= j AR(12) : ε = φ ε + u, j j= j 6 j+ 1 AR(6) ± : ε = φ ( 1) ε j + u, j= j (19) 12 j+ 1 AR(12) ± : ε = φ ( 1) j= 1 12 ε j + u. The models used for he innovaions u are he GARCH (1, 1) and EGARCH (1,1) models and he nonlinear moving average and bilinear models. The above models were chosen by AP because hey include a wide variey of paerns of serial correlaion wih boh posiive and negaive serial correlaions. We calculaed he.5 level-correced powers by simulaion for sample size T = 1 using 1, replicaions. The finie-sample criical values are simulaed using 1, replicaions. The parameer values are chosen so ha he maximum powers are 19

21 approximaely.4 and.8 for he wo parameer values considered. All models are simulaed wih an approximaely saionary sarup by aking he las T random variables from a simulaed sequence of he T + 5 random variables where sarup values are se equal o zero. Table 4 presens he.5 level-correced power of each of he ess for he AR(1), MA(1), AR(6), AR(12), AR(6)±, and AR(12)± models wih GARCH(1,1) innovaions and Table 5 does likewise for models wih EGARCH(1, 1) errors. The.5 level-correced powers for he models wih non-mds errors are repored in Tables 6 and 7. The powers in Table 4 presen a mixed picure. A comparison of he LM-based ess shows ha Exp-LM ends o have he highes power for he AR(1) and MA(1) models and he lowes power for he AR(12), AR(6)±, and AR(12)± models. For he laer models, he sup LM es has he highes power. We conclude ha he sup LM es has higher all-around power han he Exp-LM by a small margin. The same paern ends o hold when differen values of φ and θ are used in he error models. Among he BP ess, he BP6 has he highes power. The sup LM es has higher power han he BP6 es and by a considerable margin in many cases. The powers of he BP6 es end o be higher han he powers of he Exp-LM for he AR(6), AR(12), AR(6)± and AR(12)± models, especially for AR(12)±. The powers of he BP6 es are seldom higher han he powers of he Exp-LM es. Nex we compare he ess based on he powers in Table 5. Our conclusions are essenially he same as hose based on he powers in Table 4. Alhough we conclude again ha he sup LM es has higher all-around power han he Exp-LM and Exp-LM ess, his conclusion is more debaable for EGARCH (1, 1) han for GARCH (1, 1). The 2

22 BP6 again has he highes power among he BP ess. The powers of he BP6 es are lower han he powers of he sup LM and Exp-LM ess bu are someimes greaer han he powers of he Exp-LM es. Tables 6 and 7 give he powers of he ess for he models wih nonlinear moving average innovaions and wih bilinear innovaions. The paern of he power resuls is similar o hose in Tables 4 and 5. The sup LM es has higher power han he BP6 es when he ess are based on V ˆ(AIC) and V ˆ (SBC). However, he sup LM es does no have higher power han he Exp-LM and Exp-LM for AR(1) and MA(1) models. The BP6 again has he highes power among he BP ess. The powers of he BP6 es end o be higher han he powers of he Exp-LM and Exp-LM ess for he AR(6)±, and AR(12)± models wih nonlinear moving average errors and he Exp-LM es for he AR(6)±, and AR(12)± models wih bilinear errors. We conclude ha he sup LM es has higher all around power han he Exp-LM and Exp-LM ess for he non-mds models. Following AP, we also invesigaed he.5 level-correced powers where ARMA(1, 1) models are used for he errors. As previously, he innovaions u are generaed by he MDS or non-mds models considered previously. In simulaion experimens where ARMA(1,1) models are used for he errors, he sup LM ess no longer have he bes all around power compared o he Exp-LM and Exp-LM ess, alhough hey coninue o be beer han he BP6 ess. As an example, consider he following models for he error: 21

23 ARMA(1,1) : ε = φε + u + θu, 1 1 ( φθ, ) = (.1,.5),(.2,.5),(.3,.15),(.4,.3),(.1,.3), (.2,.4),(.3,.5),(.4,.8),(.1,.2),(.2,.3), (.3,.6),,(.1,.5),(.2,.5),(.3,.2),(.4,.3). (2) Table 8 presens he.5 level-correced powers for he case where he u are generaed by he EGARCH (1, 1) model. In his example, he Exp-LM and Exp-LM ess have higher power han he sup LM ess and much higher power han he BP6 ess. The Exp- LM es has he highes all around power. Once he resuls from ARMA error models are aken ino accoun, we conclude ha he sup LM, Exp-LM and Exp-LM ess all have beer power han he BP ess, bu none of he sup LM, Exp-LM and Exp-LM ess is dominan. AP considered seasonal MA models for he errors ε. The models are MA( j) : ε = u + θu j, j = 1,...,6. (21) In Table 9, we repor he.5 level correced power for hese error models wih θ =.15 where he u are generaed by he EGARCH (1, 1) model. For he seasonal models, he BP6 and B12 ess are bes of hose considered, wih BP6 es having he highes all around power. This conclusion is he same as ha reached by AP. AP implemen he ess using TR = 5 and we do so using TR = 2. We invesigaed wheher his affecs he consisency of he ess. We found ha wih Π.8 and a process wih ρ 1 =... = ρ 2, ρ 21 increasing he runcaion lag does no make any very noiceable difference o powers and hus he consisency of he ess as he sample size T increases o 15,. Furher, we simulaed he level-correced powers for T 22

24 = 1 using TR = 4. The powers of he LM-based ess for TR = 4 are essenially he same for TR = 2, and hence he conclusions from he power comparisons are unchanged. 7. EMPIRICAL APPLICATION As Campbell, Lo and MacKinlay (1997, hereafer CLM) noe, he predicabiliy in sock reurns is an acive research opic. They illusrae he empirical relevance of predicabiliy by applying he BP ess o CRSP sock reurn indexes. In his secion, heir empirical applicaion is exended in wo ways: Firs, resuls are presened for he AP ess as well as he generalized AP and BP ess; second, resuls are presened for an exension of heir sample period. CLM repor he means, sandard deviaions, he firs four sample auocorrelaions (in percen) as well as he BP 5 and BP 1 saisics in heir Table 2.4 for monhly, weekly and daily value-weighed (VWRETD) and equal-weighed (EWRETD) sock reurn indexes (NYSE/AMEX). The sample period is July 3, 1962 o December 31, We replicaed he resuls by CLM for his sample period and he sub-periods seleced by CLM. In his secion, he sample period is July 3, 1962 o December 3, 25. Resuls are repored for he sup LM, Exp-LM and Exp-LM ess, and he generalized AP and BP ess, boh for he sample period and sub-periods considered by CLM and for he exended sample period and seleced sub-periods. The skewness and kurosis saisics are also calculaed o provide a check on he normaliy of he reurns. As expeced, he kurosis saisics provide srong evidence agains normaliy. Table 1 repors resuls for he monhly equal-weighed sock reurn indexes. This able illusraes ha inferences from he generalized AP ess can conflic wih hose from 23

25 he generalized BP ess. The generalized AP ess end o rejec and he generalized BP ess end no o rejec. For he daa used by CLM, he BP ess end o no rejec a he nominal.5 level and similarly for he generalized BP ess. The generalized AP ess end o rejec a he nominal.5 and or.1 levels excep for he Ocober 31, 1978 o December 3, 1994 sub-period. The greaer number of rejecions by he generalized AP ess may be explained by he higher power of he generalized AP ess compared o he generalized BP ess. Table 1 also shows ha boh he AP and BP ess end o rejec a he nominal.5 and or.1 levels for he exended sample period and seleced sub-periods. The same is rue for he generalized AP ess. Again here are subsanially fewer rejecions by he generalized BP ess han he generalized AP ess. For he monhly value-weighed sock reurn daa, all he ess end no o rejec. Consequenly, abled resuls are no repored. Table 11 repors he resuls for weekly value-weighed sock reurn indexes. This able illusraes ha he applicaion of he generalized ess can change he inferences drawn from he daa. For he daa used by CLM, he BP1 and BP2 ess rejec a he nominal.5 level. However, he generalized BP5, BP1 and BP2 ess do no rejec a he nominal.5 level. The AP and he generalized AP ess also do no rejec a he nominal.5 level. In his case, he applicaion of he generalized ess suggess ha inferences based on he BP ess may be misleading. For he exended sample period and seleced sub-periods, Table 11 shows ha he AP and BP ess end o rejec a he nominal.5 and or.1 levels. The excepion is he July 7, 1962 o December 27, 25 period since for his period, he AP ess do no rejec 24

26 a he.5 level. By conras, he generalized ess do no rejec a he nominal.5 level excep for he January 5, 1988 o January 3, 1995 sub-period. Again he generalized ess make a difference. These resuls are consisen wih he finding from our Mone Carlo experimens ha he AP and BP ess end o over-rejec. For he weekly equal-weighed sock reurn daa, all he ess end o rejec. The excepion is for he January 5, 1988 o January 1995 sub-period. For his period, he generalized ess do no rejec. The resuls for he daily value-weighed and equal-weighed reurns show ha all he ess end o rejec a he nominal.5 or.1 levels. As a consequence, we do no repor ables for he resuls based on daily daa. These resuls suppor he hypohesis ha he sock reurn indexes are correlaed and hence predicable in he sense ha he serial correlaions are nonzero. An excepion is he daily value-weighed reurns for he January 3, 1995 o December 3, 25 sub-period. The AP es and generalized AP ess do no rejec a he nominal.5 level. By conras, he BP ess rejecs a he nominal.5 level and he generalized BP ess do no rejec a he nominal.5 level. The rejecion by he BP ess is consisen wih he resuls of our Mone Carlo experimens, which show ha he BP ess can subsanially over-rejec. The resuls of he generalized ess for his sub-period sugges ha daily value-weighed sock reurn indexes are no always serially correlaed. In ligh of our Mone Carlo experimens, he resuls repored by CLM for he BP ess are difficul o inerpre in isolaion. Alhough he BP saisics are ofen enormous for weekly and daily daa, his alone does no provide srong evidence ha he null of zero correlaion is false. This is because he BP ess end o subsanially over-rejec 25

27 when daa are generaed by uncorrelaed dependen processes such as a GARCH (1, 1) or EGARCH (1, 1) model. In paricular, he over-rejecion is mos pronounced for large sample sizes, sizes similar o hose in his empirical applicaion. The generalized AP saisics provide more compelling evidence ha he null is indeed false. I is worh sressing ha failure of he generalized AP and BP ess o rejec does no imply ha sock reurn indexes are unpredicable. If he AP or BP ess do no rejec, he convenional inference is ha he sock index reurns are no predicable. This inference is correc if he reurns are in fac independen. By design, he generalized AP or BP ess have a low probabiliy of rejecing if he reurns are uncorrelaed bu saisically dependen. Reurns ha are saisically dependen are in principle predicable. Consequenly, he failure o rejec does no imply unpredicabiliy. Of course, he generalized ess may no rejec because he reurns are independen. However, he generalized ess do no signal wheher he failure o rejec is due o uncorrelaedness or saisical independence. 8. CONCLUDING COMMENTS In his paper, he ess inroduced by AP are generalized o es he null hypohesis ha a ime series process is uncorrelaed in he presence of saisical dependence. The asympoic heory developed by AP for esing he null agains ARMA (1, 1) alernaives also applies o he generalized ess. Hence, he generalized ess use he asympoic criical values calculaed by AP. Simulaions show ha he differences beween he rue and nominal levels of he generalized AP ess are essenially zero for suiable sample sizes. Simulaions also show ha generalized AP ess have good power properies for nonseasonal alernaives 26

28 compared o he corresponding generalized Box-Pierce ess. In paricular, we recommend he Exp-LM es for nonseasonal applicaions in economics and finance. Wih respec o he covariance esimaors of V, for MDS processes we recommend using ˆ GP V alhough ˆ * V produces no oo differen resuls in he examples we consider. If he possibiliy of non-mds processes is allowed, hen V ˆ (SBC) ends o be bes, bu larger sample sizes are needed. The paper includes an empirical applicaion o sock reurn indexes ha is moivaed by he search for predicabiliy in reurns. The applicaion repors he resuls of Box-Pierce ess and generalized Box-Pierce ess as well as he AP and generalized AP ess. The resuls illusrae ha inferences from he generalized AP ess can conflic wih hose from he generalized Box-Pierce ess and ha he applicaion of he generalized AP and Box-Pierce ess can make a difference o he inferences drawn from he daa. Andrews, Liu and Ploberger (1998, hereafer ALP) exended heir approach o esing whie noise agains muliplicaive seasonal ARMA (1, 1) models. ALP developed LR, sup LM and average exponenial ess and obain asympoic resuls similar o hose in AP. In simulaion experimens, ALP compared he level-correced powers of he LR ess and heir varians wih he level-correced powers of ess commonly used in pracice such as he Box-Pierce ess. The LR es and an average exponenial es have subsanially higher all around power han ess commonly used in pracice. The LM-based ess considered by ALP can be generalized using our approach so ha hey have he correc level asympoically when he ime series is serially uncorrelaed bu saisically dependen. This is a opic for furher research. 27

29 ACKNOWLEDGEMENTS The auhors graefully acknowledge Don Andrews and John Geweke for helpful commens. REFERENCES Akaike, H. (1973), Informaion Theory and he Exension of Maximum Likelihood Principle, in Second Inernaional Symposium on Informaion Theory, eds. B. Perov and F.Casaki, Budapes: Akailseoniai-Kiudo, pp Andrews, D. W. K., and Monahan, J. C. (1992), An Improved Heeroskedasiciy and Auocorrelaion Consisen Covariance Marix Esimaor, Economerica, 6, Andrews, D.W. K., and Ploberger, W. (1994), Opimal Tess When a Nuisance Parameer Is Presen Only Under he Alernaive, Economerica, 62, Andrews, D.W. K., and Ploberger, W. (1995), Admissibiliy of he Likelihood Raio Tes When a Nuisance Parameer Is Presen Only Under he Alernaive, Annals of Saisics, 23, Andrews, D.W. K., and Ploberger, W. (1996), Tesing for Serial Correlaion Agains an ARMA (1, 1) Process, Journal of he American Saisical Associaion, 91, Andrews, D.W. K., Liu, X., and Ploberger, W. (1998), Tess for Whie Noise Agains Alernaives Wih Boh Seasonal and Nonseasonal Serial Correlaion, Biomerika, 85, Bera, A.K., and Higgins, M. L. (1997), ARCH and Bilineariy As Compeing Models For Nonlinear Dependence, Journal of Business and Economic Saisics, 15,

30 Bollerslev, T. (1986), Generalized Auoregressive Condiional Heeroskedasiciy, Journal of Economerics, 31, Box, G.E.P., and Pierce, D. A. (197), Disribuion of Residual Auocorrelaions in Auoregressive Inegraed Moving Average Time Series Models, Journal of he American Saisical Associaion, 93, Campbell, J.Y., Lo, A. W., and MacKinlay, A. C., (1997), The Economerics of Financial Markes, New Jersey: Princeon Universiy Press. Davies, R. B. (1977), Hypohesis Tesing When a Nuisance Parameer Is Presen Only Under he Alernaive, Biomerika, 64, Davies, R. B. (1987), Hypohesis Tesing When a Nuisance Parameer Is Presen Only Under he Alernaive, Biomerika, 74, Den Haan, W.J., and Levin, A. (1997), A Praciioner s Guide o Robus Covariance Marix Esimaion, in Handbook of Saisics: Robus Inference, eds. G. S. Maddala and C.R. Rao, Amserdam: Norh-Holland, pp De Jong, R. M., and Davidson, J. (2), The Funcional Cenral Limi Theorem and Weak Convergence o Sochasic Inegrals, Par 1: Weakly Dependen Processes, Economeric Theory, 16, Diebold, F. X. (1986), Tesing for Serial Correlaion in he Presence of Heeroskedasiciy, Proceedings of he American Saisical Associaion, Business and Economics Saisics Secion, Durbin, J., and Wason, G. S. (195), Tesing For Serial Correlaion in Leas Squares Regression, I, Biomerika, 37,

31 El Babsiri, M., and Zakoian, J.-M (21), Conemporaneous Asymmery in GARCH Processes, Journal of Economerics, 11, Granger, C. W. J., and Andersen, A. P. (1978), An Inroducion o Bilinear Time Series Models, Goingen: Vanenhoek and Ruprech. Guo, B. B., and Phillips, P.C. B. (1998), Tesing for Auocorrelaion and Uni Roos in he Presence of Condiional Heeroskedasiciy of Unknown Form, Manuscrip, Cowles Foundaion for Research in Economics, New Haven: Yale Universiy. Hamilon, K.G., and James, F. (1997), Acceleraion of RANLUX, Compuer Physics Communicaions, 11, Hannan, E. J., and Heyde, C.C. (1972), On Limi Theorems For Quadraic Funcions of Discree Time Series, Annals of Mahemaical Saisics, 43, Hansen, B.E. (1996), Inference When a Nuisance Parameer Is No Idenified Under he Null Hypohesis, Economerica, 64, Haas, M., Minik, S., and Paolella, M. (24), Mixed Normal Condiional Heeroskedasiciy, Journal of Financial Economerics, 2, He, C., and Teräsvira, T. (1999), Properies of Momens of a Family of GARCH Processes, Journal of Economerics, 92, He, C., Teräsvira, T., and Malmsen, H. (22), Momen Srucure of a Family of Firs- Order Exponenial GARCH models, Economeric Theory, 18, Lamber, P., and Lauren, S. (21), Modelling Financial Time Series Using GARCH- Type Models Wih a Skewed Suden Disribuion for he Innovaions, working Paper, Universie Caholique de Louvain and Universie de Liege. 3

32 Lo, A.W., and MacKinlay, A.C. (1989), The size and power of he variance raio es in finie samples: A Mone Carlo invesigaion, Journal of Economerics, 4, Lobao, I., Nankervis, J. C., and Savin, N.E. (21), Tesing for Auocorrelaion Using a Modified Box-Pierce Q Tes, Inernaional Economic Review, 42, Lobao, I., Nankervis, J. C., and Savin, N.E. (22), Tesing for Zero Auocorrelaion in he Presence of Saisical Dependence, Economeric Theory, 18, Nelson, D. (1991), Condiional Heeroskedasiciy in Asse Reurns: A New Approach, Economerica, 59, Poerba, J.M., and Summers, L.H. (1988), Mean Reversion in Sock Prices: Evidence and Implicaions, Journal of Financial Economics, 22, Pöscher, B. M. (199), Esimaion of Auoregressive Moving-Average Order Given an Infinie Number of Models and Approximaion of Specral Densiies, Journal of Time Series Analysis, 11, Romano, J.L., and Thombs, L.A. (1996), Inference for Auocorrelaions Under Weak Assumpions, Journal of American Saisical Associaion, 91, Schwarz, G. (1978), Esimaing he Dimension of a Model, Annals of Saisics, 6, Taylor, S. J. (1982), Tess of he Random Walk Hypohesis agains a Price-Trend Hypohesis, Journal of Financial and Quaniaive Analysis, 17, Taylor, S. (1984), Esimaing he Variances of Auocorrelaions Calculaed From Financial Time Series, Applied Saisics, 33,

33 Taylor, S.J. (25), Asse Price Dynamics, Volailiy, and Predicion, Princeon: Princeon Universiy Press Tong, H. (199), Nonlinear Time Series, Oxford: Oxford Universiy Press 32

34 Table 1. Asympoic Criical Values TR sup-lm Exp-LM Exp-LM 1% 5% 1% 1% 5% 1% 1% 5% 1% NOTE: Criical values obained by seing Π = {, ±.1,, ±.79, ±.8}. The number of replicaions is 15 million. 33

35 Table 2. Rejecion Probabiliies (Percen) of Nominal.5 Tess: MDS Models GARCH(1, 1) EGARCH(1, 1) ˆ V T I sup LM Exp-LM Exp-LM BP BP BP ˆ * V sup LM Exp-LM Exp-LM BP BP BP ˆ GP V sup LM Exp-LM Exp-LM BP BP BP V ˆ (AIC) sup LM Exp-LM Exp-LM BP BP BP V ˆ (SBC) sup LM Exp-LM Exp-LM BP BP BP ˆ NOTE: The ess, as funcions of V, are defined in (12) and (14) above. The esimaor ˆ * V is consisen in he diagonal MDS case; ˆ GP V is consisen for general MDS processes; he esimaors V ˆ(AIC) and V ˆ(SBC) are consisen esimaor in boh MDS and non-mds cases in he non-mds case. The number of replicaions is 25,. 34

36 Table 3. Rejecion Probabiliies (Percen) of Nominal.5 Tess: Non-MDS Models Nonlinear Moving Average Bilinear ˆ V T I sup LM Exp-LM Exp-LM BP BP BP ˆ * V sup LM Exp-LM Exp-LM BP BP BP ˆ GP V sup LM Exp-LM Exp-LM BP BP BP V ˆ (AIC) sup LM Exp-LM Exp-LM BP BP BP V ˆ (SBC) sup LM Exp-LM Exp-LM BP BP BP NOTE: See Table 2. 35