Generalized Methods of Integrated Moments for High-Frequency Data

Transcription

1 Geeralzed Methods of Itegrated Momets for Hgh-Frequecy 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 two-step 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 frst-step estmato error, a bas-correcto 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 volatlty-volume 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 Aït-Sahala, 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: [email protected]. Uversty of Chcago Booth School of Busess, 587 S. Woodlaw Aveue, Chcago, IL Emal: [email protected]. 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 hgh-frequecy (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 hgh-frequecy 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 hgh-frequecy 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, two-step 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 hgh-frequecy 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 out-of-sample performace, smplcty of terpretato ad real-tme cotrol, etc. Pseudo-true 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 fte-dmesoal model parameter s costructed as the mmzer of a sample crtero fucto of the quadratc form. Our aalyss also exteds the scope of the hgh-frequecy 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 frst-step 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 bas-correcto to the sample momet fucto ad show that the bas-corrected 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 seve-based 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 bas-corrected 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 frst-step 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 by-products. 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 rsk-eutral volatlty dyamcs wth lear mea-reverso 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 hgh-frequecy 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 fte-dmesoal 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 F-codtoally 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 σ-felds 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 sgal-plus-ose 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 oe-dmesoal 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 σ-fte measure λ o R. Moreover, for some costat r (, 1), a sequece of stoppg tmes (T m ) m 1 ad λ-tegrable 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 hgh-frequecy data has bee cosdered by, for example, Jacod, L, ad Zheg (213), who cosder Y ( ) wth a locato-scale 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 λ-tegrable 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 hgh-frequecy 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 volatlty-of-volatlty 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 log-memory 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 fte-dmesoal 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 rsk-eutral 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 locato-scale form. The prcg error a χ s troduced to capture prce compoets that stadard rsk-eutral 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 rsk-eutral 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 rsk-eutral 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 matrx-valued 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 rsk-eutral 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 rsk-eutral 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), Aït-Sahala 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 tme-varyg mea terms of a tme-chaged 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 large-t settg for daly or weakly data, the statoarty assumpto s more restrctve for hgh-frequecy 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 ose-robust 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 F-measurable, (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 Aït-Sahala (25), Hase ad Lude (26), Bad ad Russell (28), Bardorff-Nelse, 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 pre-averagg 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 cotuous-tme 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 hgh-frequecy data; see Aderse, Bollerslev, Debold, ad Labys (23), Bardorff-Nelse ad Shephard (24a), Jacod ad Protter (212) ad refereces there. As s typcal the hgh-frequecy lterature, our -fll asymptotc results requre oly mld codtos o the sample-path 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 o-ergodc, 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 bas-corrected 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 α-mxg 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 α-mxg 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 α-mxg 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 F-codtoal 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 fte-dmesoal 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 F-codtoal 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 scalar-valued 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 bas-corrected 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 sample-aalogue 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 hgh-order 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 bas-corrected 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 F-codtoal autocovarace of the sequece g (Y (β, χ ), z, v; θ ) at lag l, ad γ (β, z, v) s the correspodg log-ru 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 (F-codtoal) 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 frst-step samplg error V. We are ow ready to state the asymptotc propertes of G ( ). We the characterze the aforemetoed hgh-order bas of the raw estmator Ĝ(θ ) as a drect corollary (Corollary 1). I the sequel, we use L-s to deote F-stable covergece law 11 ad, for a geerc F-measurable postve semdefte matrx Σ, we use MN (, Σ) to deote the cetered mxed Gaussa dstrbuto wth F-codtoal 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 log-ru 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 log-ru 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 F-measurable 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 F-codtoal 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 ) L-s 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 well-behaved 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 ) L-s 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 q-dmesoal stadard ormal varable that s depedet of F G. If, addto, the F-codtoal 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 G-codtoal 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 F-codtoal 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 ch-square 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 t-statstcs (.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 closed-form 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 F-measurable (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 frst-order 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 P-ull 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 tme-varat 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 ) L-s MN (, Σ), where Σ (H ΞH) 1 H ΞΣ g ΞH(H ΞH) 1. ). 2

21 (c) Suppose, addto, Assumptos HAC ad AVAR1 (resp. AVAR2) ad let Ŝ(ˆθ ) Ŝ 1, (ˆθ ) (resp. Ŝ 2, (ˆθ )). Deote H θ G (ˆθ ), Σg, (ˆθ ) Γ (ˆθ ) + Ŝ(ˆθ ) ad Σ (HΞ H ) 1 HΞ Σg, (ˆθ )Ξ H (HΞ H ) 1. We have Σ P Σ. Commet. Proposto 1(a) shows the cosstecy of the GMIM estmator ˆθ. Part (b) shows the assocated stable covergece law, where the asymptotc dstrbuto s cetered mxed Gaussa wth (F-codtoal) asymptotc covarace matrx Σ. The asymptotc covarace matrx ca be cosstetly estmated by Σ as show by part (c). The asymptotc covarace matrx Σ has a famlar form as the classcal GMM settg (Hase (1982)), although Σ s a radom matrx here. Smlar to the well-kow result the GMM lterature, the asymptotc covarace matrx Σ s mmzed the matrx sese whe Ξ = Σ 1 g, provded that Σ g s osgular a.s. A feasble effcet GMIM estmator ca be obtaed by frst computg a prelmary GMIM estmator, say θ, wth the detty weghtg matrx ad the coduct the GMIM estmato wth the weghtg matrx Ξ = Σ g, ( θ ) 1. Here, the effcecy s wth respect to the choce of weghtg matrx whle takg the strumet ϕ( ) as gve. The choce of optmal strumet ad, as a matter of fact, the characterzato of the semparametrc effcecy boud the curret -fll settg for ostatoary depedet data rema ope questos. Effcet estmato of tegrated volatlty fuctoals of the form T g(v s)ds has bee recetly tackled by Clémet, Delattre, ad Gloter (213), Jacod ad Rosebaum (213) ad Reault, Sarsoy, ad Werker (213). Effcecy may also be mproved by cosderg a cotuum of strumets as Carrasco, Cherov, Flores, ad Ghysels (27). Extedg these results to the aalyss of GMIM appears to be very challegg ad s left to future research. Hase s (1982) overdetfcato test ca be adapted to the curret settg wth the famlar ch-square dstrbuto, as show by the followg proposto. Proposto 2. Suppose () codtos Proposto 1; () Σ g s o-sgular a.s. ad Ξ = Σ 1 g. The 1 Q (ˆθ ) L-s χ 2 q dm(θ). 4 Smulato results I ths secto, we exame the asymptotc theory above a smulato settg that mmcs the setup of our emprcal applcato Secto 5.2. Throughout the smulatos, we fx T = 21 days ad cosder two samplg frequeces: = 1 or 5 mutes. The wdow sze k the spot varace estmato s take to be 12, 15, ad 18 for the 1-mute sample, ad 4, 45, ad 5 for the 5-mute sample. The perturbato o k s reasoably large for checkg robustess. We set k = k. For each day, the trucato parameters are take as ϖ =.49 ad ᾱ = 3 BV 21

22 where BV s the daly bpower varato (Bardorff-Nelse ad Shephard (24b)). There are 2, Mote Carlo trals total. We smulate X t ad V t accordg to { dxt = (.5 +.5V t )dt + V t dw t + J X dn t +.2λ N dt, V t = exp ( 2.8+6F t ), df t = 4F t dt +.8d W t + J F dn t.2λ N dt, (4.1) wth E[dW t d W t ] =.75dt, J X N (.2,.5 2 ), J F N (.2,.2 2 ), ad N t beg a Posso process wth testy λ N = 25. Gve the path of V t, the sequece (Y ) s smulated depedetly wth the margal codtoal dstrbuto c Posso(m + m 1 V ), where c = 1, cm = 2, ad cm 1 = 8 are calbrated to data used Secto 5.2. The parameter of terest s θ = (c, cm, cm 1 ); ths reparametrzato s also used the emprcal study Secto 5.2 as Aderse (1996). We coduct ths estmato usg the frst two codtoal momets of Y t as descrbed by (2.5) Example 4. We set the strumet as ϕ(v t ) = (1, V t ), whch results four tegrated momet codtos, leavg oe degree of freedom for overdetfcato. Our goal ths exercse s to exame the fte-sample propertes of the GMIM estmator, as well as the rejecto rates of the overdetfcato test. The estmator for the asymptotc covarace matrx Σ g s take to be Γ, (ˆθ ) + Ŝ1,(ˆθ ); see commet () of Theorem 3. For comparso, we also report results for the ucorrected procedure, whch s mplemeted accordg to the classcal GMM theory but wth the spot varace estmate V treated as f t were the true spot varace V. Note that, for the ucorrected procedure, the asymptotc covarace matrx Σ g oly cotas the compoet Γ. Fgure 1 presets fte-sample dstrbutos of the effcet GMIM estmator ad the effcet ucorrected estmator. We see that the ucorrected estmators exhbt evdet bases, whle the GMIM estmators are properly cetered at the true values. Ths fdg s further cofrmed by Table 1, from whch we see that, all Mote Carlo scearos, the bas of the GMIM estmator s much smaller tha that of the ucorrected estmator ad farly close to zero. Moreover, we fd that the bas-correcto also reduces the root mea squared error (RMSE) of the estmates most cases. Fgure 2 compares fte-sample dstrbutos of the stadardzed GMIM ad the stadardzed ucorrected estmators wth the asymptotc N (, 1) dstrbuto; the stadardzato s feasble (.e., estmators of asymptotc varaces are used). As predcted by the asymptotc theory, the dstrbuto of the stadardzed GMIM estmator s well approxmated by the asymptotc N (, 1) dstrbuto for both 1-mute ad 5-mute samplg, although some dstorto ca be see for the latter. O the cotrary, the dstrbuto of the stadardzed ucorrected estmator dffers substatally from N (, 1). 22

23 Fgure 1: Hstograms of No-stadardzed Estmators 2 c, = 1 mute 2 c, = 5 mutes c m, = 1 mute c m 1, = 1 mute c m, = 5 mutes c m 1, = 5 mutes Note: Ths fgure compares the fte-sample dstrbutos of the effcet ucorrected estmators (sold) ad the effcet GMIM estmators (shaded area). The dashed les hghlght the true parameter values. The samplg terval s = 1 (left) ad 5 mutes (rght). We set T = 21 days ad k =15 ad 45 for 1-mute ad 5-mute samplg, respectvely. There are 2, Mote Carlo trals. Fally, Table 2 reports the fte-sample rejecto rates of overdetfcato tests usg the GMIM procedure, alog wth results from the ucorrected procedure as a comparso. We fd that tests based o the ucorrected procedure almost always falsely reject the ull hypothess, whch s ot surprsg vew of the fdgs above. The rejecto rates of the GMIM overdetfcato test are farly close to, although slghtly lower tha, the omal level for 1-mute samplg. For 5-mute samplg, the GMIM overdetfcato tests become more uderszed. Ths evdece suggests that small samples, the GMIM overdetfcato test teds to be coservatve, at least for the Mote Carlo settg cosdered here. 23

24 Table 1: Summary of Mote Carlo Estmato Results Pael A. = 1 mute Ucorrected GMIM k = 12 k = 15 k = 18 k = 12 k = 15 k = 18 c Bas RMSE c m Bas RMSE c m 1 Bas RMSE Pael B. = 5 mutes k = 4 k = 45 k = 5 k = 4 k = 45 k = 5 c Bas RMSE c m Bas RMSE c m 1 Bas RMSE Note: We report the bas ad the root mea squared error (RMSE) of the effcet ucorrected ad the effcet GMIM estmators the smulato for varous k values. We set T = 21 days. The samplg terval s = 1 or 5 mutes. The true parameter values are c = 1, c m = 2 ad c m 1 = 8. There are 2, Mote Carlo trals. 5 Emprcal applcatos 5.1 Applcato 1: VIX prcg models To llustrate the use of the proposed method, we frst apply t to study the specfcato of the rsk-eutral dyamcs of the stochastc volatlty process by usg traday data of the S&P 5 dex ad the VIX. Startg wth the setup, we suppose that the dyamcs of the logarthm of the S&P 5 dex X t uder the rsk-eutral measure, heceforth the Q-measure, follows X t = X + t b Q s ds + t t Vs dws Q + R ( ) z N(ds, dz) ν Q (V s, dz)ds, (5.1) 24

25 Fgure 2: Hstograms of Stadardzed Estmators.4.2 c, = 1 mute c m, = 1 mute c m, = 1 mute c, = 5 mutes c m, = 5 mutes c m, = 5 mutes Note: Ths fgure compares the fte-sample dstrbutos of the stadardzed effcet ucorrected estmators (sold) ad effcet GMIM estmators (shaded area). The N (, 1) desty fucto s plotted for comparso (dashed). The samplg terval s = 1 (left) ad 5 mutes (rght). We set T = 21 days ad k =15 ad 45 for 1-mute ad 5-mute samplg, respectvely. There are 2, Mote Carlo trals. where the drft b Q t s determed by the o-arbtrage codto, W Q t s a Browa moto uder the Q-measure, ad N(dt, dz) s the jump measure of X wth compesator ν Q (V t, dz)dt whch s allowed to deped o the spot varace. 13 We assume that the predctable compesator of the jump quadratc varato s a affe fucto the spot varace, that s, R z2 ν Q (V t, dz) = η + η 1 V t, where η ad η 1 are costats. Whle ths assumpto s commoly adopted emprcal work, 14 the dscusso below does rely o ts valdty. The ma focus of ths emprcal exercse s o the rsk-eutral dyamcs of the stochastc varace process, whch s gve by: V t = V + t κ Q ( v Q V s )ds + M Q t, (5.2) 13 See Duffe (21), Sgleto (26) ad Garca, Ghysels, ad Reault (21) for comprehesve revews of the o-arbtrage prcg theory ad related ecoometrc methods. 14 Ths assumpto s trvally satsfed f the compesator ν Q ( ) does ot deped o V t. It s also satsfed f X t has compoud Posso jumps wth ts stochastc arrval rate for jumps beg a lear fucto V t. See Chapter 15 of Sgleto (26) ad may refereces there for detaled examples. 25

26 Table 2: Comparso of Mote Carlo Null Rejecto Rates Ucorrected Procedure GMIM Procedure Level k = 12 k = 15 k = 18 k = 12 k = 15 k = 18 1 % m 5 % % Level k = 4 k = 45 k = 5 k = 4 k = 45 k = 5 1% m 5% % Note: We report the fte-sample rejecto rates (%) of the overdetfcato tests for the ucorrected procedure (left) ad the GMIM procedure (rght) at sgfcace levels 1%, 5% ad 1% for varous k values. We set T = 21 days. The samplg terval s = 1 or 5 mutes. There are 2, Mote Carlo trals. where κ Q ad v Q are model parameters ad M Q s a martgale uder the Q-measure that captures both Browa movemets ad (compesated) jumps of V t. We ote that (5.2) oly mposes a mea-revertg parametrc restrcto o the drft term whle leavg the martgale compoet M Q completely oparametrc. I partcular, we allow for geeral forms of volatlty-of-volatlty ad volatlty jumps. Ths settg allows us to focus o the specfcato of the rsk-eutral drft of the spot varace. We also ote that, sce (5.2) oly parametrzes the drft term uder the Q measure, the equvalece betwee P ad Q, whch s mpled by o-arbtrage, does ot further restrct the dyamcs of V t uder the P-measure. Ths class of rsk-eutral volatlty models has bee wdely studed emprcal opto prcg ad facal ecoometrcs. 15 Whle t has prove very useful for modelg opto prces at the daly or weekly frequecy, whether t fulflls the more challegg task of provdg a satsfactory prcg specfcato for traday data s a ope ad mportat questo. We vestgate ths emprcal questo by examg the prcg of the VIX. Below, we refer to the squared VIX as the mpled varace. As show by Jag ad Ta (25) ad Carr ad Wu 15 Examples clude those studed by Baksh, Cao, ad Che (1997), Bates (2), Cherov ad Ghysels (2), Pa (22), Eraker, Johaes, ad Polso (23), Eraker (24), Aït-Sahala ad Kmmel (27) ad Broade, Cherov, ad Johaes (27), amog others, where jumps may be drve by the compoud Posso process wth tme-varyg testy or the CGMY process (Carr, Gema, Mada, ad Yor (23)). Ths class also clude o-gaussa OU processes cosdered by Bardorff-Nelse ad Shephard (21); see also Shephard (25) for a collecto of smlar models. 26

27 (29), the theoretcal value of the mpled varace s gve by Y t E Q [ t+τ t t+τ V s ds + t R ] z 2 ν Q (V s, dz)ds F t, (5.3) where τ = 21 tradg days ad E Q s the expectato operator uder Q. To derve a aalytcal expresso for Y t, we frst observe that (5.2) mples that de Q [V s F t ]/ds = κ Q ( v Q E Q [V s F t ]). We solve ths dfferetal equato for E Q [V s F t ], whch s the plugged to (5.3), yeldg Y t = θ 1 + θ 2V t, (5.4) where θ 1 η + v Q (1 + η 1 )(1 (1 e κqτ )/κ Q τ) ad θ 2 (1 + η 1)(1 e κqτ )/κ Q τ. Equato (5.4) hghlghts the key aspect for usg the VIX to study the rsk-eutral volatlty dyamcs. The aforemetoed large class of models, although potetally very dfferet from each other wth dstct prcg mplcatos for dvdual optos, all mply a lear prcg fucto for the mpled varace. Cosequetly, specfcato tests for ths class of structural models ca be carred out by examg the lear specfcato (5.4). We do so by coductg the overdetfcato test (see Proposto 2). As descrbed Example 1, we suppose that the observed mpled varace Y t VIX 2 t s the theoretcal prce plus a prcg error such that E[Y t θ 1 θ 2 V t F] =. We mplemet the GMIM procedure wth the strumet ϕ(v t ) = (1, V t, 1/(V t + c)) for c = Note that the thrd strumet 1/(V t +c) gves more weght to low volatlty levels, whle the secod strumet V t does the opposte. Our sample perod rages from Jauary 27 to September 212, as costraed by data avalablty; the data source s TckData Ic. The VIX s updated by the CBOE roughly every 15 secods. The S&P 5 dex data s updated more frequetly. I order to reduce the asychrocty betwee the two tme seres, we resample the data at every mute. At ths frequecy, mcrostructure effects o the S&P 5 dex are eglgble our sample. We remd the reader that we allow Y t to cota geeral forms of ose (.e. prcg error), so mcrostructure ose the VIX data s readly accommodated. Days wth rregular tradg hours are elmated, resultg a sample of 1,457 days spag 23 quarters. Tug parameters are chose as follows: the trucato parameters ᾱ ad ϖ are set as the smulato, k = 15, the estmator of asymptotc covarace matrx s gve by Proposto 1(c) wth Ŝ(ˆθ ) = Ŝ1,(ˆθ ), usg m = 12 ad k = 15. For each quarter, we estmate parameters (5.4) va the effcet GMIM estmator. I the upper two paels of Fgure 3, we plot the parameter estmates of θ 1 ad θ 2 ad ther cofdece 16 Settg c >, stead of c =, facltates the verfcato of regularty codtos Proposto 2. I partcular, otce that 1/(V t + c) s three-tmes cotuously dfferetable V t wth bouded dervatves. 27

28 Fgure 3: Results for the VIX Prcg Model Itercept Slope Note: We plot the tme seres of quarterly estmates (sold) of the tercept (θ 1 ) ad slope (θ 2 ) the lear VIX prcg model, alog wth ther 9% two-sded potwse cofdece bads (shaded area). The lower boud of each cofdece bad s the 95% lower cofdece boud. The bottom pael plots the scaled overdetfcato test statstc (.e. 1 Q (ˆθ )) (astersk), ad the dashed le dcates the 95% crtcal value. We fx k = k = 15 ad m = 12. tervals. We see that these quarterly estmates exhbt substatal temporal varato. We also coduct a overdetfcato test for each quarter ad plot the value of the test statstc o the bottom pael of Fgure 3. We fd that the lear specfcato (5.4) s rejected at the 5% level for 14 out of 23 quarters. The evdece here pots away from the lear specfcato of VIX prcg ad, hece, the lear mea-reverso specfcato of the rsk-eutral volatlty dyamcs gve by (5.2), eve sample perods as short as a quarter. Ths fdg suggests that some form of olearty (e.g., a expoetal Orste Uhlebeck specfcato) eeds to be corporated the rsk-eutral drft term of the spot varace process. 5.2 Applcato 2: Volatlty-volume relatoshp I the secod applcato, we use hgh-frequecy equty data to vestgate the Mxture of Dstrbuto Hypothess (MDH). The MDH posts a jot depedece betwee returs ad volume o 28

29 a latet formato flow varable ad has spurred a szable lterature facal ecoomcs. 17 A key mplcato of the MDH s the relatoshp betwee retur volatlty ad tradg volume. I ts classcal form, the volatlty-volume relatoshp predcts that the codtoal dstrbuto of the tradg volume Y t gve the retur varace V t s N (µ MDH V t, σ 2 MDH V t), where µ MDH ad σ 2 MDH are parameters ad the ormal dstrbuto s motvated by a asymptotc argumet; see Tauche ad Ptts (1983). We refer to ths model as the stadard MDH. Aderse (1996) proposes a modfed MDH o the bass of the Gloste ad Mlgrom (1985) model. The modfed MDH features the radom arrval of uformed ad formed traders ad predcts that the codtoal dstrbuto of tradg volume gve the spot varace s scaled Posso, that s, Y t V t c Posso(m +m 1 V t ). Usg Hase s (1982) overdetfcato test o daly data, Aderse (1996) (p. 21) fds that the modfed MDH s broadly cosstet wth the data ad performs vastly better tha the stadard MDH. Motvated by the fact that the tradg actvty has creased substatally over the past decade, we take Aderse s model oe step further to address traday data. As descrbed Example 4, we set Y Y V to be the tradg volume wth the tme terval [, ( + 1) ) ad suppose that c Posso(m + m 1 V ). We mplemet the effcet GMIM estmato procedure ad, subsequetly, the overdetfcato test o the bass of the frst two codtoal momet codtos gve by (2.5). We coduct the same exercse for the stadard MDH, for whch the frst two codtoal momet codtos are gve by (2.2) wth ψ(y t, V t ; (µ MDH, σ 2 MDH)) = (Y t µ MDH V t, Y 2 t µ 2 MDHV 2 t σ 2 MDHV t ). The same strumet ϕ(v t ) = (1, V t ) s used for both the stadard MDH ad the modfed MDH, so as to mata a far comparso. Our sample comprses trasacto prce ad volume data for fve tckers: GE, IBM, JPM, MMM, ad PG; the data source s the TAQ database. The sample cotas 2 quarters from Jauary 28 to December 212. I our aalyss, each quarter s treated o ts ow. Ths sample perod cludes some of the most volatle perods moder facal hstory. Data preprocessg takes a few steps. Frst, we keep trasactos from major exchages at whch most of the tradg of these tckers take place. 18 Secod, we sample trasacto prce wth a samplg terval = 5 mutes usg the prevous-tck approach. The volume Y s the total volume wth [, (+1) ) across all exchages. To mtgate the mpact of block trades, we omt before aggregato all 17 See, for example, Clark (1973), Epps ad Epps (1976), Tauche ad Ptts (1983), Harrs (1986), Harrs (1987), Rchardso ad Smth (1994), Aderse (1996) ad refereces there. 18 These exchages clude Natoal Assocato of Securtes (ADF), NYSE, NYSE Arca, NASDAQ, Drect Edge A ad X, BATS, ad BATS Y-Exchages. NYSE s the exchage where the studed compaes are lsted, whereas the other exchages are electroc commucato etworks. Our results do ot chage qualtatvely whe usg trasactos oly from NYSE. 29

30 trasactos wth volumes exceedg 1, shares. Next, we delete the U.S. holdays ad halftradg days, as well as May 6, 21 whe the Flash Crash occurred. As the ecoomc mechasm aroud opeg ad closg auctos s very dfferet from the regular traday tradg, we remove data durg the frst ad the last 5 mutes of regular tradg hours. Overght returs ad volumes are also elmated. We do ot de-tred the tradg volume seres, as the tred (f there s ay) s ulkely to be mportat for the quarterly horzo. The ut of the volume seres s 1, shares. Tug parameters are chose as follows: the trucato parameters ᾱ ad ϖ are set as the smulato, k = 45, the estmator of asymptotc covarace matrx s gve by Proposto 1(c) wth Ŝ(ˆθ ) = Ŝ1,(ˆθ ), usg m = 12 ad k = 45. Fgure 4 plots the quarterly tme seres of the effcet GMIM estmates for the modfed MDH model. To save space, we oly plot the estmates for cm ad cm 1, that these two parameters determe the codtoal mea of volume gve the spot varace, that s, E [Y t V t ] = cm +cm 1 V t. The fdgs are summarzed as follows. Frst, cosstet wth Aderse (1996), the pot estmates of cm ad cm 1 are almost always postve. We further report Paels A ad B of Table 3 the umbers of quarters wth statstcally sgfcat estmates, whch show that the estmates are deed sgfcat most cases. Secod, whle we observe some temporal varato of the parameter estmates, the parameter stablty s ot very severe, the sese that estmates may adjacet quarters appear to be statstcally dfferet. Thrd, we fd that for all tckers, the estmates of cm 1 have qute small values durg the facal crss 28. Ths fdg suggests that the uderlyg formato flow has hgher prce mpact durg the crss perod tha ormal perods, whch s lkely due to the creased level of formato asymmetry the marketplace durg the crss. We further exame the specfcato of the modfed ad the stadard MDH for each tckerquarter usg the GMIM overdetfcato test. I Pael C of Table 3, we report the umber of quarters for whch the modfed MDH s rejected by the test. We see that the modfed MDH s rarely rejected ad the umber of rejectos s le wth the Type-I error of our test. By cotrast, we see from Pael D of Table 3 that the stadard MDH s rejected by the GMIM overdetfcato test for a majorty of quarters for all tckers. These fdgs are cosstet wth those of Aderse (1996), ad provde further support to the pror fdgs a hgh-frequecy settg. 6 Related lterature Ths paper s related to several strads of lterature. Frst, t s closely related to pror work o oparametrc ferece for tegrated volatlty fuctoals; see Aderse, Bollerslev, Debold, ad Labys (23), Bardorff-Nelse ad Shephard (24a), Jacod ad Protter (212) ad may refereces there. The most closely related paper s the recet work of Jacod ad Rosebaum 3

31 Fgure 4: Quarterly Parameter Estmates for the Modfed MDH Model 1 5 GE : c m IBM : c m JPM : c m MMM : c m PG : c m GE : c m IBM : c m JPM : c m MMM : c m PG : c m Note: For each tcker, we plot the tme seres of quarterly estmates (sold) of cm (left) ad cm 1 (rght) the modfed MDH model, alog wth ther 9% two-sded potwse cofdece bads (shaded area). The lower boud of each cofdece bad s the 95% lower cofdece boud. From top to bottom, the tckers are GE, IBM, JPM, MMM ad PG. We fx k = k = 45 ad m =

32 Table 3: Summary of Testg Results for MDH Models Sg. Level GE IBM JPM MMM PG Pael A. H : cm = vs. H 1 : cm >. 5% % Pael B. H : cm 1 = vs. H 1 : cm 1 >. 5% % Pael C. H : Modfed MDH s correctly specfed. 5% % Pael D. H : Stadard MDH s correctly specfed. 5% % Note: For each tcker, we report the umber of quarters (out of 2 quarters total) whch the ull hypothess of terest s rejected at the 5% or 1% sgfcace level. Pael A (resp. Pael B) reports oe-sded testg results for the ull hypothess wth cm = (resp. cm 1 = ). Pael C (resp. Pael D) reports overdetfcato testg results for the modfed MDH (resp. stadard MDH). (213), who use the spot volatlty estmates to costruct estmators for a large class of tegrated volatlty fuctoals of the form T g(v s)ds. The use of spot volatlty estmates ca be dated back to early work such as Foster ad Nelso (1996) ad Comte ad Reault (1998), to the best of our kowledge. Jacod ad Rosebaum (213) provde a detaled aalyss of the bas from the frst-step estmato ad propose a bas-correcto that s smlar to ours. The tegrated momet codto G( ) the curret paper has a more geeral form because t ot oly depeds o V t, but also depeds o the observable process Z t ad the uobservable process β t ; moreover, the fuctoal form of ḡ( ) s geeral ukow as t s partally determed by the ukow dstrbuto P χ. These complcatos make our aalyss otably dfferet from Jacod ad Rosebaum (213). Coceptually, the scope of our aalyss s very dfferet from the exstg lterature: 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. Secod, our semparametrc method ca be cosdered as oe wth oparametrcally geerated regressors. 19 From ths vewpot, the method ca be further compared wth the lterature 19 Although the frst-step spot varace estmato ca be cosdered as a osy measuremet of the true spot 32

33 o estmatg stochastc volatlty models usg jot -fll ad log-spa asymptotcs, see, for example, Bollerslev ad Zhou (22), Bardorff-Nelse ad Shephard (22), Bad ad Phllps (23), Corrad ad Dstaso (26), Gloter (27), Kaaya ad Krstese (21), Bad ad Reò (212) ad Todorov ad Tauche (212). These papers use realzed volatlty measures formed from hgh-frequecy data to proxy volatlty fuctoals defed cotuous tme. The realzed measures ca the be used to perform parametrc or oparametrc estmato wth a appeal to the large T asymptotcs. These methods rely crucally o the -fll approxmato error beg domated by the samplg varablty the log-spa asymptotcs, so that the former ca be cosdered eglgble for asymptotc ferece. I cotrast, the fxed T settg here allows us to explctly characterze the asymptotc bas duced by the -fll approxmato error, costruct bas-correcto, ad corporate the effect of the approxmato error to the asymptotc varace of the GMIM estmator. That beg sad, the role of the curret paper for ths lterature s completely complemetary, because ferece cocerg certa quattes, such as the drft term (ad hece the law) of a stochastc volatlty model, demads a asymptotc settg wth a log spa. Fally, whe specalzed a opto prcg settg, the curret paper ca be compared wth Aderse, Fusar, ad Todorov (213). These authors cosder a settg where the prcg errors of a large umber of opto cotracts are weakly depedet so that they ca be averaged out by vrtue of the cetral lmt theorem. Ths large cross secto settg smplfes the aalyss of opto prcg models wth multple latet factors, because the rsk factors each day ca be detfed from the large cross secto as radom parameters. Our method s lmted to prcg models wth oe volatlty factor, but does ot requre a large pael of optos wth cross-sectoally depedet or weakly depedet prcg errors. Ideed, we allow the errors prcg equatos to be arbtrarly correlated across opto cotracts, as s typcal GMM. 7 Cocluso The proposed GMIM framework exteds the classcal GMM for estmatg codtoal momet equalty models usg hgh-frequecy data. Such data have become creasgly avalable facal markets durg the past decade ad provde rch formato for studyg ecoometrc models. Our asymptotc framework s -fll wth a fxed tme spa ad allows for geeral forms of ostatoarty ad depedece. Sce the method ca be appled to relatvely short samples, t coveetly allows for tme-varyg parameters across short (e.g., quarterly) subsamples. The key to our aalyss s the dervato of the asymptotc propertes of the bas-corrected varace, the ature of our ecoometrc aalyss s very dfferet from the lterature o errors--varables models (see, e.g., Hausma, Newey, Ichmura, ad Powell (1991), Scheach (24, 27)). 33

34 sample momet fucto, whch depeds o the osy process Y, the observable semmartgale Z ad the spot varace estmate V. Our aalyss o the estmator of ts asymptotc covarace matrx s also ew. Gve these techcal ovatos, ferece methods the classcal GMM lterature, such as overdetfcato tests ad Aderso Rub type cofdece sets, ca be adapted to the GMIM settg. The theory s derved uder a reasoably geeral settg, as we allow for complcatos such as prce ad volatlty jumps, the leverage effect ad serally depedet ose the Y varable. The usefuless of the proposed method s demostrated wth two dstct emprcal examples. Refereces Aït-Sahala, Y., ad R. Kmmel (27): Maxmum Lkelhood Estmato of Stochastc Volatlty Models, Joural of Facal Ecoomcs, 83, Aït-Sahala, Y., ad J. Yu (29): Hgh Frequecy Market Mcrostructure Nose Estmates ad Lqudty Measures, Aals of Appled Statstcs, 3( ), Aderse, T. (1996): Retur Volatlty ad Tradg Volume: A Iformato Flow Iterpretato of Stochastc Volatlty, Joural of Face, 51, Aderse, T. G., T. Bollerslev, P. F. Chrstofferse, ad F. X. Debold (213): Facal Rsk Measuremet for Facal Rsk Maagemet, Hadbook of the Ecoomcs of Face, Vol.II, ed. by G. Costades, M. Harrs, ad R. Stulz, chap. 17, pp Amsterdam: Elsever Scece B.V. Aderse, T. G., T. Bollerslev, F. X. Debold, ad P. Labys (23): Modelg ad Forecastg Realzed Volatlty, Ecoometrca, 71, Aderse, T. G., N. Fusar, ad V. Todorov (213): Parametrc Iferece ad Dyamc State Recovery from Opto Paels, Dscusso paper, Northwester Uversty. Aderso, T. W., ad H. Rub (1949): Estmato of the Parameters of a Sgle Equato a Complete System of Stochastc Equatos, Aals of Mathematcal Statstcs, 2, Adrews, D. (1991): Heteroskedastcty ad Autocorrelato Cosstet Covarace Matrx Estmato, Ecoometrca, 59, Adrews, D. W. K., ad G. Soares (21): Iferece for Parameters Defed by Momet Iequaltes Usg Geeralzed Momet Selecto, Ecoometrca, 78(1),

35 Baksh, G., C. Cao, ad Z. Che (1997): Emprcal Performace of Alteratve Opto Prcg Models, The Joural of Face, 52, Bad, F., ad R. Reò (212): Tme-varyg Leverage Effects, Joural of Ecoometrcs, 169, Bad, F. M., ad P. C. B. Phllps (23): Dffuso Models, Ecoometrca, 71, Fully Noparametrc Estmato of Scalar Bad, F. M., ad J. R. Russell (28): Mcrostructure Nose, Realzed Volatlty ad Optmal Samplg, Revew of Ecoomc Studes, 75, Bardorff-Nelse, O., ad N. Shephard (22): Ecoometrc Aalyss of Realzed Volatlty ad ts Use Estmatg Stochastc Volatlty Models, Joural of the Royal Statstcal Socety, Seres B, 64. Bardorff-Nelse, O. E., P. R. Hase, A. Lude, ad N. Shephard (28): Desgg Realzed Kerels to Measure the ex post Varato of Equty Prces the Presece of Nose, Ecoometrca, 76, Bardorff-Nelse, O. E., ad N. Shephard (21): No-Gaussa Orste-Uhlebeck- Based Models ad Some of Ther Uses Facal Ecoomcs, Joural of the Royal Statstcal Socety, B, 63, (24a): Ecoometrc Aalyss of Realzed Covarato: Hgh Frequecy Based Covarace, Regresso, ad Correlato Facal Ecoomcs, Ecoometrca, 72(3), pp (24b): Power ad Bpower Varato wth Stochastc Volatlty ad Jumps (wth Dscusso), Joural of Facal Ecoometrcs, 2, Bates, D. (2): Post- 87 Crash Fears the S&P 5 Futures Opto Market, Joural of Ecoometrcs, 94, Bates, D. (212): U.S. Stock Market Crash Rsk, , Joural of Facal Ecoomcs, 15, Black, F. (1976): Studes of Stock Prce Volatlty Chages, Proceedgs of the 1976 Meetgs of the Amerca Statstcal Assocato, pp Bollerslev, T., R. Egle, ad D. Nelso (1994): ARCH Models, Hadbook of Ecoometrcs, vol. 4. Elsever. 35

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38 Hase, L. P., ad K. J. Sgleto (1982): Geeralzed Istrumetal Varables Estmato of Nolear Ratoal Expectatos Models, Ecoometrca, 5(5), pp Hase, P. R., ad A. Lude (26): Realzed Varace ad Market Mcrostructure Nose, Joural of Busess ad Ecoomc Statstcs, 24, Harrs, L. (1986): Cross-Securty Tests of the Mxture of Dstrbuto Hypothess, Joural of Facal ad Quattatve Aalyss, 21, (1987): Trasacto Data Tests of the Mxture of Dstrbuto Hypothess, Joural of Facal ad Quattatve Aalyss, 22, Hasbrouck, J. (1993): Assessg the Qualty of a Securty Market: A New Approach to Trasacto-Cost Measuremet, Revew of Facal Studes, 6, Hausma, J., W. Newey, H. Ichmura, ad J. Powell (1991): Idetfcato ad Estmato of Polyomal Errors--Varables Models, Joural of Ecoometrcs, 5, Hautsch, N. (212): Ecoometrcs of Facal Hgh-Frequecy Data. Sprger. Hesto, S. (1993): A Closed-form Soluto for Optos wth Stochastc Volatlty wth Applcatos to Bods ad Currecy Optos, Revew of Facal Studes, 6, Jacod, J., Y. L, P. A. Myklad, M. Podolskj, ad M. Vetter (29): Mcrostructure Nose the Cotuous Case: The Pre-Averagg Approach, Stochastc Processes ad Ther Applcatos, 119, Jacod, J., Y. L, ad X. Zheg (213): Statstcal Propertes of Mcrostructure Nose, Dscusso paper, Hog Kog Uversty of Scece ad Techology. Jacod, J., M. Podolskj, ad M. Vetter (21): Lmt Theorems for Movg Averages of Dscretzed Processes Plus Nose, The Aals of Statstcs, 38(3), pp Jacod, J., ad P. Protter (212): Dscretzato of Processes. Sprger. Jacod, J., ad M. Rosebaum (213): Quartcty ad Other Fuctoals of Volatlty: Effcet Estmato, Aals of Statstcs, 118, Jacod, J., ad A. N. Shryaev (23): Lmt Theorems for Stochastc Processes. Sprger- Verlag, New York, secod ed. Jag, G. J., ad Y. S. Ta (25): The Model-Free Impled Volatlty ad ts Iformato Cotet, Revew of Facal Studes, 18,

39 Kaaya, S., ad D. Krstese (21): Estmato of Stochastc Volatlty Models by Noparametrc Flterg, Dscusso paper, Uversty of Oxford. Krstese, D. (21): Noparametrc Flterg of the Realzed Spot Volatlty: A Kerel- Based Approach, Ecoometrc Theory, 26(1), pp Mac, C. (21): Dsetaglg the Jumps of the Dffuso a Geometrc Jumpg Browa Moto, Gorale dell Isttuto Italao degl Attuar, LXIV, Newey, W. K. (1994): The Asymptotc Varace of Semparametrc Estmators, Ecoometrca, 62(6), pp Newey, W. K., ad D. L. McFadde (1994): Large Sample Estmato ad Hypothess Testg, Hadbook of Ecoometrcs, vol. 4. Elsever. Newey, W. K., ad K. D. West (1987): A Smple, Postve Sem-defte, Heteroskedastcty ad Autocorrelato Cosstet Covarace Matrx, Ecoometrca, 55, Pa, J. (22): The Jump-Rsk Prema Implct Optos: Evdece from a Itegrated Tme- Seres Study, Joural of Facal Ecoomcs, 63, 3 5. Reault, E., C. Sarsoy, ad B. J. Werker (213): Effcet Estmato of Itegrated Volatlty ad Related Processes, Dscusso paper, Brow Uversty. Reò, R. (28): Noparametrc Estmato of the Dffuso Coeffcet of Stochastc Volatlty Models, Ecoometrc Theory, 24, Rchardso, M., ad T. Smth (1994): A Drect Test of the Mxture of Dstrbutos Hypothess: Measurg the Daly Flow of Iformato, The Joural of Facal ad Quattatve Aalyss, 29(1), pp Sarga, J. D. (1958): The Estmato of Ecoomc Relatoshps usg Istrumetal Varables, Ecoometrca, 26(3), Scheach, S. M. (24): Estmato of Nolear Models wth Measuremet Error, Ecoometrca, 72(1), pp (27): Istrumetal Varable Estmato of Nolear Errors--Varables Models, Ecoometrca, 75(1), pp Shephard, N. (25): Stochastc Volatlty. Oxford Uversty Press. 39

40 Sgleto, K. J. (26): Emprcal Dyamc Asset Prcg: Model Specfcato ad Ecoometrc Assessmet. Prceto Uversty Press. Stock, J. H., ad J. H. Wrght (2): GMM wth Weak Idetfcato, Ecoometrca, 68, Tauche, G., ad M. Ptts (1983): The Prce Varablty-Volume Relatoshp o Speculatve Markets, Ecoometrca, 51, Todorov, V., ad G. Tauche (212): The Realzed Laplace Trasform of Volatlty, Ecoometrca, 8, Whte, H. (1982): Maxmum Lkelhood Estmato of Msspecfed Models, Ecoometrca, 5, (21): Asymptotc Theory for Ecoometrcas. Academc Press. Xu, D. (21): Quas-Maxmum Lkelhood Estmato of Volatlty wth Hgh Frequecy Data, Joural of Ecoometrcs, 159, Yoshhara, K. (1978): Momet Iequaltes for Mxg Sequeces, Koda Mathematcal Joural, 1, Zhag, L., P. A. Myklad, ad Y. Aït-Sahala (25): A Tale of Two Tme Scales: Determg Itegrated Volatlty wth Nosy Hgh-Frequecy Data, Joural of the Amerca Statstcal Assocato, 1,

41 A Proofs The followg otatos are used throughout the proofs below. We deote the codtoal expectato operator E [ F] by E F. For a radom varable ξ ad p 1, we wrte ξ F,p = (E F ξ p ) 1/p. Recall that N [T/ ] k. We wrte for N = ad wrte,j for N,j=. We use K to deote a geerc postve costat that may vary from le to le; we sometmes wrte K u to dcate ts depedece o some costat u. As s typcal ths type of problems, by a classcal localzato argumet, we ca replace Assumpto H wth the followg assumpto wthout loss of geeralty. Assumpto SH: We have Assumpto H. Moreover, the processes β t, Z t, σ t, b t ad σ t are bouded ad, for some λ-tegrable fucto J : R R, we have δ(ω, t, z) J(z) ad δ(ω, t, z) 2 J(z), for all ω () Ω (), t ad z R. We recall some kow, but otrval, estmates that are repeatedly used below. Cosder a cotuous process X t gve by We the set, for each =,..., N, X t = X + t b s ds + t σ s dw s. V = 1 k k j=1 ( +jx ) 2, ṽ = V V. (A.1) Lemma A.1. Suppose that Assumpto SH holds for some r (, 1). Let u 1 be a costat. The for some determstc sequece a, we have Proof: E V V u K u a (2u r)ϖ+1 u, E ṽ u K u (k u/2 + (k ) (u/2) 1 ), E V u K u + K u (2u r)ϖ+1 u. (A.2) The frst equalty s by (4.8) Jacod ad Rosebaum (213). The secod equalty s by (4.11) Jacod ad Rosebaum (213) ad Jese s equalty. The thrd equalty readly follows from the frst two equaltes ad the boudedess of V t. Q.E.D. A.1 Proof of Theorem 1 We start wth a techcal lemma. 41

42 Lemma A.2. Let β t = (β t, Z t ). Suppose () Assumpto H; () for some p, we have f C(p) ad h P (2p); () f p > 1, we further assume ϖ [(p 1)/(2p r), 1/2); (v) k ad k. The (a) f( β, V P ) T f( β s, V s )ds; (b) 2 h( β, V P ). Proof: (a) By a localzato argumet, we suppose Assumpto SH wthout loss of geeralty. We frst prove the asserto uder the assumpto that f s bouded. Costruct two processes, β t + ad V t +, as follows: for each 1 ad t [( 1), ), we set β t + β ad V t + V. Observe that E T f( β, V ) f( β s, V s )ds N Kk + E f( β s +, V s + ) f( β s, V s ) ds. By Theorem Jacod ad Protter (212), we have V s + V s for each s. By the rght cotuty of the process β, we have β s + β s for each s, whch further mples ( β s +, V + P P ( β s, V s ). By the cotuty of f ( ), f( β s +, V s + ) f( β s, V s ). By the bouded covergece theorem, N E f( β s +, V s + ) f( β s, V s ) ds, yeldg T f( β, V P ) f( β s, V s )ds for bouded f. (A.3) We ow prove the asserto of part (a) wth the boudedess codto o f relaxed. φ( ) be a C fucto R + [, 1], wth 1 [1, ) (x) φ (x) 1 [1/2, ) (x), ad for m 1, we set φ m (v) φ ( v /m), φ m (v) 1 φ m (v). P s ) Let We defe f m ( β, v) f( β, v)φ m (v) ad f m( β, v) f( β, v)φ m(v), so f( β, v) = f m ( β, v) + f m( β, v). Sce f m s bouded ad cotuous, (A.3) yelds f m( β, V ) = T f m( β s, V s )ds + o p (1). Sce the process V t s bouded, T f m( β s, V s )ds = T f( β s, V s )ds for m large eough. By Proposto Jacod ad Protter (212) ad Markov s equalty, t remas to show that lm m lm sup E f m ( β, V ) =. (A.4) By codto (), for all m 1, f m ( β, V ) K(1 + V p )φ m ( V ) K V p 1 { V m/2}. Uder the same codto o X, Jacod ad Protter (212) show that (see (9.4.7)), for some determstc sequece a, E[ V p 1 { V m/2} ] Km p + K 1 p+ϖ(2p r) a. Uder codto (), we have 1 p + ϖ (2p r) ad, hece, E f m( β, V ) Km p + O (a ). From here, (A.4) follows. The proof of part (a) s ow complete. 42

43 (b) By Lemma A.1 ad h P (2p), E [ 2 ] [ h( β, V ) E 2 ] (1 + V 2p ) K + K 2 2p+ϖ(4p r) a. Note that codto () mples that 2 2p + ϖ(4p r) ad, hece, the majorat sde of the above equalty s o(1). The asserto part (b) readly follows. Q.E.D. Proof of Theorem 1: By a compoetwse argumet, we ca assume that g( ) s R-valued wthout loss of geeralty. We frst show that Ĝ(θ) P G(θ) for each θ Θ. We decompose Ĝ (θ) = Ĝ1, (θ) + Ĝ2, (θ), where Ĝ 1, (θ) ζ, ζ g(y, Z, V ; θ) ḡ(β, Z, V ; θ), Ĝ 2, (θ) ḡ(β, Z, V ; θ). Sce (β, Z, V ) s F-measurable, ḡ(β, Z, V ; θ) = E F [g(y, Z, V ; θ)] by the defto of ḡ ( ). Hece, the varables (ζ ) have zero F-codtoal mea. Uder the trasto probablty P (1), the α-mxg coeffcet of these varables s bouded by α mx ( ). Settg a ḡ,k (β, Z, V ; θ) + 1 ad ζ ζ /a, we have E F ζ k 1 for all. Sce l α mx (l) (k 2)/k <, by the mxg equalty (see, e.g., Theorem 3 Yoshhara (1978)), we have [ E F Ĝ 1, (θ) 2] K 2 (a ) 2 K + K 2 ḡ,k (β, Z, V ; θ) 2. Sce ḡ,k ( ; θ) P (p), we ca apply Lemma A.2 (wth h ( ) = ḡ,k ( ; θ) 2 P (2p)) to deduce that the majorat sde of the above dsplay s o p (1). From here, t follows that Ĝ1,(θ) = o p (1). I addto, sce ḡ ( ; θ) C (p), we ca apply Lemma A.2 aga (wth f ( ) = ḡ( ; θ)) to derve Hece, Ĝ(θ) Ĝ 2, (θ) P G (θ) for each θ Θ. T P ḡ(β s, Z s, V s ; θ)ds G(θ). To show the asserted uform covergece probablty, t remas to show that Ĝ( ) s stochastcally equcotuous. Sce g( ) LIP(p, ), we see that for θ, θ Θ, Ĝ (θ) Ĝ ( θ ) B, θ θ, where B, B (Y, Z, V ) for some fucto B ( ) as descrbed Defto 1, whch satsfes B P (p). Hece, E F [B, ] 43

44 B (β, Z, V ) K (1 + V p ). By Lemma A.1, E F [B, ] = O p (1). Sce B, s postve, we further derve B, = O p (1). From here, t follows that Ĝ( ) s stochastcally equcotuous. Q.E.D. A.2 Proof of Theorem 2 We eed two lemmas. Lemma A.3 s used to combe stable covergece ad covergece codtoal law (see Defto 2 below). Lemma A.4 geeralzes Theorem 3.2 Jacod ad Rosebaum (213). Defto 2 (Covergece codtoal law): Let ζ be a sequece of R d -valued radom varables defed o the space (Ω, F G, P) ad L be a trasto probablty from (Ω, F {, Ω (1) L F }) to a exteso of (Ω, F G, P). We wrte ζ L f ad oly f the F-codtoal characterstc fucto of ζ coverges probablty to the F-codtoal characterstc fucto L F of L. If a varable ζ defed o the exteso has F-codtoal law L, we also wrte ζ ζ. Lemma A.3. Let ξ ad ζ be two sequeces of radom vectors defed o (Ω, F G, P) ad let ξ ad ζ be varables defed o a exteso of (Ω, F G, P). Suppose that ξ s F-measurable, L-s L F ξ ξ ad ζ ζ. The (ξ, ζ ) L-s (ξ, ζ), wth ξ ad ζ beg F-codtoally depedet. Proof: The jot covergece (ξ, ζ ) L-s (ξ, ζ) s by Proposto 5 Bardorff-Nelse, Hase, Lude, ad Shephard (28). It remas to show that ξ ad ζ are F-codtoally depedet. Let f ( ) ad g ( ) be bouded cotuous fuctos ad U be a bouded F-measurable varable. It remas to verfy E [f (ξ) g (ζ) U] = E [E F [f (ξ)]e F [g (ζ)] U]. (A.5) Sce ξ L-s ξ, E[f (ξ ) E F [g (ζ)] U] E[f (ξ) E F [g (ζ)] U]. By repeated codtog, we see that the lmt cocdes wth the rght-had sde of (A.5). By the assumpto o ζ, we have P E F [g (ζ )] E F [g (ζ)]. The, by the bouded covergece theorem, E[f (ξ ) E F [g (ζ )] U] E[f (ξ ) E F [g (ζ)] U]. Sce ξ s F-measurable, E[f (ξ ) E F [g (ζ )] U] = E[f (ξ ) g (ζ ) U]. Therefore, the rght-had sde of (A.5) s also the lmt of E[f (ξ ) g (ζ ) U]. But, sce (ξ, ζ ) L-s (ξ, ζ), we see that E[f (ξ ) g (ζ ) U] also coverges to the left-had sde of (A.5). Hece, (A.5) must hold. Q.E.D. Lemma A.4. Let β t = (β t, Z t ) ad let f be a R d -valued fucto for some d 1. Suppose that () Assumpto H holds for some r (, 1); () f C 2,3 (p) for some p 3; () ϖ 44

45 (2p 1) /2(2p r); (v) Assumpto LW. The the sequece of varables 1/2 ( ( f( β, V ) 1 ) ) T 2 k vf( β, V ) V 2 f( β s, V s )ds (A.6) coverges F-stably law to MN (, Σ f ), where Σ f 2 T vf( β s, V s ) v f( β s, V s ) V 2 s ds. Proof: Ths lemma geeralzes Theorem 3.2 Jacod ad Rosebaum (213) by allowg f ( ) to deped o the addtoal process β. The proof s adapted from Jacod ad Rosebaum (213). To avod repetto, we oly emphasze the modfcatos. By localzato, we suppose that Assumpto SH holds wthout loss of geeralty. For otatoal smplcty, we set, for (β, z, v) B Z V, h(β, z, v) = 2 vf (β, z, v) v 2. Recall (A.1). The varable (A.6) ca be decomposed as 5 j=1 F j,, where F 1, 1/2 1/2 F 2, 1/2 F 3, 1/2 F 4, 1/2 k 1 ( f( β, V ) f( β, V ) ) ( h( β, V ) h( β, V ) ), (+1) v f( β, V )k 1 ( f( β, V ) f( β ) s, V s ) k u=1 ( V(+u 1) V ), ( f( β, V + ṽ ) f( β, V ) T ds 1/2 f( β s, V s )ds, (N +1) v f( β, V )ṽ k 1 h( β, V ) ), ( F 5, 1/2 k 1 v f( β k (, V ) ( +u X ) 2 ) ) V (+u 1). u=1 The proof wll be completed by showg the followg clams: (A.7) { Fj, = o p (1), for j = 1, 2, 3, 4, F 5, L-s MN (, Σ f ). (A.8) We frst cosder (A.8) for the case wth j = 1. Sce f C 2,3 (p) ad β s bouded by Assumpto SH, we have for all ad v V, v f( β, v) K(1 + v p 1 ) ad 45

46 v h( β, v) K(v + v p 1 ). Hece, by a mea value expaso, E F 1, K 1/2 [( E 1 + V p 1 + V V p 1) V V ]. As show Lemma 4.4 Jacod ad Rosebaum (213) (see case v = 1), the majorat sde of the above equalty ca be bouded by Ka (2p r)ϖ+1/2 p for some determstc sequece a. Sce ϖ (2p 1)/2(2p r), we derve (A.8) for j = 1. Now, cosder (A.8) wth j = 2. Sce f( β s, V s ) s uformly bouded, t s easy to see that 1/2 f( β s, V s )ds (N +1) Kk 1/2, T where the covergece s due to Assumpto LW. Moreover, the frst term F 2, s also o p (1) due to a stadard estmate (see, e.g., p Jacod ad Protter (212)) for the Rema approxmato error of Itô semmartgales. Next, cosder (A.8) wth j = 3. We set ζ 3, k 1 E[ζ 3, F ] ad ζ 3, ζ 3, ζ 3,. We the decompose F 3, = F 3, + F 3,, where F 3, 1/2 v f( β, V )ζ 3,, F 3, 1/2 k u=1 ( V(+u 1) V ), ζ 3, v f( β, V )ζ 3,. Uder Assumpto SH, t s easy to see E ζ 3, Kk. Sce v f( β t, V t ) s bouded, we further have E F 3, Kk 1/2. Hece, F 3, = o p(1). Moreover, by a stadard estmate for Itô semmartgales, we have, for ay u = 1,..., k, E V (+u 1) V 2 Kk. From here, a use of the Cauchy Schwarz equalty yelds E ζ 3, 2 KE ζ3, 2 Kk. By costructo, E[ζ 3, F ] = ad ζ 3, s F (+k 1) measurable. Hece, v f( β, V )ζ 3, ad v f( β l, V l )ζ 3,l are ucorrelated wheever l k. We the use the Cauchy Schwarz equalty to derve E F 3, 2 Kk E v f( β, V )ζ 3, 2 Kk 2, whch further mples F 3, = o p(1). From here, (A.8) wth j = 3 readly follows. To prove (A.8) wth j = 4, we set ζ 4, = vf( β ( ), V ) (ṽ ) 2 2k 1 V 2, ζ 4, = f( β, V + ṽ ) f( β, V ) v f( β, V )ṽ vf( β, V ) (ṽ ) 2 46

47 +k 1 h( β, V ) k 1 h( β, V ). We ca the decompose F 4, = F 4, + F 4,, where F 4, 1/2 Sce f C 2,3 (p) ad β ( [ ] ) E ζ 4, F + ζ 4,, F 4, 1/2 ( ζ 4, E [ ζ 4, F ]). s uformly bouded, we have 2 vf( β, v) K ( 1 + v p 2) ad 3 vf( β, v) K(1 + v p 3 ) for all v V. We ca the use the mea value theorem to derve ζ 4, K(1 + ṽ p 3 ) ṽ 3 + Kk 1 (1 + ṽ p 1 ) ṽ. Now, we ca use the same argumet the proof of Lemma 4.4 Jacod ad Rosebaum (213) (see case v = 4 there) to derve F 4, = o p(1). Moreover, ote that ζ 4, E[ζ 4, F ] ad ζ 4,l E[ζ 4,l F l ] are ucorrelated wheever l k, we have E F 4, ( ) 2 Kk E ζ 4, 2 Kk k 2 + k, where the frst equalty s by the Cauchy Schwarz equalty ad the secod equalty s by the secod le of (A.2). From here, t follows that E F 4, 2. Hece, F 4, = F 4, + F 4, = o p(1), as clamed (A.8). Fally, we otce that the stable covergece (A.8) follows essetally the same proof as that of Lemma 4.5 Jacod ad Rosebaum (213). (To be precse, the oly modfcato eeded s to replace the weght lm g (c,5 ) ther defto of Vt by v f( β, V ).) The proof s ow complete. Q.E.D. Now, we are ready to prove Theorem 2. Proof of Theorem 2: (a) We frst verfy that the codtos Theorem 1 hold whe replacg the fucto g(y, z, v; θ) wth h(y, z, v; θ) 2 vg(y, z, v; θ)v 2. We defe h ( ) ad h j,p ( ) va (2.7) ad (3.1) but wth h replacg g. By Assumpto S(), h(β, z, v; θ) = 2 vḡ(β, z, v; θ)v 2. Sce ḡ( ; θ) C 2,3 (p), we see that 2 vḡ( ; θ) C(p 2) ad, hece, h( ; θ) C (p). Further observe that h,k ( ; θ) = ḡ 2,k ( ; θ)v 2, whch belogs to P(p) by Assumpto D(). The codto ϖ (2p 1)/2(2p r) clearly mples that ϖ (p 1)/(2p r). Uder Assumpto LIP(), t s easy to verfy that h( ) LIP(p, ). Now, we ca apply Theorem 1 wth g( ) replaced by h( ) ad derve the frst asserto of part (a). As a result, Ĝ( ) G ( ) = o p (1) uformly o compact sets. Uder Assumptos S(), D() ad LIP(), we ca apply Theorem 1 to derve that Ĝ( ) G( ) = o p (1) uformly o compact sets. From here, the secod asserto of part (a) readly follows. (b) Step 1. We outle the proof of part (b) ths step. Wthout loss of geeralty, we suppose Assumpto SH. To smplfy otato, we suppress the appearace of θ by wrtg g(y, z, v) (resp. ḡ(β, z, v)) place of g(y, z, v; θ ) (resp. ḡ(β, z, v; θ )). We also set h (y, z, v) = 2 vg (y, z, v) v 2 ad h (β, z, v) = 2 vḡ (β, z, v) v 2. 47

48 The proof reles o the decomposto 1/2 G (θ ) = R 1, + R 2, + R 3,, where R 1, 1/2 R 2, 1/2 R 3, 1/2 k 1 ( ḡ(β, Z, V ) k 1 h(β, Z, V ) ), By Lemma A.4 wth f( ) = ḡ ( ), we have R 1, ( h(β, Z, V ) h(y, Z, V )), ( g(y, Z, V ) ḡ(β, Z, V ) ). L-s MN (, S); recall (3.5) for the defto of S. Below, we show R 2, = o p (1) step 2. We the show (recallg Defto 2 ad (3.4)) R 3, L F MN (, Γ) step 4, after preparg some prelmary results step 3. The asserto of part (b) the follows from Lemma A.3. Step 2. I ths step, we show that R 2, = o p (1). By usg a compoetwse argumet, we ca assume that R 2, s scalar wthout loss of geeralty. We set h h(β, Z, V ) h(y, Z, V ) ad rewrte R 2, = 1/2 k 1 h. By Assumpto S() ad the F- measurablty of (β, Z, V ), we have E F [ h ] =. Furthermore, sce ḡ 2,k P (p 2) (Assumpto D()), h F,k Kḡ 2,k (β, Z, V ) V 2 K(1 + V p ). (A.9) By Assumpto MIX, codtoal o F, the α-mxg coeffcet of the sequece ( h ) s bouded by α mx ( ). Observe that E F [ R 2 2, ] k 2 E F [ h h j ],j K k 2 α mx ( j ) 1 2/k h F,k h j F,k,,j (A.1) where the frst equalty s by the tragle equalty, ad the secod equalty follows from the mxg equalty. We also ote that the codto ϖ (2p 1)/2(2p r) mples ϖ (2p 1)/(4p r); hece, by the thrd le of (A.2), E V 2p K. (A.11) By (A.9) (A.11), as well as the assumpto that α mx ( ) has sze k/(k 2), we derve E[R 2 2, ] Kk 2. Therefore, R 2, = o p (1) as wated. Step 3. It remas to show R 3, L F MN (, Γ). By the Cramer Wold devce, we ca assume that R 3, s oe-dmesoal wthout loss of geeralty. I ths step, we collect some prelmary 48

49 results. For otatoal smplcty, we set ẑ = (β, Z, V ), z = (β, Z, V ) ad ξ (β, z, v) = g (Y (β, χ ), z, v) ḡ (β, z, v). We ca rewrte By (A.2), R 3, = 1/2 ξ (ẑ ). E V V 2 Kā, where ā (4 r)ϖ 1 (A.12) + k 1 + k. (A.13) Sce ϖ (2p 1)/2(2p r), r < 1 ad p 3, t s easy to see (4 r)ϖ > 1. Hece, ā. Uder Assumpto SH, we have E z j z 2 K(1 j ) by a stadard estmate for Itô semmartgales. By (A.13), we further have E ẑ j ẑ 2 K(ā + 1 j ). Recall from Assumpto D the costat κ (, 1]. The we have, by Jese s equalty, E [ ẑ j ẑ 2κ] K(ā + 1 j ) κ. (A.14) Observe that, for, j, E F [ξ (ẑ ) ξ j(ẑ j ) ξ (z ) ξ j(z )] Kα mx (j) 1 2/k ξ (ẑ ) F,k ξ j (ẑ j ) ξ j(z ) F,k +Kα mx (j) 1 2/k ξ j (z ) F,k ξ (ẑ ) ξ (z ) F,k Kα mx (j) 1 2/k ( ḡ,k (ẑ )ρ k(ẑ j, z ) + ḡ,k(z )ρ k (ẑ, z ) ), (A.15) where the frst equalty s obtaed by usg the tragle equalty ad the the mxg equalty; the secod equalty follows from ξ ( ) F,k Kḡ,k ( ) ad (3.1). Note that Assumpto D mples ḡ,k ( ) P (p/2) ad ρ k (ẑ j, z p/2 ) K(1 + V ( j) ) ẑ j z κ. Therefore, (A.15) mples E F [ξ (ẑ ) ξ j (ẑ j) ξ (z ) ξ j (z )] ( ) ( ) Kα mx (j) 1 2/k p/2 p/2 ( ẑ 1 + V 1 + V ( j) j z κ + ẑ z κ). By the Cauchy Schwarz equalty, (A.11) ad (A.14), we further deduce that (A.16) E E F [ξ (ẑ ) ξ j (ẑ j) ξ (z ) ξ j (z )] Kα mx (j) 1 2/k (ā + 1 j ) κ/2. (A.17) 49

50 Next, we set [ Γ E F ξ (ẑ ) 2] N N [ + 2 E F ξ (ẑ ) ξ j (ẑ j) ], Γ j=1 =j N [ E F ξ (z ) 2] N + 2 E F [ξ (z ) ξ j (z )]. j=1 =j (A.18) By (A.17) ad l α mx(l) 1 2/k <, we deduce E Γ Γ N N Kā κ/2 + K α mx (j) 1 2/k (ā + 1 j ) κ/2 j=1 =j N Kā κ/2 + K κ/2 j κ/2 α mx (j) 1 2/k. j=1 As metoed above, ā. Moreover, by Kroecker s lemma, κ/2 N j=1 jκ/2 α mx (j) 1 2/k. Hece, Γ Γ P. (A.19) We ow show Γ P Γ. (A.2) To smplfy otato, we deote γ l,s γ l (β s, Z s, V s ) ad γ s γ(β s, Z s, V s ) for l ad s. We ote that ḡ(z ) because of (2.2). Hece, we ca rewrte Γ as Γ N = γ, + 2 γ j,, j=1 =j where empty sums are set to zero by coveto. Therefore, Γ Γ = ( N = ) T γ, γ,s ds N T + 2 γ j, γ j,s ds. j=1 =j (A.21) By a argumet smlar to (deed smpler tha) (A.16), t s easy to see that γ j (β, z, v) s cotuous (β, z, v). Uder Assumpto SH, the process (γ j,t ) t s càdlàg ad uformly bouded. Hece, by vokg the Rema approxmato, we deduce that, for each j, 5

51 N =j γ j, T γ j,sds. Moreover, observe that N T γ j, γ j,s ds K sup j=1 =j t [,T ] ḡ,k (β t, Z t, V t ) 2 K, where the frst equalty s by the mxg equalty ad j 1 α mx (j) 1 2/k <, ad the secod equalty holds because (β t, Z t, V t) s bouded uder Assumpto SH ad ḡ,k ( ) s bouded o bouded sets. Ths domace codto allows us to use the domated covergece theorem to obta the lmt of the rght-had sde of (A.21). From here, (A.2) readly follows. Fally, we ote that E F [R 2 3, ] = Γ. Combg (A.19) ad (A.2), we derve E F [ R 2 3, ] P Γ. (A.22) Step 4. We ow show that R 3, L F MN (, Γ). Cosder a subset Ω of Ω gve by Ω { Γ > } ad let Ω c be the complemet of Ω. Clearly, Ω s F-measurable. I restrcto to Ω c, E F [ R 2 3, ] = o p (1) ad, thus, the F-codtoal law of R,3 coverges to the degeerate dstrbuto at zero. We ow restrct atteto o the evet Ω, so we ca assume Γ >. We cosder a arbtrary subsequece N 1 N. By the subsequece characterzato of covergece probablty, t s eough to show that there exsts a further subsequece N 2 N 1 such that, as alog N 2, the F-codtoal dstrbuto fucto of R 3, coverges uformly to the F-codtoal dstrbuto fucto of MN (, Γ) o P-almost every path Ω. [ ] By (A.22), we ca extract a subsequece N 2 N 1 such that, alog N 2, E F R 2 3, Γ > for almost every path Ω. Recall from (A.12) that R 3, = 1/2 ξ (ẑ ). Uder Assumpto MIX, ξ (ẑ ) forms a sequece wth zero mea ad α-mxg coeffcets bouded by α mx ( ) uder the trasto probablty P (1). Moreover, E ξ (ẑ ) k KE ḡ,k (ẑ ) k K, where the frst equalty s by repeated codtog, Mkowsk s ad Jese s equaltes; the secod equalty s by ḡ,k P(2p/k) ad (A.11). We are ow ready to apply Theorem 5.2 Whte (21) ad Pólya s theorem uder the trasto probablty P (1) ad deduce that, alog N 2, the F-codtoal dstrbuto fucto of R 3, coverges uformly to the F-codtoal dstrbuto fucto of MN (, Γ) for almost every path Ω. As metoed the prevous paragraph, we ca use a L F subsequece argumet to further deduce that R 3, MN (, Γ). As dscussed step 1, the proof of Theorem 2(b) s ow complete. Q.E.D. 51

52 A.3 Proof of Theorem 3 Proof of Theorem 3: (a) As s typcal ths type of problem, by a polarzato argumet, we ca cosder a oe-dmesoal settg wthout loss of geeralty. We heceforth suppose that g( ) s scalar-valued. By localzato, we also suppose that Assumpto SH holds. To smplfy otato, we set ĝ (θ) = g(y, Z, V ; θ) ad g (θ) = g(y, Z, V ; θ) for ad θ Θ. We ca rewrte (3.8) as N m N Γ (ˆθ ) = ĝ (ˆθ ) w (j, m ) ĝ (ˆθ )ĝ j(ˆθ ). = We cosder a progressve lst of approxmatos to Γ (ˆθ ) gve by Γ (1) Γ (2) Γ (3) j=1 N m N g (θ ) w (j, m ) g (θ ) g j (θ ), = N j=1 E F [g (θ ) 2] m N [ + 2 w (j, m ) E F g (θ ) g j (θ ) ], = N = j=1 [ E F g (θ ) 2] N + 2 j=1 =j =j =j =j N [ E F g (θ ) g j (θ ) ]. We ote that g (θ ) s detcal to ξ (z ) defed step 3 of the proof of Theorem 2, because ḡ(β, Z, V ; θ ) = as a result of (2.8). Therefore, has the same form as Γ defed (A.18) after replacg ξ (ẑ ) ad ξ j(ẑ j ) the latter wth ξ (z ) ad ξ j(z j ), respectvely. From here, we ca use a argumet that s smlar to that step 3 of the proof of Theorem 2 to show that Γ (3) value V. To prove Γ (ˆθ ) are o p (1). Frst cosder Γ (ˆθ ) Γ (ˆθ ) m K K P Γ (3) Γ; ths s actually smpler to prove because V P Γ, t remas to verfy Γ (1) (ˆθ ) Γ, Γ (1) N j= =j m N j= (1) Γ. Observe that ĝ (ˆθ )ĝ j(ˆθ ) g (θ ) g j (θ ) =j s replaced wth the true (1) (2) (2) (3) Γ Γ ad Γ Γ ( ) ĝ (ˆθ ) ĝ j(ˆθ ) g j (θ ) + ĝ (ˆθ ) g (θ ) g j (θ ) ( ( 1/2 ( ) Km ĝ (ˆθ ) 2 + g (θ ) 2)) 1/2 ĝ (ˆθ ) g (θ ) 2, (A.23) 52

53 where the frst equalty s from the tragle equalty ad the boudedess of the kerel fucto w (, ); the secod equalty s from the tragle equalty; the thrd equalty follows from the Cauchy Schwarz equalty. We further observe that, for, ĝ (ˆθ ) g (θ ) 2 K ĝ (ˆθ ) ĝ (θ ) 2 + K ĝ (θ ) g (θ ) 2 K B (Y, Z, V ) 2 ˆθ θ 2 + K ĝ (θ ) g (θ ) 2, (A.24) where the frst equalty s from the tragle equalty ad the secod equalty follows from Assumpto LIP() (recall Defto 1). Uder Assumpto LIP(), we see E[B (Y, Z, V ) 2 ] = E[ B (β, Z, V ) 2 ] K, where the frst equalty s obtaed by repeated codtog, ad the secod equalty s from B P (p) ad (A.11). Sce ˆθ θ = O p ( 1/2 ) by assumpto, B (Y, Z, V ) 2 ˆθ θ 2 = O p ( ). Moreover, by Assumpto D(), for each, (A.25) E F ĝ (θ ) g (θ ) 2 ρ k ((β, Z, V ) 2 ), (β, Z, V ) K(1 + V p 2 ) V 2κ V ( K V V 2κ + V ) p V. (A.26) By (A.2) ad Jese s equalty, { u 1 E V V u K u ( (2u r)ϖ+1 u < u < 1 E V V u K u ( (2 r)ϖu + k u/2 + (k ) (u/2) 1 ), + k u/2 + (k ) u/2 ). (A.27) Note that (2p r) ϖ + 1 p 1/2, (2 r)ϖ 1/4 ad 1/2 (A.27), we derve From (A.24), (A.25) ad (A.28), we derve Kk 1. The, by (A.26) ad E ĝ (θ ) g (θ ) 2 Kk κ. (A.28) ĝ (ˆθ ) g (θ ) 2 = O p (k κ ). (A.29) 53

54 It s easy to see that E g (θ ) 2 K. By (A.29), we further have ( ĝ (ˆθ ) 2 + g (θ ) 2) = O p (1). (A.3) Combg (A.23), (A.29) ad (A.3), as well as Assumpto HAC(), we have Next, we cosder Γ (ˆθ ) N =j ζ j,. We ca the rewrte Γ (1) = O p (m k κ/2 ) = o p (1). (1) (2) Γ Γ. We deote ζj, g (θ )g j (θ ) E F [g (θ )g j (θ )] ad ζ j Γ (1) (2) Γ = ζ m + 2 w (j, m ) ζ j. j=1 (A.31) Note that, codtoal o F, the sequece (g (θ )) s α-mxg wth sze k/(k 2). By the mxg equalty, for, j ad l, [ E F ζ j, ζj,l ] ( Kα mx (l j) + ) 1 2/k ζ j, F,k ζj,l F,k, where ( ) + deotes the postve part. By the Cauchy-Schwarz equalty, ζ j, F,k Kḡ,2k (β, Z, V ; θ )ḡ,2k (β ( j), Z ( j), V ( j) ; θ ). Sce ḡ,2k ( ) s bouded o bouded set (Assumpto HAC()), we further have E F [ζ j, ζ j,l ] Kα mx ((l j) + ) 1 2/k. From here, t follows that E [ ( ζ j ) 2] N N 2 2 E [ E F ζ j, ζj,l] =j l= N N ( α mx (l j) + ) 1 2/k K 2 =j l= K (j + 1). The, by the tragle equalty ad Jese s equalty, as well as the boudedess of the kerel fucto w (, ), we derve from (A.31) that E Γ (1) Γ (2) m K 1/2 j= (j + 1) 1/2 = O( 1/2 m 3/2 ). 54

55 Sce m = o(k κ/2 ) by Assumpto HAC() ad k K 1/2 m = o( 1/4 ). Hece, Fally, we show that (1) (2) Γ Γ Γ (2) Γ (3) (2) Γ = 2 = o p (1). Γ (3) N j=m +1 = o p (1). Note that [ E F g (θ ) g j (θ ) ] N =j by Assumpto LW, we have m N [ +2 (1 w (j, m )) E F g (θ ) g j (θ ) ]. j=1 =j Observe E 2 N j=m +1 [ E F g (θ ) g j (θ ) ] K N =j N j=m +1 α mx (j) 1 2/k, where the equalty s by the tragle equalty ad the mxg equalty ad the covergece s due to j 1 α mx(j) 1 2/k < ad m. Smlarly, m E 2 N [ (1 w (j, m )) E F g (θ ) g j (θ ) ] m K 1 w (j, m ) α mx (j) 1 2/k. j=1 =j Note that for each j, 1 w (j, m ) as. Sce j 1 1 w (j, m ) α mx (j) 1 2/k K j 1 α mx(j) 1 2/k <, the majorat sde of the above equalty coverges to zero as (2) (3) by the domated covergece theorem. From here, t follows that Γ Γ = o p (1) as clamed. The proof of part (a) s ow complete. (b) By a polarzato argumet, we cosder the oe-dmesoal settg wthout loss of geeralty. We set j=1 η v ḡ(β, Z, V ; θ ), S 2 (η ) 2 2 V Note that ḡ( ) C 2,3 (p) mples that the fucto (β, v, z) 2 v ḡ(β, z, v; θ ) 2 v 2 s C(2p). Moreover, the codto ϖ (2p 1)/2(2p r) mples that ϖ (2p 1)/(4p r). Hece, by applyg Lemma A.2 to the fucto (β, v, z) 2 v ḡ(β, z, v; θ ) 2 v 2 P, we derve S S. It remas to show that Ŝ1,(ˆθ P ) S. Below, we complete the proof by showg Ŝ1,(ˆθ ) Ŝ1,(θ ) ad Ŝ1,(θ ) S are o p (1). By the tragle equalty, we see that Ŝ1,(ˆθ ) Ŝ1,(θ ) K (SR 1, + SR 2, ), where SR 1, ˆη (ˆθ ) ˆη (θ ) 2 2 V,. 55

56 SR 2, ˆη (ˆθ ) ˆη (θ ) ˆη (θ ) V 2 Recall the otatos Defto 1. By Assumpto LIP(), SR 1, D 1, ˆθ θ 2, where D 1, 1 k k 1 j= Note that E F [ D 2(p 1) 1, ] K (1 + V ) V 2. B 1 (Y (+j), Z (+j), V ) o p (1), SR 1, = o p (1). By the Cauchy Schwarz equalty, SR 2, SR 1/2 1, 2 V 2.. By (A.11), D1, = O p (1). Sce ˆθ θ = 1/2 D 2,, where D 2, ˆη (θ ) 2 V 2. Note that Assumpto AVAR1() mples that ḡ 1,2 ( ; θ ) P (p 1). Hece, E F [ˆη (θ ) 2 ] K(1 + V 2(p 1) ). By repeated codtog ad (A.11), we further deduce that D 2, = O p (1). Hece, SR 2, s also o p (1). We have Ŝ1,(ˆθ ) Ŝ1,(θ ) = o p (1) as wated. Fally, we show Ŝ1,(θ P ) S. Observe that, by the tragle equalty, E F Ŝ 1, (θ ) S K We set for each, j, ( η E F ˆη (θ ) η + E F ˆη (θ ) η 2) V 2. ( ζ,j v g Y (+j), Z (+j), V ; θ ) ( v ḡ β (+j), Z (+j), V ; θ ), η 1 k 1 ( k v ḡ β (+j), Z (+j), V ; θ ). j= (A.32) By Assumpto S(), E F [ζ,j ] =. We ca wrte ˆη (θ ) η = (1/k ) k 1 j= ζ,j. Hece, E F ˆη (θ ) η 2 1 k 2 K 1 k 2 k 1 K(1 + j,l= k 1 [ E F ζ,j ζ,l] ( α mx ( l j ) 1 2/k 1 + j,l= V p 2 )/k, ) p 2 V (A.33) where the frst equalty s by the tragle equalty; the secod equalty s obtaed by frst usg the mxg equalty ad the the assumpto that ḡ 1,k ( ; θ ) P ((p 2)/2); the thrd equalty s by Assumpto MIX. We set, for each, D = (1/k ) k 1 j= ( β (+j) β + Z (+j) Z ). Note that, 56

57 by a mea-value expaso ad the assumpto that β v ḡ( ; θ ) ad z v ḡ( ; θ ) are P (p/2 1), η η K(1 + V p/2 1 )D. (A.34) Sce η ad η are F-measurable, we combe (A.33) ad (A.34) to derve E F ˆη (θ ) η 2 K(1 + V p 2 )((D ) 2 + 1/k ). (A.35) Next, ote that uder Assumptos S() ad AVAR1(), v ḡ ( ; θ ) P (p/2 1). By (A.32) ad (A.35), E F Ŝ 1, (θ ) S K ( 1/ k + D + (D ) 2) ( 1 + V p ). (A.36) Further ote that E D 2 + E D 4 Kk. Hece, by the Cauchy Schwarz equalty, (A.36) ad (A.11), we derve E Ŝ1,(θ ) S K( k + 1/ k ). Hece, Ŝ1,(θ ) S = o p (1). The proof of part (b) s ow complete. (c) Deote f (z, v; θ) 2 ϕ(z, v; θ ) ϕ(z, v; θ ) v 2. Sce ϕ ( ; θ ) C(p 1), f C (2p). By Lemma A.2, Ŝ 2, (θ P ) S. Sce ϕ ( ; θ ) C (p 1) ad ϕ ( ) LIP(p 1, ), t s easy to see that sup θ Θ ϕ( ; θ) P (p 1) for ay compact subset Θ that cotas θ. Hece, wth probablty approachg oe, Ŝ2,(ˆθ ) Ŝ2,(θ ) p 1 K (1 + V ) ϕ(z, V ; ˆθ ) ϕ(z, V ; θ ) V 2 ( ) 2p K 1 + V ˆθ θ. By (A.11) ad ˆθ θ = o p (1), we see Ŝ2,(ˆθ ) Ŝ2,(θ ) = o p (1). From here, the asserto part (c) readly follows. Q.E.D. Proof of Corollary 2: Part (a) follows from Theorems 2, 3 ad the cotuous mappg theorem. To show part (b), we frst show that cv,1 α (θ P ) cv 1 α. Fx a arbtrary subsequece N 1 N. By Theorem 3, Σ g, (θ P ) Σ g ad, hece, there exsts a further subsequece N 2 N 1, alog whch Σ g, (θ ) a.s. Σ g. Cosder a path ω Ω o whch L(, ) s cotuous at Σ g ad Σ g, (θ ) Σ g holds alog N 2 ; such paths form a P-full evet. By the cotuous mappg theorem, o path ω, the F G-codtoal dstrbuto fucto of L( Σ g, (θ ) 1/2 U, Σ g, (θ )) coverges weakly to the F-codtoal dstrbuto of L(ξ, Σ g ). By assumpto, 1 α s a cotuty pot of the F- codtoal quatle fucto of L(ξ, Σ g ). Hece, o path ω, we have cv,1 α cv 1 α alog N 2. By a subsequece characterzato of covergece probablty, we deduce that cv,1 α (θ P ) cv 1 α. Ths result, combed wth that part (a), mples P (L (θ ) cv,1 α (θ )) 1 α. 57 Q.E.D.

58 A.4 Proof of Propostos 1 ad 2 Proof of Proposto 1: (a) Let Q(θ) G(θ) P ΞG (θ). By Theorem 2(a), G ( ) G( ) ad, P hece, Q ( ) Q( ) uformly over Θ. It s easy to see from Assumpto LIP() that G ( ) s cotuous ad so s Q( ). Uder Assumpto GMIM, Q( ) s uquely mmzed at θ. Sce Θ s compact, ˆθ P θ follows from a stadard argumet (see, e.g., Theorem 2.1 Newey ad McFadde (1994)). (b) Uder Assumptos S ad D, for each θ Θ, the fuctos (β, z, v) θ g(β, z, v; θ) ad (β, z, v) θ 2 vg (β, z, v; θ) v 2 satsfy codto () of Theorem 1. Moreover, by Assumpto LIP(), the fuctos θ g (y, z, v; θ) ad θ 2 vg(y, z, v; θ)v 2 belog to LIP(p, ). By Theorem 1, we have, θ G (θ) I partcular, for ay sequece θ that satsfes θ T P θ ḡ (β s, Z s, V s ; θ) ds, uformly θ Θ. (A.37) uder Assumpto GMIM, a route mapulato yelds, P θ, we have θ G ( θ ) = H + o p (1). The, 1/2 (ˆθ θ ) = (H ΞH) 1 H Ξ 1/2 G (θ ) + o p (1). (A.38) The asserto the follows from Theorem 2(b). P (c) By (A.37), H H. Sce 1/2 (ˆθ θ ) = O p (1) from part (b), the asserto of part (c) readly follows from Theorem 3. Proof of Proposto 2: Deote A (I q H (H ΞH) 1 H Ξ)Σ 1/2 g. Observe that 1/2 G (ˆθ ) = 1/2 G (θ ) + H 1/2 (ˆθ θ ) + o p (1) = AΣ 1/2 g 1/2 G (θ ) + o p (1), where the frst equalty s by a mea-value expaso, ˆθ Q.E.D. P θ ad the uform covergece gve by (A.37); the secod equalty s obtaed by usg the asymptotc lear represetato (A.38). Note that Σ 1/2 G (θ ) L-s N (, I q ) by Theorem 2. It s also straghtforward to show that g 1/2 A ΞA s dempotet wth rak q dm (θ). The asserto of the proposto readly follows.q.e.d. 58