Fixed-smoothing Asymptotics and Asymptotic F and t Tests in the Presence of Strong Autocorrelation

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1 Fixed-moohing Aymoic and Aymoic F and e in he Preence of Srong Auocorrelaion Yixiao Sun Dearmen of Economic, Univeriy of California, San Diego May 26, 24 Abrac New aymoic aroximaion are eablihed for he Wald and aiic in he reence of unknown bu rong auocorrelaion. he aymoic heory exend he uual xed-moohing aymoic under weak deendence o allow for near uni roo and weak uni roo rocee. A he localiy arameer ha characerize he neighborhood of he auoregreive roo increae from zero o in niy, he new xed-moohing aymoic diribuion change moohly from he uni-roo xed-moohing aymoic o he uual xed-moohing aymoic under weak deendence. Simulaion how ha he new aroximaion i more accurae han he uual xed-moohing aroximaion. JEL Clai caion: C3; C4; C32; C5 Keyword: Auocorrelaion Robu e, Fixed-moohing Aymoic, Local-o-Uniy, Srong Auocorrelaion, Weak Uni Roo. yiun@ucd.edu. Correondence o: Dearmen of Economic, Univeriy of California, San Diego, 95 Gilman Drive, La Jolla, CA he idea in he aer were r reened in my dicuion of Müller (24) a he 23 ASSA meeing in San Diego, CA. For helful commen, he auhor hank Yoooon Chang, Joon Park, Peer Philli, an anonymou referee and he arician of he 4h Advance in Economeric Conference a Souhern Mehodi Univeriy in 23.

2 Inroducion In order o robuify aiical inference o heerokedaiciy and auocorrelaion, i i now a andard racice o make ue of heerokedaiciy and auocorrelaion robu (HAR) andard error in ime erie analyi. he la decade of reearch on hi oic ha been largely focued on more accurae aroximaion o he e aiic conruced on he bai of HAR andard error and variance. here are now wo ye of aymoic aroximaion: he convenional increaing-moohing aymoic aroximaion and he relaively new xed-moohing aymoic aroximaion. For kernel HAR inference, he xed-moohing aymoic i he o-called xed-b aymoic of Kiefer, Vogelang and Bunzel (2, KVB hereafer) and Kiefer and Vogelang (22a, 22b, 25, KV hereafer). For erie HAR inference, he xed-moohing aymoic i he o-called xed-k aymoic of Sun (2, 23a). While he xed-moohing aymoic aroximaion i more accurae han he convenional increaing-moohing aymoic aroximaion, i.e. he chi-quare aroximaion (Janon, 24; Sun, Philli and Jin, 28), he qualiy of he xed-moohing aroximaion i no comleely aifacory when he underlying ime erie ha rong auocorrelaion. o confron wih hi roblem, he aer develo a new xed-moohing aymoic ha accommodae rong auocorrelaion. We ar by conidering a mulivariae ime erie whoe mean value i he arameer of inere. Each comonen of he ime erie follow an auoregreive roce wih he auoregreive arameer changing wih he amle ize : More eci cally, = c m = for ome oiive equence c m : Deending on he limiing hough exerimen emloyed, c m may be held xed a he amle ize! or grow wih he amle ize. he former cae i he convenional local-o-uni-roo eci caion and he laer cae eci e a moderae deviaion from he uni roo. Park (23) refer o hee wo cae a near uni roo and weak uni roo, reecively. For more dicuion, ee Giraii and Philli (26), Philli and Magdalino (27), and Philli, Magdalino and Giraii (2). We eablih he xed-moohing aymoic for he Wald and aiic when c m i held xed. hi lead o our ivoal near-uni-roo xed-moohing aymoic. A c m! ; we how ha he near-uni-roo xed-moohing aymoic converge o he xed-moohing aymoic under weak deendence. Deending on he value of c m ; he near-uni-roo xed-moohing aymoic hu rovide a mooh raniion from he uual aionary xed-moohing aymoic o he uni-roo xed-moohing aymoic. he near-uni-roo xed-moohing aymoic diribuion i nonandard bu can be imulaed. he criical value from hi diribuion are larger han he correonding one from he uual xed-moohing aymoic diribuion, which in urn are larger han hoe

3 from he convenional chi-quare diribuion. A direc imlicaion i ha aiical inference baed on he chi-quare aroximaion and aionary xed-moohing aroximaion may lead o he nding of aiical igni cance ha doe no acually exi. In he cae of erie variance eimaion, we can judiciouly deign a e of bai funcion uch ha he near-uni-roo xed-moohing aymoic diribuion become a andard F or diribuion. he deigning roce involve rojecion and orhogonalizaion. he F and aroximaion are very handy in emirical alicaion, a he F and criical value are readily available from aiical rogram and ofware ackage. here i no need o imulae nonandard criical value. he oibiliy of deriving a andard aroximaion under he xed-moohing aymoic i an advanage of uing erie variance eimaor. For hi ye of eimaor, we have he comlee freedom in chooing he bai funcion. hi i in conra wih he kernel variance eimaor where he bai funcion are imlicily given and canno be changed. Mone Carlo imulaion how ha he chi-quare e have he large ize diorion, he e baed on he near-uni-roo xed-moohing aymoic have he malle ize diorion, and he e baed on he uual xed-moohing aymoic are in he middle. he near-uni-roo xed-moohing e, which include e baed on F and aroximaion, achieve rile robune in he following ene: i i aymoically valid regardle of wheher auocorrelaion i reen or no; wheher he auocorrelaion i rong or no and wheher he level of moohing i held xed or i allowed o increae wih he amle ize. he re of he aer i organized a follow. Secion 2 decribe he baic eing and he roblem a hand. Secion 3 develo he xed-moohing aymoic in he reence of near-uni roo. hi ecion alo eablihe he behavior of he near-uniroo xed-moohing aymoic diribuion a c m!. Secion 4 reen he F and aroximaion baed on a e of judiciouly deigned bai funcion. Secion 5 dicue he alicabiliy of our xed-moohing aroximaion for locaion model o more general model. he ubequen ecion reor imulaion evidence, and he la ecion rovide ome concluding dicuion. Proof of he main reul are given in he Aendix. 2 he Baic Seing and he Problem Aume ha -dimenional ime erie y follow he roce: y = + e ; = ; 2; : : :; () 2

4 where y = (y ; : : : ; y ), = ( ; : : : ; ), and e = (e ; : : : ; e ) i a zero mean roce. he locaion model, a imle a i i, i emirically relevan in a number of iuaion. For examle, he daa migh coni of a mulivariae ime erie of forecaing lo ha are roduced by di eren forecaing mehod. We can e equal redicive accuracy of hee forecaing mehod by examining wheher he lo di erenial erie ha mean zero. here i alo a large and acive lieraure on inference for he mean of imulaed ime erie. See for examle Alexooulo (26, 27) and reference herein. More imoranly, our oin can be made more clearly in he imle locaion model. We are inereed in eing he null H : = again he alernaive H : 6= : he OLS eimaor of i he average of fy g ; i.e., ^ = y := P = y : he F e verion of he Wald aiic baed on he OLS eimaor i hen given by F = D ^ ^ D ^ = = D ^ ^ 2 ^ = where D i a real-valued caling facor ha characerize he rae of convergence of ^ o ; and ^ i an eimaor of he aymoic variance of D (^ ): Uually D = bu he exac magniude of D i no imoran in racice, a i will be canceled ou. When = ; we can conruc he -aiic a follow: D ^ = ^ Many nonarameric eimaor of he aymoic variance marix are available in he lieraure. In hi aer, we conider a cla of quadraic variance eimaor, which include he convenional kernel variance eimaor of Andrew (99), Newey and We (987), Polii (2), and he erie variance eimaor of Philli (25), Müller (27), and Sun (26, 2, 23a) a ecial cae. he quadraic variance eimaor ake he following form: ^ = D2 2 X = = X Q h ; : ^e ^e (2) where ^e = y ^ = e e for e = P = e and Q h (r; ) i a weighing funcion ha deend on he moohing arameer h: A reaonable choice of Q h (r; ) hould aify: (i) Q h ( ; ) decay o zero a j j = aroache ; (ii) Q h( ; ) increae o a j j = aroache : he eed of change i conrolled by h: For convenional kernel variance eimaor, Q h (r; ) = k ((r ) =b) and we ake h = =b; where k () i a kernel funcion. For he erie variance eimaor, Q h (r; ) = K P K j= j (r) j () and we ake h = K; where f j (r)g are bai funcion on L 2 [; ] aifying R j (r) dr = : We aramerize h in uch a way ha h indicae he level or amoun of moohing in boh cae. 3

5 De ne Q ;h (r; ) = Q h (r; ) X = Q h ; X Q h r; = X X = 2 = Q h ; 2 ; hen ^ = D2 2 X X = = Q ;h ; e e ; (3) where he demeaning oeraion on e and e ha been moved o he weighing funcion. he Wald aiic i hen equal o F =! " X X e = Similarly, he -aiic become X = = Q h ; # e e! X e =: = = P = e h P P = = Q h ; i =2 : e e Noe ha he caling facor D ha been canceled ou in boh F and : he queion i how o aroximae he amling diribuion of F and : If =2 P [ r] = e converge weakly o a Brownian moion roce, hen under ome condiion on Q h ; i can be hown ha, for a xed h : F! d F := W () Q h (r; ) dw (r) dw () W () =; (4)! d W () := q R R ; (5) Q h (r; ) dw (r) dw () where W (r) i a vecor of andard Wiener rocee and Q h (r; ) = Q h (r; ) Q h ( ; ) d Q h (r; 2 ) d 2 + Q h ( ; 2 ) d d 2 i he coninuou verion of Q ;h (r; ). See for examle, Kiefer and Vogelang (25) for he kernel cae and Sun (23a) for he erie cae. For boh variance eimaor, W () i indeenden of R R Q h (r; ) dw (r) dw () : So F reemble an F diribuion in he ene ha boh can be wrien a a quadraic form in andard normal wih an indeenden weighing marix. Similarly, reemble a diribuion in he ene ha boh can be wrien a a andard normal caled by an indeenden random variable. 4

6 he aymoic aroximaion obained under a xed h a! i he xedmoohing aymoic aroximaion. hi aroximaion imrove he radiional chiquare or normal aroximaion, which i obained under he increaing-moohing aymoic where h increae wih bu a a lower rae. However, he xed-moohing aymoic aroximaion i ill no aifacory when he underlying roce ha rong auocorrelaion. Our goal i o eablih a furher imroved aroximaion when e may be rongly auocorrelaed. 3 Fixed-Smoohing Aymoic under he Local-o-Uniy Seci caion o model rong auocorrelaion, we mainain he following aumion on fe g : Aumion (i) For ome oiive equence fc m g e = ;m e + u where e = O () and ;m = c m ; (ii) P = P = keu u k i bounded uniformly in ; (iii) fu g ai e a FCL: [ r] X u! d W (r); a! ; (6) = where W (r) i a vecor of andard Wiener rocee and = =2 i he marix quare roo of he long run variance marix of u : = = X j= Eu u j: he daa generaing roce in Aumion i imilar o ha ued in Philli, Magdalinou and Giraii (2), which eablihe a mooh raniion beween he convenional uni roo diribuion and he andard norm diribuion. When c m i xed, each comonen of fe g ha a local-o-uniy roo in he convenional ene (Philli, 987). When c m! a! ; each comonen of fe g ha a moderae uni roo in he ene ha he roo belong o a larger neighborhood of uniy han he convenional local-o-uniy roo. We could allow for di eren c m for di eren comonen of fe g : For noaional imliciy, we aume ha all comonen have he ame local-o-moderae uni roo. he FCL aumion hold for erially correlaed and heerogeneouly diribued daa ha aify cerain regulariy condiion on momen and he deendence rucure over ime. 5

7 hee rimiive regulariy condiion can be found in Philli and Durlauf (986), Philli and Solo (992), Davidon (994), among oher. Under Aumion, we have e [ r]! J cm (r) for each xed c m ; where J cm (r) i he Ornein-Uhlenbeck (OU) roce de ned by dj cm (r) = c m J cm (r) dr + dw (r) wih J cm () = or J cm (r) = R r e cm(r ) dw () : Aumion 2 (i) For kernel variance eimaor, he kernel funcion k () ai e: for any b 2 (; ], k b (x) := k(x=b) i ymmeric, coninuou, iecewie monoonic, and iecewie coninuouly di ereniable on [ ; ]. (ii) For he erie variance eimaor, f j ()g are iecewie monoonic, coninuouly di ereniable, and orhonormal in L 2 [; ] and R j (x) dx = for each j: Aumion 2 i imilar o Aumion in Sun (23b). Under hi aumion, we can relace Q h (r; ) in (3) by Q h (r; ) and all our aymoic reul coninue o hold. For our heoreical develomen, we can aume ha uch a relacemen ha been made. Furhermore, under Aumion 2, Q h (r; ) ha he following uni ed rereenaion for boh ye of variance eimaor we conider: Q h (r; ) = X j j (r) j () ; (7) j= where f j (r)g i a equence of coninuouly di ereniable funcion aifying R j (r) dr =. he righ hand ide of (7) converge aboluely and uniformly over (r; ) 2 [; ] [; ]: he rereenaion hold rivially for he cae of erie variance eimaion, a Q h (r; ) = Q h (r; ) = K KX j (r) j () ; j= and o we can ake f j g = K ; : : : ; K ; ; : : : ; ; : : : and j = j : For he cae of kernel variance eimaion, Sun (23b) rove ha he rereenaion hold wih he following choice of j and j : j (r) = ( co 2 jr ; j i even in 2 (j + ) r R in 2 (j + ) d; j i odd (8) 6

8 and j = ( ~j=2 ; j i even ~ (j+)=2 ; j i odd where f ~ j g are he Fourier ranform of k b () ; i.e., ~ j = R k(x=b) co (jx) dx: For eay reference, we de ne (r) : heorem Le Aumion and 2 hold. hen for xed c m and h; F! d F (c m ) := J cm (r) dr Q h (r; ) J c m (r) Jc m () drd If we furher aume ha Q h (r; ) i oiive de nie, hen J cm (r) dr =:! d (c m ) := R J c m (r) dr h R R i =2 : Q h (r; ) J c m (r) Jc m () drd Since cov (J cm (r); J cm ()) = 2c m fex [ c m jr j] ex [ c m (r + )]g I ; where I i he ideniy marix, we have cov j (r) J cm (r) dr; j2 (r) J cm (r) dr = 2c m j (r) j2 () fex [ c m jr j] ex [ c m (r + )]g drd I for all j and j 2 : In general, he covariance marix i no zero. A direc imlicaion i ha he weighing marix R R Q h (r; ) J c m (r) Jc m () drd i no indeenden of R J c m (r) dr: Hence he limiing diribuion F (c m ) canno be wrien a a quadraic form in andard normal wih an indeenden random weighing marix. hi i in conra wih F which ake uch a form. he limiing diribuion deend no only on he kernel/bai funcion, he moohing arameer h; he number of join hyohee bu alo on he local-o-uni-roo arameer c m. In he exreme cae when c m = ; e ha uni roo and F (c m ) become F () := W (r)dr Q h (r; ) W (r) W () drd W (r) dr =: hi cae eem o be of only heoreical inere. he emirically more relevan cae i when c m i large. he heorem below characerize he behavior of F (c m ) and (c m ) a c m! : 7

9 5% CV c = c = 2 c = 3 c = b Figure : Criical value from F (c) wih = and he Barle kernel heorem 2 A c m! for a xed h; we have F (c m ) = F () + o () and (c m ) = () + o () where F () := F and () := de ned in (4) and (5) reecively. heorem 2 how ha he equenial limi when! followed by c m! lead o he uual xed-moohing aymoic under weak deendence. Deending on he value of c m ; he limiing diribuion F (c m ) and (c m ) rovide a mooh raniion from he uual xed-moohing aymoic o he uni-roo xed moohing aymoic. Figure 3 reen he 5% criical value from F (c m ) wih = for wo kernel variance eimaor and one erie variance eimaor. For he erie variance eimaor, we ake K o be even and ue 2j (x) = 2 co(2jx), 2j (x) = 2 in(2jx); j = ; : : : ; K=2 a he bai funcion. In he gure, c = correond o he uual xed-moohing aymoic, and he olid line indicae he criical value from he 2 diribuion. For any given variance eimaor, he criical value increae monoonically a c m aroache zero. ha i, he more erien he underlying rocee are, he higher he criical value will be. While he criical value from he uual xed-moohing aymoic diribuion are larger han hoe from he chi-quare diribuion, hey are ill no large enough when he roce i highly auocorrelaed. Figure 3 alo demonrae he mooh raniion from he near-uni-roo xed-moohing aymoic o he uual xed-moohing aymoic. 8

10 5% CV 5% CV c = c = 2 c = 3 c = b Figure 2: Criical value from F (c) wih = and he Parzen kernel c = c = 2 c = 3 c = K Figure 3: Criical value from F (c) wih = and he erie LRV eimaor 9

11 4 F Aroximaion in he Cae of Serie Variance Eimaion In hi ecion, we conider he erie variance eimaion and how ha F (c m ) can be aroximaed by an F diribuion for ome (ranformed) bai funcion. In he cae of erie variance eimaion, we have F (c m ) = (r) J cm (r) dr ( KX ` (r) J cm (r) dr K `= ) ` (r) J cm (r) dr (r) J cm (r) dr = for (r) : We wan o elec he bai funcion uch ha R ` (r) J cm (r) dr iidn(; I ) acro ` = ; ; : : :; K: Noe ha R ` (r) J cm (r) dr for ` = ; ; : : :; K are joinly normal, i u ce o elec f` (r) ; ` = ; : : :; Kg uch ha f R ` (r) J cm (r) dr; ` = ; ; : : :; Kg are no correlaed wih each oher. Bu where cov ` (r) J cm (r) dr; = `2 (r) J cm (r) dr ` (r) `2 () cm (r; )drd I cm (r; ) := fex [ c m jr j] ex [ c m (r + )]g = (2c m ) i he covariance kernel of J cm () : When c m = ; we de ne cm (r; )j cm= = lim c m! c m (r; ) = min(r; ); which i he covariance kernel of he andard Brownian moion. So i u ce o elec he bai funcion o aify: ` (r) `2 () cm (r; )drd = ``2 for all `; `2 = ; : : :; K; (9) ` cm (r; )drd = for all ` = ; 2; : : :; K; () he la e of equaion mainain he zero mean condiion. ` (r) dr = for all ` = ; 2; : : :; K: () Inead of direcly earching for he aroriae bai funcion, for comuaional reaon we conider he dicree analogue of (9) (). Le A = (a ij ) be he marix

12 wih elemen a ij = cm (i=; j= ): By de niion, A i a oiive de nie ymmeric marix. For any wo vecor `; `2 2 R, we de ne he inner roduc: h`; `2i = `A`2= 2 : (2) hen R i a Hilber ace wih he above weighed inner roduc. Le ` = [` (= ) ; ` (2= ) ; : : :; ` (= )] be he bai vecor aociaed wih he bai funcion ` () for ` = ; ; : : :; K: he nie amle verion of (9) () are: `; `2 = ``2 for all `; `2 = ; : : :; K; (3) `A = for all ` = ; 2; : : :; K; (4) ` = for all ` = ; 2; : : :; K: (5) Noe ha (3) i di eren from he uual orhonormaliy in he Euclidian ene. In general, he bai vecor f`g do no aify (3) even if he bai funcion f`g are orhonormal in L 2 [; ]: However, given any candidae bai funcion or vecor, we can make hem aify he above condiion via ranformaion and orhogonalizaion. follow he e below: (i) Le V = [ ; A ] be a 2 marix and comue for ` = ; : : :; K: h ~` = I V V V V (ii) Emloy he Gram-Schmid cheme o orhogonalize he vecor f ~`g K`= under he inner roduc h; i : More eci cally, we le i ` We ~q = ~ ; ~q 2 = ~ 2 h ~ 2 ;~q i h~q ;~q i ~q ;... ~q K = ~ K h ~ K ;~q K i h~q K ;~q K i ~q K : : : h ~ K ;~q i h~q ;~q i ~q : Noe ha ` ~ i he rojecion of ` ono he orhogonal comlemen of he ace anned by and A : By conrucion ~ `A = and ~ ` = : Since ~q` i a linear combinaion of `, ~ we have ~q `A = and ~q ` = : Alo by conrucion, ~q`; ~q`2 = for ` 6= `2: Le q` = ~q`= h~q`; ~q`i; hen fq ; : : :; q K g i a e of bae in R ha ai e he condiion in (3) (5).

13 Le := ~ ~ ; ~ 2 ; : : :; ~ K uer riangular marix given by 2 h ~ 2 ;~q i h~q ;~q i ~R = 6 4 and ~ Q := (~q ; ~q 2 ; : : :; ~q K ). hen ~ = ~ Q ~ R where R i he h ~ K ;~q i h~q ;~q i... h ~ K ;~q K i h~q K ;~q K i o rereen ~ in erm of Q := (q ; q 2 ; : : :; q K ), we le D = diag(h~q ; ~q i =2 ; : : :; h~q K ; ~q K i =2 ) be a K K diagonal marix. hen ~ Q = QD and ~ = QR where R = D ~ R: he above decomoiion i relaed o he QR decomoiion bu he column of Q are no orhonormal in he uual Euclidean ene bu inead hey are orhonormal under he inner roduc de ned in (2). In a marix rogramming environmen, we can comue Q eaily. Le R be he uer riangular facor of he Choleky decomoiion of ~ A ~ = 2 ; ha i, ~ A ~ = 2 = (R ) R : hen we can le Q = ~ (R ), which ai e a deired. Q AQ= 2 = R ~ A ~ (R ) = 2 = R (R ) R (R ) = I K ; Uing he bai vecor fq ; : : :; q K g, we can conruc he variance eimaor:!! ^ q = D2 KX X X 2 q`^e q`^e K `= = =!! = D2 KX X X 2 q` e q` e K `= = where q` i an elemen of q` o ha q` = (q` ; : : :; q` ) and he econd equaliy hold = becaue P = q` = q ` = : he aociaed F aiic i F ;q = D ^ ^ q D ^ = = X =! 2 e 4 K KX j= X =! e q` X =! 3 e q` 5 X = e! = and he aiic i D ^ ;q = = q^q " X = e # 2 4 K KX j= X =! 2 3 e q` 5 =2 : 2

14 A! for a xed c m and h; we have h ; i X = e ; X = q e ; : : :; X = q K e!! d ( ; ; : : :; K ) where ( ; ; : : :; K ) are joinly normal. Since hq`; i = for ` = ; :::; K and q`; q`2 = `;`2, we know ha ` iidn(; I ) for ` = ; ; : : :; K: Hence ( h ; i F ;q! d F (c m ) = KX `) ` =: K Noe ha F (c m ) follow Hoelling ;K 2 diribuion. Uing he relaionhi beween he 2 diribuion and he F diribuion, we have F ;q := K + K! d K + K h ; i F ;q `= F (c m ) = d K + ;K 2 = d F ;K + ; (6) K where F ;K + i he F diribuion wih degree of freedom (; K + ) : Similarly, ;q := h ; i ;q! d K ; (7) where K i he diribuion wih degree of freedom K: Noe ha h ; i =! X 2 = = X cm ; cm (r; ) drd = c 2 m 2c 3 m e 2cm 4e cm + 3 ; a! ; we can relace h ; i by he above limi in he de niion of F;q and ;q and ill ge he ame limi diribuion. Our aymoic develomen jui e he ue of andard F and aroximaion in hyohei eing, even if he underlying roce ha rong auocorrelaion. More ecifically, we can emloy he modi ed e aiic F ~ and a andard F diribuion a he reference diribuion o e join hyohee. When here i a ingle hyohei, we can emloy he modi ed aiic and he andard diribuion. hi lead o our aymoic F e and e. he limiing diribuion F ;K + and K are exacly he ame a wha Sun (23a) obained in he abence of rong auocorrelaion. More eci cally, when i a xed conan le han in abolue value, Sun (23a) emloy orhonormal bai funcion, 3

15 ay f o`()g ; in L 2 [; ] wih R o`(r)dr = o conruc he erie variance eimaor, and eablihe he following weak convergence reul for he aociaed F and aiic: F! d F () and! d () where K + F () = d F ;K + and () = d K : K ha i K + F! d F ;K + and! d K : K o relae he above aroximaion o hoe in (6) and (7), we conider he cae when c m i large. A c m! ; we have o` (r) o`2()c 2 m cm (r; )drd = o` (r) c 2 m cm (r; )drd = o` (r) o`2 (r) dr + o () ; o` (r) dr + o(): So if f o` ()g i a e of orhonormal bai funcion in L 2 [; ] wih R o`(r)dr = ; hen fc m o` (r)g will aify (9) () aroximaely. ha i, a c m! ; he e ec of ranformaion and orhogonalizaion on fc m o` ()g become negligible. We can ju ue f o` ()g a he bai funcion o conruc he erie variance eimaor, leading o he modi ed F aiic F ; o. We have where F o (c m ) = F ; o := K + K J cm (r) dr ( KX K = `= h ; i F ;o! d K + F o (c m ) K ) o` (r) J cm (r) dr o` (r) J cm (r) dr 3 c 2 m cm (r; )drd 2 4 q R R c mj cm (r) dr R c2 m cm (r; )drd ( KX ) o` (r) c m J cm (r) dr o` (r) c m J cm (r) dr K `= 2 R 3 4 c mj cm (r) dr q 5 R R = c2 m cm (r; )drd (! d KX `) ` = = d F () K `= 5 J cm (r) dr = 4

16 R R a c m! ; where we have ued lim cm! c2 m cm (r; )drd = : So F; o converge weakly o F ;K + under he equenial limi. hi i conien wih heorem 2. 5 Exenion o a General Seing In he reviou ecion, we ue he imle mulivariae locaion model o highligh he e ec of rong auocorrelaion on diribuional aroximaion. Hyohei eing in locaion model, a imle a i eem, include more general eing roblem a ecial cae. Conider an M-eimaor, ^, of a n arameer vecor ha ai e ^ = arg min Q () = arg min 2 2 X (; Z ) where i a comac arameer ace, and (; Z ) i he crierion funcion baed on obervaion Z. M-eimaor are a broad cla of eimaor and include, for examle, he maximum likelihood eimaor (MLE), ordinary lea quare (OLS) eimaor, quanile regreion eimaor a ecial cae. Suoe we wan o e he null hyohei ha H : = again H : 6= : hen by he uual ideni caion aumion for he M-eimaor, under he null hyohei and addiional regulariy aumion, = i he unique minimizer of ha i Q() = E(; Z ): = E () = where () Z if and only if = : So he null hyohei H : = i equivalen o he hyohei ha he mulivariae roce ( ) ha mean zero. We have ju convered a general eing roblem ino eing for zero mean of a mulivariae roce. he laer roblem i exacly he eing roblem we conider in he reviou ecion. All reul here remain valid if he mulivariae roce ( ) ai e he aumion imoed on y : he above exenion alie o hyohei eing ha involve he whole arameer vecor : Suoe we are only inereed in ome linear combinaion of uch ha he null hyohei i H : R = r and he alernaive hyohei i H : R 6= r; where R i a n marix. Under he uual regulariy condiion, we have R^ r = X = RH ( ) ( ) for H () = X 5

17 and ome value beween and ^ : Le ~ ( ; 2 ) = RH ( ) ( 2 ) be he ranformed core roce, hen he F-e verion of he Wald aiic i ~F = R^ r ~ R^ r =; (8) where ~ = X X = = Q h ; ~ ^ ; ^ ~ ^ ; ^ Under he aumion given in KV (25) and Sun (23a), ~ F equivalen o where e = " F = = = X = # " e ^ X = RH ( ) ( ), H( ) = lim! H ( ); and ^ = X X Q h ; (e e) (e e) : i aymoically e # =; (9) I follow ha he F aiic can be viewed a a Wald aiic for eing wheher he mean of he mulivariae roce e i zero. So he aymoic aroximaion in heorem alie o he Wald aiic. he ame obervaion remain valid for he aiic. More generally, conider a andard GMM framework wih he momen condiion Ef (Z ; ) = ; = ; 2; : : :; where 2 R n and f () i an m vecor of wice coninuouly di ereniable funcion wih m n and rank E [@f (Z ; ) =@ ] = n: he GMM eimaor of i hen given by!! X X ^ = arg min f (Z ; ) W f (Z ; ) 2 = where W i an m m oiive emide nie weighing marix and lim! W = W : Suoe we e H : R = r again he alernaive H : R 6= r: Le G () = h lim P i! (Z ; ) =@ : hen under he uual regulariy condiion for GMM eimaion, we have R^ r = X = A before, we le ~ ( ; 2 ) = R [G ( ) W G ( )] = h i R G (^ ) W G ^ G (^ ) W f (Z ; ) + o () (2) G ( ) W f (Z ; 2 ) : hen he Wald aiic can be comued in he ame way a in (8), which i aymoically equivalen o F in (8) wih e = R G W G G W f(z ; ): Wih hi new de niion of e ; he aymoic aroximaion in he reviou ecion are alicable o boh he Wald aiic and aiic. 6

18 6 Imlemenaion and Simulaion 6. Imlemenaion he near-uni-roo xed-moohing aymoic aroximaion F (c m ) and (c m ) deend on he near-uni-roo arameer c m ; which canno be conienly eimaed. here are many way o gauge he value of c m in he lieraure on oimal uni roo eing. Sun (24) conruc con dence inerval for c m and ue he mehod of Bonferroni bound o obain e wih good ize roerie. Here for imliciy we ue he OLS eimaor: ^ i = P =2 ^e i^e i; P =2 ^e2 i; where for he locaion model, we le ^e i = y i y i;, for he model eimaed by he M eimaor, we le ^e i be he i-h comonen of RH (^ ) (^ ); for he model eimaed by he GMM eimaor, we le ^e i be he i-h comonen of R h ^G W ^G i ^G W f(z ; ^ ) wih ^G = G (^ ): Given he average of he eimaed auoregreive arameer: ^ = P i= ^ i; we ake c m o aify ^c m = = ^; ha i, ^c m = ( ^) : o reduce he randomne of ^ and hence ^c m ; we can dicreize he inerval [ ; ] and ue he grid oin ha i cloe o ^ a he auoregreive arameer. Le ~ be hi grid oin, we can le ^c m = ( ~) : I i imoran o oin ou ha while we rooe uing F (c m ) or (c m ) a he reference diribuion o erform hyohei eing, i doe no mean ha we lierally rea c m a xed. Wheher we hold c m xed or le i increae wih he amle ize can be viewed a di eren aymoic eci caion o obain aroximaion o he nie amle diribuion. he near-uni-roo xed-moohing aymoic doe no require ha we x he value of c m in nie amle. In fac, if e i aionary wih a xed auoregreive coe cien, hen wih robabiliy aroaching one, ^c m will increae wih he amle ize. hi will no caue any roblem, a we have hown ha a c m increae, F (c m ) will become very cloe o F () ; he xed-moohing aymoic diribuion under weak deendence. So he near-uni-roo xed-moohing aroximaion i aymoically valid under weak deendence. o ome exen, F (c m ) i a more robu aroximaion han F () : 6.2 Simulaion For he Mone Carlo exerimen, we r conider a imle mulivariae locaion model wih 4 ime erie: y = + u (2) 7

19 where he error u follow eiher a VAR() or VMA() roce: u = Au + " or u = A" + " where A = I 4 ; " = (v + f ; v 2 + f ; :::; v n + f ) = + 2 and (v ; f ) i a mulivariae Gauian whie noie roce wih uni variance. In he cae of VAR(), we e u N(; I 4 ): Under hi eci caion, he four ime erie all follow he ame VAR() or VMA() roce wih " iidn(; ) for = + 2 I J 4; where J 4 i a 4 4 marix of one. he arameer deermine he degree of deendence among he ime erie conidered. When = ; he four erie are uncorrelaed wih each oher. he large i, he larger he correlaion i. For he model arameer, we ake = :5; ; :; :3; :5; :7; :9; and :95 and e = and : We e he inerce o zero a he e we conider are invarian o hem. For each e, we conider wo igni cance level = 5% and = % and wo di eren amle ize = 2 and 4: We conider he following null hyohee: H : = ; H 2 : = 2 = ; H 3 : = 2 = 3 = ; H 4 : = 2 = 3 = 4 = ; where = ; 2; 3; 4; reecively. he correonding marix R i he r row of he ideniy marix I 4 : We examine he nie amle erformance of hree di eren grou of e. he r grou of e coni of he andard xed-moohing Wald e of KVB (2) and Sun (23a). he KVB e emloy he Barle kernel variance eimaor baed wih b =, and he e in Sun (23a) emloy he erie variance eimaor wih K = 6; 2; and 24. he bai funcion ued are 2j (x) = 2 co(2jx), 2j (x) = 2 in(2jx); j = ; : : : ; K=2: For boh he KVB e and he e in Sun (23a), we ue andard xedmoohing criical value, which are obained from imulaion (he KVB e) or direcly from he andard F diribuion (he e in Sun (23a)). he econd grou of e i imilar o he r grou bu ue criical value from he near-uni-roo xed-moohing aymoic diribuion develoed in Secion 3. he criical value are obained via imulaion wih 999 imulaion relicaion. he hird grou of e i baed on he F aroximaion in Secion 4. he bai funcion/vecor are ranformed and orhogonalized 8

20 verion of 2j (x) = 2 co(2jx), 2j (x) = 2 in(2jx); j = ; : : : ; K=2. hee hree grou of e will be referred o a he Saionary Fixed-moohing e, Near-Uniy Fixed-moohing e, and Near-Uniy F e. We could add he convenional chiquare e bu a large lieraure ha already hown ha he chi-quare e have much larger ize diorion han he correonding xed-moohing e. able reor he emirical null rejecion robabiliy of he hree grou of e for VAR() error. he amle ize i 2 and he number of imulaion relicaion i. A few obervaion can be drawn from he able. Fir, i i clear ha he aionary xedmoohing aymoic aroximaion work very well when he rocee are no rongly auocorrelaed. When he rocee become more auocorrelaed, he ize diorion ar o increae. hi i eecially rue when he number of join hyohee i more han one. he ize diorion of he aionary xed-moohing e can be a high a.795. hi haen when K = 24 and = 4: Even for he KVB e wih b = ; he ize diorion i.488 when = 4: Second, he near-uniy xed-moohing aymoic e ouerform he aionary xed-moohing e in erm of ize accuracy. When he rocee are no rongly auocorrelaed, he near-uniy xed-moohing aymoic e have more or le he ame ize roerie a he aionary xed-moohing aymoic e. However, when he rocee aroach uni roo nonaionariy, he near-uniy xed-moohing aymoic e ucceed in reducing he ize diorion. Comaring he e in he econd grou, he erie e wih K = 6 and he KVB e have he mo accurae ize. hird, he near-uniy aymoic F e are alo more accurae han he aionary xed-moohing e, which in he cae of erie variance eimaion ue alo he F criical value. When K i relaively mall, he near-uniy aymoic F e i a accurae a he e baed on nonandard imulaed criical value. However, when K i large, he near-uniy aymoic F e ha omewha larger ize diorion. able 2 i he ame a able exce ha he amle ize i = 4: he qualiaive obervaion we make for able aly o able 2. A execed, all e become more accurae. We omi he able for he VMA() cae a all hree grou of e have imilar good ize roerie for boh amle ize = 2 and 4: here i no advere e ec of uing he near-uniy xed-moohing aymoic diribuion when he underlying rocee are no rongly erien. hi i conien wih our heoreical nding. Figure 4 i a rereenaive gure for ize-adjued ower curve. he amle ize i 2 and he number of join hyohee i 2. Of coure, ize-adjumen i no feaible in racice; and hi i exacly he reaon we wan o develo an aymoically valid e ha ha accurae ize in nie amle. Since e baed on he ame aiic have he ame ize-adjued ower, a e in he r grou ha he ame ower a he correonding 9

21 e in he econd grou. So i u ce o reor he ower curve of he e in he r and econd grou. In Figure 4, K6, K2, K24 and KVB are he e in he r grou while K6*, K2*, K24* are he e in he econd grou. I i clear from he gure ha he ower of he erie e increae wih K, a hown in Sun (23a). Wha i new here i ha he ower of he xed-moohing nonandard e i cloe o ha of he xed-moohing F e. ranformaion and orhogonalizaion underlying he F e do no lead o ower lo. 7 Concluion he aer develo a new xed-moohing aymoic heory ha accommodae rongly auocorrelaed ime erie while alo working very well in he abence of rong auocorrelaion. Our rooed near-uniy xed-moohing e achieve rile robune in he following ene: hey are aymoically valid regardle of wheher he amoun of moohing i xed or increae wih he amle ize; wheher here i emoral deendence of unknown form or no; and wheher he emoral deendence i rong or no when reened. he near-uniy xed-moohing aymoic diribuion i nonandard. By chooing he bai funcion aroriaely, he nonandard diribuion can be reduced o a andard F aroximaion. See Sun and Kim (23) for an imlemenaion of hi idea in a di eren eing. An alernaive way o characerize he rong auocorrelaion i o model he momen roce a a fracional roce. In hi cae, we can follow Sun (24) o develo a new aymoic aroximaion. Our eing rocedure can be combined wih rewhiening. hi may lead o a e wih oenially very accurae ize, even if he underlying rocee are highly auocorrelaed. We leave all hee o fuure reearch. 2

22 able : Emirical ize of 5% Fixed-moohing Aymoic e wih = 2 under VAR() Error Saionary Fixed-moohing Near-Uniy Fixed-moohing Near-Uniy F e K6 K2 K24 KVB K6 K2 K24 KVB K6* K2* K24* = = = =

23 able 2: Emirical ize of 5% Fixed-moohing Aymoic e wih = 4 under VAR() Error Saionary Fixed-moohing Near-Uniy Fixed-moohing Near-Uniy F e K6 K2 K24 KVB K6 K2 K24 KVB K6* K2* K24* = = = =

24 Power Power Power Power ρ = ρ = δ 2 K6 K2 K24 KVB K6* K2* K24* K6 K2 K24 KVB K6* K2* K24* δ 2 ρ =.7 ρ = δ δ 2 Figure 4: Size-adjued Power of Di eren eing Procedure for VAR() Error wih amle ize = 2 and he number of join hyohee = 2 ( K6, K2, K24 and KVB are he Near-Uniy Fixed-moohing e while K6*, K2*, K24* are he Near-Uniy F e) 23

25 8 Aendix he following lemma will be ued in our roof. I i a ligh modi caion of Lemma in Sun (23b) and can be roved in he ame way. he deail are omied here. Lemma Suoe! = ;M + ;M and! doe no deend on M: Aume ha here exi ;M and ; uch ha (i) P ( ;M < ) P ;M < = o() for each xed M and each 2 R a! ; (ii) P ;M < P ; < = o() for each 2 R a M! ; (iii) he CDF of ; i coninuou on R, (iv) For every > ; u P (j ;M j > )! a M! : hen P (! < ) = P ; < + o() for each 2 R a! : Proof of heorem. Under he null hyohei, we have X = y = X Z e! d J c (r) dr (22) = by he coninuou maing heorem. Under Aumion 2, i i no hard o how ha 3 X X = = Q h I remain o how ha X X 3 Q h ; = = ; (y y) (y y) = 3 X (y y) (y y)! d X = = Q h ; (y y) (y y) +o () : Q h (r; ) J c m (r) J c m () drd (23) joinly wih (22) and R R Q h (r; ) J c m (r) Jc m () drd i noningular wih robabiliy one. We focu on roving he marginal convergence in (23), a he join convergence reul can be roved uing he Cramer-Wold device and he noningulariy of he limiing marix can be roved eaily. 3 Noe ha (23) i equivalen o X X = = Q h ; (e e) A (e e)! d Q h (r; ) J c m (r) AJ cm () drd (24) for all ymmeric marix A: In view of he ecral decomoiion of A = P `a`a `, he above hold if 3 X X = = Q h ; (e e) aa (e e)! d 24 Q h (r; ) J c m (r) aa J cm () drd (25)

26 for all vecor a: Bu 3 X X = = = 3 = 3 = 3 = 3 X Q h X = = X X = = X X = = X X = = ; (e e) aa (e e) Q h ; Q h Q h Q h So i u ce o how ha! := 3 X X = = for all vecor a: Q h Q h ; ; e aa e + O ; e aa e + O Q h 3 ; e aa e + O : ; e aa e! d ; + Q h (; ) e aa e X = = X = X e aa e ke k! 2 kak 2 Q h (r; ) J c m (r) aa J cm () drd Given ha P j j j (r) j () converge uniformly over (r; ) 2 [; ] 2 o Q h (r; ) ; we have, for any " > ; here exi an M > uch ha ; X M j j j " Q h for all ; and all : Now! = 3 = X X X = = = = X Q h j= 2 X MX 4 j j 2 X = = j= 4Q h := ;M + ;M : ; e aa e ; j 3 5 e aa e M X j= j j j 3 5 e aa e 25

27 We ue Lemma o comlee he roof. Condiion (i) of Lemma hold becaue for each xed M; we have ;M = MX j= j " X j = e a # " X j = e a X M! d j j (r) Jc m (r) adr j () Jc m () a d = j= MX j j (r) j () Jc m (r) aa J cm () d := ;M: j= o verify condiion (ii) of Lemma, we le 2 3 X ; = 4 j j (r) j () 5 Jc m (r) aa J cm () d hen = ;M ; = j= Q h (r; ) J c m (r) aa J cm () d: 2 4 X j=m+ 3 j j (r) j () 5 Jc m (r) aa J cm () drd: Since P M j= j j (r) j () converge o Q h (r; ) uniformly in (r; ) a M! ; we have: for any " > ; here exi M > uch ha P j=m+ j j (r) j () < " for all M > M : So when M i large enough, ;M ; " J c m (r) aa J cm () drd " J cm (r) a 2 2 dr " kj cm (r)k dr kak 2 " kj cm (r)k 2 dr kak = "O () : Since " i arbirary, we have ;M ; = o () a M! : Hence condiion (ii) of Lemma hold. I i eay o ee ha Condiion (iii) of Lemma alo hold. I remain o verify he la condiion in Lemma. We have E j ;M j " 3 = " 3 X X = = E e aa e " 3 ( X he e a i ) 2 2 =2 : = X = = X he e a i 2 =2 he a i 2 =2 e # 26

28 In view of e a = X j ;m u ja + ;me a; j= and de ning u (j j 2 ) := Eu j u j 2 ; we have e var a X = X X j = j 2 = X X j = j 2 = X X j = j 2 = j= j ;m u ja A + 2 var (e ) j ;m j 2 ;m a u j u j 2 a + O a u(j k u (j j 2 )a + O j 2 )k kak 2 + O = O() uniformly in : Hence ( E j ;M j C " ) 2 X 3 =2 C" = for ome conan C ha doe no deend on : hi imlie ha, for every > ; u P (j ;M j > )! a M! : Given ha all condiion in Lemma hold, we have X X Q h ; e aa e! d ; :=! := 3 Conequenly, F = a deired. = =! " X y = X! d J c (r) dr X 3 Q h = = ; # (y y) (y y) Q h (; ) J c (r) J c () drd he roof for -aiic i imilar and i omied here. Q h (r; ) J c m (r) aa J cm () d: Proof of heorem 2. In view of J cm () = R e cm( ) dw () ; we have c m j (r) J cm (r) dr Z r = c m j (r) e cm(r = c m j (r) e cm(r ) dw () dr ) dr dw () : X = y! = J c (r) dr = 27

29 Bu and o c m j (r) e cm(r = j (r) d he ) dr cm(r )i = j (r) e cm(r ) j + = j () j () e cm( ) + c m j (r) J cm (r) dr = + e cm(r e cm(r ) j (r) dr ) j (r) dr; j () dw () j () e cm( ) dw () e cm(r ) j (r) dr dw () : We now how ha he la wo erm in he above exreion are o () : Fir, ince E R e cm( ) dw () = and var we have e cm( uniformly over j = ; ; : : : Second, E R hi imlie ha ) dw () = j () e 2cm( ) d = 2c m e 2cm! a c m! ; e cm( ) dw () = j () o () R e cm(r ) j (r) dr dw () = and var = e cm(r ) j (r) dr dw () 2 e cm(r ) j (r) dr d = 2c m 2c m e 2cm(r e 2cm(r e cm(r ) dr ) dr d e 2cm( ) d j (r) 2 dr : j (r) 2 dr d j (r) 2 dr j (r) 2 dr ) j (r) dr dw () = j (r) 2 o () : 28

30 When j (r) 6= for large enough j a in he kernel cae, we need o renghen he above reul. In hi cae, When j (r) = and Hence j (r) = ( 2 j in 2jr; j i even 2 (j + ) co 2r (j + ) ; j i odd ` in `r for j = 2`; we have e cm(r ) (in (`r)) dr h c 2 m + 2`2 c m e cm( ) in ` + `e cm( ) co `i + c 2 m + 2`2 [c m in ` + ` co `] 2 (c m + `) c 2 m + 2`2 e cm(r e cm(r 2 ) in (kr) dr d 4 (c m + `) 2 (c 2 m + 2`2) 2 8 c 2 m + 2`2 : 2 ) j (r) dr d Similarly, when j (r) = ` co r` for j = 2` ; we have e cm(r ) (co r`) dr c m (co `) e cm( ) ` (in `) e c 2 m + 2`2 + c m (co `) ` (in `) c 2 m + 2`2 2 (c m + `) c 2 m + 2`2 : 82`2 c 2 m + 2`2 = 2 2 j 2 2 j 2 + 4c 2 : (26) m cm( ) herefore 2 e cm(r ) 2 (j + ) j (r) dr 2 d 2 (j + ) 2 : (27) + 4c 2 m Combining (26) wih (27), we have! e cm(r ) j (r) dr dw () = O 2 j 2 2 j 2 + 4c 2 : m o um u, we have roved he following c m j (r) J cm (r) dr = h j () dw () + j j ()j + j (r) i 2 o 2 () ; 29

31 which hold for boh he erie cae and he kernel cae. In he kernel cae, we have roved: Z! c m j (r) J cm (r) dr = j () dw () + j j ()j o () + O 2 j 2 2 j 2 + 4c 2 : m Boh reul hold uniformly over j = ; 2; : : :: I follow ha c m j (r) J cm (r) dr = j () dw () + o () a c m! uniformly over j = ; ; : : : for boh he erie cae and he kernel cae. herefore, = = = = = Q h (r; ) c mj cm (r) c m J c m () drd X j j (r) j () c m J cm (r) c m Jc m () drd j= X j c m j= X j j= A a conequence, j (r) J cm (r) dr c m j () J cm () d 2 j (r) dw (r) 3 3 X j () dw () + 4 j j j5 o () 2 X 4 j j (r) j () 5 dw (r) dw () + o () j= Q h (r; ) dw (r) dw () + o (): j= F (c m ) = () dw () + o () () dw () + o () = = F () + o () a c m! : he roof for (c m ) i imilar and i omied here. Q h (r; ) dw (r) dw () + o () 3

32 Reference [] Alexooulo, C. (26): A Comrehenive Review of Mehod For Simulaion Ouu Analyi. Proceeding of he 26 Winer Simulaion Conference. [2] Alexooulo, C. (27): Saiical Analyi of Simulaion Ouu: Sae of he Ar. Proceeding of he 27 Winer Simulaion Conference. [3] Davidon, J. (994): Sochaic Limi heory: An Inroducion for Economericican. Oxford Univeriy Pre. [4] Giraii, L., and Philli, P.C.B. (26): Uniform limi heory for aionary auoregreion. Journal of ime Serie Analyi 27, 5 6. [5] Janon, M. (24): On he Error of Rejecion Probabiliy in Simle Auocorrelaion Robu e, Economerica 72, [6] Kiefer, N.M.,.J. Vogelang and H. Bunzel (2): Simle Robu eing of Regreion Hyohee. Economerica 68, [7] Kiefer, N.M. and.j. Vogelang (22a): Heerokedaiciy-auocorrelaion Robu eing Uing Bandwidh Equal o Samle Size. Economeric heory 8, [8] (22b): Heerokedaiciy-auocorrelaion Robu Sandard Error Uing he Barle Kernel wihou runcaion. Economerica 7, [9] (25): A New Aymoic heory for Heerokedaiciy-Auocorrelaion Robu e. Economeric heory 2, [] Park, J. (23): Weak Uni Roo. Working aer, Dearmen of Economic, Indiana Univeriy. [] Müller, U.K. (27): A heory of Robu Long-run Variance Eimaion. Journal of Economeric 4, [2] Müller, U.K. (24): HAC Correcion for Srongly Auocorrelaed ime Serie. Journal of Buine & Economic Saiic, forhcoming. [3] Newey, W.K. and K.D. We (987): A Simle, Poiive Semide nie, Heerokedaiciy and Auocorrelaion Conien Covariance Marix. Economerica 55, [4] Philli, P.C.B. (987): oward a Uni ed Aymoic heory for Auoregreion. Biomerika 74,

33 [5] Philli, P.C.B. (25): HAC Eimaion by Auomaed Regreion. Economeric heory 2, [6] Philli, P.C.B. and S. N. Durlauf (986): Mulile Regreion wih Inegraed Procee. Review of Economic Sudie 53, [7] Philli, P.C.B. and. Magdalino ( 27): Limi heory for Moderae Deviaion from a Uni Roo. Journal of Economeric 36, 5 3. [8] Philli, P.C.B.,. Magdalino and L. Giraii (2): Smoohing Local-o-moderae Uni Roo heory. Journal of Economeric 58, [9] Philli, P.C.B. and V. Solo (992): Aymoic for Linear Procee. Annal of Saiic, 2(2), 97. [2] Polii, D. (2): Higher-order Accurae, Poiive Semi-de nie Eimaion of Largeamle Covariance and Secral Deniy Marice. Economeric heory 27, [2] Sun, Y. (24): A Convergen -aiic in Suriou Regreion. Economeric heory 2, [22] Sun, Y. (2): Robu rend Inference wih Serie Variance Eimaor and eingoimal Smoohing Parameer. Journal of Economeric 64(2), [23] Sun, Y. (23a): A Heerokedaiciy and Auocorrelaion Robu F e Uing Orhonormal Serie Variance Eimaor. Economeric Journal 6, 26. [24] Sun, Y. (23b): Fixed-moohing Aymoic in a wo-e GMM Framework. Working aer, Dearmen of Economic, UC San Diego. [25] Sun, Y. (24): Commen on HAC Correcion for Srongly Auocorrelaed ime Serie by U.K. Müller, forhcoming in Journal of Buine & Economic Saiic. [26] Sun, Y. and M.S. Kim (23): Aymoic F e in a GMM Framework wih Cro Secional Deendence. Review of Economic and Saiic, forhcoming. [27] Sun, Y., P.C.B. Philli, and S. Jin (28): Oimal Bandwidh Selecion in Heerokedaiciy-Auocorrelaion Robu eing. Economerica 76,