Strong orthogonal decompositions and non-linear impulse...

The Canadian Journal of Statistics 453Vol 34 No 3 2006 Pages 453ndash473La revue canadienne de statistique

Strong orthogonal decompositions andnon-linear impulse response functionsfor infinite-variance processesJonathan B HILL

Key words and phrases Banach spaces infinite variance non-linear impulse response function orthogonaldecomposition projection iteration smooth transition autoregression

MSC 2000 Primary 60G25 secondary 47N30

Abstract The author proves that Wold-type decompositions with strong orthogonal prediction innovationsexist in smooth reflexive Banach spaces of discrete time processes if and only if the projection opera-tor generating the innovations satisfies the property of iterations His theory includes as special cases allprevious Wold-type decompositions of discrete time processes completely characterizes when non-linearheavy-tailed processes obtain a strong-orthogonal moving average representation and easily promotes atheory of non-linear impulse response functions for infinite-variance processes The author exemplifies histheory by developing a non-linear impulse response function for smooth transition threshold processes anddiscusses how to test decomposition innovations for strong orthogonality and whether the proposed modelrepresents the best predictor A data set on currency exchange rates allows him to illustrate his methodology

Decompositions orthogonales fortes et fonctions reponsesa impulsion non lineaire pour des processus a variance infinieResume Lrsquoauteur demontre que pour qursquoun espace de Banach reflexif lisse de processus a temps discretadmette une decomposition de type Wold a innovations previsionnelles fortement orthogonales il faut et ilsuffit que lrsquooperateur projectif engendrant les innovations possede la propriete drsquoiterations Son resultat quienglobe comme cas particuliers toutes les decompositions de type Wold obtenues anterieurement pour desprocessus a temps discret caracterise completement les situations ou les processus non lineaires a queueslourdes possedent une representation en moyenne mobile fortement orthogonale en plus de conduire natu-rellement a une theorie des fonctions reponses a impulsion non lineaire pour les processus a variance infinieIl se sert de cette theorie pour developper une fonction a impulsion non lineaire pour des processus a seuilde transition lisse et montre comment tester lrsquoorthogonalite forte des innovations de la decomposition etverifier si le modele propose est bel et bien le meilleur predicteur Un jeu de donnees sur des taux de changede devises lui permet drsquoillustrer son propos

1 INTRODUCTION

This paper presents a complete theory of Wold-type orthogonal decompositions in smooth re-flexive Banach spaces of discrete time processes Such Banach spaces include Hilbert spacesthe stable laws and Lp-spaces they contain linear and non-linear processes and include manyprocesses with a stochastic recurrence representation (eg GARCH and FIGARCH processessee Basrak Davis amp Mikosch 2001 2002) An immediate application is the construction ofnon-linear impulse response functions (IRFs) for possibly non-stationary andor long memoryheavy-tailed processes based on moving average representations

In particular we provide necessary and sufficient conditions for the existence of orthogonaldecompositions in the time-domain with asymmetric strong orthogonal innovations For someBanach space stochastic process Xt = Xt minusinfin lt t ltinfin we consider the decomposition

Xn =

infinsumi=0

ψniZnminusi + Vn (1)

454 HILL Vol 34 No 3

for some set of orthogonal innovations Zt and a ldquoresidualrdquo Vn The seminal work of Wold(1938) provides a foundation for characterizing stationary finite-variance processes with covari-ance orthogonal innovations In the classic setting the innovations necessarily satisfy the strongorthogonality condition

sp(Zt+i Zt)perp sp(Xtminus1 Xtminusj) foralli ge 0 forallj ge 1 (2)

where sp denotes the closed linear span In general Banach spaces however the ldquocovariancerdquomay not exist conditional expectations and the best predictor may not equate metric projectionoperators need not be linear and innovations may not satisfy (2) although they will satisfy (3)below A related decomposition theory with strong orthogonal innovations for processes withan unbounded variance or for any process based on metric projection other than minimizing themean squared error is relatively limited and the most promising contributions to the literaturefocus entirely on closed linear spans

Let spt = sp (Xs s le t) denote the closed linear span of Xt and denote by Pt a metricprojection operator (eg Ptminus1 spt rarr sptminus1) Urbanic (1964 1967) considered decomposi-tions of strictly stationary infinite-variance processes which admit independent metric projectioninnovations Faulkner amp Huneycutt (1978) consider decompositions with innovations Zt thatonly satisfy a weak asymmetric orthogonality condition

sp (Zt)perp sp (Xtminus1 Xtminusj) forallj ge 1 (3)

Miamee amp Pourahmadi (1988) develop a weak-orthogonal decomposition theory for p-stationaryprocesses based on innovations in spt minus Ptminus1spt The theory however fundamentally exploitsthe codimension one property of closed linear spans spt = sp (Xt sptminus1)

Similarly Cambanis Hardin amp Weron (1988) establish an asymmetric decomposition theoryfor Lp(ΩFt μ) processes in spt The authors prove (i) projection operator linearity (ii) iteratedprojections and (iii) the existence of strong orthogonal innovations are equivalent when theinnovations are restricted to the space spt minus Ptminus1spt Specifically the authors prove (i)rArr (iii)rArr (ii)rArr (i) and operator linearity is expedited by the fact that sptminus1 is codimension one in spt

In the literature therefore either independence is assumed only weak orthogonality isproven or explicit properties of closed linear spans are exploited to promote strong orthogo-nality For projections into arbitrary (non-linear) Lp-spaces Cambanis Hardin amp Weron (1988)point out that operator linearity sufficiently renders strong orthogonal innovations (2) This re-sult however is trivial see Theorem 3 below Moreover no result exists (that we know of)characterizing strong orthogonal decompositions for finite-variance processes based on best Lp-metric projection p lt 2

The construction and use of orthogonal innovation spaces sptminusPtminus1spt in order to promotestrong orthogonality is not a trivial simplification however The explicit omission of non-linearbest predictors and orthogonal innovations must be viewed critically in light of developments inthe theory and empirical methods associated with non-linear stochastic processes For examplemoving average forms have been utilized to characterize linear dependence within processeswith regularly varying tails see Davis amp Resnick (1985ab) and Kokoszka amp Taqqu (19941996) The innovations in this literature are typically assumed to be independent and identicallydistributed hence strongly orthogonal to far more than subspaces of spt Except for the specialcase of symmetric stable process (Cambanis Hardin amp Weron 1988) nowhere in this literatureare necessary and sufficient conditions for the existence of such moving averages derived

Moreover we see the use of moving averages a la IRFs in time series settings in whichamassed evidence suggests non-linear data generating processes with heavy tails See egHols amp de Vries (1991) Cheung (1993) Gallant Rossi amp Tauchen (1993) Phillips McFar-land amp McMahon (1996) Lin (1997) Mikosch amp Starica (2000) Falk amp Wang (2003) andHill (2005b) In the economics and finance literatures the implied ldquoimpulsesrdquo are predominantlyassumed to be independent and identically distributed finite-variance innovations computed from

2006 INFINITE-VARIANCE PROCESSES 455

inherently linear vector autoregression [VAR] representations (eg Sims 1980) A linear struc-ture ensures symmetry with respect to how positive and negative shocks persist over time andrenders shocks independent of the history of the process If we wish to track heavy-tailed shockswith asymmetric impacts on the level process based on best (non-linear) forecasts then a decom-position theory that goes substantially beyond the extant literature is required

Toward this end Gallant Rossi amp Tauchen (1993) and Koop Pesaran amp Potter (1996) de-velop non-parametric representations of impulse responses for general non-linear processes inthe Hilbert space L2(ΩFt μ) The impulses are assumed to be independent and the responsesare simply defined as differences between conditional expectations As stated above the condi-tional expectations may not be the best predictor in a general Banach space (eg Lp(ΩFt μ)p lt 2) Gourieroux amp Jasiak (2003) develop a parametric Volterra-type expansion of indepen-dent and identically distributed Gaussian innovations for strongly stationary square-integrableprocesses that do not display long memory properties In this case the level process has a finitevariance and limited memory and the innovations are assumed to be symmetrically distributed

In this paper we extend orthogonal decomposition theory to its arguable limit For anysmooth reflexive Banach space Bt we prove in Theorem 3 (the main result) that the property ofiterated projections is necessary and sufficient for the existence of a decomposition with asym-metrically strong orthogonal innovations sp (Zt+i Zt)perpBtminus1 for every i ge 1 Using anarbitrary metric projection mapping Ptminus1 Bt rarr Btminus1 our results do not exploit operatorlinearity in general and they specifically do not rely on properties of the closed linear spanTheorem 3 allows for a simple characterization of a non-linear IRF based on best Lp-metric pro-jection Our results include as special cases Wold decompositions of Hilbert space processes ofLp-space processes of processes in Banach spaces which do not admit a linear metric projectionoperator of long memory or non-stationary or non-square integrable processes of processeswith asymmetrically distributed innovationsimpulses and does not restrict projection mappingsto closed linear spans Moreover our primitive result linking iterated projections to strong or-thogonality holds for any appropriate Lp-metric projection operator even if the process belongsto L2 For example our theory fully characterizes when the best L1-predictor of a finite-varianceprocess generates strong orthogonal errors

If the operator Pt does not iterate then a strong orthogonal moving average does not existWe lose moving average-based non-linear IRFs with adequately noisy impulses and theories oflinear dependence for moving averages with independent innovations do not apply Converselyif a strong orthogonal moving average form does not exist then the projection operator does notiterate PsPt = Ps for some or all s lt t In this case we lose an array of prediction-based resultswhich rely on iterated projections including iterative multi-step ahead forecasts and non-linearIRFs based on Lp-metric projection

We make the theory concrete by constructing in Corollary 4 a parametric decomposition ofthe form (1) with solutions for ψni In Section 4 we then develop a theory of non-linear im-pulse response functions based on best Lp-metric projection and the properties of strong orthog-onality and iterated projections We construct in Section 5 an extended example demonstratingthe decomposition of a non-linear smooth transition threshold model and associated non-linearimpulse response function Although Theorem 3 characterizes the dual relationship betweenprized prediction characteristics it says nothing about when they will hold or how to verify thatthey hold This is compounded by the inherent difficulty associated with computing the bestLp-predictor We therefore focus our attention on the empirical task of verifying whether thenon-linear model actually presents the conditional expectation andor the best Lp-predictor andwhether the proposed decomposition innovations are strong orthogonal We apply the methodsto daily returns of the Yen Euro and British Pound exchange rates against the US Dollar Wefind significant evidence that the threshold model adequately characterizes the best Lp-predictorfor some p lt 2 for some exchange rates

The rest of paper is organized as follows Section 2 contains a preliminary metric projection


theory Section 3 contains the main results we develop a theory of non-linear impulse responsefunctions in Section 4 and Section 5 contains an example and an application The Appendixcontains formal proofs

In the sequel we employ the following notation and definition conventions Denote byBt equiv B(ΩFt μ middot ) a closed smooth reflexive Banach measure space of nondetermin-istic stochastic processes Xτ τ le t endowed with the norm middot measure μ and σ-fieldFt = σ(Xτ τ le t) Denote

BequivB(ΩF μ middot ) =⋃tisinZ

Bt F=⋃tisinZ

Ft

It is understood that x lt infin for any x isin Bt and Ftminus1 sub Ft Let Lt equiv Lp(ΩFt μ)p le 2 We denote the signed power sgn (z)|z|a as z〈a〉 a isin IR Denote by perp any orthogonalitycondition in Bt and let Bperp

t denote the orthogonal complement of BtFor closed linear subspaces of Bt say S1 Sn n gt 1 we write S1 + middot middot middot+ Sn to denote

the stochastic spacesumn

i=1 Zi Zi isin Si

For orthogonal subspaces S1 Sn n gt 1 the

space Sn oplus Snminus1 oplus middot middot middot oplus S1 (synonymouslyoplusnminus1

i=0 Snminusi) denotes the spacesumnminus1

i=0 Si where

minus1sumi=0

Snminusi perpnminus1sumi=

Snminusi

for all 1 le lt n In general orthogonality is not symmetric For spacesoplusnminus1

i=0 Snminusi we saythe subspaces St are strong orthogonal Similarly whenever StperpSs for every s lt t we say thesubspaces St are weak orthogonal Clearly strong orthogonality implies weak orthogonality

2 PROJECTION OPERATORS AND ORTHOGONALITY IN BANACH SPACE

The subsequent decomposition theory is based on orthogonal innovation Banach spaces Forbackground theory see Singer (1970) Lindenstrauss amp Tzafriri (1977) Giles (1967 2000) andMegginson (1998) For arbitrary random variables (x y) isin B we work with the property ofJames Orthogonality see James (1947) y is James orthogonal to x whenever y + λx ge yfor every real scalar λ isin IR denoted yperpx Banach space norms middot may be supported byarbitrarily many semi-inner products [ middot middot ] However for smooth spaces B and (x y) isin B if yis orthogonal to x there exists one inner product that supports [y x] = 0 (see eg Giles 1967Singer 1970)

LEMMA 1 Let B be a Banach space with norm middot For any subspaces U V sube B suchthat UperpV there exists a semi-inner product [ middot middot ] that supports middot such that [U V ] = 0for each u isin U and v isin V the inner product [ middot middot ] satisfies [u v] = 0 and [u u]12 = u[v v]12 = v Moreover if B is smooth then [ middot middot ] is unique

21 Metric projection operators

Consider arbitrary subspaces U V sube B σ(V ) sub σ(U) where σ(V ) denotes the sigma algebrainduced by the elements of V For some element u isin U we say v isin V is the ldquobest predictorrdquo ofu with respect to V if and only if

uminus v le uminus vfor every element v isin V Because the space B is reflexive the predictor v exists and isunique We define then the metric projection operator that maps P U rarr V as P (u |V ) = vthe projection P (u |V ) is identically the ldquobest predictorrdquo of u The projection P (u |V ) iscontinuous bounded and idempotent although not in general linear see below For sub-spaces U V sube B the notation P (U |V ) is understood to represent the projection spaceP (U |V ) = P (u |V ) u isin U


22 Iterated projections and operator linearity

We say that the property of iterated projections holds in V1 sube B for some projection operatorP U rarr V1 when for any subspace V0 sube V1 sube B P

P (u |V1) |V0

= P (u |V0) We say that a

projection operator P which maps P U rarr V is a linear operator on U supe V if for any elementsu1 u2 isin U and any real numbers a b isin IR P (au1 + bu2 |V ) = aP (u1 |V ) + bP (u2 |V ) ahomogenous additive function of u1 and u2 If a projection operator is a linear operator theniterated projections holds see Lemma 2 part viii

23 Metric projection

In the following assume that u is an arbitrary element of U and denote by E (u |V ) the expec-tation of u conditioned on σ(V ) For a proof see Lemma 2 of Hill (2005a) or consult Singer(1970) Giles (2000) and Megginson (1998)

LEMMA 2 (i) Orthogonality the element v isin V satisfies P (u |V ) = v if and only if (uminusv)perpVif and only if [uminus v v] = 0 for a unique [ middot middot ] and every v isin V (ii) Pu = 0 if and only if uperpVif and only if [u v] = 0 for every v isin V (iii) for any v isin V P (v |V ) = v (iv) quasi-linearityfor any z isin V and any u isin U P (u + z |V ) = P (u |V ) + z (v) norm-boundedness for anyelement u isin U P (u |V ) le ku for some scalar 0 lt k lt infin (vi) unbiasedness in Lp ifU V sube Lp 1 lt p le 2 and if E (u |V ) isin V then P (u |V ) = E (u |V ) = v with probabilityone if and only if (uminusv)perpV and Vperp(uminusv) (vii) scalar-homogeneity P (au |V ) = aP (u |V )for every real scalar a isin IR and (viii) if P is a linear operator on U supe V then P satisfies theproperty of iterated projections

Remark 1 Property (ii) implies that if uperpV then P (u |V0) = 0 forallV0 sube V Property (iii)identically implies idempotence PP = P The conditions for property (vi) are non-trivialthe conditional expectations E (u |V ) may not be an element of the space V For examplesuppose V = sp (v1 vn) the closed linear span of stable random variables (vi)

ni=1 with tail

index α lt 2 and suppose u is a stable random variable with tail index α Then P (u |V ) isin Vby construction yet E (u |V ) need not be linear see for example Hardin Samorodnitsky ampTaqqu (1991) Of course for non-Gaussian processes in L2(ΩFt μ) the best L2-predictorE (u |V ) need not be linear

3 MAIN RESULTS

Denote by Pts the metric projection mapping P Bt rarr Bs s le t Construct the followingspaces

Bminusinfin =⋂n

Bn B+infin =⋃n

Bn

We assume the spaces Bt contain only non-deterministic processes such that Ftminus1 sub Ft andBtminus1 sub Bt Consequently prediction generates non-trivial errors Xt minus PtsXt gt 0 forallXt isinBt foralls lt t

THEOREM 3 For any space Bn there exists a sequence of subspaces Nnminusiinfini=0 Nt sube Btsuch that

Bn =

( infinsumi=0

Nnminusi

)+ Bminusinfin (4)

where NtperpBtminus1 and NtperpNs for every s lt t le n Moreover the following are equivalent

(i) Bn =(oplusinfin

i=0 Nnminusi

)oplusBminusinfin whereopluskminus1

i=0 NnminusiperpBnminusk forallk ge 1

(ii) PttminusPttminusk = Pttminusk for every t k le


Furthermore provided (i) holds every element Y isin oplusinfin

i=0 Nnminusi obtains a unique norm-convergent expansion Y =

suminfin

i=0 ξnminusi for some ξt isin Nt

Remark 2 Cambanis Hardin amp Weron (1988) point out that operator linearity implies result(i) for processes in Lp(Ωn μ) and for projection into arbitrary Lp(Ωn μ)-spaces Thisresult however is trivial and does not anticipate the dual relationship between orthogonalityand iterated projections (i) hArr (ii) without invoking operator linearity Assume Pts is a linearoperator on Bt and consider any element

sumi=k

ξnminusi isinsum

i=k

Nnminusi ξnminusi isin Nnminusi 0 le k le

Because NnminusiperpBnminusiminus1 by construction and by Lemma 2 part ii we have Pnnminusiminus1ξnminusi = 0By operator linearity we conclude

Pnnminusminus1

sumi=k

ξnminusi =

sumi=k

Pnnminusminus1ξnminusi = 0

hencesum

i=k ξnminusiperpBnminusminus1 (see Lemma 2 part ii) Because the elementsum

i=k ξnminusi isinsumi=k Nnminusi is arbitrary we deduce

sumi=k NnminusiperpBnminusminus1 for any 0 le k le This identically

implies strong orthogonality of the innovation spaces Nt

Remark 3 Theorem 3 characterizes the existence of a decomposition for any process in a smoothreflexive Banach space based on any appropriate metric-projection operator This will be partic-ularly useful if evidence suggests that a chosen linear or non-linear model of a finite-varianceprocess does not represent the best L2-predictor but does characterize the best Lp-predictor forsome p lt 2 see Section 4

Because any element Y isinoplusinfin

i=0 Nnminusi obtains a unique norm-convergent series representa-tion we may write elements Xn isin Bn in a straightforward moving-average form See Corol-lary 4 of Hill (2005a)

COROLLARY 4 Consider any Banach space Bn such that a Wold decomposition exists

i For every Xn isin Bn there exists a sequence of orthogonal subspaces Nnminusiinfini=0 Nt subeBt a sequence of stochastic elements Zt Zt isin Nt an element Vn isin Bminusinfin and realnumbers ψniinfini=0 such that

Xn =

infinsumi=0

ψniZnminusi + Vn

where the seriessuminfin

i=0 ψniZnminusi is norm-convergent and the innovations Zt are strongorthogonal in the sense that ZtperpBtminus1 and

sp (Zn Znminus1 )perpBminusinfin sp (Zt+i Zt)perpBtminus1 forallt le n foralli ge 0

if and only if PttminusPttminusk = Pttminus for every t k le

ii Moreover ψn0 = 1 and the coefficients ψni uniquely satisfy the recursive relationshipfor i = 1 2

ψni =[Znminusi Xn]

[Znminusi Znminusi]minus

iminus1sumj=0

ψnj[Znminusiminusj Zn]

[Znminusi Znminusi]


Remark 4 Although Zn equiv Xnminus Pnnminus1Xn by definition for an arbitrary process Xn wecannot in general say Znminusi = Xnminusiminus Pnminusinminusiminus1Xnminusi the best one-step ahead predictionerror of Xnminusi A well-known exception holds for causal-invertible ARMA processes see egCline (1983) If iterated projections hold (equivalently if the innovations are strong orthogonal)then Znminusi equiv PnnminusiXn minus Pnnminusiminus1Xn = PnnminusiXn minus Pnnminusiminus1PnnminusiXn the innovationbased on a one-step ahead projection of the i-step ahead forecast

Remark 5 Theorem 5 of Hill (2005a) characterizes necessary and sufficient conditions for theinnovations to be symmetrically strong orthogonal for processes in Lp

(ΩFt μ

) Essentially

Ltminus1perpNt Pttminus1Xt = E(Xt |Ftminus1

) and E

[Xt minus E(Xt |Ftminus1

)〈pminus1〉 |Ftminus1] = 0 Ftminus1-ae all identically imply the innovations with be symmetrically strong orthogonal The secondproperty identically implies Pttminus1Xt minus Pttminus1(Xt)〈pminus1〉 = 0 the latter property is simply amartingale difference property and implies E


)〈pminus1〉Ytminus1

]= 0 for every

Ftminus1-measurable random variable Ytminus1 See also Cambanis Hardin amp Weron (1988)

4 NON-LINEAR IMPULSE RESPONSE FUNCTIONS IN Lt

In the following we develop a general theory of non-linear impulse response functions (IRFs)based on strong orthogonal decomposition innovations Let Vt be an Lt-valued random variabledefine the sequence of spaces Lt = Lt oplus Vt+1 and let Lminusinfin = 0 for simplicity Definethe h-step ahead non-linear impulse response function

I(h VtLtminus1) = P (xt+h | Ltminus1)minus P (xt+h |Ltminus1)

The above definition simply generalizes the expectations based format of Koop Pesaran ampPotter (1996) the response at horizon h is the best h-step ahead prediction response toa random shock Vt at time t conditioned on all past histories Ltminus1 We could writeI(h vt ωtminus1) = P (xt+h |ωtminus1 vt) minus P (xt+h |ωtminus1) to make explicit a particular historyωtminus1 equiv xtminus1 xtminus2 and particular shock vt in the manner of Koop Pesaran amp Pot-ter (1996) The impulse response function I(h VtLtminus1) is an tminus1-measurable random vari-able and I(h vt ωtminus1) is simply a realization We may compute I(h VtLtminus1) for a largenumber of draws vt ωtminus1 from the joint distribution of Vt and Xtminus1 Xtminus2 An em-pirical distribution function and confidence bands of the responses I(h vt ωtminus1) can then beestimated see Section 5

THEOREM 5 Assume the process xτ τ le t lies in Lp(Ωt μ) and obtains a strongorthogonal decomposition xt =

suminfin

i=0 ψtiZttminusi with respect to the subspaces Lτ τ le tminus1Assume the metric projection operator P Lt rarr Ltminus1 iterates from Ltminusk to Ltminuskminus1 for any kThen

I(h VtLtminus1) = ψt+hhP (Zt+ht | Ltminus1) (5)

Remark 6 Strong orthogonality is required because the line of proof exploits iterated projectionssee Theorem 3

Remark 7 An h-step ahead ldquoimpulse responserdquo is simply a scaled predicted strong orthogonalinnovation where the prediction exploits information contained in the random impulse Vt In astandard linear setting the xt =

suminfin

i=0 ψiVtminusi Vt are independent and identically distributed Itis easy to show that (5) reduces to a classic representation for any particular history ωtminus1 andimpulse vt I(h vt ωtminus1) = ψhvt

Remark 8 In L2 the non-linear IRF I(h VtLtminus1) is identically the generalized impulse re-sponse function characterized by Equation (9) in Koop Pesaran amp Potter (1996) as long as the


projection operator minimizes the mean squared error Otherwise (5) characterizes a further gen-eralization of the generalized IRF of Koop Pesaran and Potter (1996) to best Lp-projection offinite-variance processes

Remark 9 Koop Pesaran amp Potter (1996) characterize non-parametric and bootstrap methodsfor estimating the conditional expectations based on draws from the empirical distributions ofxtminus1 xtminus2 and Vt It is beyond the scope of the present paper to consider such comparablebootstrap methods for approximating a best Lp-predictor In the sequel we estimate ψn+hh

and P (Zn+hn | Lnminus1) directly using in-sample information and either imputed or simulatedimpulses under assumed stationarity (eg ψn+hh = ψnh = ψh for all n)

Requiring the operator to iterate from Ltminusk to Ltminuskminus1 ie

PP (xt |Ltminusk) | Ltminuskminus1 = P (xt | Ltminuskminus1)

does not diminish the generality of the result by very much For example if t equiv σ(xτ τ le t)= σ(ετ τ le t) for some stochastic process εt and the impulses Vt are simply εt then theassumption holds because

Lt = Lt oplus Vt+1 = Lt oplus εt+1 = Lt+1

This will hold for infinitely large classes of linear and non-linear processes see Section 5 for anexample

LEMMA 6 Let t equiv σ(xτ τ le t) = σ(ετ τ le t) and Vt = εt for all t isin Z Then

PP (xt |Ltminusk) | Ltminuskminus1

= P (xt | Ltminuskminus1)

for all k ge 0 Additionally if xt admits a strong orthogonal decomposition then

PP (xt | Ltminusk) |Ltminusk

= P (xt |Ltminusk)

5 THRESHOLD MODELS AND EMPIRICAL APPLICATION

In practice the analyst will need to verify whether a particular decomposition actually generatesstrong orthogonal innovations and indeed whether the predictor used to generate the innovationsactually represents the best predictor The verification of such properties is required as a neces-sary foundation for generating an exact non-linear IRF which requires iterated projections seeTheorems 3 and 5 In this section we focus our attention entirely on a simple threshold modelfor the sake of brevity

Due to linearity and iteration properties the predominant practice in the literature is to as-sume that a particular model represents the conditional mean which may not be the best Lp-predictor for some or finitely many p gt 0 As a nod toward convention and practical simplicitywe explore a conditional expectations-based decomposition and discuss model specification teststo verify whether the conjectured model represents the best Lp-predictor for any p isin (1 2] andwhether the resulting prediction errors are strong orthogonal We then derive a sample non-linear IRF for the particular threshold model and apply the model and specification tests to thedaily returns of currency exchange rates Proofs of each result in this section can be found inHill (2005a)

51 Threshold model and orthogonal decomposition

A growing literature suggests that the returns to many macroeconomic and financial time serieshave heavy tails are serially uncorrelated and have some form of non-linear structure See Tong(1990) Kees amp Kool (1992) Loretan amp Phillips (1994) Franses amp van Dijk (2000) LundberghTerasvirta amp van Dijk (2003) and Lundbergh amp Terasvirta (2005) to name a few In particular


the daily log-returns of many currency exchange rates appear to be serially uncorrelated to havean infinite kurtosis or infinite variance and to display serially asymmetric extremes see Hols ampde Vries (1991) Hill (2005b) and the citations therein Moreover the non-linear structure ofexchange rates has an intuitive regime transitional form based on ldquobandingrdquo policies in economicunions see eg Lundbergh amp Terasvirta (2006)

Together the characteristics of noisy returns persistent extremes and currency policies sug-gest daily exchange rate returns may be governed by smooth transition autoregression [STAR]data generating process see Saikkonen amp Luukkonen (1988) Terasvirta (1994) MichealNobay amp Peel (1997) and Lundbergh amp Terasvirta (2006) Denote by xt the log-return Δln yt

of a daily exchange rate yt Let xt isin Lp(Ωt P ) = Lt t = σ(xs s le t) and let α bethe moment supremum of εt E |εt|p lt infin for all p lt α Assume α gt 1 and consider any1 lt p lt min(α α4 + 1) Simple STAR models which capture the above stylized traits includethe exponential and logistic STAR

xt = φxtminus1gtminus1(γ c) + εt |φ| lt 1 γ ge 0 c gt 0 (6)

where εt is strictly stationary

E (εt | tminus1) = 0 gtminus1(γ c) = expminusγ(|εtminus1| minus c)2in the ESTAR case and in the LSTAR case

gtminus1(γ c) =[1 + expminusγ(|εtminus1| minus c)]minus1 minus 1 + exp(γc)minus1

Whether E (εt | tminus1) = 0 is supported in practice will be considered below There are manyavailable variations on this theme and numerous alternative choices for the threshold variable(here we use the previous periodrsquos shock |εtminus1|) see van Dijk Terasvirta amp Franses (2000)

The ESTAR form naturally articulates ldquoinnerrdquo (|εtminus1| asymp c) and ldquoouterrdquo (|εtminus1| = c) regimeswhen the previous periodrsquos shock |εtminus1| is near c xt asymp φxtminus1 + εt hence the return is seri-ally persistent when |εtminus1| is far from c xt asymp εt such that the return is noisy The LSTARcharacterizes ldquolowerrdquo and ldquoupperrdquo regimes respectively when |εtminus1| asymp 0 then xt asymp εt and as|εtminus1| rarr infin then xt = φ1+exp(minusγc)minus1xtminus1 +εt As the scale γ rarrinfin the LSTAR modelconverges to a Self Exciting Threshold Autoregression xt = φxtminus1I(|εtminus1| gt c) + εt TheLSTAR model naturally implies that extremes are persistent and non-extremes are noisy TheESTAR model can also capture this asymmetry if c is extremely large the returns will be noisyif |εtminus1| is far from c which will predominantly occur when |εtminus1| lt c

Although the ESTAR model has traditionally been used to capture symmetric banding poli-cies for exchange rate levels (eg Micheal Nobay amp Peel 1997 and Lundbergh amp Terasvir-ta 2006) its ability to capture extremal asymmetries is noteworthy because daily returns likelyillicit asymmetric responses from traders and policy makers large deviations may suggest amarket crisis whereas small deviations may not be noteworthy See Engle amp Ng (1993) Suchvolatility asymmetries have been recently modeled as smooth transition GARCH processes seeGonzalez-Rivera (1998) and McMillan amp Speight (2002)

We may decompose xt by straightforward backward substitution

THEOREM 7 Assume that the stochastic process xτ τ le t lies in Lp(Ωt μ) and that(6) holds Then xt =

suminfin

i=0 φiεtminusi

prodij=1 gtminusj(γ c) as t = σ(ετ τ le t) and xt =suminfin

i=0 ψiZtminusi + Vt where Vt = bφ(1minus aφ) Zt = εt ψ0 = 1 and for i ge 1

Ztminusi = εtminusigtminusi(γ c)minus b+ gtminusi(γ c)minus a timesinfinsum

j=i+1

φjminusiεtminusj

jprodk=i+1

gtminusk(γ c)

ψi = φiaiminus1 i = 1 2


where

a = E gt(γ c) isin [minus1 1] b = E εtgt(γ c) E (Ztminusi | tminusiminus1) = 0

If Zt is Lp-strong orthogonal for some p le 2 then P (xt+h | t) =suminfin

i=h ψt+hiZt+ht+hminusi

Remark 10 It is straightforward to show plimNrarrinfinE (xt | tminusN ) = E (xt) = bφ(1 minus aφ)where the limit holds almost surely If εt is symmetrically distributed then b = 0 If γ = 0 suchthat xt is a simple AR(1) then a = b = 0 Ztminusi = εtminusi and ψi = φi If φ = 0 then triviallyxt = Zt = εt

Remark 11 If Zt is strong orthogonal we deduce the h-step ahead forecast

P (xt+h | t) =infinsum

i=0

φi+hai+hminus1[

εtminusigtminusi(γ c)minus b

+gtminusi(γ c)minus a

times

infinsumj=i+1

φjminusiεtminusj

jprodk=i+1

gtminusk(γ c)]

52 Verifying weak and strong orthogonality

For any t write

Zt = εtgt(γ c)minus b+ gt(γ c)minus a timesinfinsum

j=1

φjεtminusj

jprodk=1

gtminusk(γ c) (7)

Exploiting E (εt | tminus1) = 0 and the definitions of a and b we know E (Zt | tminus1) = 0 suchthat Zt is weakly orthogonal to Ltminus1 in some sense see Cambanis Hardin amp Weron (1988)However if we allow E (εt | tminus1) = 0 for both non-linear forms gt(γ c) such that the STARmodel does not represent the best L2-predictor it may nonetheless represent the best Lp-predictorfor some p lt 2 and the innovations Zt may be weak andor strong orthogonal in the sense ofSection 3

The decomposition innovations Zt are weakly orthogonal to Ltminus1 if and only if

P (Zt | tminus1) = 0 (see Lemma 1 part ii) which in Lp is true if and only if E (Z〈pminus1〉t | tminus1) = 0

This may be easily tested for any chosen p gt 1 For example Hong amp White (1995) develop anuisance parameter-free consistent non-parametric test of functional form based on the observa-tion that if Ytminus1 equiv E (Z

〈pminus1〉t | tminus1) = 0 then E (Z

〈pminus1〉t Ytminus1) = Y 2

tminus1 gt 0 Essentially any

non-parametric estimator Ytminus1 may be substituted for Ytminus1 including use of Fourier series aflexible Fourier form regression splines etc See Section 54 From p lt α4 + 1 Minkowskirsquosinequality stationarity the fact that |gt(γ c)| le 1 with probability one and (7) it follows that

Zt4(pminus1) le |b|+ (1 + |a|)| times |εt4(pminus1)(1minus φ)minus1 ltinfin

hence E |Z〈pminus1〉t |4 ltinfin and

Z〈pminus1〉t minus E

[Z

〈pminus1〉t | tminus1]4(pminus1) lt 2times Z〈pminus1〉

t 4(pminus1) ltinfin

such that the moment conditions of Hong amp White (1995) are satisfied Along with fairly stan-dard regulatory assumptions the test statistic is simple to compute and is based on the samplemoment nminus1

sumnt=1 Z

〈pminus1〉t Ytminus1 for some plug-in Zt to be detailed below The statistic has an

asymptotic standard normal null distribution A multitude of alternative consistent parametricand non-parametric model specification tests exist see Bierens amp Ploberger (1997) and the cita-tions therein


For strong orthogonality in Lp we need

E

( hsumk=0

πkZt+k

)〈pminus1〉 ∣∣∣ tminus1

= 0 forallh ge 0 forallπ isin IRh

A simple method follows randomly generate π isin IRh for various h = 1 2 perform the

non-parametric test on the resulting(sumh

k=0 πkZt+k

)〈pminus1〉 repeat by generating a large number

of sequences πkhk=0 and subsequent test statistics and average the resulting P -values

In practice an estimated Zt will be used as an obvious plug-in for Zt Because εt is unob-servable simply assume εt = 0 forallt le 0 and x0 = 0 for the ESTAR model eg

x1 = ε1

x2 = φx1 expminusγ(|x1| minus c)2+ ε2

x3 = φx2 expminusγ(|x2 minus φx1 expminusγx2

1| minus c)2

+ ε3

etc Other methods for handling the first period may be considered as well After construct-ing the regressor the threshold model (6) can then be estimated straightforwardly by M -estimation or Lp-GMM for various 1 lt p lt 2 using standard iterative estimation techniques

generating εt and Zt The moment condition is simply E ε〈pminus1〉t partpartθft(θ) = 0 where

ft(θ) = φxtminus1gtminus1(γ c) and θ = (φ γ c) See Arcones (2000) de Jong amp Han (2002)and Han amp de Jong (2004) For brevity we assume all conditions which ensure the Lp-GMMestimator is consistent and asymptotically normally distributed hold see de Jong amp Han (2002)

53 Non-linear impulse response function

Assume Zt forms a sequence of strong orthogonal innovations and set vn = εn FromTheorems 5 and 7 and Lemma 6 the h-step ahead non-linear impulse response functionI(h VnLnminus1) is ψn+hhP (Zn+hn | Lnminus1) hence

I(h VnLnminus1) = ψn+hhP (Zn+hn | Lnminus1)

= φhahminus1[Vngn(γ c)minus b+ gn(γ c)minus a

timesinfinsum

j=1

φjεnminusj

jprodk=1

gnminusk(γ c)]

The response to the ldquosolerdquo random impulse Vn = εn is history εnminusiinfini=1 dependent and asym-metric with respect to the sign of Vn through gn(γ c) See Koop Pesaran amp Potter (1996) forfurther commentary on path dependence in non-linear IRFs

If we use estimated residuals εt generated from estimates φ γ and c and sample estimatorsa = nminus1

sumnt=1 gt(γ c) and b = nminus1

sumnt=1 εtgt(γ c) we obtain a sample IRF based on one

history and one impulse εtnt=1

I(h εn εtnminus1t=1 ) = φhahminus1

[εngn(γ c)minus b+gn(γ c)minus atimes

nminus1sumj=1

φj εnminusj

jprodk=1

gnminusk(γ c)

]

Multiple alternative strategies for handling the random history εnminusiinfini=1 and impulse Vn areavailable For example we may randomly draw a history εnminusinminus1

i=1 and impulse vn fromthe empirical distribution of the sample path εtnt=1 or simulate independent impulses if thefinite distributions of εt are known (eg Pareto stable t normal) We may repeat eithermethod J-times generating sequences of histories andor impulses and sequences of IRFs foreach horizon h Subsequent confidence bands and kernel densities may then be computed


54 Empirical study

Finally we perform a limited empirical application of the STAR model to currency exchangerates We study log returns xt = Δ ln yt to the YenDollar EuroDollar and British-PoundDollardaily spot exchange rates yt for the period extending from January 1 2000 to August 312005 The data were obtained from the New York Federal Reserve Bank statistical releasesObservations with missing values are removed (eg weekends holidays) leaving a sample of1424 daily returns We filter each series through a standard daily dummy regression in order tocontrol for day effects We assume the extreme tails are regularly varying with shape P (xt lt ε)= εminusαL1(ε) and P (xt gt ε) = εminusαL2(ε) where Li(ε) are slowly varying and α gt 0 denotesthe tail index Consult Bingham Goldie amp Teugels (1987) We study the tail shape of each seriesby computing the tail estimator α due to B M Hill (1975) We apply asymptotic theory and aNeweyndashWest-type kernel estimator of the asymptotic variance of α for dependent heterogenousprocesses see J B Hill (2005bc) Consult J B Hill (2005b) for a method for determining thesample tail fractile for computing the Hill estimator

The STAR models are estimated by identity matrix weighted Lp-GMM for p = 11 and 15500 estimated decomposition innovations Zt are computed according to

Zt = εtgt(γ c)minus b+ gt(γ c)minus a timesnminus1sumj=1

φj εtminusj timesjprod

k=1

gtminusk(γ c)

and the sample IRF I(h VnLnminus1) is computed accordinglyWe use the non-parametric method of Hong amp White (1995) to test the daily returns xt

for evidence of E (xt | tminus1) = 0 and P (xt | tminus1) = 0 (ie E (x〈pminus1〉t | tminus1) = 0) We also

test E (εt |Ltminus1) = 0 and E (Zt |Ltminus1) = 0 for evidence the STAR model represents the bestL2-predictor We then test the STAR residuals εt and the estimated decomposition innovationsZt for evidence of weak and strong orthogonality by using ε〈pminus1〉

t and Z〈pminus1〉t For tests

of strong orthogonality we use the series length h = 10 and 20 and randomly select π isin IRhFor a non-parametric estimator of the conditional mean of ε〈pminus1〉

t and Z〈pminus1〉t we exploit

Corollary 1 of Bierens (1990) which states the conditional mean of any YtE )Y 2t ) ltinfin satisfies

the Fourier series expansion (Bierens 1990)

Y lowasttminus1 equiv E (Yt | tminus1) = θ0 +

infinsumi=1

θi exp

( infinsumj=1

τjxtminusj

)

with probability one for some sequence θi We use

Y lowasttminus1 = θ0 +

100sumi=1

θi exp

(tminus1sumj=1

τjxtminusj

)

where the τj are randomly selected from IR We repeat the HongndashWhite test for 100 randomlyselected values of π and 100 randomly selected values of τ (10000 repetitions) and report theaverage P -value

The test of the hypothesis E (xt | tminus1) = 0 due to Hong amp White (1995) requires E (x4t ) lt

infin which likely fails to hold for each exchange rate return xt (see Table 1) Similarly the finite-variance assumptions of Bierens (1990) may not hold for xt in particular for the Yen Thus somecaution should be taken when interpreting a test of this hypothesis The same is true for tests ofE (εt | tminus1) = 0 and E (Zt | tminus1) = 0 The test of P (xt | tminus1) = 0 however requiresE (x

〈pminus1〉t )4 lt infin which we assume holds for small enough p lt 2 The smallest estimated tail

index is 255 plusmn70 (for the Yen) and p lt 2554 + 1 = 16375 is satisfied when p = 11 or 15The lower bound of the 95 interval is 185 and 1854 + 1 = 14875 hence estimation and


test results for the Yen when p = 15 should be interpreted with some caution The precedingdiscourse identically applies to tests of weak and strong orthogonality of εt and Zt

TABLE 1 Lp-GMM estimates (p = 15) and tail indices

ESTAR

Parameter Yen Euro BP

φ minus089 (027)a 296 (073) 099 (026)

γ 233 (137) 430 (119) 155 (757)

c 013 (004) 015 (003) 020 (009)

max(|εt|) 031 025 021

LSTAR


φ minus093 (053) minus043 (814) 130 (107)

γ 407 (267) 683 (159) 672 (291)

c 003 (001) 001 (003) 020 (000)

max(|εt|) 030 027 021

Yen Euro BP

E (xt | tminus1) = 0b 058 096 020

P(xt | tminus1) = 0c 061 072 097

min(xt) minus030 minus025 minus019

max(xt) 025 027 021

αm 255plusmn70d 337plusmn120 296plusmn103

a Parameter estimate (heteroscedasticity robust standard error)

b The test of Hong White (1995) for conditional mean mis-specification values denote p-values

c The null is E(x〈pminus1〉

t| tminus1) = 0 ie P (xt | tminus1) = 0

d Tail index estimator and 95 interval length based on a Newey-West kernel estimator with Bartlett kernel

see Hill (2005b)

Tables 1 and 2 contain tail index estimates Lp-GMM parameter estimates of the STAR mod-els and HongndashWhite test results We comment only on the most pertinent results with respect tothe concerns of this paper First only the Yen provides unambiguous evidence for heavy-tails forthe Yen Euro and Pound the tail estimates and 95 interval widths are respectively 255plusmn 70337plusmn 120 and 296plusmn 103 Second for brevity we omit all results concerning the case p = 11because parameter estimates are uniformly insignificant Third significant evidence suggests thedaily log return xt has some form of unmodeled (non)linear structure E (xt | tminus1) = 0 andP (xt | tminus1) = 0 when p = 15 (see Table 1)

Fourth when the LSTAR model is used it is only the Pound for which evidence sug-gests both E (εt | tminus1) = 0 and E (Zt | tminus1) = 0 (Tests 1 and 3 of Table 2) as well asP (εt+k | tminus1) = 0 P (Zt+k | tminus1) = 0 and P

(sumhk=0 πkZt+k|tminus1

)= 0 when p = 15

(Tests 2 4ndash6 of Table 2) Thus the LSTAR model reasonably articulates the best L15-predictorof the Pound resulting in strong orthogonal decomposition innovations in L15 Theorem 3 canbe immediately used to justify iterated projections for the Pound based on an LSTAR best L15-predictor and Theorem 5 and Lemma 6 justify the existence of a non-linear IRF Finally for


the LSTAR model of the Pound notice the estimated threshold c is 02 and the largest shockin absolute value is 021 coupled with the large estimated scale (γ = 672) the LSTAR modelsuggests extremes of the Pound are persistent non-extremes are noisy and regime change occursquickly matching evidence found in Hill (2005b)

TABLE 2 HongndashWhite testsa

ESTAR LSTAR

Test Null hypothesis hd Yen Euro BP Yen Euro BP

1 E (εt | tminus1) = 0 - 264 096 384 258 078 100

2 P (εt+k | tminus1) = 0b - 941 091 540 582 088 126

3 E (Zt | tminus1) = 0 - 099 000 249 188 197 109

4 P (Zt+k | tminus1) = 0 - 161 813 159 401 465 217

5 P (P

h

k=0πkεt+k | tminus1) = 0c 10 191 127 799 075 199 159

20 233 108 100 076 702 386

6 P (P

h

k=0πkZt+k | tminus1) = 0 10 091 831 292 937 351 179

20 114 339 119 332 332 376

a The P -value of the test of Hong amp White (1995)

b The null is E(ε〈pminus1〉

t+k| tminus1) = 0

c The null is E(P

h

k=0πkεt+k)〈pminus1〉 | tminus1 = 0

d The ldquohrdquo inP

h

k=0πkεt+k

FIGURE 1 ESTAR IRF kernel densities (randomized impulses) ldquohrdquo reflects the number of steps ahead

Figures 1 and 2 display (Gaussian) kernel density functions of a sequence of non-linear andlinear IRFs We use the estimated residuals εtnminus1

t=1 as one ldquodrawrdquo for the history and randomlydraw 500 independent and identically distributed impulses vnj500j=1 from a Pareto distributionP (vnj gt v) = vminusα and P (vnj lt minusv) = (minusv)minusα v gt 0 where α is used as a plug-infor α Sequences of IRFs are then generated for horizons h = 0 and 1 (when h ge 2 nearlyall of the probability mass of the IRFs occurs at zero) Figure 1 plots kernel densities of theIRFs for each h for the British Pound based on the LSTAR model A prominent characteristicis the extremely heavy-tailed nature of the IRF empirical distribution the estimated tail index ofI(0 vnj εtnminus1

t=1 )500j=1 is 69plusmn 15 due simply to the non-linear multiplicative presence of theshock history εtnminus1

t=1 By comparison a linear AR(1) IRF is φhεn which is Pareto distributedwith index α = 296 The estimated tail index of the sequence of linear IRFs is 324plusmn 30 (whichcontains 296) See Figure 2


FIGURE 2 AR(1) IRF kernel densities (randomized impulses)

APPENDIX

Proof of Theorem 3 By Lemma A1 below for arbitrary integer k gt 0 the finite decompositionholds

Bn =

(kminus1sumi=0

Nnminusi

)+ Bnminusk Nt equiv Bt minus Pttminus1Bt

Consider an arbitrary element Xn isin Bn and define the sequences Ynk isinsumkminus1

i=0 Nnminusi and Vnk

isin Bnminusk such thatXn = Ynk + Vnk

By orthogonality Vnkperpsumkminus1

i=0 Nnminusi norm-boundedness Xn lt infin and the triangular in-equality for any k ge 1

Vnk le Vnk + Ynk = Xn ltinfin

Ynk = Xn minus Vnk le 2Xn ltinfin

Because the sequences Ynk and Vnk are norm bounded in a reflexive Banach space Bnthey have simultaneously weakly convergent subsequences say Ynki

and Vnki see the

BolzanondashWeierstrass Theorem In particular define the stochastic limits as

limkirarrinfin

Ynki= Yn lim

kirarrinfinVnki

= Vn

Now because the equality Xn = Ynk + Vnk holds for any integer k gt 0 and the sequencesYnki

isin sumkiminus1i=0 Nnminusi and Vnki

isin Bnminuskiconverge we deduce by continuity for arbitrary

Xn isin Bn Xn = Yn +Vn where clearly Yn isinsuminfin

i=0 Nnminusi and Vn isin Bminusinfin Because Xn isin Bn

is arbitrary (4) is provedThe proof of (i) hArr (ii) follows in a manner identical to the line of proof of Lemma A1

belowFinally the claim that every element Y isin oplusinfin

i=0 Nnminusi obtains a unique norm-convergentexpansion Y =

suminfin

i=0 ξnminusi ξt isin Nt follows from a direct application of Lemma A2 below

LEMMA A1 For any Banach space Bn there exists a sequence of subspaces Nnminusikminus1i=0

Nt sube Bt such that Bn =sumkminus1

i=0 Nnminusi + Bnminusk where Nnminusikminus1i=0 are weak orthogonal in that

NnminusiperpBnminusiminus1 and NnminusiperpNnminusj for every 0 le i lt j le k minus 1 Moreover the following areequivalent for any integer k gt 0 (i) Bn =

(opluskminus1i=0 Nnminusi

) oplus Bnminusk and (ii) PttminusPttminusk =Pttminus for every t k le


LEMMA A2 Consider a sequence of orthogonal subspaces Mnminusiinfini=0 Mtminus1 sube Mt sube Btsuch that

oplusinfin

i=0 Mnminusi exists and consider a sequence of elements xj xj isin Mj The spacext xt isin Mt forms a Schauder basis for its closed linear span Consequently every elementX isin oplusinfin

i=0 Mnminusi obtains a unique norm convergent expansion X =suminfin

i=0 aixnminusi xt isin Mtfor some sequence of real constants at

Proof of Lemma A1 Define the sequence Nt Nt equiv Btminus Pttminus1Bt where NtperpBtminus1(see Lemma 2 part i) and Pttminus1Bt = Btminus1 due to Btminus1 sub Bt Because Banach spacesare linear and Pttminus1Bt = Btminus1 sub Bt we deduce Nt sube Bt We obtain the tautologicalexpression

Bn = Nn + Pnnminus1Bn

Recursively decomposing Bnminus1 etc it follows that for arbitrary k ge 1

Bn = Nn + Pnnminus1Bn = Nn + Bnminus1

= Nn + Nnminus1 + Bnminus2 = middot middot middot

=kminus1sumi=0

Nnminusi + Bnminusk

where for each t le n NtperpBtminus1 Observe that given the orthogonality property NnminusiperpBnminusiminus1

and Nnminusj sube Bnminusj sube Bnminusi for every 0 le i lt j le k it follows that NnminusiperpNnminusj 0 le i ltj le k

Assume (i) holds Then for any k ge 1 Bn =(opluskminus1

i=0 Nnminusi

)oplusBnminusk hence we may writefor any Xn isin Bn

Xn =kminus1sumi=0

ξnminusi + Vnk

where ξi isin Ni and Vnk isin Bnminusk Thus by quasi-linearity (see Lemma 2 part iv) we deducefor any 0 le t le k

PnnminustXn = Pnnminust

(kminus1sumi=0

ξnminusi + Vnk

)

= Pnnminust

(tminus1sumi=0

ξnminusi

)+

kminus1sumi=t

ξnminusi + Vnk

=

kminus1sumi=t

ξnminusi + Vnk

where Pnnminust

(sumtminus1i=0 ξnminusi

)= 0 (see Lemma 2 part ii) due to

sumtminus1i=0 ξnminusi isin

oplustminus1i=0 Nnminusi andoplustminus1

i=0 NnminusiperpBnminust by assumption Similarly for any 1 le s le t le k

PnnminustPnnminussXn = PnnminustPnnminuss

(kminus1sumi=0

ξnminusi + Vnk

)

= Pnnminust

(Pnnminuss

sminus1sumi=0

ξnminusi +

kminus1sumi=s

ξnminusi + Vnk

)

= Pnnminust

(kminus1sumi=s

ξnminusi + Vnk

)


= Pnnminust

(tminus1sumi=s

ξnminusi

)+

kminus1sumi=t

ξnminusi + Vnk

=

kminus1sumi=t

ξnminusi + Vnk

This proves PnnminustXn = PnnminustPnnminussXn for arbitrary Xn in Bn and any s t such that 0 les le t le k Because Xn isin Bn is arbitrary we deduce the operators satisfy PnnminustPnnminuss =Pnnminust for any 0 le s le t le k hence (i)rArr (ii)

Next assume (ii) holds It suffices to prove for any k gt 0 the subspaces Nnminusikminus1i=0 and

Bnminusk are strong orthogonal such that for every 1 le j lt k

jminus1sumi=0

Nnminusiperpkminus1sumi=j

Nnminusi

kminus1sumi=0

NnminusiperpBnminusk

Consider any elementsumkminus1

i=0 ξnminusi isinsumkminus1

i=0 Nnminusi By iterated projections PnnminustPnnminuss =Pnnminust for arbitrary 0 le s le t le k

Pnnminusk

(kminus1sumi=0

ξnminusi

)= PnnminuskPnnminus1

(kminus1sumi=0

ξnminusi

)

= Pnnminusk

(Pnnminus1ξn +

kminus1sumi=1

ξnminusi

)

= Pnnminusk

(kminus1sumi=1

ξnminusi

)

where Pnnminus1ξn = 0 (see Lemma 2 part ii) due to ξn isin Nn and NnperpBnminus1 by constructionProceeding with subsequent Pnnminush h = 1 k we obtain

Pnnminusk

(kminus1sumi=0

ξnminusi

)= Pnnminusk

(kminus1sumi=1

ξnminusi

)

= Pnnminusk

PBnminus2

(kminus1sumi=1

ξnminusi

)

= Pnnminusk

(PBnminus2

ξnminus1 +

kminus1sumi=2

ξnminusi

)

= Pnnminusk

(kminus1sumi=2

ξnminusi

)= middot middot middot = 0

Thereforesumkminus1

i=0 ξnminusiperpBnminusk (see Lemma 2 part ii) for any integer k gt 0 and any ele-ments ξt isin Nt Because the elements ξt isin Nt are arbitrary we deduce for every k gt 0sumkminus1

i=0 NnminusiperpBnminusk

Finally becausesumkminus1

i=j Nnminusi is a subspace of Bnminusj for any 1 le j le k we conclude

thatsumjminus1

i=0 Nnminusiperpsumkminus1

i=j Nnminusi It follows thatsumkminus1

i=0 Nnminusi =opluskminus1

i=0 Nnminusi and Bn =(opluskminus1i=0 Nnminusi

)oplusBnminusk which proves (ii)rArr (i)

Proof of Lemma A2 Consider an arbitrary sequence of elements xj xj isin Mj and recalloplusinfin

i=0 Mnminusi exists A necessary and sufficient condition for a sequence of Banach space ele-ments xj to form a Schauder basis (hereafter referred to as a basis) is the existence of some


scalar constant 0 lt K lt infin such that for all scalar real-valued sequences λj and integerss le t ∥∥∥∥ ssum

i=0

λjxnminusj

∥∥∥∥ le K

∥∥∥∥ tsumi=0

λjxnminusj

∥∥∥∥ (8)

See eg Proposition 4124 of Megginson (1998) see also Singer (1970) In our case by theexistence of the space

oplusinfin

i=0 Mnminusi the subspaces Mj are strong orthogonal by construction forany s lt t it follows that

soplusi=0

Mnminusiperptoplus

i=s+1

Mnminusi

Synonymously for all scalar real-valued sequences λj and components xj isinMj

ssumi=0

λjxnminusjperptsum

i=s+1

λjxnminusj (9)

By the definition of James orthogonality it follows from (9) that∥∥∥∥ ssumi=0

λjxnminusj + a

tsumi=s+1

λjxnminusj

∥∥∥∥ ge ∥∥∥∥ ssumi=0

λjxnminusj

∥∥∥∥for all real scalars a For a = 1 we conclude that for all s lt t∥∥∥∥ tsum

i=0

λjxnminusj

∥∥∥∥ =

∥∥∥∥ ssumi=0

λjxnminusj +

tsumi=s+1

λjxnminusj

∥∥∥∥ ge ∥∥∥∥ ssumi=0

λjxnminusj

∥∥∥∥The inequality in (8) follows with K = 1 which proves the result

Proof of Theorem 5 By assumption xt =suminfin

i=0 ψtiZttminusi where the Zttminusi are strong orthog-onal to Ltminusiminus1 and the projection operator iterates from Ltminusk to Ltminuskminus1 oplus vtminusk for any t andk ge 0 Project xt separately onto Ltminus1 and Ltminus1 using quasi-linearity and homogeneity andobserving Lt sube Lt P

(xt |Ltminus1

)=suminfin

i=1 ψtiZttminusi and

P (xt | Ltminus1) =infinsum

i=1

ψtiZttminusi + ψt0P (Ztt | Ltminus1)

hence the 0-step ahead non-linear impulse response I(0 vtLtminus1) at time t is simply

I(0 vtLtminus1) = ψt0P (Ztt | Ltminus1)

For the 1-step ahead impulse response I(1 vtLtminus1) by strong orthogonality and quasi-linearity

I(1 vtLtminus1) = P (xt+1 | Ltminus1)minus P(xt+1 |Ltminus1

)=

infinsumi=2

ψt+1iZt+1t+1minusi + P(Zt+1t+1 + ψt+11Zt+1t | Ltminus1

)minus

infinsumi=2

ψt+1iZt+1tminusi

= P(Zt+1t+1 + ψt+11Zt+1t | Ltminus1

)


By iterated projections quasi-linearity homogeneity and orthogonality

P(Zt+1t+1 + ψt+11Zt+1t | Ltminus1

)= P

P(Zt+1t+1 + ψt+11Zt+1t |Lt

) | Ltminus1

= P

ψt+11Zt+1t + P

(Zt+1t+1 |Lt

) | Ltminus1

= ψt+11P (Zt+1t | Ltminus1)

Similarly using iterated projections and Ltminus1 sube Lt sube Lt+1 the 2-step ahead impulse responsefunction is

I(2 vtLtminus1) = P(xt+2 | Ltminus1

) minus P(xt+2 |Ltminus1

)=

infinsumi=3

ψt+2iZt+2t+2minusi

+ P(Zt+2t+2 + ψt+21Zt+2t+1 + ψt+22Zt+2t | Ltminus1

)minus

infinsumi=3

ψt+2iZt+2t+2minusi

= P(Zt+2t+2 + ψt+21Zt+2t+1 + ψt+22Zt+2t | Ltminus1

)= P

[PP(Zt+2t+2 + ψt+21Zt+2t+1 + ψt+22Zt+2t |Lt+1

) |Lt

| Ltminus1

]= ψt+22P

(Zt+2t | Ltminus1

)

and so on Therefore I(h vtLtminus1) = ψt+hhP (Zt+ht | Ltminus1)

Proof of Lemma 6 Clearly Ltminuskminus1 = Ltminuskminus1 oplus εtminusk = Ltminusk Hence

PP (xt |Ltminusk)|Ltminuskminus1

= PP (xt |Ltminusk) |Ltminusk

= P (xt |Ltminusk) = P (xt | Ltminuskminus1)

If additionally xt admits a strong orthogonal decomposition then by Theorem 3 and the identityLtminuskminus1 = Ltminuskminus1 oplus εtminusk = Ltminusk we deduce


= PP (xt |Ltminusk+1) |Ltminusk

= P (xt |Ltminusk)

ACKNOWLEDGEMENTS

The author would like to thank an anonymous referee an Associate Editor and the Editor for their commentsand suggestions that led to substantial improvements All errors of course are the authorrsquos alone Anearlier version of this paper was written while the author was a visitor at the Department of EconomicsUniversity of California at San Diego whose kind hospitality is gratefully acknowledged Finally thispaper is dedicated to the memory of the authorrsquos father Walter Howard Hill (1922ndash1996)

REFERENCES

M A Arcones (2000) M-Estimators converging to a stable limit Journal of Multivariate Analysis 74193ndash221

B Basrak R A Davis amp T Mikosch (2001) Regular Variation and GARCH Processes Unpublishedrepport Forsikringsmatematisk Laboratorium Koslashbenhavns Universitet Copenhagen Denmark

B Basrak R A Davis amp T Mikosch (2002) A characterization of multivariate regular variation TheAnnals of Applied Probability 12 908ndash920

H J Bierens (1990) A consistent conditional moment test of functional form Econometrica 58 1443ndash1458

H J Bierens amp W Ploberger (1997) Asymptotic theory of integrated conditional moment tests Econo-metrica 65 1129ndash1151

N H Bingham C M Goldie amp J L Teugels (1987) Regular Variation Cambridge University Press


S Cambanis D Hardin amp A Weron (1988) Innovations and Wold decompositions of stable sequencesProbability Theory and Related Fields 79 1ndash27

Y W Cheung (1993) Long memory in foreign exchange rates Journal of Business and Economic Statis-tics 11 93ndash101

D B H Cline (1983) Estimation and Linear Prediction for Regression Autoregression and ARMA withInfinite Variance Data Unpublished doctoral dissertation Department of Statistics Colorado StateUniversity Fort Collins CO

R A Davis amp S Resnick (1985a) Limit theory for moving averages of random variables with regularlyvarying tail probabilities The Annals of Probability 13 179ndash195

R A Davis amp S Resnick (1985b) More limit theory for the sample correlation-function of moving aver-ages Stochastic Processes and Their Applications 20 257ndash279

R de Jong amp C Han (2002) Properties of Lp-GMM estimators Econometric Theory 18 491ndash504R Engle amp V Ng (1993) Measuring and testing the impact of news on volatility Journal of Finance 48

1749ndash1778B Falk amp C Wang (2003) Testing long-run PPP with infinite-variance returns Journal of Applied Econo-

metrics 18 471ndash484G D Faulkner amp J E Huneycutt Jr (1978) Orthogonal decompositions of isometries in a Banach Space

Proceedings of the American Mathematical Society 69 125ndash128P H Franses amp D van Dijk (2000) Non-linear Time Series Models in Empirical Finance Cambridge

University PressR Gallant P Rossi amp G Tauchen (1993) Nonlinear dynamic structures Econometrica 61 871ndash907J R Giles (1967) Classes of semi-inner-product spaces Transactions of the American Mathematical

Society 129 436ndash446J R Giles (2000) Introduction to the Analysis of Normed Linear Spaces Cambridge University PressG Gonzalez-Rivera (1998) Smooth-transition GARCH models Studies in Nonlinear Dynamics and

Econometrics 3 61ndash78C Gourieroux amp J Jasiak (2003) Nonlinear Innovations and Impulse Responses with an Application to

VaR Sensitivity Unpublished report Centre de recherche en economie et statistique Institut nationalde la statistique et des etudes economiques Paris France

C Han amp R de Jong (2004) Closest moment estimation under general conditions Annales drsquoeconomie etde statistique 74 1ndash13

C Hardin G Samorodnitsky amp M Taqqu (1991) Non-linear regression of stable random variables TheAnnals of Applied Probability 1 582ndash612

B M Hill (1975) A simple general approach to inference about the tail of a distribution The Annals ofMathematical Statistics 3 1163ndash1174

J B Hill (2005a) Strong Orthogonal Decompositions and Projection Iterations in Banach Space Availableat httpwwwfiuedu hilljonastrong orthog banach wppdf Department of Economics Florida Interna-tional University Miami

J B Hill (2005b) Gaussian Tests of ldquoExtremal White Noiserdquo for Dependent Heterogeneous Heavy TailedTime Series with an Application Available at httpwwwfiuedu hilljonaco-relation-testpdf Depart-ment of Economics Florida International University Miami

J B Hill (2005c) On Tail Index Estimation Using Dependent Heterogenous Data Unpublished reportavailable at httpeconwpawustledu80epsempapers05050505005pdf Department of EconomicsFlorida International University Miami

M C A B Hols amp C G de Vries (1991) The limiting distribution of extremal exchange rate returnsJournal of Applied Econometrics 6 287ndash302

Y Hong amp H White (1995) Consistent specification testing via nonparametric series regression Econo-metrica 63 1133ndash1159

R C James (1947) Orthogonality and linear functionals in a normed linear space Transactions of theAmerican Mathematical Society 61 265ndash292

G K Kees amp C J M Kool (1992) Tail estimates of East European exchange rates Journal of Businessand Economic Statistics 10 83ndash96

P S Kokoszka amp M S Taqqu (1994) Infinite variance stable ARMA processes Journal of Time SeriesAnalysis 15 203ndash220


P S Kokoszka amp M S Taqqu (1996) Infinite variance stable moving averages with long memory Journalof Econometrics 73 79ndash99

G Koop M H Pesaran amp S Potter (1996) Impulse response analysis in nonlinear multivariate modelsJournal of Econometrics 4 119ndash147

W L Lin (1997) Impulse response function for conditional volatility in GARCH models Journal ofBusiness and Economic Statistics 15 15ndash25

J Lindenstrauss amp L Tzafriri (1977) Classical Banach Spaces Springer New YorkM Loretan amp P C B Phillips (1994) Testing the covariance stationarity of heavy-tailed economic time

series An overview of the theory with applications to financial data sets Journal of Empirical Finance1 211ndash248

S Lundbergh T Terasvirta amp D van Dijk (2003) Time-varying smooth transition autoregressive modelsJournal of Business and Economic Statistics 21 104ndash121

S Lundbergh amp T Terasvirta (2006) A time series model for an exchange rate in a target zone withapplications Journal of Econometrics 131 579ndash609

D McMillan amp A Speight (2002) Nonlinear dependence in inter-war exchange rates Some further evi-dence Applied Economics Letters 9 359ndash364

R Megginson (1998) An Introduction to Banach Space Theory Springer New YorkA G Miamee amp M Pourahmadi (1988) Wold decompositions prediction and parameterization of station-

ary processes with infinite variance Probability and Related Fields 79 145ndash164P Micheal A R Nobay amp D A Peel (1997)Transaction costs and nonlinear adjustment in real exchange

rates An empirical investigation Journal of Political Economy 105 862ndash879T Mikosch amp C Starica (2000) Limit theory for the sample autocorrelations and extremes of a GARCH

(1 1) process The Annals of Statistics 28 1427ndash1451P C B Phillips J McFarland amp P McMahon (1996) Robust tests of forward exchange market efficiency

with empirical evidence from the 1920s Journal of Applied Econometrics 11 1ndash22P Saikkonen amp R Luukkonen (1988) Lagrange multiplies tests for testing nonlinearities in time series

models Scandinavian Journal of Statistics 15 55ndash68C Sims (1980) Macroeconomics and reality Econometrica 48 1ndash48I Singer (1970) Bases in Banach Spaces Springer New YorkT Terasvirta (1994) Specification estimation and evaluation of smooth transition autoregressive models

Journal of the American Statistical Association 89 208ndash218H Tong (1990) Nonlinear Time Series A Dynamical System Approach Oxford University PressK Urbanic (1964) Prediction of strictly stationary sequences Colloquium Mathematicum 12 115ndash129K Urbanic (1967) Some prediction problems for strictly stationary processes In Proceedings of the Fifth

Berkeley Symposium on Mathematical Statistics and Probability June 21ndashJuly 18 1965 December 271965ndashJanuary 7 1966 (L M Le Cam amp J Neyman eds) University of California Press BerkeleyVol II Contributions to Probability Part 1 pp 235ndash258

D van Dijk T Terasvirta amp P H Franses (2000) Smooth Transition Autoregressive ModelsmdashA Surveyof Recent Developments Econometric Institute Research Report EI2000-23A Erasmus UniversiteitRotterdam The Netherlands

H O A Wold (1938) A Study in the Analysis of Stationary Time Series Uppsala Almqvist and Wiksell

Received 31 July 2004 Jonathan B HILL jonathanhillfiuedu

Accepted 15 December 2005 Department of EconomicsFlorida International University

University Park DM-307b Miami FL 33199 USA


for some set of orthogonal innovations Zt and a ldquoresidualrdquo Vn The seminal work of Wold(1938) provides a foundation for characterizing stationary finite-variance processes with covari-ance orthogonal innovations In the classic setting the innovations necessarily satisfy the strongorthogonality condition

sp(Zt+i Zt)perp sp(Xtminus1 Xtminusj) foralli ge 0 forallj ge 1 (2)

where sp denotes the closed linear span In general Banach spaces however the ldquocovariancerdquomay not exist conditional expectations and the best predictor may not equate metric projectionoperators need not be linear and innovations may not satisfy (2) although they will satisfy (3)below A related decomposition theory with strong orthogonal innovations for processes withan unbounded variance or for any process based on metric projection other than minimizing themean squared error is relatively limited and the most promising contributions to the literaturefocus entirely on closed linear spans

Let spt = sp (Xs s le t) denote the closed linear span of Xt and denote by Pt a metricprojection operator (eg Ptminus1 spt rarr sptminus1) Urbanic (1964 1967) considered decomposi-tions of strictly stationary infinite-variance processes which admit independent metric projectioninnovations Faulkner amp Huneycutt (1978) consider decompositions with innovations Zt thatonly satisfy a weak asymmetric orthogonality condition

sp (Zt)perp sp (Xtminus1 Xtminusj) forallj ge 1 (3)

Miamee amp Pourahmadi (1988) develop a weak-orthogonal decomposition theory for p-stationaryprocesses based on innovations in spt minus Ptminus1spt The theory however fundamentally exploitsthe codimension one property of closed linear spans spt = sp (Xt sptminus1)

Similarly Cambanis Hardin amp Weron (1988) establish an asymmetric decomposition theoryfor Lp(ΩFt μ) processes in spt The authors prove (i) projection operator linearity (ii) iteratedprojections and (iii) the existence of strong orthogonal innovations are equivalent when theinnovations are restricted to the space spt minus Ptminus1spt Specifically the authors prove (i)rArr (iii)rArr (ii)rArr (i) and operator linearity is expedited by the fact that sptminus1 is codimension one in spt

In the literature therefore either independence is assumed only weak orthogonality isproven or explicit properties of closed linear spans are exploited to promote strong orthogo-nality For projections into arbitrary (non-linear) Lp-spaces Cambanis Hardin amp Weron (1988)point out that operator linearity sufficiently renders strong orthogonal innovations (2) This re-sult however is trivial see Theorem 3 below Moreover no result exists (that we know of)characterizing strong orthogonal decompositions for finite-variance processes based on best Lp-metric projection p lt 2

The construction and use of orthogonal innovation spaces sptminusPtminus1spt in order to promotestrong orthogonality is not a trivial simplification however The explicit omission of non-linearbest predictors and orthogonal innovations must be viewed critically in light of developments inthe theory and empirical methods associated with non-linear stochastic processes For examplemoving average forms have been utilized to characterize linear dependence within processeswith regularly varying tails see Davis amp Resnick (1985ab) and Kokoszka amp Taqqu (19941996) The innovations in this literature are typically assumed to be independent and identicallydistributed hence strongly orthogonal to far more than subspaces of spt Except for the specialcase of symmetric stable process (Cambanis Hardin amp Weron 1988) nowhere in this literatureare necessary and sufficient conditions for the existence of such moving averages derived

Moreover we see the use of moving averages a la IRFs in time series settings in whichamassed evidence suggests non-linear data generating processes with heavy tails See egHols amp de Vries (1991) Cheung (1993) Gallant Rossi amp Tauchen (1993) Phillips McFar-land amp McMahon (1996) Lin (1997) Mikosch amp Starica (2000) Falk amp Wang (2003) andHill (2005b) In the economics and finance literatures the implied ldquoimpulsesrdquo are predominantlyassumed to be independent and identically distributed finite-variance innovations computed from












Bt F=⋃tisinZ

Ft




i=1 Zi Zi isin Si




i=0 Si where

minus1sumi=0


Snminusi












P (u |V1) |V0







ni=1 with tail


3 MAIN RESULTS


Bminusinfin =⋂n

Bn B+infin =⋃n

Bn



Bn =

( infinsumi=0

Nnminusi

)+ Bminusinfin (4)


(i) Bn =(oplusinfin

i=0 Nnminusi







suminfin



sumi=k

ξnminusi isinsum

i=k



Pnnminusminus1

sumi=k

ξnminusi =

sumi=k


hencesum










Xn =

infinsumi=0

ψniZnminusi + Vn






ψni =[Znminusi Xn]


iminus1sumj=0


[Znminusi Znminusi]




(ΩFt μ

) Essentially


) and E





]= 0 for every







suminfin





suminfin

















= P (xt |Ltminusk)



















suminfin

i=0 φiεtminusi




j=i+1

φjminusiεtminusj

jprodk=i+1

gtminusk(γ c)



where







i=0

φi+hai+hminus1[



times

infinsumj=i+1

φjminusiεtminusj

jprodk=i+1

gtminusk(γ c)]


For any t write


j=1

φjεtminusj

jprodk=1

gtminusk(γ c) (7)












[Z




sumnt=1 Z





E

( hsumk=0

πkZt+k





k=0 πkZt+k




x1 = ε1



1| minus c)2

+ ε3








timesinfinsum

j=1

φjεnminusj

jprodk=1

gnminusk(γ c)]








nminus1sumj=1

φj εnminusj

jprodk=1

gnminusk(γ c)

]




54 Empirical study





k=1

gtminusk(γ c)










infinsumi=1

θi exp

( infinsumj=1

τjxtminusj

)



100sumi=1

θi exp

(tminus1sumj=1

τjxtminusj

)









ESTAR


φ minus089 (027)a 296 (073) 099 (026)

γ 233 (137) 430 (119) 155 (757)

c 013 (004) 015 (003) 020 (009)

max(|εt|) 031 025 021

LSTAR


φ minus093 (053) minus043 (814) 130 (107)

γ 407 (267) 683 (159) 672 (291)

c 003 (001) 001 (003) 020 (000)

max(|εt|) 030 027 021

Yen Euro BP

E (xt | tminus1) = 0b 058 096 020

P(xt | tminus1) = 0c 061 072 097


max(xt) 025 027 021







see Hill (2005b)




)= 0 when p = 15





ESTAR LSTAR


1 E (εt | tminus1) = 0 - 264 096 384 258 078 100

2 P (εt+k | tminus1) = 0b - 941 091 540 582 088 126

3 E (Zt | tminus1) = 0 - 099 000 249 188 197 109

4 P (Zt+k | tminus1) = 0 - 161 813 159 401 465 217

5 P (P

h

k=0πkεt+k | tminus1) = 0c 10 191 127 799 075 199 159

20 233 108 100 076 702 386

6 P (P

h

k=0πkZt+k | tminus1) = 0 10 091 831 292 937 351 179

20 114 339 119 332 332 376



t+k| tminus1) = 0

c The null is E(P

h



h

k=0πkεt+k








APPENDIX


Bn =

(kminus1sumi=0

Nnminusi










and Vnki see the


limkirarrinfin

Ynki= Yn lim

kirarrinfinVnki

= Vn









suminfin










oplusinfin









=kminus1sumi=0

Nnminusi + Bnminusk




i=0 Nnminusi


Xn =kminus1sumi=0

ξnminusi + Vnk



(kminus1sumi=0

ξnminusi + Vnk

)

= Pnnminust

(tminus1sumi=0

ξnminusi

)+

kminus1sumi=t

ξnminusi + Vnk

=

kminus1sumi=t

ξnminusi + Vnk

where Pnnminust







(kminus1sumi=0

ξnminusi + Vnk

)

= Pnnminust

(Pnnminuss

sminus1sumi=0

ξnminusi +

kminus1sumi=s

ξnminusi + Vnk

)

= Pnnminust

(kminus1sumi=s

ξnminusi + Vnk

)


= Pnnminust

(tminus1sumi=s

ξnminusi

)+

kminus1sumi=t

ξnminusi + Vnk

=

kminus1sumi=t

ξnminusi + Vnk




jminus1sumi=0


Nnminusi

kminus1sumi=0





Pnnminusk

(kminus1sumi=0

ξnminusi


(kminus1sumi=0

ξnminusi

)

= Pnnminusk

(Pnnminus1ξn +

kminus1sumi=1

ξnminusi

)

= Pnnminusk

(kminus1sumi=1

ξnminusi

)


Pnnminusk

(kminus1sumi=0

ξnminusi

)= Pnnminusk

(kminus1sumi=1

ξnminusi

)

= Pnnminusk

PBnminus2

(kminus1sumi=1

ξnminusi

)

= Pnnminusk

(PBnminus2

ξnminus1 +

kminus1sumi=2

ξnminusi

)

= Pnnminusk

(kminus1sumi=2

ξnminusi


Thereforesumkminus1





thatsumjminus1










i=0

λjxnminusj

∥∥∥∥ le K


λjxnminusj

∥∥∥∥ (8)


oplusinfin


soplusi=0

Mnminusiperptoplus

i=s+1

Mnminusi


ssumi=0

λjxnminusjperptsum

i=s+1

λjxnminusj (9)


λjxnminusj + a

tsumi=s+1

λjxnminusj

∥∥∥∥ ge ∥∥∥∥ ssumi=0

λjxnminusj


i=0

λjxnminusj

∥∥∥∥ =


λjxnminusj +

tsumi=s+1

λjxnminusj

∥∥∥∥ ge ∥∥∥∥ ssumi=0

λjxnminusj




(xt |Ltminus1

)=suminfin



i=1






)=

infinsumi=2


)minus

infinsumi=2

ψt+1iZt+1tminusi


)




)= P


) | Ltminus1

= P

ψt+11Zt+1t + P

(Zt+1t+1 |Lt

) | Ltminus1





)=

infinsumi=3

ψt+2iZt+2t+2minusi


)minus

infinsumi=3

ψt+2iZt+2t+2minusi


)= P


) |Lt

| Ltminus1

]= ψt+22P

(Zt+2t | Ltminus1

)









= P (xt |Ltminusk)

ACKNOWLEDGEMENTS


REFERENCES

































































Bt F=⋃tisinZ

Ft




i=1 Zi Zi isin Si




i=0 Si where

minus1sumi=0


Snminusi












P (u |V1) |V0







ni=1 with tail


3 MAIN RESULTS


Bminusinfin =⋂n

Bn B+infin =⋃n

Bn



Bn =

( infinsumi=0

Nnminusi

)+ Bminusinfin (4)


(i) Bn =(oplusinfin

i=0 Nnminusi







suminfin



sumi=k

ξnminusi isinsum

i=k



Pnnminusminus1

sumi=k

ξnminusi =

sumi=k


hencesum










Xn =

infinsumi=0

ψniZnminusi + Vn






ψni =[Znminusi Xn]


iminus1sumj=0


[Znminusi Znminusi]




(ΩFt μ

) Essentially


) and E





]= 0 for every







suminfin





suminfin

















= P (xt |Ltminusk)



















suminfin

i=0 φiεtminusi




j=i+1

φjminusiεtminusj

jprodk=i+1

gtminusk(γ c)



where







i=0

φi+hai+hminus1[



times

infinsumj=i+1

φjminusiεtminusj

jprodk=i+1

gtminusk(γ c)]


For any t write


j=1

φjεtminusj

jprodk=1

gtminusk(γ c) (7)












[Z




sumnt=1 Z





E

( hsumk=0

πkZt+k





k=0 πkZt+k




x1 = ε1



1| minus c)2

+ ε3








timesinfinsum

j=1

φjεnminusj

jprodk=1

gnminusk(γ c)]








nminus1sumj=1

φj εnminusj

jprodk=1

gnminusk(γ c)

]




54 Empirical study





k=1

gtminusk(γ c)










infinsumi=1

θi exp

( infinsumj=1

τjxtminusj

)



100sumi=1

θi exp

(tminus1sumj=1

τjxtminusj

)









ESTAR


φ minus089 (027)a 296 (073) 099 (026)

γ 233 (137) 430 (119) 155 (757)

c 013 (004) 015 (003) 020 (009)

max(|εt|) 031 025 021

LSTAR


φ minus093 (053) minus043 (814) 130 (107)

γ 407 (267) 683 (159) 672 (291)

c 003 (001) 001 (003) 020 (000)

max(|εt|) 030 027 021

Yen Euro BP

E (xt | tminus1) = 0b 058 096 020

P(xt | tminus1) = 0c 061 072 097


max(xt) 025 027 021







see Hill (2005b)




)= 0 when p = 15





ESTAR LSTAR


1 E (εt | tminus1) = 0 - 264 096 384 258 078 100

2 P (εt+k | tminus1) = 0b - 941 091 540 582 088 126

3 E (Zt | tminus1) = 0 - 099 000 249 188 197 109

4 P (Zt+k | tminus1) = 0 - 161 813 159 401 465 217

5 P (P

h

k=0πkεt+k | tminus1) = 0c 10 191 127 799 075 199 159

20 233 108 100 076 702 386

6 P (P

h

k=0πkZt+k | tminus1) = 0 10 091 831 292 937 351 179

20 114 339 119 332 332 376



t+k| tminus1) = 0

c The null is E(P

h



h

k=0πkεt+k








APPENDIX


Bn =

(kminus1sumi=0

Nnminusi










and Vnki see the


limkirarrinfin

Ynki= Yn lim

kirarrinfinVnki

= Vn









suminfin










oplusinfin









=kminus1sumi=0

Nnminusi + Bnminusk




i=0 Nnminusi


Xn =kminus1sumi=0

ξnminusi + Vnk



(kminus1sumi=0

ξnminusi + Vnk

)

= Pnnminust

(tminus1sumi=0

ξnminusi

)+

kminus1sumi=t

ξnminusi + Vnk

=

kminus1sumi=t

ξnminusi + Vnk

where Pnnminust







(kminus1sumi=0

ξnminusi + Vnk

)

= Pnnminust

(Pnnminuss

sminus1sumi=0

ξnminusi +

kminus1sumi=s

ξnminusi + Vnk

)

= Pnnminust

(kminus1sumi=s

ξnminusi + Vnk

)


= Pnnminust

(tminus1sumi=s

ξnminusi

)+

kminus1sumi=t

ξnminusi + Vnk

=

kminus1sumi=t

ξnminusi + Vnk




jminus1sumi=0


Nnminusi

kminus1sumi=0





Pnnminusk

(kminus1sumi=0

ξnminusi


(kminus1sumi=0

ξnminusi

)

= Pnnminusk

(Pnnminus1ξn +

kminus1sumi=1

ξnminusi

)

= Pnnminusk

(kminus1sumi=1

ξnminusi

)


Pnnminusk

(kminus1sumi=0

ξnminusi

)= Pnnminusk

(kminus1sumi=1

ξnminusi

)

= Pnnminusk

PBnminus2

(kminus1sumi=1

ξnminusi

)

= Pnnminusk

(PBnminus2

ξnminus1 +

kminus1sumi=2

ξnminusi

)

= Pnnminusk

(kminus1sumi=2

ξnminusi


Thereforesumkminus1





thatsumjminus1










i=0

λjxnminusj

∥∥∥∥ le K


λjxnminusj

∥∥∥∥ (8)


oplusinfin


soplusi=0

Mnminusiperptoplus

i=s+1

Mnminusi


ssumi=0

λjxnminusjperptsum

i=s+1

λjxnminusj (9)


λjxnminusj + a

tsumi=s+1

λjxnminusj

∥∥∥∥ ge ∥∥∥∥ ssumi=0

λjxnminusj


i=0

λjxnminusj

∥∥∥∥ =


λjxnminusj +

tsumi=s+1

λjxnminusj

∥∥∥∥ ge ∥∥∥∥ ssumi=0

λjxnminusj




(xt |Ltminus1

)=suminfin



i=1






)=

infinsumi=2


)minus

infinsumi=2

ψt+1iZt+1tminusi


)




)= P


) | Ltminus1

= P

ψt+11Zt+1t + P

(Zt+1t+1 |Lt

) | Ltminus1





)=

infinsumi=3

ψt+2iZt+2t+2minusi


)minus

infinsumi=3

ψt+2iZt+2t+2minusi


)= P


) |Lt

| Ltminus1

]= ψt+22P

(Zt+2t | Ltminus1

)









= P (xt |Ltminusk)

ACKNOWLEDGEMENTS


REFERENCES


























































Bt F=⋃tisinZ

Ft




i=1 Zi Zi isin Si




i=0 Si where

minus1sumi=0


Snminusi












P (u |V1) |V0







ni=1 with tail


3 MAIN RESULTS


Bminusinfin =⋂n

Bn B+infin =⋃n

Bn



Bn =

( infinsumi=0

Nnminusi

)+ Bminusinfin (4)


(i) Bn =(oplusinfin

i=0 Nnminusi







suminfin



sumi=k

ξnminusi isinsum

i=k



Pnnminusminus1

sumi=k

ξnminusi =

sumi=k


hencesum










Xn =

infinsumi=0

ψniZnminusi + Vn






ψni =[Znminusi Xn]


iminus1sumj=0


[Znminusi Znminusi]




(ΩFt μ

) Essentially


) and E





]= 0 for every







suminfin





suminfin

















= P (xt |Ltminusk)



















suminfin

i=0 φiεtminusi




j=i+1

φjminusiεtminusj

jprodk=i+1

gtminusk(γ c)



where







i=0

φi+hai+hminus1[



times

infinsumj=i+1

φjminusiεtminusj

jprodk=i+1

gtminusk(γ c)]


For any t write


j=1

φjεtminusj

jprodk=1

gtminusk(γ c) (7)












[Z




sumnt=1 Z





E

( hsumk=0

πkZt+k





k=0 πkZt+k




x1 = ε1



1| minus c)2

+ ε3








timesinfinsum

j=1

φjεnminusj

jprodk=1

gnminusk(γ c)]








nminus1sumj=1

φj εnminusj

jprodk=1

gnminusk(γ c)

]




54 Empirical study





k=1

gtminusk(γ c)










infinsumi=1

θi exp

( infinsumj=1

τjxtminusj

)



100sumi=1

θi exp

(tminus1sumj=1

τjxtminusj

)









ESTAR


φ minus089 (027)a 296 (073) 099 (026)

γ 233 (137) 430 (119) 155 (757)

c 013 (004) 015 (003) 020 (009)

max(|εt|) 031 025 021

LSTAR


φ minus093 (053) minus043 (814) 130 (107)

γ 407 (267) 683 (159) 672 (291)

c 003 (001) 001 (003) 020 (000)

max(|εt|) 030 027 021

Yen Euro BP

E (xt | tminus1) = 0b 058 096 020

P(xt | tminus1) = 0c 061 072 097


max(xt) 025 027 021







see Hill (2005b)




)= 0 when p = 15





ESTAR LSTAR


1 E (εt | tminus1) = 0 - 264 096 384 258 078 100

2 P (εt+k | tminus1) = 0b - 941 091 540 582 088 126

3 E (Zt | tminus1) = 0 - 099 000 249 188 197 109

4 P (Zt+k | tminus1) = 0 - 161 813 159 401 465 217

5 P (P

h

k=0πkεt+k | tminus1) = 0c 10 191 127 799 075 199 159

20 233 108 100 076 702 386

6 P (P

h

k=0πkZt+k | tminus1) = 0 10 091 831 292 937 351 179

20 114 339 119 332 332 376



t+k| tminus1) = 0

c The null is E(P

h



h

k=0πkεt+k








APPENDIX


Bn =

(kminus1sumi=0

Nnminusi










and Vnki see the


limkirarrinfin

Ynki= Yn lim

kirarrinfinVnki

= Vn









suminfin










oplusinfin









=kminus1sumi=0

Nnminusi + Bnminusk




i=0 Nnminusi


Xn =kminus1sumi=0

ξnminusi + Vnk



(kminus1sumi=0

ξnminusi + Vnk

)

= Pnnminust

(tminus1sumi=0

ξnminusi

)+

kminus1sumi=t

ξnminusi + Vnk

=

kminus1sumi=t

ξnminusi + Vnk

where Pnnminust







(kminus1sumi=0

ξnminusi + Vnk

)

= Pnnminust

(Pnnminuss

sminus1sumi=0

ξnminusi +

kminus1sumi=s

ξnminusi + Vnk

)

= Pnnminust

(kminus1sumi=s

ξnminusi + Vnk

)


= Pnnminust

(tminus1sumi=s

ξnminusi

)+

kminus1sumi=t

ξnminusi + Vnk

=

kminus1sumi=t

ξnminusi + Vnk




jminus1sumi=0


Nnminusi

kminus1sumi=0





Pnnminusk

(kminus1sumi=0

ξnminusi


(kminus1sumi=0

ξnminusi

)

= Pnnminusk

(Pnnminus1ξn +

kminus1sumi=1

ξnminusi

)

= Pnnminusk

(kminus1sumi=1

ξnminusi

)


Pnnminusk

(kminus1sumi=0

ξnminusi

)= Pnnminusk

(kminus1sumi=1

ξnminusi

)

= Pnnminusk

PBnminus2

(kminus1sumi=1

ξnminusi

)

= Pnnminusk

(PBnminus2

ξnminus1 +

kminus1sumi=2

ξnminusi

)

= Pnnminusk

(kminus1sumi=2

ξnminusi


Thereforesumkminus1





thatsumjminus1










i=0

λjxnminusj

∥∥∥∥ le K


λjxnminusj

∥∥∥∥ (8)


oplusinfin


soplusi=0

Mnminusiperptoplus

i=s+1

Mnminusi


ssumi=0

λjxnminusjperptsum

i=s+1

λjxnminusj (9)


λjxnminusj + a

tsumi=s+1

λjxnminusj

∥∥∥∥ ge ∥∥∥∥ ssumi=0

λjxnminusj


i=0

λjxnminusj

∥∥∥∥ =


λjxnminusj +

tsumi=s+1

λjxnminusj

∥∥∥∥ ge ∥∥∥∥ ssumi=0

λjxnminusj




(xt |Ltminus1

)=suminfin



i=1






)=

infinsumi=2


)minus

infinsumi=2

ψt+1iZt+1tminusi


)




)= P


) | Ltminus1

= P

ψt+11Zt+1t + P

(Zt+1t+1 |Lt

) | Ltminus1





)=

infinsumi=3

ψt+2iZt+2t+2minusi


)minus

infinsumi=3

ψt+2iZt+2t+2minusi


)= P


) |Lt

| Ltminus1

]= ψt+22P

(Zt+2t | Ltminus1

)









= P (xt |Ltminusk)

ACKNOWLEDGEMENTS


REFERENCES

























































P (u |V1) |V0







ni=1 with tail


3 MAIN RESULTS


Bminusinfin =⋂n

Bn B+infin =⋃n

Bn



Bn =

( infinsumi=0

Nnminusi

)+ Bminusinfin (4)


(i) Bn =(oplusinfin

i=0 Nnminusi







suminfin



sumi=k

ξnminusi isinsum

i=k



Pnnminusminus1

sumi=k

ξnminusi =

sumi=k


hencesum










Xn =

infinsumi=0

ψniZnminusi + Vn






ψni =[Znminusi Xn]


iminus1sumj=0


[Znminusi Znminusi]




(ΩFt μ

) Essentially


) and E





]= 0 for every







suminfin





suminfin

















= P (xt |Ltminusk)



















suminfin

i=0 φiεtminusi




j=i+1

φjminusiεtminusj

jprodk=i+1

gtminusk(γ c)



where







i=0

φi+hai+hminus1[



times

infinsumj=i+1

φjminusiεtminusj

jprodk=i+1

gtminusk(γ c)]


For any t write


j=1

φjεtminusj

jprodk=1

gtminusk(γ c) (7)












[Z




sumnt=1 Z





E

( hsumk=0

πkZt+k





k=0 πkZt+k




x1 = ε1



1| minus c)2

+ ε3








timesinfinsum

j=1

φjεnminusj

jprodk=1

gnminusk(γ c)]








nminus1sumj=1

φj εnminusj

jprodk=1

gnminusk(γ c)

]




54 Empirical study





k=1

gtminusk(γ c)










infinsumi=1

θi exp

( infinsumj=1

τjxtminusj

)



100sumi=1

θi exp

(tminus1sumj=1

τjxtminusj

)









ESTAR


φ minus089 (027)a 296 (073) 099 (026)

γ 233 (137) 430 (119) 155 (757)

c 013 (004) 015 (003) 020 (009)

max(|εt|) 031 025 021

LSTAR


φ minus093 (053) minus043 (814) 130 (107)

γ 407 (267) 683 (159) 672 (291)

c 003 (001) 001 (003) 020 (000)

max(|εt|) 030 027 021

Yen Euro BP

E (xt | tminus1) = 0b 058 096 020

P(xt | tminus1) = 0c 061 072 097


max(xt) 025 027 021







see Hill (2005b)




)= 0 when p = 15





ESTAR LSTAR


1 E (εt | tminus1) = 0 - 264 096 384 258 078 100

2 P (εt+k | tminus1) = 0b - 941 091 540 582 088 126

3 E (Zt | tminus1) = 0 - 099 000 249 188 197 109

4 P (Zt+k | tminus1) = 0 - 161 813 159 401 465 217

5 P (P

h

k=0πkεt+k | tminus1) = 0c 10 191 127 799 075 199 159

20 233 108 100 076 702 386

6 P (P

h

k=0πkZt+k | tminus1) = 0 10 091 831 292 937 351 179

20 114 339 119 332 332 376



t+k| tminus1) = 0

c The null is E(P

h



h

k=0πkεt+k








APPENDIX


Bn =

(kminus1sumi=0

Nnminusi










and Vnki see the


limkirarrinfin

Ynki= Yn lim

kirarrinfinVnki

= Vn









suminfin










oplusinfin









=kminus1sumi=0

Nnminusi + Bnminusk




i=0 Nnminusi


Xn =kminus1sumi=0

ξnminusi + Vnk



(kminus1sumi=0

ξnminusi + Vnk

)

= Pnnminust

(tminus1sumi=0

ξnminusi

)+

kminus1sumi=t

ξnminusi + Vnk

=

kminus1sumi=t

ξnminusi + Vnk

where Pnnminust







(kminus1sumi=0

ξnminusi + Vnk

)

= Pnnminust

(Pnnminuss

sminus1sumi=0

ξnminusi +

kminus1sumi=s

ξnminusi + Vnk

)

= Pnnminust

(kminus1sumi=s

ξnminusi + Vnk

)


= Pnnminust

(tminus1sumi=s

ξnminusi

)+

kminus1sumi=t

ξnminusi + Vnk

=

kminus1sumi=t

ξnminusi + Vnk




jminus1sumi=0


Nnminusi

kminus1sumi=0





Pnnminusk

(kminus1sumi=0

ξnminusi


(kminus1sumi=0

ξnminusi

)

= Pnnminusk

(Pnnminus1ξn +

kminus1sumi=1

ξnminusi

)

= Pnnminusk

(kminus1sumi=1

ξnminusi

)


Pnnminusk

(kminus1sumi=0

ξnminusi

)= Pnnminusk

(kminus1sumi=1

ξnminusi

)

= Pnnminusk

PBnminus2

(kminus1sumi=1

ξnminusi

)

= Pnnminusk

(PBnminus2

ξnminus1 +

kminus1sumi=2

ξnminusi

)

= Pnnminusk

(kminus1sumi=2

ξnminusi


Thereforesumkminus1





thatsumjminus1










i=0

λjxnminusj

∥∥∥∥ le K


λjxnminusj

∥∥∥∥ (8)


oplusinfin


soplusi=0

Mnminusiperptoplus

i=s+1

Mnminusi


ssumi=0

λjxnminusjperptsum

i=s+1

λjxnminusj (9)


λjxnminusj + a

tsumi=s+1

λjxnminusj

∥∥∥∥ ge ∥∥∥∥ ssumi=0

λjxnminusj


i=0

λjxnminusj

∥∥∥∥ =


λjxnminusj +

tsumi=s+1

λjxnminusj

∥∥∥∥ ge ∥∥∥∥ ssumi=0

λjxnminusj




(xt |Ltminus1

)=suminfin



i=1






)=

infinsumi=2


)minus

infinsumi=2

ψt+1iZt+1tminusi


)




)= P


) | Ltminus1

= P

ψt+11Zt+1t + P

(Zt+1t+1 |Lt

) | Ltminus1





)=

infinsumi=3

ψt+2iZt+2t+2minusi


)minus

infinsumi=3

ψt+2iZt+2t+2minusi


)= P


) |Lt

| Ltminus1

]= ψt+22P

(Zt+2t | Ltminus1

)









= P (xt |Ltminusk)

ACKNOWLEDGEMENTS


REFERENCES

























































suminfin



sumi=k

ξnminusi isinsum

i=k



Pnnminusminus1

sumi=k

ξnminusi =

sumi=k


hencesum










Xn =

infinsumi=0

ψniZnminusi + Vn






ψni =[Znminusi Xn]


iminus1sumj=0


[Znminusi Znminusi]




(ΩFt μ

) Essentially


) and E





]= 0 for every







suminfin





suminfin

















= P (xt |Ltminusk)



















suminfin

i=0 φiεtminusi




j=i+1

φjminusiεtminusj

jprodk=i+1

gtminusk(γ c)



where







i=0

φi+hai+hminus1[



times

infinsumj=i+1

φjminusiεtminusj

jprodk=i+1

gtminusk(γ c)]


For any t write


j=1

φjεtminusj

jprodk=1

gtminusk(γ c) (7)












[Z




sumnt=1 Z





E

( hsumk=0

πkZt+k





k=0 πkZt+k




x1 = ε1



1| minus c)2

+ ε3








timesinfinsum

j=1

φjεnminusj

jprodk=1

gnminusk(γ c)]








nminus1sumj=1

φj εnminusj

jprodk=1

gnminusk(γ c)

]




54 Empirical study





k=1

gtminusk(γ c)










infinsumi=1

θi exp

( infinsumj=1

τjxtminusj

)



100sumi=1

θi exp

(tminus1sumj=1

τjxtminusj

)









ESTAR


φ minus089 (027)a 296 (073) 099 (026)

γ 233 (137) 430 (119) 155 (757)

c 013 (004) 015 (003) 020 (009)

max(|εt|) 031 025 021

LSTAR


φ minus093 (053) minus043 (814) 130 (107)

γ 407 (267) 683 (159) 672 (291)

c 003 (001) 001 (003) 020 (000)

max(|εt|) 030 027 021

Yen Euro BP

E (xt | tminus1) = 0b 058 096 020

P(xt | tminus1) = 0c 061 072 097


max(xt) 025 027 021







see Hill (2005b)




)= 0 when p = 15





ESTAR LSTAR


1 E (εt | tminus1) = 0 - 264 096 384 258 078 100

2 P (εt+k | tminus1) = 0b - 941 091 540 582 088 126

3 E (Zt | tminus1) = 0 - 099 000 249 188 197 109

4 P (Zt+k | tminus1) = 0 - 161 813 159 401 465 217

5 P (P

h

k=0πkεt+k | tminus1) = 0c 10 191 127 799 075 199 159

20 233 108 100 076 702 386

6 P (P

h

k=0πkZt+k | tminus1) = 0 10 091 831 292 937 351 179

20 114 339 119 332 332 376



t+k| tminus1) = 0

c The null is E(P

h



h

k=0πkεt+k








APPENDIX


Bn =

(kminus1sumi=0

Nnminusi










and Vnki see the


limkirarrinfin

Ynki= Yn lim

kirarrinfinVnki

= Vn









suminfin










oplusinfin









=kminus1sumi=0

Nnminusi + Bnminusk




i=0 Nnminusi


Xn =kminus1sumi=0

ξnminusi + Vnk



(kminus1sumi=0

ξnminusi + Vnk

)

= Pnnminust

(tminus1sumi=0

ξnminusi

)+

kminus1sumi=t

ξnminusi + Vnk

=

kminus1sumi=t

ξnminusi + Vnk

where Pnnminust







(kminus1sumi=0

ξnminusi + Vnk

)

= Pnnminust

(Pnnminuss

sminus1sumi=0

ξnminusi +

kminus1sumi=s

ξnminusi + Vnk

)

= Pnnminust

(kminus1sumi=s

ξnminusi + Vnk

)


= Pnnminust

(tminus1sumi=s

ξnminusi

)+

kminus1sumi=t

ξnminusi + Vnk

=

kminus1sumi=t

ξnminusi + Vnk




jminus1sumi=0


Nnminusi

kminus1sumi=0





Pnnminusk

(kminus1sumi=0

ξnminusi


(kminus1sumi=0

ξnminusi

)

= Pnnminusk

(Pnnminus1ξn +

kminus1sumi=1

ξnminusi

)

= Pnnminusk

(kminus1sumi=1

ξnminusi

)


Pnnminusk

(kminus1sumi=0

ξnminusi

)= Pnnminusk

(kminus1sumi=1

ξnminusi

)

= Pnnminusk

PBnminus2

(kminus1sumi=1

ξnminusi

)

= Pnnminusk

(PBnminus2

ξnminus1 +

kminus1sumi=2

ξnminusi

)

= Pnnminusk

(kminus1sumi=2

ξnminusi


Thereforesumkminus1





thatsumjminus1










i=0

λjxnminusj

∥∥∥∥ le K


λjxnminusj

∥∥∥∥ (8)


oplusinfin


soplusi=0

Mnminusiperptoplus

i=s+1

Mnminusi


ssumi=0

λjxnminusjperptsum

i=s+1

λjxnminusj (9)


λjxnminusj + a

tsumi=s+1

λjxnminusj

∥∥∥∥ ge ∥∥∥∥ ssumi=0

λjxnminusj


i=0

λjxnminusj

∥∥∥∥ =


λjxnminusj +

tsumi=s+1

λjxnminusj

∥∥∥∥ ge ∥∥∥∥ ssumi=0

λjxnminusj




(xt |Ltminus1

)=suminfin



i=1






)=

infinsumi=2


)minus

infinsumi=2

ψt+1iZt+1tminusi


)




)= P


) | Ltminus1

= P

ψt+11Zt+1t + P

(Zt+1t+1 |Lt

) | Ltminus1





)=

infinsumi=3

ψt+2iZt+2t+2minusi


)minus

infinsumi=3

ψt+2iZt+2t+2minusi


)= P


) |Lt

| Ltminus1

]= ψt+22P

(Zt+2t | Ltminus1

)









= P (xt |Ltminusk)

ACKNOWLEDGEMENTS


REFERENCES

























































(ΩFt μ

) Essentially


) and E





]= 0 for every







suminfin





suminfin

















= P (xt |Ltminusk)



















suminfin

i=0 φiεtminusi




j=i+1

φjminusiεtminusj

jprodk=i+1

gtminusk(γ c)



where







i=0

φi+hai+hminus1[



times

infinsumj=i+1

φjminusiεtminusj

jprodk=i+1

gtminusk(γ c)]


For any t write


j=1

φjεtminusj

jprodk=1

gtminusk(γ c) (7)












[Z




sumnt=1 Z





E

( hsumk=0

πkZt+k





k=0 πkZt+k




x1 = ε1



1| minus c)2

+ ε3








timesinfinsum

j=1

φjεnminusj

jprodk=1

gnminusk(γ c)]








nminus1sumj=1

φj εnminusj

jprodk=1

gnminusk(γ c)

]




54 Empirical study





k=1

gtminusk(γ c)










infinsumi=1

θi exp

( infinsumj=1

τjxtminusj

)



100sumi=1

θi exp

(tminus1sumj=1

τjxtminusj

)









ESTAR


φ minus089 (027)a 296 (073) 099 (026)

γ 233 (137) 430 (119) 155 (757)

c 013 (004) 015 (003) 020 (009)

max(|εt|) 031 025 021

LSTAR


φ minus093 (053) minus043 (814) 130 (107)

γ 407 (267) 683 (159) 672 (291)

c 003 (001) 001 (003) 020 (000)

max(|εt|) 030 027 021

Yen Euro BP

E (xt | tminus1) = 0b 058 096 020

P(xt | tminus1) = 0c 061 072 097


max(xt) 025 027 021







see Hill (2005b)




)= 0 when p = 15





ESTAR LSTAR


1 E (εt | tminus1) = 0 - 264 096 384 258 078 100

2 P (εt+k | tminus1) = 0b - 941 091 540 582 088 126

3 E (Zt | tminus1) = 0 - 099 000 249 188 197 109

4 P (Zt+k | tminus1) = 0 - 161 813 159 401 465 217

5 P (P

h

k=0πkεt+k | tminus1) = 0c 10 191 127 799 075 199 159

20 233 108 100 076 702 386

6 P (P

h

k=0πkZt+k | tminus1) = 0 10 091 831 292 937 351 179

20 114 339 119 332 332 376



t+k| tminus1) = 0

c The null is E(P

h



h

k=0πkεt+k








APPENDIX


Bn =

(kminus1sumi=0

Nnminusi










and Vnki see the


limkirarrinfin

Ynki= Yn lim

kirarrinfinVnki

= Vn









suminfin










oplusinfin









=kminus1sumi=0

Nnminusi + Bnminusk




i=0 Nnminusi


Xn =kminus1sumi=0

ξnminusi + Vnk



(kminus1sumi=0

ξnminusi + Vnk

)

= Pnnminust

(tminus1sumi=0

ξnminusi

)+

kminus1sumi=t

ξnminusi + Vnk

=

kminus1sumi=t

ξnminusi + Vnk

where Pnnminust







(kminus1sumi=0

ξnminusi + Vnk

)

= Pnnminust

(Pnnminuss

sminus1sumi=0

ξnminusi +

kminus1sumi=s

ξnminusi + Vnk

)

= Pnnminust

(kminus1sumi=s

ξnminusi + Vnk

)


= Pnnminust

(tminus1sumi=s

ξnminusi

)+

kminus1sumi=t

ξnminusi + Vnk

=

kminus1sumi=t

ξnminusi + Vnk




jminus1sumi=0


Nnminusi

kminus1sumi=0





Pnnminusk

(kminus1sumi=0

ξnminusi


(kminus1sumi=0

ξnminusi

)

= Pnnminusk

(Pnnminus1ξn +

kminus1sumi=1

ξnminusi

)

= Pnnminusk

(kminus1sumi=1

ξnminusi

)


Pnnminusk

(kminus1sumi=0

ξnminusi

)= Pnnminusk

(kminus1sumi=1

ξnminusi

)

= Pnnminusk

PBnminus2

(kminus1sumi=1

ξnminusi

)

= Pnnminusk

(PBnminus2

ξnminus1 +

kminus1sumi=2

ξnminusi

)

= Pnnminusk

(kminus1sumi=2

ξnminusi


Thereforesumkminus1





thatsumjminus1










i=0

λjxnminusj

∥∥∥∥ le K


λjxnminusj

∥∥∥∥ (8)


oplusinfin


soplusi=0

Mnminusiperptoplus

i=s+1

Mnminusi


ssumi=0

λjxnminusjperptsum

i=s+1

λjxnminusj (9)


λjxnminusj + a

tsumi=s+1

λjxnminusj

∥∥∥∥ ge ∥∥∥∥ ssumi=0

λjxnminusj


i=0

λjxnminusj

∥∥∥∥ =


λjxnminusj +

tsumi=s+1

λjxnminusj

∥∥∥∥ ge ∥∥∥∥ ssumi=0

λjxnminusj




(xt |Ltminus1

)=suminfin



i=1






)=

infinsumi=2


)minus

infinsumi=2

ψt+1iZt+1tminusi


)




)= P


) | Ltminus1

= P

ψt+11Zt+1t + P

(Zt+1t+1 |Lt

) | Ltminus1





)=

infinsumi=3

ψt+2iZt+2t+2minusi


)minus

infinsumi=3

ψt+2iZt+2t+2minusi


)= P


) |Lt

| Ltminus1

]= ψt+22P

(Zt+2t | Ltminus1

)









= P (xt |Ltminusk)

ACKNOWLEDGEMENTS


REFERENCES




































































= P (xt |Ltminusk)



















suminfin

i=0 φiεtminusi




j=i+1

φjminusiεtminusj

jprodk=i+1

gtminusk(γ c)



where







i=0

φi+hai+hminus1[



times

infinsumj=i+1

φjminusiεtminusj

jprodk=i+1

gtminusk(γ c)]


For any t write


j=1

φjεtminusj

jprodk=1

gtminusk(γ c) (7)












[Z




sumnt=1 Z





E

( hsumk=0

πkZt+k





k=0 πkZt+k




x1 = ε1



1| minus c)2

+ ε3








timesinfinsum

j=1

φjεnminusj

jprodk=1

gnminusk(γ c)]








nminus1sumj=1

φj εnminusj

jprodk=1

gnminusk(γ c)

]




54 Empirical study





k=1

gtminusk(γ c)










infinsumi=1

θi exp

( infinsumj=1

τjxtminusj

)



100sumi=1

θi exp

(tminus1sumj=1

τjxtminusj

)









ESTAR


φ minus089 (027)a 296 (073) 099 (026)

γ 233 (137) 430 (119) 155 (757)

c 013 (004) 015 (003) 020 (009)

max(|εt|) 031 025 021

LSTAR


φ minus093 (053) minus043 (814) 130 (107)

γ 407 (267) 683 (159) 672 (291)

c 003 (001) 001 (003) 020 (000)

max(|εt|) 030 027 021

Yen Euro BP

E (xt | tminus1) = 0b 058 096 020

P(xt | tminus1) = 0c 061 072 097


max(xt) 025 027 021







see Hill (2005b)




)= 0 when p = 15





ESTAR LSTAR


1 E (εt | tminus1) = 0 - 264 096 384 258 078 100

2 P (εt+k | tminus1) = 0b - 941 091 540 582 088 126

3 E (Zt | tminus1) = 0 - 099 000 249 188 197 109

4 P (Zt+k | tminus1) = 0 - 161 813 159 401 465 217

5 P (P

h

k=0πkεt+k | tminus1) = 0c 10 191 127 799 075 199 159

20 233 108 100 076 702 386

6 P (P

h

k=0πkZt+k | tminus1) = 0 10 091 831 292 937 351 179

20 114 339 119 332 332 376



t+k| tminus1) = 0

c The null is E(P

h



h

k=0πkεt+k








APPENDIX


Bn =

(kminus1sumi=0

Nnminusi










and Vnki see the


limkirarrinfin

Ynki= Yn lim

kirarrinfinVnki

= Vn









suminfin










oplusinfin









=kminus1sumi=0

Nnminusi + Bnminusk




i=0 Nnminusi


Xn =kminus1sumi=0

ξnminusi + Vnk



(kminus1sumi=0

ξnminusi + Vnk

)

= Pnnminust

(tminus1sumi=0

ξnminusi

)+

kminus1sumi=t

ξnminusi + Vnk

=

kminus1sumi=t

ξnminusi + Vnk

where Pnnminust







(kminus1sumi=0

ξnminusi + Vnk

)

= Pnnminust

(Pnnminuss

sminus1sumi=0

ξnminusi +

kminus1sumi=s

ξnminusi + Vnk

)

= Pnnminust

(kminus1sumi=s

ξnminusi + Vnk

)


= Pnnminust

(tminus1sumi=s

ξnminusi

)+

kminus1sumi=t

ξnminusi + Vnk

=

kminus1sumi=t

ξnminusi + Vnk




jminus1sumi=0


Nnminusi

kminus1sumi=0





Pnnminusk

(kminus1sumi=0

ξnminusi


(kminus1sumi=0

ξnminusi

)

= Pnnminusk

(Pnnminus1ξn +

kminus1sumi=1

ξnminusi

)

= Pnnminusk

(kminus1sumi=1

ξnminusi

)


Pnnminusk

(kminus1sumi=0

ξnminusi

)= Pnnminusk

(kminus1sumi=1

ξnminusi

)

= Pnnminusk

PBnminus2

(kminus1sumi=1

ξnminusi

)

= Pnnminusk

(PBnminus2

ξnminus1 +

kminus1sumi=2

ξnminusi

)

= Pnnminusk

(kminus1sumi=2

ξnminusi


Thereforesumkminus1





thatsumjminus1










i=0

λjxnminusj

∥∥∥∥ le K


λjxnminusj

∥∥∥∥ (8)


oplusinfin


soplusi=0

Mnminusiperptoplus

i=s+1

Mnminusi


ssumi=0

λjxnminusjperptsum

i=s+1

λjxnminusj (9)


λjxnminusj + a

tsumi=s+1

λjxnminusj

∥∥∥∥ ge ∥∥∥∥ ssumi=0

λjxnminusj


i=0

λjxnminusj

∥∥∥∥ =


λjxnminusj +

tsumi=s+1

λjxnminusj

∥∥∥∥ ge ∥∥∥∥ ssumi=0

λjxnminusj




(xt |Ltminus1

)=suminfin



i=1






)=

infinsumi=2


)minus

infinsumi=2

ψt+1iZt+1tminusi


)




)= P


) | Ltminus1

= P

ψt+11Zt+1t + P

(Zt+1t+1 |Lt

) | Ltminus1





)=

infinsumi=3

ψt+2iZt+2t+2minusi


)minus

infinsumi=3

ψt+2iZt+2t+2minusi


)= P


) |Lt

| Ltminus1

]= ψt+22P

(Zt+2t | Ltminus1

)









= P (xt |Ltminusk)

ACKNOWLEDGEMENTS


REFERENCES



































































suminfin

i=0 φiεtminusi




j=i+1

φjminusiεtminusj

jprodk=i+1

gtminusk(γ c)



where







i=0

φi+hai+hminus1[



times

infinsumj=i+1

φjminusiεtminusj

jprodk=i+1

gtminusk(γ c)]


For any t write


j=1

φjεtminusj

jprodk=1

gtminusk(γ c) (7)












[Z




sumnt=1 Z





E

( hsumk=0

πkZt+k





k=0 πkZt+k




x1 = ε1



1| minus c)2

+ ε3








timesinfinsum

j=1

φjεnminusj

jprodk=1

gnminusk(γ c)]








nminus1sumj=1

φj εnminusj

jprodk=1

gnminusk(γ c)

]




54 Empirical study





k=1

gtminusk(γ c)










infinsumi=1

θi exp

( infinsumj=1

τjxtminusj

)



100sumi=1

θi exp

(tminus1sumj=1

τjxtminusj

)









ESTAR


φ minus089 (027)a 296 (073) 099 (026)

γ 233 (137) 430 (119) 155 (757)

c 013 (004) 015 (003) 020 (009)

max(|εt|) 031 025 021

LSTAR


φ minus093 (053) minus043 (814) 130 (107)

γ 407 (267) 683 (159) 672 (291)

c 003 (001) 001 (003) 020 (000)

max(|εt|) 030 027 021

Yen Euro BP

E (xt | tminus1) = 0b 058 096 020

P(xt | tminus1) = 0c 061 072 097


max(xt) 025 027 021







see Hill (2005b)




)= 0 when p = 15





ESTAR LSTAR


1 E (εt | tminus1) = 0 - 264 096 384 258 078 100

2 P (εt+k | tminus1) = 0b - 941 091 540 582 088 126

3 E (Zt | tminus1) = 0 - 099 000 249 188 197 109

4 P (Zt+k | tminus1) = 0 - 161 813 159 401 465 217

5 P (P

h

k=0πkεt+k | tminus1) = 0c 10 191 127 799 075 199 159

20 233 108 100 076 702 386

6 P (P

h

k=0πkZt+k | tminus1) = 0 10 091 831 292 937 351 179

20 114 339 119 332 332 376



t+k| tminus1) = 0

c The null is E(P

h



h

k=0πkεt+k








APPENDIX


Bn =

(kminus1sumi=0

Nnminusi










and Vnki see the


limkirarrinfin

Ynki= Yn lim

kirarrinfinVnki

= Vn









suminfin










oplusinfin









=kminus1sumi=0

Nnminusi + Bnminusk




i=0 Nnminusi


Xn =kminus1sumi=0

ξnminusi + Vnk



(kminus1sumi=0

ξnminusi + Vnk

)

= Pnnminust

(tminus1sumi=0

ξnminusi

)+

kminus1sumi=t

ξnminusi + Vnk

=

kminus1sumi=t

ξnminusi + Vnk

where Pnnminust







(kminus1sumi=0

ξnminusi + Vnk

)

= Pnnminust

(Pnnminuss

sminus1sumi=0

ξnminusi +

kminus1sumi=s

ξnminusi + Vnk

)

= Pnnminust

(kminus1sumi=s

ξnminusi + Vnk

)


= Pnnminust

(tminus1sumi=s

ξnminusi

)+

kminus1sumi=t

ξnminusi + Vnk

=

kminus1sumi=t

ξnminusi + Vnk




jminus1sumi=0


Nnminusi

kminus1sumi=0





Pnnminusk

(kminus1sumi=0

ξnminusi


(kminus1sumi=0

ξnminusi

)

= Pnnminusk

(Pnnminus1ξn +

kminus1sumi=1

ξnminusi

)

= Pnnminusk

(kminus1sumi=1

ξnminusi

)


Pnnminusk

(kminus1sumi=0

ξnminusi

)= Pnnminusk

(kminus1sumi=1

ξnminusi

)

= Pnnminusk

PBnminus2

(kminus1sumi=1

ξnminusi

)

= Pnnminusk

(PBnminus2

ξnminus1 +

kminus1sumi=2

ξnminusi

)

= Pnnminusk

(kminus1sumi=2

ξnminusi


Thereforesumkminus1





thatsumjminus1










i=0

λjxnminusj

∥∥∥∥ le K


λjxnminusj

∥∥∥∥ (8)


oplusinfin


soplusi=0

Mnminusiperptoplus

i=s+1

Mnminusi


ssumi=0

λjxnminusjperptsum

i=s+1

λjxnminusj (9)


λjxnminusj + a

tsumi=s+1

λjxnminusj

∥∥∥∥ ge ∥∥∥∥ ssumi=0

λjxnminusj


i=0

λjxnminusj

∥∥∥∥ =


λjxnminusj +

tsumi=s+1

λjxnminusj

∥∥∥∥ ge ∥∥∥∥ ssumi=0

λjxnminusj




(xt |Ltminus1

)=suminfin



i=1






)=

infinsumi=2


)minus

infinsumi=2

ψt+1iZt+1tminusi


)




)= P


) | Ltminus1

= P

ψt+11Zt+1t + P

(Zt+1t+1 |Lt

) | Ltminus1





)=

infinsumi=3

ψt+2iZt+2t+2minusi


)minus

infinsumi=3

ψt+2iZt+2t+2minusi


)= P


) |Lt

| Ltminus1

]= ψt+22P

(Zt+2t | Ltminus1

)









= P (xt |Ltminusk)

ACKNOWLEDGEMENTS


REFERENCES























































where







i=0

φi+hai+hminus1[



times

infinsumj=i+1

φjminusiεtminusj

jprodk=i+1

gtminusk(γ c)]


For any t write


j=1

φjεtminusj

jprodk=1

gtminusk(γ c) (7)












[Z




sumnt=1 Z





E

( hsumk=0

πkZt+k





k=0 πkZt+k




x1 = ε1



1| minus c)2

+ ε3








timesinfinsum

j=1

φjεnminusj

jprodk=1

gnminusk(γ c)]








nminus1sumj=1

φj εnminusj

jprodk=1

gnminusk(γ c)

]




54 Empirical study





k=1

gtminusk(γ c)










infinsumi=1

θi exp

( infinsumj=1

τjxtminusj

)



100sumi=1

θi exp

(tminus1sumj=1

τjxtminusj

)









ESTAR


φ minus089 (027)a 296 (073) 099 (026)

γ 233 (137) 430 (119) 155 (757)

c 013 (004) 015 (003) 020 (009)

max(|εt|) 031 025 021

LSTAR


φ minus093 (053) minus043 (814) 130 (107)

γ 407 (267) 683 (159) 672 (291)

c 003 (001) 001 (003) 020 (000)

max(|εt|) 030 027 021

Yen Euro BP

E (xt | tminus1) = 0b 058 096 020

P(xt | tminus1) = 0c 061 072 097


max(xt) 025 027 021







see Hill (2005b)




)= 0 when p = 15





ESTAR LSTAR


1 E (εt | tminus1) = 0 - 264 096 384 258 078 100

2 P (εt+k | tminus1) = 0b - 941 091 540 582 088 126

3 E (Zt | tminus1) = 0 - 099 000 249 188 197 109

4 P (Zt+k | tminus1) = 0 - 161 813 159 401 465 217

5 P (P

h

k=0πkεt+k | tminus1) = 0c 10 191 127 799 075 199 159

20 233 108 100 076 702 386

6 P (P

h

k=0πkZt+k | tminus1) = 0 10 091 831 292 937 351 179

20 114 339 119 332 332 376



t+k| tminus1) = 0

c The null is E(P

h



h

k=0πkεt+k








APPENDIX


Bn =

(kminus1sumi=0

Nnminusi










and Vnki see the


limkirarrinfin

Ynki= Yn lim

kirarrinfinVnki

= Vn









suminfin










oplusinfin









=kminus1sumi=0

Nnminusi + Bnminusk




i=0 Nnminusi


Xn =kminus1sumi=0

ξnminusi + Vnk



(kminus1sumi=0

ξnminusi + Vnk

)

= Pnnminust

(tminus1sumi=0

ξnminusi

)+

kminus1sumi=t

ξnminusi + Vnk

=

kminus1sumi=t

ξnminusi + Vnk

where Pnnminust







(kminus1sumi=0

ξnminusi + Vnk

)

= Pnnminust

(Pnnminuss

sminus1sumi=0

ξnminusi +

kminus1sumi=s

ξnminusi + Vnk

)

= Pnnminust

(kminus1sumi=s

ξnminusi + Vnk

)


= Pnnminust

(tminus1sumi=s

ξnminusi

)+

kminus1sumi=t

ξnminusi + Vnk

=

kminus1sumi=t

ξnminusi + Vnk




jminus1sumi=0


Nnminusi

kminus1sumi=0





Pnnminusk

(kminus1sumi=0

ξnminusi


(kminus1sumi=0

ξnminusi

)

= Pnnminusk

(Pnnminus1ξn +

kminus1sumi=1

ξnminusi

)

= Pnnminusk

(kminus1sumi=1

ξnminusi

)


Pnnminusk

(kminus1sumi=0

ξnminusi

)= Pnnminusk

(kminus1sumi=1

ξnminusi

)

= Pnnminusk

PBnminus2

(kminus1sumi=1

ξnminusi

)

= Pnnminusk

(PBnminus2

ξnminus1 +

kminus1sumi=2

ξnminusi

)

= Pnnminusk

(kminus1sumi=2

ξnminusi


Thereforesumkminus1





thatsumjminus1










i=0

λjxnminusj

∥∥∥∥ le K


λjxnminusj

∥∥∥∥ (8)


oplusinfin


soplusi=0

Mnminusiperptoplus

i=s+1

Mnminusi


ssumi=0

λjxnminusjperptsum

i=s+1

λjxnminusj (9)


λjxnminusj + a

tsumi=s+1

λjxnminusj

∥∥∥∥ ge ∥∥∥∥ ssumi=0

λjxnminusj


i=0

λjxnminusj

∥∥∥∥ =


λjxnminusj +

tsumi=s+1

λjxnminusj

∥∥∥∥ ge ∥∥∥∥ ssumi=0

λjxnminusj




(xt |Ltminus1

)=suminfin



i=1






)=

infinsumi=2


)minus

infinsumi=2

ψt+1iZt+1tminusi


)




)= P


) | Ltminus1

= P

ψt+11Zt+1t + P

(Zt+1t+1 |Lt

) | Ltminus1





)=

infinsumi=3

ψt+2iZt+2t+2minusi


)minus

infinsumi=3

ψt+2iZt+2t+2minusi


)= P


) |Lt

| Ltminus1

]= ψt+22P

(Zt+2t | Ltminus1

)









= P (xt |Ltminusk)

ACKNOWLEDGEMENTS


REFERENCES
























































E

( hsumk=0

πkZt+k





k=0 πkZt+k




x1 = ε1



1| minus c)2

+ ε3








timesinfinsum

j=1

φjεnminusj

jprodk=1

gnminusk(γ c)]








nminus1sumj=1

φj εnminusj

jprodk=1

gnminusk(γ c)

]




54 Empirical study





k=1

gtminusk(γ c)










infinsumi=1

θi exp

( infinsumj=1

τjxtminusj

)



100sumi=1

θi exp

(tminus1sumj=1

τjxtminusj

)









ESTAR


φ minus089 (027)a 296 (073) 099 (026)

γ 233 (137) 430 (119) 155 (757)

c 013 (004) 015 (003) 020 (009)

max(|εt|) 031 025 021

LSTAR


φ minus093 (053) minus043 (814) 130 (107)

γ 407 (267) 683 (159) 672 (291)

c 003 (001) 001 (003) 020 (000)

max(|εt|) 030 027 021

Yen Euro BP

E (xt | tminus1) = 0b 058 096 020

P(xt | tminus1) = 0c 061 072 097


max(xt) 025 027 021







see Hill (2005b)




)= 0 when p = 15





ESTAR LSTAR


1 E (εt | tminus1) = 0 - 264 096 384 258 078 100

2 P (εt+k | tminus1) = 0b - 941 091 540 582 088 126

3 E (Zt | tminus1) = 0 - 099 000 249 188 197 109

4 P (Zt+k | tminus1) = 0 - 161 813 159 401 465 217

5 P (P

h

k=0πkεt+k | tminus1) = 0c 10 191 127 799 075 199 159

20 233 108 100 076 702 386

6 P (P

h

k=0πkZt+k | tminus1) = 0 10 091 831 292 937 351 179

20 114 339 119 332 332 376



t+k| tminus1) = 0

c The null is E(P

h



h

k=0πkεt+k








APPENDIX


Bn =

(kminus1sumi=0

Nnminusi










and Vnki see the


limkirarrinfin

Ynki= Yn lim

kirarrinfinVnki

= Vn









suminfin










oplusinfin









=kminus1sumi=0

Nnminusi + Bnminusk




i=0 Nnminusi


Xn =kminus1sumi=0

ξnminusi + Vnk



(kminus1sumi=0

ξnminusi + Vnk

)

= Pnnminust

(tminus1sumi=0

ξnminusi

)+

kminus1sumi=t

ξnminusi + Vnk

=

kminus1sumi=t

ξnminusi + Vnk

where Pnnminust







(kminus1sumi=0

ξnminusi + Vnk

)

= Pnnminust

(Pnnminuss

sminus1sumi=0

ξnminusi +

kminus1sumi=s

ξnminusi + Vnk

)

= Pnnminust

(kminus1sumi=s

ξnminusi + Vnk

)


= Pnnminust

(tminus1sumi=s

ξnminusi

)+

kminus1sumi=t

ξnminusi + Vnk

=

kminus1sumi=t

ξnminusi + Vnk




jminus1sumi=0


Nnminusi

kminus1sumi=0





Pnnminusk

(kminus1sumi=0

ξnminusi


(kminus1sumi=0

ξnminusi

)

= Pnnminusk

(Pnnminus1ξn +

kminus1sumi=1

ξnminusi

)

= Pnnminusk

(kminus1sumi=1

ξnminusi

)


Pnnminusk

(kminus1sumi=0

ξnminusi

)= Pnnminusk

(kminus1sumi=1

ξnminusi

)

= Pnnminusk

PBnminus2

(kminus1sumi=1

ξnminusi

)

= Pnnminusk

(PBnminus2

ξnminus1 +

kminus1sumi=2

ξnminusi

)

= Pnnminusk

(kminus1sumi=2

ξnminusi


Thereforesumkminus1





thatsumjminus1










i=0

λjxnminusj

∥∥∥∥ le K


λjxnminusj

∥∥∥∥ (8)


oplusinfin


soplusi=0

Mnminusiperptoplus

i=s+1

Mnminusi


ssumi=0

λjxnminusjperptsum

i=s+1

λjxnminusj (9)


λjxnminusj + a

tsumi=s+1

λjxnminusj

∥∥∥∥ ge ∥∥∥∥ ssumi=0

λjxnminusj


i=0

λjxnminusj

∥∥∥∥ =


λjxnminusj +

tsumi=s+1

λjxnminusj

∥∥∥∥ ge ∥∥∥∥ ssumi=0

λjxnminusj




(xt |Ltminus1

)=suminfin



i=1






)=

infinsumi=2


)minus

infinsumi=2

ψt+1iZt+1tminusi


)




)= P


) | Ltminus1

= P

ψt+11Zt+1t + P

(Zt+1t+1 |Lt

) | Ltminus1





)=

infinsumi=3

ψt+2iZt+2t+2minusi


)minus

infinsumi=3

ψt+2iZt+2t+2minusi


)= P


) |Lt

| Ltminus1

]= ψt+22P

(Zt+2t | Ltminus1

)









= P (xt |Ltminusk)

ACKNOWLEDGEMENTS


REFERENCES























































54 Empirical study





k=1

gtminusk(γ c)










infinsumi=1

θi exp

( infinsumj=1

τjxtminusj

)



100sumi=1

θi exp

(tminus1sumj=1

τjxtminusj

)









ESTAR


φ minus089 (027)a 296 (073) 099 (026)

γ 233 (137) 430 (119) 155 (757)

c 013 (004) 015 (003) 020 (009)

max(|εt|) 031 025 021

LSTAR


φ minus093 (053) minus043 (814) 130 (107)

γ 407 (267) 683 (159) 672 (291)

c 003 (001) 001 (003) 020 (000)

max(|εt|) 030 027 021

Yen Euro BP

E (xt | tminus1) = 0b 058 096 020

P(xt | tminus1) = 0c 061 072 097


max(xt) 025 027 021







see Hill (2005b)




)= 0 when p = 15





ESTAR LSTAR


1 E (εt | tminus1) = 0 - 264 096 384 258 078 100

2 P (εt+k | tminus1) = 0b - 941 091 540 582 088 126

3 E (Zt | tminus1) = 0 - 099 000 249 188 197 109

4 P (Zt+k | tminus1) = 0 - 161 813 159 401 465 217

5 P (P

h

k=0πkεt+k | tminus1) = 0c 10 191 127 799 075 199 159

20 233 108 100 076 702 386

6 P (P

h

k=0πkZt+k | tminus1) = 0 10 091 831 292 937 351 179

20 114 339 119 332 332 376



t+k| tminus1) = 0

c The null is E(P

h



h

k=0πkεt+k








APPENDIX


Bn =

(kminus1sumi=0

Nnminusi










and Vnki see the


limkirarrinfin

Ynki= Yn lim

kirarrinfinVnki

= Vn









suminfin










oplusinfin









=kminus1sumi=0

Nnminusi + Bnminusk




i=0 Nnminusi


Xn =kminus1sumi=0

ξnminusi + Vnk



(kminus1sumi=0

ξnminusi + Vnk

)

= Pnnminust

(tminus1sumi=0

ξnminusi

)+

kminus1sumi=t

ξnminusi + Vnk

=

kminus1sumi=t

ξnminusi + Vnk

where Pnnminust







(kminus1sumi=0

ξnminusi + Vnk

)

= Pnnminust

(Pnnminuss

sminus1sumi=0

ξnminusi +

kminus1sumi=s

ξnminusi + Vnk

)

= Pnnminust

(kminus1sumi=s

ξnminusi + Vnk

)


= Pnnminust

(tminus1sumi=s

ξnminusi

)+

kminus1sumi=t

ξnminusi + Vnk

=

kminus1sumi=t

ξnminusi + Vnk




jminus1sumi=0


Nnminusi

kminus1sumi=0





Pnnminusk

(kminus1sumi=0

ξnminusi


(kminus1sumi=0

ξnminusi

)

= Pnnminusk

(Pnnminus1ξn +

kminus1sumi=1

ξnminusi

)

= Pnnminusk

(kminus1sumi=1

ξnminusi

)


Pnnminusk

(kminus1sumi=0

ξnminusi

)= Pnnminusk

(kminus1sumi=1

ξnminusi

)

= Pnnminusk

PBnminus2

(kminus1sumi=1

ξnminusi

)

= Pnnminusk

(PBnminus2

ξnminus1 +

kminus1sumi=2

ξnminusi

)

= Pnnminusk

(kminus1sumi=2

ξnminusi


Thereforesumkminus1





thatsumjminus1










i=0

λjxnminusj

∥∥∥∥ le K


λjxnminusj

∥∥∥∥ (8)


oplusinfin


soplusi=0

Mnminusiperptoplus

i=s+1

Mnminusi


ssumi=0

λjxnminusjperptsum

i=s+1

λjxnminusj (9)


λjxnminusj + a

tsumi=s+1

λjxnminusj

∥∥∥∥ ge ∥∥∥∥ ssumi=0

λjxnminusj


i=0

λjxnminusj

∥∥∥∥ =


λjxnminusj +

tsumi=s+1

λjxnminusj

∥∥∥∥ ge ∥∥∥∥ ssumi=0

λjxnminusj




(xt |Ltminus1

)=suminfin



i=1






)=

infinsumi=2


)minus

infinsumi=2

ψt+1iZt+1tminusi


)




)= P


) | Ltminus1

= P

ψt+11Zt+1t + P

(Zt+1t+1 |Lt

) | Ltminus1





)=

infinsumi=3

ψt+2iZt+2t+2minusi


)minus

infinsumi=3

ψt+2iZt+2t+2minusi


)= P


) |Lt

| Ltminus1

]= ψt+22P

(Zt+2t | Ltminus1

)









= P (xt |Ltminusk)

ACKNOWLEDGEMENTS


REFERENCES

























































ESTAR


φ minus089 (027)a 296 (073) 099 (026)

γ 233 (137) 430 (119) 155 (757)

c 013 (004) 015 (003) 020 (009)

max(|εt|) 031 025 021

LSTAR


φ minus093 (053) minus043 (814) 130 (107)

γ 407 (267) 683 (159) 672 (291)

c 003 (001) 001 (003) 020 (000)

max(|εt|) 030 027 021

Yen Euro BP

E (xt | tminus1) = 0b 058 096 020

P(xt | tminus1) = 0c 061 072 097


max(xt) 025 027 021







see Hill (2005b)




)= 0 when p = 15





ESTAR LSTAR


1 E (εt | tminus1) = 0 - 264 096 384 258 078 100

2 P (εt+k | tminus1) = 0b - 941 091 540 582 088 126

3 E (Zt | tminus1) = 0 - 099 000 249 188 197 109

4 P (Zt+k | tminus1) = 0 - 161 813 159 401 465 217

5 P (P

h

k=0πkεt+k | tminus1) = 0c 10 191 127 799 075 199 159

20 233 108 100 076 702 386

6 P (P

h

k=0πkZt+k | tminus1) = 0 10 091 831 292 937 351 179

20 114 339 119 332 332 376



t+k| tminus1) = 0

c The null is E(P

h



h

k=0πkεt+k








APPENDIX


Bn =

(kminus1sumi=0

Nnminusi










and Vnki see the


limkirarrinfin

Ynki= Yn lim

kirarrinfinVnki

= Vn









suminfin










oplusinfin









=kminus1sumi=0

Nnminusi + Bnminusk




i=0 Nnminusi


Xn =kminus1sumi=0

ξnminusi + Vnk



(kminus1sumi=0

ξnminusi + Vnk

)

= Pnnminust

(tminus1sumi=0

ξnminusi

)+

kminus1sumi=t

ξnminusi + Vnk

=

kminus1sumi=t

ξnminusi + Vnk

where Pnnminust







(kminus1sumi=0

ξnminusi + Vnk

)

= Pnnminust

(Pnnminuss

sminus1sumi=0

ξnminusi +

kminus1sumi=s

ξnminusi + Vnk

)

= Pnnminust

(kminus1sumi=s

ξnminusi + Vnk

)


= Pnnminust

(tminus1sumi=s

ξnminusi

)+

kminus1sumi=t

ξnminusi + Vnk

=

kminus1sumi=t

ξnminusi + Vnk




jminus1sumi=0


Nnminusi

kminus1sumi=0





Pnnminusk

(kminus1sumi=0

ξnminusi


(kminus1sumi=0

ξnminusi

)

= Pnnminusk

(Pnnminus1ξn +

kminus1sumi=1

ξnminusi

)

= Pnnminusk

(kminus1sumi=1

ξnminusi

)


Pnnminusk

(kminus1sumi=0

ξnminusi

)= Pnnminusk

(kminus1sumi=1

ξnminusi

)

= Pnnminusk

PBnminus2

(kminus1sumi=1

ξnminusi

)

= Pnnminusk

(PBnminus2

ξnminus1 +

kminus1sumi=2

ξnminusi

)

= Pnnminusk

(kminus1sumi=2

ξnminusi


Thereforesumkminus1





thatsumjminus1










i=0

λjxnminusj

∥∥∥∥ le K


λjxnminusj

∥∥∥∥ (8)


oplusinfin


soplusi=0

Mnminusiperptoplus

i=s+1

Mnminusi


ssumi=0

λjxnminusjperptsum

i=s+1

λjxnminusj (9)


λjxnminusj + a

tsumi=s+1

λjxnminusj

∥∥∥∥ ge ∥∥∥∥ ssumi=0

λjxnminusj


i=0

λjxnminusj

∥∥∥∥ =


λjxnminusj +

tsumi=s+1

λjxnminusj

∥∥∥∥ ge ∥∥∥∥ ssumi=0

λjxnminusj




(xt |Ltminus1

)=suminfin



i=1






)=

infinsumi=2


)minus

infinsumi=2

ψt+1iZt+1tminusi


)




)= P


) | Ltminus1

= P

ψt+11Zt+1t + P

(Zt+1t+1 |Lt

) | Ltminus1





)=

infinsumi=3

ψt+2iZt+2t+2minusi


)minus

infinsumi=3

ψt+2iZt+2t+2minusi


)= P


) |Lt

| Ltminus1

]= ψt+22P

(Zt+2t | Ltminus1

)









= P (xt |Ltminusk)

ACKNOWLEDGEMENTS


REFERENCES

























































ESTAR LSTAR


1 E (εt | tminus1) = 0 - 264 096 384 258 078 100

2 P (εt+k | tminus1) = 0b - 941 091 540 582 088 126

3 E (Zt | tminus1) = 0 - 099 000 249 188 197 109

4 P (Zt+k | tminus1) = 0 - 161 813 159 401 465 217

5 P (P

h

k=0πkεt+k | tminus1) = 0c 10 191 127 799 075 199 159

20 233 108 100 076 702 386

6 P (P

h

k=0πkZt+k | tminus1) = 0 10 091 831 292 937 351 179

20 114 339 119 332 332 376



t+k| tminus1) = 0

c The null is E(P

h



h

k=0πkεt+k








APPENDIX


Bn =

(kminus1sumi=0

Nnminusi










and Vnki see the


limkirarrinfin

Ynki= Yn lim

kirarrinfinVnki

= Vn









suminfin










oplusinfin









=kminus1sumi=0

Nnminusi + Bnminusk




i=0 Nnminusi


Xn =kminus1sumi=0

ξnminusi + Vnk



(kminus1sumi=0

ξnminusi + Vnk

)

= Pnnminust

(tminus1sumi=0

ξnminusi

)+

kminus1sumi=t

ξnminusi + Vnk

=

kminus1sumi=t

ξnminusi + Vnk

where Pnnminust







(kminus1sumi=0

ξnminusi + Vnk

)

= Pnnminust

(Pnnminuss

sminus1sumi=0

ξnminusi +

kminus1sumi=s

ξnminusi + Vnk

)

= Pnnminust

(kminus1sumi=s

ξnminusi + Vnk

)


= Pnnminust

(tminus1sumi=s

ξnminusi

)+

kminus1sumi=t

ξnminusi + Vnk

=

kminus1sumi=t

ξnminusi + Vnk




jminus1sumi=0


Nnminusi

kminus1sumi=0





Pnnminusk

(kminus1sumi=0

ξnminusi


(kminus1sumi=0

ξnminusi

)

= Pnnminusk

(Pnnminus1ξn +

kminus1sumi=1

ξnminusi

)

= Pnnminusk

(kminus1sumi=1

ξnminusi

)


Pnnminusk

(kminus1sumi=0

ξnminusi

)= Pnnminusk

(kminus1sumi=1

ξnminusi

)

= Pnnminusk

PBnminus2

(kminus1sumi=1

ξnminusi

)

= Pnnminusk

(PBnminus2

ξnminus1 +

kminus1sumi=2

ξnminusi

)

= Pnnminusk

(kminus1sumi=2

ξnminusi


Thereforesumkminus1





thatsumjminus1










i=0

λjxnminusj

∥∥∥∥ le K


λjxnminusj

∥∥∥∥ (8)


oplusinfin


soplusi=0

Mnminusiperptoplus

i=s+1

Mnminusi


ssumi=0

λjxnminusjperptsum

i=s+1

λjxnminusj (9)


λjxnminusj + a

tsumi=s+1

λjxnminusj

∥∥∥∥ ge ∥∥∥∥ ssumi=0

λjxnminusj


i=0

λjxnminusj

∥∥∥∥ =


λjxnminusj +

tsumi=s+1

λjxnminusj

∥∥∥∥ ge ∥∥∥∥ ssumi=0

λjxnminusj




(xt |Ltminus1

)=suminfin



i=1






)=

infinsumi=2


)minus

infinsumi=2

ψt+1iZt+1tminusi


)




)= P


) | Ltminus1

= P

ψt+11Zt+1t + P

(Zt+1t+1 |Lt

) | Ltminus1





)=

infinsumi=3

ψt+2iZt+2t+2minusi


)minus

infinsumi=3

ψt+2iZt+2t+2minusi


)= P


) |Lt

| Ltminus1

]= ψt+22P

(Zt+2t | Ltminus1

)









= P (xt |Ltminusk)

ACKNOWLEDGEMENTS


REFERENCES
























































APPENDIX


Bn =

(kminus1sumi=0

Nnminusi










and Vnki see the


limkirarrinfin

Ynki= Yn lim

kirarrinfinVnki

= Vn









suminfin










oplusinfin









=kminus1sumi=0

Nnminusi + Bnminusk




i=0 Nnminusi


Xn =kminus1sumi=0

ξnminusi + Vnk



(kminus1sumi=0

ξnminusi + Vnk

)

= Pnnminust

(tminus1sumi=0

ξnminusi

)+

kminus1sumi=t

ξnminusi + Vnk

=

kminus1sumi=t

ξnminusi + Vnk

where Pnnminust







(kminus1sumi=0

ξnminusi + Vnk

)

= Pnnminust

(Pnnminuss

sminus1sumi=0

ξnminusi +

kminus1sumi=s

ξnminusi + Vnk

)

= Pnnminust

(kminus1sumi=s

ξnminusi + Vnk

)


= Pnnminust

(tminus1sumi=s

ξnminusi

)+

kminus1sumi=t

ξnminusi + Vnk

=

kminus1sumi=t

ξnminusi + Vnk




jminus1sumi=0


Nnminusi

kminus1sumi=0





Pnnminusk

(kminus1sumi=0

ξnminusi


(kminus1sumi=0

ξnminusi

)

= Pnnminusk

(Pnnminus1ξn +

kminus1sumi=1

ξnminusi

)

= Pnnminusk

(kminus1sumi=1

ξnminusi

)


Pnnminusk

(kminus1sumi=0

ξnminusi

)= Pnnminusk

(kminus1sumi=1

ξnminusi

)

= Pnnminusk

PBnminus2

(kminus1sumi=1

ξnminusi

)

= Pnnminusk

(PBnminus2

ξnminus1 +

kminus1sumi=2

ξnminusi

)

= Pnnminusk

(kminus1sumi=2

ξnminusi


Thereforesumkminus1





thatsumjminus1










i=0

λjxnminusj

∥∥∥∥ le K


λjxnminusj

∥∥∥∥ (8)


oplusinfin


soplusi=0

Mnminusiperptoplus

i=s+1

Mnminusi


ssumi=0

λjxnminusjperptsum

i=s+1

λjxnminusj (9)


λjxnminusj + a

tsumi=s+1

λjxnminusj

∥∥∥∥ ge ∥∥∥∥ ssumi=0

λjxnminusj


i=0

λjxnminusj

∥∥∥∥ =


λjxnminusj +

tsumi=s+1

λjxnminusj

∥∥∥∥ ge ∥∥∥∥ ssumi=0

λjxnminusj




(xt |Ltminus1

)=suminfin



i=1






)=

infinsumi=2


)minus

infinsumi=2

ψt+1iZt+1tminusi


)




)= P


) | Ltminus1

= P

ψt+11Zt+1t + P

(Zt+1t+1 |Lt

) | Ltminus1





)=

infinsumi=3

ψt+2iZt+2t+2minusi


)minus

infinsumi=3

ψt+2iZt+2t+2minusi


)= P


) |Lt

| Ltminus1

]= ψt+22P

(Zt+2t | Ltminus1

)









= P (xt |Ltminusk)

ACKNOWLEDGEMENTS


REFERENCES
























































oplusinfin









=kminus1sumi=0

Nnminusi + Bnminusk




i=0 Nnminusi


Xn =kminus1sumi=0

ξnminusi + Vnk



(kminus1sumi=0

ξnminusi + Vnk

)

= Pnnminust

(tminus1sumi=0

ξnminusi

)+

kminus1sumi=t

ξnminusi + Vnk

=

kminus1sumi=t

ξnminusi + Vnk

where Pnnminust







(kminus1sumi=0

ξnminusi + Vnk

)

= Pnnminust

(Pnnminuss

sminus1sumi=0

ξnminusi +

kminus1sumi=s

ξnminusi + Vnk

)

= Pnnminust

(kminus1sumi=s

ξnminusi + Vnk

)


= Pnnminust

(tminus1sumi=s

ξnminusi

)+

kminus1sumi=t

ξnminusi + Vnk

=

kminus1sumi=t

ξnminusi + Vnk




jminus1sumi=0


Nnminusi

kminus1sumi=0





Pnnminusk

(kminus1sumi=0

ξnminusi


(kminus1sumi=0

ξnminusi

)

= Pnnminusk

(Pnnminus1ξn +

kminus1sumi=1

ξnminusi

)

= Pnnminusk

(kminus1sumi=1

ξnminusi

)


Pnnminusk

(kminus1sumi=0

ξnminusi

)= Pnnminusk

(kminus1sumi=1

ξnminusi

)

= Pnnminusk

PBnminus2

(kminus1sumi=1

ξnminusi

)

= Pnnminusk

(PBnminus2

ξnminus1 +

kminus1sumi=2

ξnminusi

)

= Pnnminusk

(kminus1sumi=2

ξnminusi


Thereforesumkminus1





thatsumjminus1










i=0

λjxnminusj

∥∥∥∥ le K


λjxnminusj

∥∥∥∥ (8)


oplusinfin


soplusi=0

Mnminusiperptoplus

i=s+1

Mnminusi


ssumi=0

λjxnminusjperptsum

i=s+1

λjxnminusj (9)


λjxnminusj + a

tsumi=s+1

λjxnminusj

∥∥∥∥ ge ∥∥∥∥ ssumi=0

λjxnminusj


i=0

λjxnminusj

∥∥∥∥ =


λjxnminusj +

tsumi=s+1

λjxnminusj

∥∥∥∥ ge ∥∥∥∥ ssumi=0

λjxnminusj




(xt |Ltminus1

)=suminfin



i=1






)=

infinsumi=2


)minus

infinsumi=2

ψt+1iZt+1tminusi


)




)= P


) | Ltminus1

= P

ψt+11Zt+1t + P

(Zt+1t+1 |Lt

) | Ltminus1





)=

infinsumi=3

ψt+2iZt+2t+2minusi


)minus

infinsumi=3

ψt+2iZt+2t+2minusi


)= P


) |Lt

| Ltminus1

]= ψt+22P

(Zt+2t | Ltminus1

)









= P (xt |Ltminusk)

ACKNOWLEDGEMENTS


REFERENCES























































= Pnnminust

(tminus1sumi=s

ξnminusi

)+

kminus1sumi=t

ξnminusi + Vnk

=

kminus1sumi=t

ξnminusi + Vnk




jminus1sumi=0


Nnminusi

kminus1sumi=0





Pnnminusk

(kminus1sumi=0

ξnminusi


(kminus1sumi=0

ξnminusi

)

= Pnnminusk

(Pnnminus1ξn +

kminus1sumi=1

ξnminusi

)

= Pnnminusk

(kminus1sumi=1

ξnminusi

)


Pnnminusk

(kminus1sumi=0

ξnminusi

)= Pnnminusk

(kminus1sumi=1

ξnminusi

)

= Pnnminusk

PBnminus2

(kminus1sumi=1

ξnminusi

)

= Pnnminusk

(PBnminus2

ξnminus1 +

kminus1sumi=2

ξnminusi

)

= Pnnminusk

(kminus1sumi=2

ξnminusi


Thereforesumkminus1





thatsumjminus1










i=0

λjxnminusj

∥∥∥∥ le K


λjxnminusj

∥∥∥∥ (8)


oplusinfin


soplusi=0

Mnminusiperptoplus

i=s+1

Mnminusi


ssumi=0

λjxnminusjperptsum

i=s+1

λjxnminusj (9)


λjxnminusj + a

tsumi=s+1

λjxnminusj

∥∥∥∥ ge ∥∥∥∥ ssumi=0

λjxnminusj


i=0

λjxnminusj

∥∥∥∥ =


λjxnminusj +

tsumi=s+1

λjxnminusj

∥∥∥∥ ge ∥∥∥∥ ssumi=0

λjxnminusj




(xt |Ltminus1

)=suminfin



i=1






)=

infinsumi=2


)minus

infinsumi=2

ψt+1iZt+1tminusi


)




)= P


) | Ltminus1

= P

ψt+11Zt+1t + P

(Zt+1t+1 |Lt

) | Ltminus1





)=

infinsumi=3

ψt+2iZt+2t+2minusi


)minus

infinsumi=3

ψt+2iZt+2t+2minusi


)= P


) |Lt

| Ltminus1

]= ψt+22P

(Zt+2t | Ltminus1

)









= P (xt |Ltminusk)

ACKNOWLEDGEMENTS


REFERENCES
























































i=0

λjxnminusj

∥∥∥∥ le K


λjxnminusj

∥∥∥∥ (8)


oplusinfin


soplusi=0

Mnminusiperptoplus

i=s+1

Mnminusi


ssumi=0

λjxnminusjperptsum

i=s+1

λjxnminusj (9)


λjxnminusj + a

tsumi=s+1

λjxnminusj

∥∥∥∥ ge ∥∥∥∥ ssumi=0

λjxnminusj


i=0

λjxnminusj

∥∥∥∥ =


λjxnminusj +

tsumi=s+1

λjxnminusj

∥∥∥∥ ge ∥∥∥∥ ssumi=0

λjxnminusj




(xt |Ltminus1

)=suminfin



i=1






)=

infinsumi=2


)minus

infinsumi=2

ψt+1iZt+1tminusi


)




)= P


) | Ltminus1

= P

ψt+11Zt+1t + P

(Zt+1t+1 |Lt

) | Ltminus1





)=

infinsumi=3

ψt+2iZt+2t+2minusi


)minus

infinsumi=3

ψt+2iZt+2t+2minusi


)= P


) |Lt

| Ltminus1

]= ψt+22P

(Zt+2t | Ltminus1

)









= P (xt |Ltminusk)

ACKNOWLEDGEMENTS


REFERENCES

























































)= P


) | Ltminus1

= P

ψt+11Zt+1t + P

(Zt+1t+1 |Lt

) | Ltminus1





)=

infinsumi=3

ψt+2iZt+2t+2minusi


)minus

infinsumi=3

ψt+2iZt+2t+2minusi


)= P


) |Lt

| Ltminus1

]= ψt+22P

(Zt+2t | Ltminus1

)









= P (xt |Ltminusk)

ACKNOWLEDGEMENTS


REFERENCES



























































































































Date post:	26-Sep-2020
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Strong orthogonal decompositions and non-linear impulse...

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