Generalized Methods of Integrated Moments for HighFrequency Data


 Eric Carroll
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1 Geeralzed Methods of Itegrated Momets for HghFrequecy Data Ja L Duke Uversty Dacheg Xu Chcago Booth Ths Verso: February 14, 214 Abstract We study the asymptotc ferece for a codtoal momet equalty model usg hghfrequecy data sampled wth a fxed tme spa. The model volves the latet spot varace of a asset as a covarate. We propose a twostep semparametrc ferece procedure by frst oparametrcally recoverg the volatlty path from asset returs ad the coductg ferece by matchg tegrated momet codtos. We show that, due to the frststep estmato error, a bascorrecto s eeded for the sample momet codto to acheve asymptotc (mxed) ormalty. We provde feasble ferece procedures for the model parameter ad establsh ther asymptotc valdty. Emprcal applcatos o VIX prcg ad the volatltyvolume relatoshp are provded to llustrate the use of the proposed method. Keywords: hgh frequecy data; semmartgale; VIX; spot volatlty; bas correcto; GMM. JEL Codes: C22. Ths paper supersedes our workg paper prevously crculated uder the ttle Spot Varace Regressos, wth substatally more geeral results. We are grateful to Yace AïtSahala, Torbe Aderse, Federco Bad, Ala Bester, Tm Bollerslev, Federco Bug, Chrs Hase, Olver Lto, Per Myklad, Adrew Patto, Erc Reault, Jeff Russell, Ele Tamer, George Tauche, Vktor Todorov, La Zhag, as well as may semar ad coferece partcpats at the Uversty of Chcago, Brow Uversty, the 212 Tragle Ecoometrcs Coferece, the 213 Facal Ecoometrcs Coferece at Toulouse School of Ecoomcs, the 6th Aual SoFE Coferece, ad the 213 workshop o Measurg ad Modelg Facal Rsk wth Hgh Frequecy Data at EUI for ther helpful commets. L s work was partally supported by NSF Grat SES Xu s work was supported part by the FMC Faculty Scholar Fud at the Uversty of Chcago Booth School of Busess. Box 997, Duke Uversty, Durham, NC, Emal: Uversty of Chcago Booth School of Busess, 587 S. Woodlaw Aveue, Chcago, IL Emal: 1
2 1 Itroducto Iferece methods based o momet equaltes have bee a powerful tool emprcal ecoomsts arseal sce the veto of the geeralzed method of momets (GMM) (Hase (1982), Hase ad Sgleto (1982)). I ther applcato, momet codtos ofte arse from codtoal momet equaltes as orthogoalty codtos betwee strumets ad radom dsturbaces. Asymptotc propertes of these methods are determed by the propertes of sample momets, whch are well kow (Whte (21)) the classcal large T settg wth a asymptotcally expadg tme spa. I ths paper, we study a ovel varat of the GMM for estmatg codtoal momet equalty models usg hghfrequecy (traday) data that are sampled wth a relatvely short sample perod. We derve a asymptotc theory a settg where data are sampled at asymptotcally creasg frequeces wth a fxed tme spa, allowg for geeral forms of depedece ad heterogeety the data. Our study s maly motvated by facal applcatos such as the estmato of certa types of opto prcg models ad market mcrostructure models, where hghfrequecy data are rapdly becomg more readly avalable. A mportat aspect of facal models s that they ofte volve volatlty processes of facal tme seres. Ths s ot surprsg sce volatlty s the prmary measure of rsk moder face (Egle (24)). Sce volatlty s uobservable, ts appearace the model poses a substatal challege for ferece. The commo soluto to the latet volatlty problem s to mpose auxlary parametrc restrctos o volatlty dyamcs; see Bollerslev, Egle, ad Nelso (1994), Ghysels, Harvey, ad Reault (1995) ad Shephard (25) for revews. Sce a correct parametrc specfcato of the auxlary model may affect the ferece of the prmary model, t s prudet to cosder a oparametrc approach as a complemet. 1 Ideed, a large lterature o oparametrc ferece for volatlty has emerged durg the past decade by haressg the rch formato hghfrequecy data; see Jacod ad Protter (212), Hautsch (212) ad Aderse, Bollerslev, Chrstofferse, ad Debold (213) for recet revews. Ths paper proposes a smple, yet geeral, twostep semparametrc procedure for estmatg codtoal momet equalty models that clude volatlty as a latet varable. I the frst step, we oparametrcally recover the volatlty process from hghfrequecy asset returs va a spot realzed varace estmator (Foster ad Nelso (1996), Comte ad Reault (1998)) wth trucato for prce jumps (Mac (21), Jacod ad Protter (212)). I the secod step, we costruct sample versos of strumeted codtoal momet equaltes. Ulke the classcal GMM, the populato momet codto here takes form of a tegrated stochastc process that volves the 1 Although t s subject to the rsk of msspecfcato, a tght parametrc specfcato may have several advatages over a oparametrc approach, such as better statstcal effcecy, better fte ad outofsample performace, smplcty of terpretato ad realtme cotrol, etc. Pseudotrue parameters (Whte (1982)) for msspecfed parametrc models may be worth cosderg practce as well. 2
3 spot varace ad other state varables over a fxed tme spa, stead of a ucodtoal momet. We thus refer to the proposed framework as the geeralzed method of tegrated momets (GMIM). The GMIM estmator for a ftedmesoal model parameter s costructed as the mmzer of a sample crtero fucto of the quadratc form. Our aalyss also exteds the scope of the hghfrequecy lterature o volatlty estmato: whle pror work focused o the ferece of the volatlty tself, we treat ts estmato oly as a prelmary step ad maly cosder the subsequet ferece of parameters ecoomc models. Sce we treat the volatlty process a oparametrc maer, our method s semparametrc ths partcular aspect. The key dstctve feature of our semparametrc procedure s that the oparametrc object here (.e., the volatlty process) s a osmooth stochastc process rather tha a smooth determstc fucto. Ideed, the sample path of the volatlty process a typcal stochastc volatlty model (Hesto (1993), Duffe, Pa, ad Sgleto (2)) s owhere dfferetable because of Browa volatlty shocks ad s ofte dscotuous due to volatlty jumps. Ths feature gves rse to a terestg theoretcal result: the frststep volatlty estmato leads to a large bas the sample momet fucto, the sese that the bas caot be made asymptotcally eglgble the dervato of cetral lmt theorems by just restrctg the asymptotc behavor of tug parameters. We hece cosder a explct bascorrecto to the sample momet fucto ad show that the bascorrected sample momet fucto ejoys a cetral lmt theorem. Ths result exteds the theory of Jacod ad Rosebaum (213) ad s oe of our ma techcal cotrbutos. I cotrast, typcal kerel or sevebased methods, the bas from the oparametrc estmato ca be tued to be asymptotcally small by udersmoothg (or overfttg) the ukow fucto, uder the assumpto that the fucto s suffcetly smooth; see, for example, Newey (1994) ad Gaglard, Gouréroux, ad Reault (211). The GMIM estmator s costructed usg the bascorrected sample momet fucto. We show that the GMIM estmator s cosstet ad has a mxed Gaussa asymptotc dstrbuto. The asymptotc covarace matrx s radom ad cossts of two addtve compoets. The frst compoet s due to the radom dsturbaces (e.g., prcg errors a opto prcg model) that mplctly defe the codtoal momet equaltes. We allow the radom dsturbace to be serally weakly depedet ad propose a heteroskedastcty ad autocorrelato cosstet (HAC) estmator for t. The HAC estmator s ostadard (cf. Newey ad West (1987)) due to ts volvemet wth dscretzed processes cludg, partcular, the latet volatlty process, a fll asymptotc settg. The secod compoet s cotrbuted by the frststep estmato error, for whch ew cosstet estmators are also provded closed form. Overdetfcato tests (Hase (1982)) ad Aderso Rub type cofdece sets (Aderso ad Rub (1949), Stock ad Wrght (2), Adrews ad Soares (21)) are also dscussed as byproducts. We llustrate the proposed method wth two emprcal applcatos. The frst applcato 3
4 cocers the prcg of the CBOE volatlty dex (VIX). We explot a smple dea: a large (but far from exhaustve) class of structural models for the rskeutral volatlty dyamcs wth lear meareverso mples that the squared VIX s lear the spot varace of the S&P 5 dex. We test the specfcato of ths class of models va the GMIM overdetfcato test ad fd that these models are rejected 14 out of 23 quarters (27Q1 212Q3) at the 5% sgfcace level. I the secod applcato, we vestgate the relatoshp betwee retur varace ad tradg volume for stock data. Usg daly data, Aderse (1996) foud that a codtoal Posso model for tradg volume s broadly cosstet wth data ad outperforms early models cosdered by Tauche ad Ptts (1983) ad Harrs (1986). We estmate ad coduct specfcato tests for these models usg hghfrequecy data uder the GMIM framework ad fd further support for the fdgs of Aderse (1996). Ths paper s orgazed as follows. Secto 2 presets the settg. Secto 3 presets the ma theory. Secto 4 shows smulato results, followed by two emprcal applcatos Secto 5. We dscuss related lterature Secto 6. Secto 7 cocludes. The appedx cotas all proofs. 2 Geeralzed method of tegrated momets 2.1 The settg We observe a data sequece (X t, Z t, Y t ) at dscrete tmes t =,, 2,... wth a fxed tme spa [, T ], wth the samplg terval asymptotcally. I applcatos, X t typcally deotes the (logarthmc) asset prce, Z t deotes observable state varables ad Y t deotes depedet varables such as prces of dervatve cotracts, tradg volumes, etc. I ths subsecto, we formalze the probablstc settg uderlyg our aalyss, wth cocrete emprcal examples gve Secto 2.2. Let (Ω (), F, (F t ) t, P () ) be a fltered probablty space. Wthout further meto, we assume that all processes defed o ths space are càdlàg (.e., rght cotuous wth left lmt) adapted ad take values some ftedmesoal real space. We edow ths probablty space wth the processes X t, Z t ad β t that, respectvely, take values X, Z ad B. The process β t s ot observable; stead, we observe Y = Y (β, χ ), =,..., [T/ ], (2.1) where χ s a radom dsturbace, Y ( ) s a determstc trasform takg values a ftedmesoal real space Y ad [T/ ] s the teger part of T/. We shall assume the radom dsturbaces (χ ) to be Fcodtoally statoary ad weakly depedet. To be precse, we descrbe the formal settg as follows. We cosder aother probabl 4
5 ty space (Ω (1), G, P (1) ) that s edowed wth a statoary ergodc sequece (χ ) Z, where Z deotes the set of tegers ad χ takes value a Polsh space wth ts margal law deoted by P χ. We stress from the outset that we do ot assume the sequece (χ ) to be serally depedet. Let Ω = Ω () Ω (1) ad P = P () P (1). Processes defed o each space, Ω () or Ω (1), are exteded the usual way to the product space (Ω, F G, P), whch serves as the probablty space uderlyg our aalyss. For the sake of otatoal smplcty, we detfy the σfelds F ad F t wth ther trval extesos F {, Ω (1)} ad F t {, Ω (1)} o the product space. By costructo, the sequece (χ ) Z s depedet of F. We ote that the varable Y s a osy trasform of β wth χ beg the cofoudg radom dsturbace. I ts smplest form, (2.1) may have a sgalplusose appearace: Y = β +χ. That oted, (2.1) ofte takes more complcated forms may applcatos, as llustrated by the examples Secto 2.2. Heurstcally, the formulato (2.1) hghlghts two dstct model compoets for the sequece (Y ) : formato sde the formato set F (e.g., F codtoal temporal heterogeety) s captured by the process β t ad formato outsde F s captured by (χ ). 2 The basc regularty codto for the uderlyg processes s the followg. Assumpto H: () The process X t s a oedmesoal Itô semmartgale o (Ω (), F, (F t ) t, P () ) wth the form X t = X + t b s ds + t σ s dw s + t R δ (s, z) µ (ds, dz), where the process b t s locally bouded; the process σ t s strctly postve; W t s a stadard Browa moto; δ : Ω R + R R s a predctable fucto ad µ s a Posso radom measure wth compesator ν of the form ν (dt, dz) = dt λ (dz) for some σfte measure λ o R. Moreover, for some costat r (, 1), a sequece of stoppg tmes (T m ) m 1 ad λtegrable determstc fuctos (J m ) m 1, we have δ(ω (), t, z) r 1 J m (z) for all ω () Ω (), t T m ad z R. () The process Z t (β t, Z t, σ t) s also a Itô semmartgale o (Ω (), F, (F t ) t, P () ) wth the form t Z t = Z + t + t + R R t bs ds + σ s d W s δ (s, z) 1 { δ(s,z) 1} (µ ν) (ds, dz) δ (s, z) 1 { δ(s,z) >1} µ (ds, dz), 2 Ths formal settg for troducg weakly depedet radom dsturbaces to hghfrequecy data has bee cosdered by, for example, Jacod, L, ad Zheg (213), who cosder Y ( ) wth a locatoscale form. 5
6 where b t ad σ t are locally bouded processes, W t s a (multvarate) Browa moto ad δ s a predctable fucto such that for some determstc λtegrable fucto J m : R R, δ(ω (), t, z) 2 1 J m (z) for all ω () Ω (), t T m ad z R. The key codto Assumpto H s that the process X t s a Itô semmartgale. I applcatos, X t s typcally the (logarthmc) prce of a asset ad σ t s ts stochastc volatlty process. We set V t σt 2 ad refer to t as the spot varace process; t takes values V (, ). Assumpto H accommodates may models face ad s commoly used for dervg fll asymptotc results for hghfrequecy data; see, for example, Jacod ad Protter (212) ad the refereces there. There s o statoarty requremet o the processes X t, β t, Z t ad σ t. Although the sequece χ s statoary, the sequece Y s allowed to be hghly ostatoary through ts depedece o β. Assumpto H also allows for prce ad volatlty jumps ad mposes o restrcto o the depedece amog varous compoets of studed processes. I partcular, the Browa shocks dw t ad d W t ca be correlated, whch accommodates the leverage effect (Black (1976)). The costat r Assumpto H() serves as a upper boud for the geeralzed Blumethal Getoor dex, or the actvty, of jumps. Assumpto H() also restrcts the processes β t, Z t ad σ t to be Itô semmartgales. We ote that ths assumpto accommodates stochastc volatlty models wth multple factors (see, e.g., Cherov, Gallat, Ghysels, ad Tauche (23)), provded that each factor s a Itô semmartgale. Ths assumpto also allows geeral forms for volatltyofvolatlty ad volatlty jumps, where the latter may have fte actvty ad eve fte varato. Whle Assumpto H() admts may volatlty models face, t does exclude a mportat class of logmemory volatlty models that are drve by fractoal Browa moto; see Comte ad Reault (1996, 1998). The geeralzato ths drecto seems to deserve a focused research o ts ow ad s left to future study. 2.2 The codtoal momet equalty model ad examples The prmary terest of ths paper s the asymptotc ferece for a ftedmesoal parameter θ that satsfes the followg codtoal momet equalty: E [ψ (Y, Z, V ; θ ) F] =, almost surely (a.s.), (2.2) where ψ : Y Z V R q 1, q 1 1, s a measurable fucto wth a kow fuctoal form up to the ukow parameter θ, ad the codtoal expectato tegrates out the radom dsturbace χ. We suppose that the true parameter θ s determstc ad takes value a compact parameter space Θ R dm(θ ). I the sequel, we use θ to deote a geerc elemet Θ. The traspose of a matrx A s deoted by A. To motvate model (2.2), we cosder a few emprcal examples. 6
7 Example 1 (Lear regresso model): Let X t deote the logarthm of the S&P 5 dex ad let VIX t deote the CBOE volatlty dex. We set Y t VIX 2 t. For a large (but far from exhaustve) class of rskeutral dyamcs for the spot varace process V t, the theoretcal value of the squared VIX has a lear form θ1 + θ 2 V t; see Secto 5.1 for detals. Emprcally, we ca model the observed Y as the theoretcal prce plus a prcg error a χ, that s, Y = θ1 + θ2v + a χ, E[χ F] =, E[χ 2 F] = 1, (2.3) where we allow the scalg factor a t of the prcg error to be stochastc wth the codto E[χ 2 F] = 1 beg a ormalzato. Note that (2.3) ca be wrtte the form of (2.1) wth β t (θ1 + θ 2 V t, a t ), where Y ( ) takes a locatoscale form. The prcg error a χ s troduced to capture prce compoets that stadard rskeutral prcg models do ot ted to capture. The prcg errors ca be serally depedet as we allow the process a t ad the sequece (χ ) both to be serally depedet a oparametrc maer; allowg for geeral statstcal structure o the prcg errors s mportat, as emphaszed by Bates (2). By settg ψ (Y t, V t ; θ) = Y t θ 1 θ 2 V t, we verfy (2.2). Example 2 (Nolear regresso model): Let X t be the prce process of a uderlyg asset ad Y t be the prce vector of q 1 optos wrtte o t. We set Z t = (t, X t, r t, d t ) where r t s the short terest rate ad d t s the dvded yeld. If, uder the rskeutral measure, the process (Z t, V t ) s Markova, 3 the the theoretcal prces of the collecto of q 1 optos ca be wrtte as a R q 1 valued fucto f (Z t, V t ; θ ), where θ arses from the rskeutral model for the dyamcs of the state varables. Emprcally, t s commo to model the observed opto prce vector Y t as the theoretcal prce plus a prcg error, that s, Y = f (Z, V ; θ ) + a χ, E[χ F] =, E[χ χ F] = I q 1, (2.4) where a t s a q 1 q 1 matrxvalued process that deotes the stochastc covolatlty of the prcg errors wth the codto E[χ χ F] = I q 1 beg a ormalzato. Note that (2.4) ca be wrtte the form of (2.1) wth β t (β 1,t, β 2,t ), β 1,t f(z t, V t ; θ ) ad β 2,t vec(a t ), where vec( ) deotes the vectorzato operator. Settg ψ (Y t, Z t, V t ; θ) = Y t f (Z t, V t ; θ), we verfy (2.2). 3 Assumg that V t s the oly uobservable Markov state varable excludes dervatve prcg models wth multple volatlty factors uder the rskeutral measure, whch have bee cosdered by, for example, Chrstofferse, Hesto, ad Jacobs (29), Bates (212) ad Aderse, Fusar, ad Todorov (213). Note that ths assumpto does ot mply (Z t, V t) s Markov uder the physcal measure (.e. P), as the equvalece betwee measures mposes lttle restrcto o drft ad jump compoets of (Z t, V t). Hece, t s useful to cosder the geeral Itô semmartgale settg (Assumpto H) uder the physcal measure eve f oe mposes addtoal restrctos uder the rskeutral measure. 7
8 Example 3 (Parametrzed codtoal heteroskedastcty): Cosder the same settg as Example 2. The process A t = vec(a t a t ) s a ecoomcally relevat quatty as t ca be terpreted as a summary measure of market qualty (Hasbrouck (1993), AïtSahala ad Yu (29)). To vestgate whether A t depeds o other state varables, oe may further model A t as A t = h (Z t, V t ; θ ) for some determstc fucto h ( ). 4 The we ca verfy (2.2) by settg ψ (Y t, Z t, V t ; θ) = ( Y t f (Z t, V t ; θ) vec((y t f (Z t, V t ; θ)) (Y t f (Z t, V t ; θ)) ) h (Z t, V t ; θ) ). Example 4 (Scaled Posso regresso model): Aderse (1996) proposes a Posso model for the volatlty volume relatoshp for daly data, whch the codtoal dstrbuto of daly volume gve the retur varace s a scaled Posso dstrbuto. Here, we cosder a verso of hs model for traday data. Let Y the terval [, ( + 1) ). Suppose that Y V deote the tradg volume of a asset wth θ 1 Posso(θ 2 + θ 3 V ). To cast ths model the form (2.1), we represet the Posso dstrbuto wth tmevaryg mea terms of a tmechaged Posso process: let χ = (χ (β)) β be a stadard Posso process dexed by β ad the set β t θ 2 + θ 3 V t ad Y = θ 1 χ (β ). I Secto 5.2, we estmate ths model by usg the frst two codtoal momets of Y t. Ths amouts to settg ψ (Y t, V t ; θ) = whch readly verfes (2.2). ( Y t θ 1 (θ 2 + θ 3 V t ) Y 2 t θ 2 1 (θ 2 + θ 3 V t ) 2 θ 2 1 (θ 2 + θ 3 V t ) ), (2.5) As show the above examples, the codtoal momet equalty model (2.2) arses a varety of emprcal settgs. These settgs aturally volve the spot varace process V t, but are agostc regardg the precse form of ts dyamcs (uder the physcal measure). Ths reaffrms the relevace of cludg V t (2.2) ad treatg t oparametrcally our ecoometrc theory. We also ote that t s desrable to allow the studed processes to be ostatoary these emprcal settgs. For example, opto prcg usually cludes tme ad the uderlyg asset prce as observed state varables, both of whch reder the process Z t ostatoary. Moreover, whle t may be reasoable to assume that the stochastc volatlty process s statoary the classcal larget settg for daly or weakly data, the statoarty assumpto s more restrctve for hghfrequecy data due to tradaly seasoaltes. Fally, we ote that whle X do allow Y as Examples 1 3, Y s assumed to be observed wthout mcrostructure ose, we to be osy a qute geeral fasho. I partcular, opto prcg settgs such has the form of a semmartgale plus a ose (.e., prcg error) term, 4 Upo a reparametrzato, we ca assume that f( ) ad h( ) share the same parameter wthout loss of geeralty. 8
9 whch s commoly used the study of oserobust estmatos of tegrated volatlty. 5 Our asymmetrc treatmet for mcrostructure ose X ad Y s reasoably realstc as the opto market s less lqud tha the stock market, so mcrostructure effects play a less mportat role for the latter tha the former Itegrated momet equaltes ad the GMIM estmator Our ferece s based o matchg a set of tegrated momet equaltes that are mpled by (2.2). To costruct these tegrated momet codtos, we cosder a measurable fucto ϕ : Z V Θ R q 2 for some q 2 1. Below, we refer to ϕ ( ) as the strumet. We set q = q 1 q 2 ad cosder a R q valued fucto g (y, z, v; θ) ψ (y, z, v; θ) ϕ (z, v; θ), (2.6) wth whch we assocate ḡ (β, z, v; θ) g (Y (β, χ), z, v; θ) P χ (dχ). (2.7) Sce Z ad V are Fmeasurable, (2.2) mples that E [g(y, Z, V ; θ ) F] = or, equvaletly, ḡ (β, Z, V ; θ ) =, =,..., [T/ ]. (2.8) If ḡ (β, z, v; θ) s cotuous (β, z, v), the the process (ḡ (β t, Z t, V t ; θ)) t s càdlàg, so we ca defe G (θ) T ḡ (β s, Z s, V s ; θ) ds, θ Θ. (2.9) By (2.8) ad a Rema approxmato, we obta a vector of tegrated momet equaltes gve by G (θ ) =. (2.1) I Secto 3.2, we costruct a estmator G ( ) for the radom fucto G ( ) ad show that G ( ) coverges probablty toward G ( ) uformly. Followg Sarga (1958) ad Hase (1982), 5 See, for example, Zhag, Myklad, ad AïtSahala (25), Hase ad Lude (26), Bad ad Russell (28), BardorffNelse, Hase, Lude, ad Shephard (28), Jacod, L, Myklad, Podolskj, ad Vetter (29) ad Xu (21). 6 We ote that our aalyss s based o geeral tegrated volatlty fuctoals, for whch lttle s kow osy settgs the curret lterature. To the best of our kowledge, the most geeral class of estmators s the preaveragg method of Jacod, Podolskj, ad Vetter (21), whch ca be used to estmate tegrated volatlty fuctoals of the form T V j s ds for postve teger j. Ths class of tegrated volatlty polyomals, however, s qute restrctve for our purpose of estmatg geeral olear models. Sce estmatg geeral tegrated volatlty fuctoals the osy settg s a very challegg task by tself, we leave the exteso wth osy X to future research, so as to focus o the ma dea of the curret paper. 9
10 we estmate θ by makg G (θ) as close to zero as possble accordg to some metrc. precsely, we cosder a sequece Ξ of weghtg matrces ad defe the GMIM estmator ˆθ as ˆθ argm θ Θ More Q (θ), where Q (θ) G (θ) Ξ G (θ). (2.11) The GMIM estmator clearly resembles the classcal GMM estmator. Moreover, trasformg the codtoal momet equalty (2.2) to the tegrated momet equalty (2.1) s aalogous to the commo practce of estmatg codtoal momet equalty models by formg ucodtoal momet codtos. That beg sad, there are fudametal dffereces betwee the two settgs. The classcal GMM settg requres a large sample wth a expadg tme spa order to recover the varat dstrbuto of the studed processes. I the fll settg here, we do ot requre the exstece of a varat dstrbuto. I the cotuoustme lmt, the tegrated momet fucto G ( ), rather tha beg a ucodtoal momet, arses aturally as the lmtg, or populato, verso of the sample momet codto. The pheomeo that stochastc lmts take the form of temporally tegrated quattes s commo the ecoometrcs for hghfrequecy data; see Aderse, Bollerslev, Debold, ad Labys (23), BardorffNelse ad Shephard (24a), Jacod ad Protter (212) ad refereces there. As s typcal the hghfrequecy lterature, our fll asymptotc results requre oly mld codtos o the samplepath regularty of the processes β t, Z t, X t ad V t (see Assumpto H), whle allowg for geeral forms of ostatoarty ad depedece; the curret settg s actually oergodc, as the tegrated momet fucto G ( ) s tself a radom fucto. 3 Asymptotc theory I Secto 3.1, we dscuss regularty codtos. I Sectos 3.2 ad 3.3, we preset the key theoretcal results of the curret paper, that s, the asymptotc propertes of the bascorrected sample momet fucto (Secto 3.2) ad cosstet estmators of ts asymptotc covarace matrx (Secto 3.3). Asymptotc results for the GMIM estmator the follow straghtforwardly ad are preseted Secto Assumptos I ths subsecto, we collect ad dscuss some regularty codtos that are used repeatedly the sequel. Ths subsecto s techcal ature ad may be skpped by readers terested our ma results durg ther frst readg. 1
11 Assumpto MIX: The sequece (χ ) Z s statoary ad αmxg wth mxg coeffcet α mx ( ) of sze k/ (k 2) for some k > 2. 7 Assumpto MIX mposes a mxg codto o the sequece (χ ) Z so that, codtoal o F, the sequece (Y ) s also αmxg wth mxg coeffcets bouded by α mx ( ). Note that Assumpto MIX oly cocers (χ ) Z. We do ot eed processes defed o (Ω (), F, P () ) to be mxg. Our use of αmxg coeffcets s oly for cocreteess; other types of mxg cocepts ca also be used. The degree of depedece s cotrolled by the costat k. A larger value of k makes Assumpto MIX weaker, but demads stroger domace codtos as show below (see Assumpto D). We eed some otato for troducg addtoal assumptos. Let deote the Eucldea orm. For j, p 1, θ Θ, β, β B, z, z Z ad v, v V, we set ( 1/p ḡ j,p (β, z, v; θ) vg(y j (β, χ), z, v; θ) p P χ (dχ)), ( ρ p (β, z, v), (β, z, v ) ) (3.1) ( 1/p g(y (β, χ), z, v; θ ) g(y (β, χ), z, v ; θ ) p P χ (dχ)), provded that the jth partal dervatve vg j exsts. The fuctos ḡ j,p ( ) compute the L p orms of g (Y (, χ ),, )) ad ts partal dervatves. The fucto ρ p (, ) computes the L p dstace betwee g (Y (β, χ ), z, v; θ ) ad g (Y (β, χ ), z, v ; θ ) uder the probablty measure P (1). Ths semmetrc s useful for cosderg the smoothess of the Fcodtoal momets (such as the covarace ad autocovarace) of the sequece (g (Y (β, χ ), z, v; θ )) as fuctos of (β, z, v). It s also coveet to troduce a few classes of fuctos. Let A be the collecto of all measurable fuctos that are defed o B Z V ad take values some ftedmesoal real space. For p, we set { } f A : for each bouded set K B Z, there exsts a costat K >, P(p) such that f (β, z, v) K(1 + v p ) for all (β, z) K ad v V ad C (p) {f P(p) : f s cotuous}. We deote by C 2,3 the subclass of fuctos A that are twce cotuously dfferetable (β, z) B Z ad three tmes cotuously dfferetable v V. We the set, for p 3, { } f C C 2,3 2,3 : for each bouded set K B Z, there exsts a costat K >, such (p) that vf j. (β, z, v) K(1 + v p j ) for all (β, z) K, v V ad j =, 1, 2, 3 7 The mxg coeffcets are of sze a, a >, f they decay at polyomal rate a+ε for some ε >. See Defto 3.45 Whte (21). 11
12 The costat K the deftos of P(p) ad C 2,3 (p) s uform wth respect to β ad z, but ths requremet s ot strog, because we oly eed the uformty to hold over a bouded set K ad we allow K to deped o K. 8 The key restrcto o P(p), C (p) ad C 2,3 (p) s that ther member fuctos, as well as the dervatves of these fuctos wth respect to v for the thrd, have at most polyomal growth v. I our aalyss, the argumet v ofte takes value at some estmate of the spot varace, ad the polyomal growth codto s used for cotrollg the effect of approxmato error betwee the spot varace ad ts estmate. Our ma regularty codtos o g ( ) are gve by Assumptos S, D ad LIP below. Assumpto S: () The fucto g (y, z, v; θ) s cotuously dfferetable θ ad twce cotuously dfferetable v; () for some p 3 ad each θ Θ, we have ḡ( ; θ) C 2,3 (p), θ ḡ ( ; θ) C (p) ad θ 2 vḡ ( ; θ) C (p 2); () for each θ Θ ad (β, z, v) B Z V, we have j vḡ(β, z, v; θ) = j vg(y (β, χ), z, v; θ)p χ (dχ) ad θ j vḡ(β, z, v; θ) = θ j vg(y (β, χ), z, v; θ)p χ (dχ) for j =, 1, 2. Assumpto S maly cocers smoothess. Assumpto S() specfes the basc smoothess requremet o the fucto g ( ). Assumpto S() mposes addtoal smoothess codtos o ḡ( ; θ). We cosder ḡ ( ) drectly because, as a tegrated verso of g( ) (recall (2.7)), t s ofte smooth eve f the latter s ot. Assumpto S() s a mld codto that allows us to chage the order betwee dfferetato ad tegrato. We do ot elaborate prmtve codtos for t, because they are well kow. I Assumpto D below, the fucto θ g j,k (, θ) s defed by (3.1) wth g( ) replaced by θ g( ). Assumpto D: For some k > 2, p 3 ad κ (, 1], we have () ḡ,k ( ; θ) P ((p/2) (2p/k)), θ g,k (, θ) P(p) ad ḡ 2,k ( ; θ), θ g 2,k ( ; θ) P (p 2) for each θ Θ; () for ay bouded set K B Z, there exsts a fte costat K > such that, ρ k ( z, z ) K(1 + v p/2 1 + v p/2 1 ) z z κ for all z, z K V wth z z 1, where z (β, z, v) ad z (β, z, v ). Assumpto D s of the domace type. Assumpto D() restrcts the kth Fcodtoal absolute momets to have at most polyomal growth the spot varace ad s maly eeded for usg mxg equaltes. Assumpto D() s a local domace codto for the semmetrc ρ k (, ). Ths codto s weaker whe the Hölder expoet κ s closer to zero. The multplcatve factor K(1 + v p/2 1 + v p/2 1 ) s uform (β, z, β, z ) o bouded sets ad has at most polyomal growth the argumets that correspod to the spot varace. Defto 1 (Class LIP): Let j, p be tegers such that j p. A fucto (y, z, v, θ) g (y, z, v; θ) o Y Z V Θ s sad to be the class LIP(p, j) f, for each j, there exsts a 8 Our theory does ot eed the processes β t ad Z t to be bouded. However, by a localzato argumet, we ca assume these processes to be bouded wthout loss of geeralty whe dervg lmt theorems. 12
13 fucto B (y, z, v) such that vg (y, z, v; θ) vg (y, z, v; θ ) B (y, z, v) θ θ for all θ, θ Θ ad (y, z, v) Y Z V, ad the fucto (β, z, v) B (β, z, v) B (Y (β, χ), z, v) 2 P χ (dχ) belogs to P(p ). Assumpto LIP: () g ( ) LIP(p, 2); () θ g ( ) LIP(p, 2). Assumpto LIP mposes a type of Lpschtz codto for g ( ; θ) ad ts partal dervatves. Ths codto s used for establshg uform (w.r.t. θ) covergece probablty of varous sample momet fuctos. It s also used to show that the effect of replacg the true parameter value wth ts estmate s asymptotcally eglgble the HAC estmato. For cocreteess, we llustrate how to verfy the above regularty codtos the settg of Example 4, whch s the ma focal pot of our umercal work Sectos 4 ad 5. Focusg o ths example s structve because t llustrates the key techcal argumet whch s commo to may applcatos. Example 4 Cotued: To smplfy the dscusso, we take the costat k Assumptos MIX ad D as a teger. We use K to deote a postve costat whch may vary from le to le. We cosder a strumet of the form ϕ(v) = v ι for some teger ι, whle otg that settg ϕ( ) to be scalarvalued s wthout loss of geeralty for the purpose of verfyg Assumptos S, D ad LIP. It s easy to see g(y, v; θ) = ḡ(β, v; θ) = ( ( y θ 1 (θ 2 + θ 3 v) y 2 θ 2 1 (θ 2 + θ 3 v) 2 θ 2 1 (θ 2 + θ 3 v) θ 1 β θ 1 (θ 2 + θ 3 v) ) v ι, θ 2 1 (β + β2 ) θ 2 1 (θ 2 + θ 3 v) 2 θ 2 1 (θ 2 + θ 3 v) Assumpto S s verfed for ay p max{3, ι + 2} by drect specto. ) v ι. By propertes of the Posso dstrbuto, E[ Y t k F] K( β t + β t k ). It s the easy to see that ḡ j,k ( ; θ) P(ι + 2 j) for j {, 1, 2}, so Assumpto D() s verfed for p max{2, k/2}(ι + 2). I addto, for β ad β a bouded set wth β β 1, we have E χ (β) χ (β ) 2k K β β. By the Cauchy Schwarz equalty, E χ (β) 2 χ (β ) 2 k K β β 1/2. It s the easy to see that ρ k ((β, v), (β, v )) K(1 + v ι+1 + v ι+1 )( β β 1/2k + v v ). Hece, Assumpto D() s verfed for κ = 1/2k ad p 2(ι + 2). Turg to Assumpto LIP, we ote that j vg(y, v; θ) j vg (y, v; θ ) K(1 + v ι+2 j ) θ θ for θ, θ the compact set Θ. Assumpto LIP() s verfed for p ι+2. Assumpto LIP() ca be verfed smlarly. To sum up, for ay k > 2, Assumptos S, D ad LIP are verfed for p max{2, k/2}(ι + 2) ad κ = 1/2k. 13
14 3.2 The bascorrected sample momet fucto ad ts asymptotc propertes I ths subsecto, we costruct a sample momet fucto G ( ) for estmatg the tegrated momet fucto G ( ) (2.9). We the preset the asymptotc propertes of G ( ). We frst oparametrcally recover the spot varace V by usg a spot trucated realzed varato estmator. To ths ed, we cosder a sequece k of tegers wth k ad k, whch plays the role of the local wdow for spot varace estmato. The spot varace estmate s gve as follows: for each =,..., [T/ ] k, V 1 k k j=1 ( +j X ) 2 1{ +j X ᾱ ϖ }, where +jx X (+j) X (+j 1), ad ᾱ >, ϖ (, 1/2) are costats that specfy the trucato threshold. Ths estmator s a localzed verso of the estmator proposed by Mac (21), where the trucato s eeded so that the spot varace estmate V s robust to jumps X. 9 Below, we deote N [T/ ] k. We start wth a (seemgly) atural sampleaalogue estmator for G(θ), whch s gve by N ( Ĝ (θ) g Y, Z, V ) ; θ, θ Θ. = Theorem 1 shows that Ĝ( ) s a cosstet estmator for G( ) uder the uform metrc. Theorem 1. Suppose () Assumptos H ad MIX hold for some r (, 1) ad k > 2; () for some p ad each θ Θ, ḡ ( ; θ) C (p) ad ḡ,k ( ; θ) P(p); () f p > 1, we further assume that ϖ (p 1)/(2p r); (v) g( ) LIP(p, ); (v) k ad k. The Ĝ( ) P G ( ) uformly o compact sets. We also eed a cetral lmt theorem for the sample momet fucto (evaluated at θ ), whch s useful for coductg asymptotc ferece. It turs out that the raw sample aalogue Ĝ(θ) does ot admt a cetral lmt theorem due to a hghorder bas; see Corollary 1 below for a formal statemet. Nevertheless, Theorem 1 s useful for establshg the cosstecy of varous estmators, such as that of the asymptotc varace. We hece cosder a bascorrected sample momet fucto gve by G (θ) Ĝ(θ) 1 N ( B (θ), where B (θ) 2 k vg Y, Z, V ) ; θ V 2. (3.2) Ths sample momet fucto s used for defg the GMIM estmator (2.11). As show 9 The estmato of spot varace ca be dated at least back to Foster ad Nelso (1996) ad Comte ad Reault (1998), a settg wthout jumps. Also see Reò (28), Krstese (21), ad refereces there. = 14
15 Theorem 2 below, 1/2 G (θ ) ejoys a cetral lmt theorem wth a mxed Gaussa asymptotc dstrbuto. To descrbe the asymptotc covarace matrx, we eed more otato. For each l, we deote the jot dstrbuto of (χ, χ l ) by P χ,l ad set, for (β, z, v) B Z V, γ l (β, z, v) g(y (β, χ), z, v; θ )g(y (β, χ ), z, v; θ ) P χ,l (dχ, dχ ). (3.3) We the set γ (β, z, v) γ (β, z, v) + Γ T γ (β s, Z s, V s ) ds. (γ l (β, z, v) + γ l (β, z, v) ), l=1 Here, γ l (β, z, v) s the Fcodtoal autocovarace of the sequece g (Y (β, χ ), z, v; θ ) at lag l, ad γ (β, z, v) s the correspodg logru covarace matrx. 1 Fally, we set (3.4) T S 2 v ḡ(β s, Z s, V s ; θ ) v ḡ(β s, Z s, V s ; θ ) Vs 2 ds. (3.5) The (Fcodtoal) asymptotc covarace matrx of 1/2 G (θ ) s gve by Σ g Γ + S, (3.6) where Γ arses from the serally depedet radom dsturbaces (χ ) ad S arses from the frststep samplg error V. We are ow ready to state the asymptotc propertes of G ( ). We the characterze the aforemetoed hghorder bas of the raw estmator Ĝ(θ ) as a drect corollary (Corollary 1). I the sequel, we use Ls to deote Fstable covergece law 11 ad, for a geerc Fmeasurable postve semdefte matrx Σ, we use MN (, Σ) to deote the cetered mxed Gaussa dstrbuto wth Fcodtoal covarace matrx Σ. We shall assume the followg for the local wdow k. Assumpto LW: k 2 ad k 3. 1 The process γ (β t, Z t, V t) may be more properly referred to as the local logru covarace matrx, as t s evaluated locally at tme t. It arses from a large umber of adjacet observatos that are serally depedet (through χ ), but all these observatos are sampled from a asymptotcally shrkg tme wdow. I other words, γ (β t, Z t, V t) s logru tck tme, but local caledar tme. The seres (3.4) s absolutely coverget. Ideed, uder Assumpto MIX, by the mxg equalty, γ (β, z, v) + l=1 γ l(β, z, v) Kḡ,k (β, z, v; θ ) 2. Therefore, γ (β, z, v) s fte wheever ḡ,k (β, z, v; θ ) s fte, for whch Assumpto D suffces. 11 Stable covergece law s slghtly stroger tha the usual oto of weak covergece. It requres that the covergece holds jotly wth ay bouded Fmeasurable radom varable defed o the orgal probablty space. Its mportace for our problem stems from the fact that the lmtg varable of our estmator s a F codtoally Gaussa varable ad stable covergece allows for feasble ferece usg a cosstet estmator for ts Fcodtoal varace. See Jacod ad Shryaev (23) for further detals o stable covergece. 15
16 Theorem 2. Suppose () Assumptos H, MIX, S, D ad LW hold for some r (, 1), k > 2 ad p 3; () ϖ (2p 1)/2(2p r). The (a) uder Assumpto LIP(), B P (θ) T 2 vḡ(β s, Z s, V s ; θ)vs 2 P ds ad G (θ) G(θ), uformly θ o compact sets; (b) 1/2 G (θ ) Ls MN (, Σ g ). Corollary 1. Uder the codtos Theorem 2, k Ĝ (θ ) P T 2 vḡ(β s, Z s, V s ; θ )V 2 s ds. Commets. () Theorem 2(a) shows the uform cosstecy of G ( ). Ths result s a smple cosequece of Theorem 1 ad s used for establshg the cosstecy of the GMIM estmator. () Theorem 2(b) characterzes the stable covergece of 1/2 G (θ ). The rate of covergece s parametrc, as s typcal semparametrc problems. Note that G (θ ) s cetered at zero because of (2.1). We oly cosder G ( ) evaluated at the true value θ because ths s eough for coductg asymptotc ferece o the bass of (2.1). () I the specal case where g (y, z, v; θ ) does ot deped o y ad z, Theorem 2(b) cocdes wth Theorem 3.2 of Jacod ad Rosebaum (213), whch cocers the estmato of tegrated volatlty fuctoals of the form T g(v s)ds. For the same techcal reasos as here, Jacod ad Rosebaum (213) (see (3.6) there) also adopt Assumpto LW to restrct the rage of rates at whch k grows to fty. Jacod ad Rosebaum (213) show 12 that 1/2 Ĝ (θ ) cotas several bas terms of order O p (k ) whch arse from border effects, dffusve movemet of the spot varace process, ad volatlty jumps, wth the latter two beg very dffcult (f possble) to correct. As a cosequece, the codto k 2 s eeded to make these bas terms asymptotcally eglgble. However, a addtoal bas term (whch s characterzed by Corollary 1) remas 1/2 Ĝ (θ ), whch s of the order O p (1/k ) ad s explosve whe k 2. Ths bas term has to be explctly corrected for the purpose of dervg a wellbehaved lmt theorem; the correcto term k 1 B ( ) (3.2) exactly fulflls ths task. 3.3 Estmato of asymptotc covarace matrces I ths subsecto, we descrbe estmators for the asymptotc covarace matrx Σ g. These estmators are essetal for coductg feasble ferece. We start wth the estmato of Γ (recall (3.4)). Let ˆθ be a prelmary estmator of θ. We cosder the sample aalogue of T γ l (β s, Z s, V s ; θ ) ds, l, gve by N Γ l, (ˆθ ) g (Y, Z, V ; ˆθ ) ( g Y ( l), Z ( l), V ( l) ; ˆθ ). (3.7) =l 12 See Theorem 3.1 Jacod ad Rosebaum (213). 16
17 Followg Newey ad West (1987), we cosder a kerel fucto w (j, m ) ad a badwdth sequece m of tegers. The estmator for Γ s the gve by Γ (ˆθ ) Γ m, (ˆθ ) + w (j, m ) ( Γj, (ˆθ ) + Γ j, (ˆθ ) ). (3.8) We eed the followg codto for studyg the asymptotcs of Γ (ˆθ ). j=1 Assumpto HAC: () The kerel fucto w (, ) s uformly boud ad for each j 1, lm m w (j, m) = 1; () m ad m k κ/2, where κ (, 1] s the costat gve Assumpto D; () the fucto ḡ,2k ( ; θ ) s bouded o bouded sets. As Newey ad West (1987), whe the kerel fucto w (, ) s chose properly, Γ (ˆθ ) s postve semdefte fte samples; oe example s to take w (j, m) = 1 j/(m + 1), that s, the Bartlett kerel. I ths paper, we restrct atteto to kerels wth bouded support. It s possble to cosder estmators wth more geeral forms as cosdered by Adrews (1991). Sce the effcet estmato of the asymptotc covarace matrx s ot the prmary focus of the curret paper, we leave ths complcato to a future study. We cosder two estmators for S. The frst estmator s applcable a geeral settg. We choose a sequece of tegers k ad set ˆη (ˆθ ) 1 k 1 k v g (Y (+j), Z (+j), V ; ˆθ ),. j= The varable ˆη (ˆθ ) serves as a approxmato of v ḡ(β, Z, V ; θ ). We the set N ([T/ ] k +1) Ŝ 1, (ˆθ ) 2 = ˆη (ˆθ )ˆη (ˆθ ) V 2. We eed Assumpto AVAR1 below for the cosstecy of Ŝ1,(ˆθ ) toward S. Assumpto AVAR1: k ad k. () ḡ 1,k ( ; θ ), β v ḡ( ; θ ) ad z v ḡ( ; θ ) belog to P (p/2 1); () Assumpto AVAR1() mposes domace codtos for the momets, as well as ther dervatves wth respect to β ad z, of v g(y (, χ),, ; θ ). Assumpto AVAR1() mposes mld codtos o the sequece k. Whle k s allowed to be dfferet from k, settg k = k s a coveet choce. The secod estmator for S s desged to explot a specal structure of regresso models, whch s formalzed by the followg assumpto. 17
18 Assumpto AVAR2: There exsts a fucto (z, v; θ) ϕ (z, v; θ) wth a kow fuctoal form such that for some p 1, () v ḡ (β t, Z t, V t ; θ ) = ϕ (Z t, V t ; θ ) for all t [, T ]; () ϕ( ; θ ) C(p 1); () ϕ( ) LIP(p 1, ). Assumpto AVAR2() posts that the value of v ḡ (β t, Z t, V t ; θ ) ca be computed from the realzatos of Z t ad V t, provded that θ s kow. Assumpto AVAR2() mposes some mld smoothess requremets o ϕ( ; θ ). Assumpto AVAR2() says that ϕ( ) s smooth θ, so that replacg θ wth ts prelmary estmator results a asymptotcally eglgble effect. The example below shows that, a olear regresso settg such as Example 2, Assumpto AVAR2 mposes essetally o addtoal restrctos beyod Assumptos S() ad LIP(). Example 2 Cotued: Uder the settg of Example 2, t s easy to see that v ḡ (β t, Z t, V t ; θ) = (β 1,t f (Z t, V t ; θ)) v ϕ (Z t, V t ; θ) v f(z t, V t ; θ) ϕ (Z t, V t ; θ). We set ϕ(z, v; θ) v f(z, v; θ) ϕ (z, v; θ) ad ote that Assumpto AVAR2() readly follows because β 1,t f (Z t, V t ; θ ). Assumptos AVAR2() ad AVAR2() are related to ad are somewhat weaker tha Assumptos S() ad LIP(), respectvely. To see the coecto, we ote that v ḡ (β, z, v; θ ) = (β 1 f(z, v; θ )) v ϕ (z, v; θ ) + ϕ(z, v; θ ), v g (y, z, v; θ) = (y f (z, v; θ)) v ϕ (z, v; θ) + ϕ(z, v; θ). Whle Assumptos S() ad LIP() mply that v ḡ( ; θ ) C(p 1) ad v g ( ) LIP(p 1, ), Assumptos AVAR2() ad AVAR2() oly requre the secod compoet each of the two dsplayed decompostos above to satsfy the same regularty codto. The secod estmator for S s gve by N Ŝ 2, (ˆθ ) 2 = ϕ(z, V ; ˆθ ) ϕ(z, V ; ˆθ ) V 2. Theorem 3. Suppose () the codtos Theorem 2 () 1/2 (ˆθ θ ) = O p (1). The (a) uder Assumpto HAC, Γ (ˆθ ) Γ; (b) uder Assumpto AVAR1, Ŝ1,(ˆθ ) S; (c) uder Assumpto AVAR2, Ŝ2,(ˆθ P ) S. P P Cosequetly, uder Assumptos HAC ad AVAR1 (resp. AVAR2), Σ g, (ˆθ ) Γ (ˆθ ) + Ŝ 1, (ˆθ ) (resp. Σ g, (ˆθ ) Γ (ˆθ ) + Ŝ2,(ˆθ )) s a cosstet estmator of Σ g. Commets. () The prelmary estmator ˆθ s assumed to be 1/2 cosstet. The GMIM estmator satsfes ths codto; see Proposto 1 below. () The HAC estmator Γ (ˆθ ) s vald uder the assumpto that (χ ) s weakly depedet. If t s kow a pror that (χ ) forms a depedet sequece, the Γ = T γ (β s, Z s, V s ) ds, 18
19 whch ca be cosstetly estmated by Γ, (ˆθ ). Ideed, a termedate step of the proof of Theorem 3(a) s to show that Γ P l, (ˆθ ) T γ l (β s, Z s, V s ) ds for each l. As a drect cosequece of Theorems 2 ad 3, we ca costruct Aderso Rub type cofdece sets for θ by vertg tests. To ths ed, we cosder a fucto L(, ) : R q R q q R ad a test statstc of the form L (θ) L( 1/2 G (θ), Σ g, (θ)), where Σ g, ( ) s gve by Theorem 3. We let α (, 1) deote the sgfcace level. Corollary 2. Suppose () the codtos Theorem 3 hold; () the fucto (u, A) L(u, A) s cotuous at (u, A) for all u R q ad for almost every A uder the dstrbuto of Σ g. The (a) L (θ ) Ls L(ξ, Σ g ), where the varable ξ s defed o a exteso of the space (Ω, F G, P) ad, codtoal o F, s cetered Gaussa wth covarace matrx Σ g. (b) Let U be a geerc qdmesoal stadard ormal varable that s depedet of F G. If, addto, the Fcodtoal dstrbuto of L(ξ, Σ g ) s cotuous ad strctly creasg at ts 1 α quatle cv 1 α, the the 1 α quatle of the F Gcodtoal dstrbuto of L( Σ g, (θ ) 1/2 U, Σ g, (θ )), deoted by cv,1 α (θ ), coverges probablty to cv 1 α. Cosequetly, P (L (θ ) cv,1 α (θ )) 1 α. Corollary 2(a) establshes the asymptotc dstrbuto of the test statstc L (θ ). Corollary 2(b) further shows that cv,1 α forms a asymptotcally vald sequece of crtcal values, as t cosstetly estmates the 1 α Fcodtoal quatle of the lmt varable L (ξ, Σ g ). We ca the costruct a sequece of cofdece sets CS {θ Θ : L (θ) cv,1 α (θ)}. Sce P (L (θ ) cv,1 α (θ )) 1 α, we have P(θ CS ) 1 α. That s, CS forms a sequece of cofdece sets for θ wth asymptotc level 1 α. The cofdece set CS s smlar to that proposed by Stock ad Wrght (2) whe the test statstc takes a quadratc form (.e. L (u, A) = u A 1 u). I ths case, the dstrbuto of the lmt varable L(ξ, Σ g ) s chsquare wth degree of freedom q ad, hece, the crtcal value ca be chose as a costat. Sce CS s of the Aderso Rub type, t s asymptotcally vald eve f θ s oly weakly detfed, wth the lack of detfcato cosdered as a extreme form of weak detfcato. The test statstc may also take other forms, such as the maxmum of tstatstcs (.e., L(u, A) = max 1 j q u j /A 1/2 jj ), as cosdered by Adrews ad Soares (21). I geeral, the crtcal value cv,1 α (θ) depeds o θ ad does ot have a closedform expresso, but t ca be easly computed by smulato. 3.4 Asymptotc propertes of the GMIM estmator We ow descrbe the asymptotc behavor of the GMIM estmator ˆθ defed by (2.11). Wth the lmt theorems for sample momet fuctos (Theorems 1 ad 2) had, we ca derve 19
20 the asymptotcs of ˆθ by usg stadard techques from the classcal GMM lterature (see, e.g., Hase (1982), Newey ad McFadde (1994) ad Hall (25)). Below, we collect a stadard set of assumptos, wth some slght modfcatos made so as to accommodate the curret settg. Assumpto GMIM: () Θ s compact; () θ s the teror of Θ; () Ξ P Ξ, where Ξ s a Fmeasurable (radom) matrx that s postve semdefte a.s.; (v) ΞG(θ) = a.s. oly f θ = θ ; (v) for H T θḡ (β s, Z s, V s ; θ ) ds, the radom matrx H ΞH s osgular a.s. Assumpto GMIM() mposes compactess o the parameter space. Ths codto s used to establsh the cosstecy of the GMIM estmator. Assumpto GMIM() allows us to derve a lear represetato for the GMIM estmator through a Taylor expaso for the frstorder codto of the mmzato problem (2.11). Assumpto GMIM() specfes the lmtg behavor of the weghtg matrx Ξ. Ulke the stadard GMM settg, the lmt Ξ may be radom, whch s mportat because the lmtg optmal weghtg matrx s typcally radom the curret settg. Assumpto GMIM(v) s a detfcato codto, whch guaratees the uqueess of θ as a mmzer of the populato GMIM crtero fucto Q(θ) G (θ) ΞG (θ), up to a Pull set. Ths codto s a jot restrcto o the populato momet fucto G( ) ad the weghtg matrx Ξ. I partcular, whe Ξ has full rak, Assumpto GMIM(v) amouts to sayg that θ s the uque soluto to G(θ) =. Ths codto s commoly used to specfy detfcato a GMM settg, but t takes a somewhat ostadard form here because the populato momet fucto G ( ) s tself a radom fucto. It s structve to further llustrate the ature of ths codto the smple settg of Example 1: f we set the strumet ϕ(v) to be (1, v) as for ordary least squares, the G (θ) = ( T T V sds T V sds T V s 2 ds ) ( θ 1 θ 1 θ 2 θ 2 We see that θ s the uque soluto to G(θ) f ad oly f T T V s 2 ds ( T V sds) 2. By the Cauchy Schwarz equalty, T T V s 2 ds ( T V sds) 2 ad the equalty s strct uless the process V t s tmevarat over [, T ]. I other words, the detfcato s acheved as soo as the process V t s ot colear, a pathwse sese, wth the costat term. Fally, Assumpto GMIM(v) s used to derve a asymptotc lear represetato of the GMIM estmator. The asymptotc behavor of the GMIM estmator ˆθ s summarzed by Proposto 1 below. Proposto 1. Suppose () Assumptos H, MIX, S, D, LIP, LW ad GMIM hold for some r (, 1), k > 2 ad p 3; () ϖ (2p 1)/2(2p r). The (a) ˆθ P θ. (b) 1/2 (ˆθ θ ) Ls MN (, Σ), where Σ (H ΞH) 1 H ΞΣ g ΞH(H ΞH) 1. ). 2
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