Robust Trend Inference with Series Variance Estimator and Testing-optimal Smoothing Parameter

Transcription

1 obus rend Inference wih Series Variance Esimaor and esing-oimal Smoohing Parameer Yixiao Sun Dearmen of Economics, Universiy of California, San Diego he auhor graefully acnowledges arial research suor from NSF under Gran No. SES He hans wo anonymous referees for helful commens and suggesion. Corresondence o: Dearmen of Economics, Universiy of California, San Diego, 95 Gilman Drive, La Jolla, CA

2 ABSAC he aer develos a novel esing rocedure for hyoheses on deerminisic rends in a mulivariae rend saionary model. he rends are esimaed by he OLS esimaor and he long run variance (LV) marix is esimaed by a series ye esimaor wih carefully seleced basis funcions. egardless of wheher he number of basis funcions is xed or grows wih he samle size, he Wald saisic converges o a sandard disribuion. I is shown ha criical values from he xed- asymoics are second order correc under he large- asymoics. A new racical aroach is roosed o selec ha addresses he cenral concern of hyohesis esing: he seleced smoohing arameer is esing-oimal in ha i minimizes he ye II error while conrolling for he ye I error. Simulaions indicae ha he new es is as accurae in size as he nonsandard es of Vogelsang and Franses (5) and as owerful as he corresonding Wald es based on he large- asymoics. he new es herefore combines he advanages of he nonsandard es and he sandard Wald es while avoiding heir main disadvanages (ower loss and size disorion, resecively). JEL Classi caion: C3; C4; C3; C5 eywords: Asymoic exansion, F-disribuion, Hoelling s disribuion, long-run variance, robus sandard error, series mehod, esing-oimal smoohing arameer choice, rend inference, ye I and ye II errors.

3 Inroducion rend regression is one of simle and ye imoran regressions in economic and climaic ime series analysis. In his aer, we consider a linear rend regression wih mulile deenden variables. For examle, he deenden variables may consis of GDPs from a number of counries. Vogelsang and Franses (5) rovide more emirical examles. Esimaion of he rends is relaively easy as he equaion-by-equaion OLS esimaor is asymoically as e cien as he sysem GLS esimaor. Hence, for oin esimaion, here is no need o ae error auocorrelaion ino accoun in large samles. However, rend inference is suble as he variance of he OLS rend esimaor deends on he long run variance (LV) of he error rocess. Since he LV is roorional o he secral densiy of he error rocess evaluaed a zero, many nonarameric secral densiy mehods can be used o esimae he LV. Commonly used mehods are mosly ernel-based. In his aer, we consider esimaing he LV using nonarameric series mehods. he resuling series LV esimaor is he samle variance of regression coe ciens in a nonarameric series regression. he smoohing arameer in he series LV esimaor is he number of basis funcions emloyed. When he number of basis funcions is xed, he LV esimaor is inconsisen and converges o a scaled Wishar disribuion. he underlying scale cancels ou in he limiing disribuion he asymoic variance of he rend esimaor. Hence, when is xed, he Wald saisic converges o a ivoal nonsandard disribuion. he xed- asymoics is in he siri of he xed-b asymoics as in iefer and Vogelsang (a, b, 5). his ye of asymoics caures he randomness of he LV esimaor and ess based on i ofen have beer nie samle size roeries han hose based on consisen LV esimaes. Jansson (4) and Sun, Phillis and Jin (8, SPJ hereafer) rovide some heoreical jusi caions for he non-sandard asymoic heory. We design a se of basis funcions so ha he xed- asymoic disribuion becomes he sandard F disribuion. For hese basis funcions, he series LV esimaor is asymoically invarian o he inerces and rend arameers. As a resul, i does no su er from he bias arising from he esimaion uncerainy of model arameers. his is in conras wih he convenional ernel LV esimaors where he esimaion uncerainy gives rise o a demeaning bias. See, for examle, Hannan (957) and Ng and Perron (994). By selecing he basis funcions aroriaely, we comleely remove his ye of bias o he order we care abou. his is a desirable roery as we generally refer an esimaor wih fewer bias erms, esecially in hyohesis esing. Anoher advanage of using he new series LV esimaor lies in is convenience in racical use as criical values from he xed- asymoics are readily available from saisical ables and sofware acages. While he LV esimaor is inconsisen when is nie, i becomes consisen when grows wih he samle size a a cerain rae. he smoohing arameer is an imoran uning arameer ha deermines he asymoic roeries of he LV esimaor. Following he convenional aroaches (e.g., Andrews, 99, and ey and Wes, 987, 994), Phillis (5) chooses he smoohing arameer o minimize he asymoic of he LV esimaor. Such a choice of he smoohing arameer is designed o be oimal in he sense for he oin esimaion of he LV, bu is no necessarily bes suied for semiarameric esing. hrough is e ec on he LV esimaor, he smoohing arameer a ecs he ye I and ye II errors of he associaed es. I is hus sensible ha he choice of should ae hese roeries ino accoun.

4 o develo an oimal choice of for semiarameric esing, we rs have o decide on which es o use. We can emloy he radiional Wald es, which is based he Wald saisic and uses a chi-square disribuion as he reference disribuion. Alernaively, we can emloy he new F es given in his aer, which is based on a modi ed Wald saisic and uses an F-disribuion as he reference disribuion. We nd ha criical values from he F-disribuion are higher order correc under he convenional large- asymoics. A direc imlicaion is ha he F es generally has smaller size disorion han he radiional Wald es. On he basis of his heoreical resul and he emhasis on he size conrol in he economerics lieraure, we emloy he F es o conduc inference on he rend arameers. One of he main conribuions of he aer is o develo an oimal rocedure for selecing he smoohing arameer ha addresses he cenral concern of semiarameric esing. he ulimae goal of any esing roblem is o achieve smaller ye I and ye II errors. However, hese wo yes of errors ofen move in oosie direcions. We can conrol one ye of error while rying o minimize he oher ye of error. In his aer, we roose o choose o minimize he ye II error subjec o he consrain ha he ye I error is bounded. he resuling oimal is said o be esing-oimal for he given bound. he bound is de ned o be ; where is he nominal ye I error and > is he arameer ha caures he user s olerance on he discreancy beween he nominal and rue ye I errors. he roosed aroach o selecing he esing-oimal requires asymoic measuremens of ye I and ye II errors of he F es. hese measuremens are rovided by means of high order asymoic exansions of he nie samle disribuion of he F saisic under he null and local alernaive hyoheses. In a ransformed sace, he null hyohesis is a xed oin while he alernaive hyohesis we consider is a random oin uniformly disribued on he shere cenered a he xed null. he radius of he shere is chosen so ha he ower of he es is 75% under he rs order asymoics. his sraegy is similar o ha used in he oimal esing lieraure. In he absence of a uniformly mos owerful es, i is ofen recommended o ic a reasonable oin under he alernaive and consruc an oimal es agains his aricular oin alernaive. I is hoed ha he resuling es, alhough no uniformly mos owerful, is reasonably close o he ower enveloe. Here we use he same idea and selec he radius of he shere according o he ower requiremen. We hoe ha he smoohing arameer ha is oimal for he chosen radius also wors well for oher oins under he alernaive hyohesis. his is con rmed by our Mone Carlo sudy. he esing-oimal ha maximizes he local asymoic ower while reserving size in large samles is fundamenally di eren from he -oimal : he esing-oimal deends on he sign of he nonarameric bias, he hyohesis under consideraion and he ermied olerance for he ye I error while he -oimal does no. When he ermied olerance becomes su cienly small, he esing-oimal is of smaller order han he -oimal : Our crierion for selecion is a esing-focused crierion in ha i aims a he esing roblem and aes he seci c hyohesis ino consideraion. he aer ha is mos closely relaed o he resen aer is SPJ where robus inference for he mean of a scalar ime series is considered. In SPJ, he oimal smoohing arameer minimizes a loss funcion ha is de ned o be a weighed sum of he ye I and ye II errors. Our rocedure can also be cas in his framewor wih he Lagrange mulilier

5 for he consrained minimizaion roblem as he relaive weigh. he main di erence is ha our weigh is imlicily de ned hrough he olerance arameer. For a given ; he weigh may be di eren across di eren daa generaing rocesses. In conras, in he SPJ rocedure, he weigh is seci ed a riori and is hus xed. Boh rocedures require a user-chosen arameer: he olerance arameer or he weigh. he olerance arameer is ofen easier o choose as i involves only he ye I error while he weigh is more di cul o choose as i deends on boh ye I and ye II errors. his is an advanage of he new rocedure roosed here. he same rocedure is used in Sun, Phillis, and Jin () for robus mean inference wih exoneniaed ernels. he series LV esimaor has been considered in he lieraure under di eren names. I belongs o he class of muli-window or muli-aer esimaors (homson, 98) and he class of ler-ban esimaors (Soica and Moses, 5, ch. 5). In he simulaion and signal rocessing lieraure, he weighed area esimaor of Foley and Goldman (999) is a series LV esimaor wih aricular basis funcions. In economerics, Phillis (5) embeds his esimaor in a framewor of auomaed regression. Müller (7) moivaes i from he ersecive of robus LV esimaion. he xed- ye of asymoics has some recursors in he lieraure. Foley and Goldman (999) aroximae he disribuion of heir auocorrelaion robus -saisic by a disribuion. As we show laer, he -disribuion belongs o he class of xed- asymoic disribuions. For some basis funcions, he woring aer version of Müller (7) conains he xed- asymoics and F aroximaion. However, he basis funcions considered here are di eren from he exising lieraure. hey do no consiue a comlee basis sysem, and are designed o eliminae he demeaning e ec and he derending e ec a he same ime. o he bes of my nowledge, he aer is he rs o exlore he relaionshi beween he xed- asymoics and he convenional large- asymoics in rend esimaion and inference. I is also he rs o roose a esing-oimal smoohing arameer choice in his seing. he res of he aer is organized as follows. Secion describes he basic seing and he limiing disribuion of he rend esimaor. Secion 3 moivaes he series LV esimaor and esablishes is asymoic roeries under he xed- and large- asymoics. Secion 4 invesigaes he limiing disribuion of he Wald saisic under boh xed- and large- asymoics. Secion 5 gives a higher order exansion of he nie samle disribuion of he modi ed Wald saisic. On he basis of his exansion, Secion 6 rooses a selecion rule for ha is mos suiable for imlemenaion in semiarameric esing. he nex secion reors simulaion evidence on he erformance of he new rocedure. he las secion rovides some concluding discussion. Proofs are given in he Aendix. he Model and Preliminaries Consider n rend-saionary ime series denoed by (y ; :::; y n ) wih ; ; :::;. We assume ha he daa generaing rocess is y i i + i + u i ; ; ; :::; ; i ; ; :::; n; () where u i is a wealy deenden rocess wih zero mean. Our focus of ineres is on he inference abou he rend arameers f i g : 3

6 Assumion Le u (u ; :::; u n ) ; we assume ha u C(L)" X C j " j ; j where " s iid(; ); E " v < for some v 4, X j a C j < for a > 3; C()C () > j and is he marix Euclidean norm. Under he above assumion, he rocess u admis he following BN (Beveridge and Nelson, 98) decomosiion u C()" + ~u ~u for ~u X ~C j " j ; Cj ~ j X sj+ C s ; () where P j C ~ j < : Using his decomosiion and following Phillis and Solo (99), we can rove ha [ r] X u! d W n (r); as! ; (3) where W n (r) is an n vecor of sandard indeenden Wiener rocesses and [C()C() ] is he marix square roo of he long run variance marix of u : X j Eu u j C()C() : o reresen he OLS esimaor of he model arameers, we inroduce he following noaion: he OLS esimaor of is hen given by y i (y i ; :::; y i ) ; Y (y ; y ; :::; y n ) u i (u i ; :::; u i ) ; u (u ; :::; u n ) X (; ) ; X X; :::; X ( ; ; :::; n ) wih i ( i ; i ) : ^OLS (X X) X Y: If he errors are second-order saionary, hen he OLS esimaor is asymoically equivalen o he GLS esimaor. In addiion, because () is a seemingly unrelaed regression (SU) wih he same regressors in each equaion, he OLS esimaor is equivalen o he SU esimaor, which is he GLS esimaor ha accouns for conemoraneous correlaion across he series. hus, he simle OLS esimaor has some nice oimaliy roeries. Vogelsang and Franses (5) mae he same oin. 4

7 Le D diag ; 3. hen, for u de ned in Assumion, DX P! XD P P! d rdr 3 rdr P! DX u u! P u! dw n (r) d rdw n (r) : herefore D ^OLS! rdr d rdr r dr! dw n (r) rdw n (r)! r dr! ; 6 3 r! dwn (r) r dw n (r) : So he OLS esimaor ^ OLS of sais es 3 ^OLS! d Z r dw n (r) d N(; ): 3 Series LV Esimaor and is Asymoic Proeries o conduc inference regarding ; we need o rs esimae he LV marix : In he nex subsecion, we moivae he series LV esimaor we use in his aer. 3. Moivaion of Series LV Esimaor Consider he ernel-based esimaor roosed by Phillis, Sun and Jin (6, 7, PSJ hereafer): ^ P SJ X X r s ^u ^u s; r where (r s) [(r s)] for some second-order ernel funcion (). his esimaor is consisen when! a a cerain rae. Assume ha () is even, coninuous and osiive semide nie. By Mercer s heorem (Mercer, 99), we can wrie (r s) X (r) (s) ; (4) where f g is a sequence of eigenvalues and f (r)g is an orhonormal sequence of eigenfuncions corresonding o he eigenvalues : I can be shown ha X X and O as! : (5) Wih his reresenaion of (r s) ; we can wrie " X ^ # " X r # X s P SJ ^u r ^u s r s X : ^ (6) 5

8 where ^ ^ ^ and ^ X ^u : In he above exression, decays o zero as increases. he inuiion is ha, as increases, he eigenfuncion (r) becomes more concenraed on high frequency comonens and we should imose rogressively less weigh on hese comonens in order o caure he long run (low frequency) roeries of he underlying ime series. In addiion, for each ;! as! : Imlicily, he PSJ esimaor emloys a sof hresholding mehod where he weigh aroaches o zero bu is no equal o zero for any given : Insead of sof hresholding, we can also consider he hard hresholding esimaor: ^ X ^ (7) where is a osiive ineger. his esimaor runcaes he in nie sum in (6) and assigns equal weighs o he remaining erms. In oher words, he in nie sequence ( ; :::; ; :::) is relaced by (; ; :::; ; ; :::) : For his sequence, P and P : Comaring he squared sum wih ha in (5), we can see ha lays he role of in he PSJ esimaor. his can also be seen by comaring he asymoic biases and he asymoic variances of hese wo esimaors. As will be shown below, wih aroriaely chosen ; each of he summand ^ is an asymoically unbiased esimaor of : We refer o he LV esimaors of he form ^ as direc LV esimaors so ha ^ is an average of direc LV esimaors. Noe ha ^ is aroximaely he regression coe cien obained by regressing he ime series ^u on he regressor ( ) : ^ is ar of he oal sum of squares P ^u ^u ha is exlained by he basis funcion () : his exlained sum of squares may be regarded as anoher ways of hining abou he long run variance marix he conribuions o he variaion of ^u ha are due o low frequency variaions in he series. o obain more exible esimaors of he form ^; we can use any orhogonal basis funcions o consruc ^: For examle, we may use olynomial basis funcions. In addiion, he basis funcions do no have o be a comlee basis sysem. In fac, we use incomlee basis funcions below in order o remove he demeaning and derending e ecs. We have hus obained a general class of LV esimaors. For convenience, we refer o hem as series LV esimaors as hey are based on nonarameric series regressions. he series LV esimaor has di eren inerreaions. Firs, i can be regarded as a mulile-window esimaor wih window funcion ( ), see homson (98) and Percival and Walden (993). In he economerics lieraure, Sun (6) alies he mulilewindow esimaor o he esimaion of realized volailiy. he robus long-run variance esimaors derived by Müller (7) also belong o he class of mulile-window esimaors. In a di eren conex and for a di eren model, Müller (7) has esablished he xed- asymoics given in Secion 3.. Phillis (5) gives an alernaive moivaion of he mulile-window esimaor and esablishes is asymoic roeries. Second, when ( x) (x); we can wrie ^ P ( )^u ; which can be regarded as ouu from alying a linear ler o he residual rocess ^u : he ransfer funcion of he linear ler is H (!) X ( ) ex(i!): 6

9 o caure he long run behavior of he rocess, we require ha H (!) be concenraed around he origin. ha is, H (!) resembles a band ass ler ha asses low frequencies wihin a cerain range and rejecs (aenuaes) frequencies ouside ha range. Hence, ^ can also be regarded as a ler-ban esimaor and ^ is a simle average of hese lerban esimaors. Finally, ^ can be regarded as he samle variance of regression coe ciens f^ ; ; ; :::; g: By consrucion, i is auomaically osiive semide nie, a desirable roery for racical use. Many series LV esimaors can be obained by choosing di eren basis funcions. However, in nonarameric series esimaion, i is a convenional wisdom ha he choice of basis funcions is ofen less imoran han he choice of he smoohing arameer. For his reason, we emloy he basis funcions ha are mos convenien for racical use and focus on he roblem of selecing he smoohing arameer : 3. Fixed- Asymoics In his subsecion, we esablish he asymoic disribuion of ^ under he assumion ha is xed. Le ^u y y ( ) ^; hen [ r] X ^u [ r] X Z r! [W n (r) rw n ()] : V n (r); h u u ( ) ^ s ds i " Z s ds# Z s dw n (s) where V n (r) W n (r) rw n () 6r (r ) Z dw n (): Using summaion and inegraion by ars and invoing he coninuous maing heorem, we obain, for xed and under Assumion : ^! d : X Z X Z Z (r) dv n (r) (s) dv n (s) Z ~ (r) dw n (r) ~ (s) dw n (s) X ; (8) where ~ (r) (r) Z is he ransformed basis funcion and Z (s) ds (s) s Z ~ (r) dw n (r): ds r ; (9) 7

10 We call he above asymoics he xed- asymoics. his is similar o he xed-b asymoics of iefer and Vogelsang (5). Common choices of are he sine and cosine rigonomeric olynomials. In fac, using a simle Fourier exansion and assuming ha () is even, we can show ha he eigenfuncions in (4) are he sine and cosine funcions. A subse of he cosine funcions (r) cos r; ; ; ::: enjoys he desirable roery ha ~ (r) (r) for ; ; 4; ::: So no only f (r)g are orhonormal bu also are heir ransforms as de ned in (9). Noe ha he rs basis wih is redundan as P ^u. We herefore ae cos ; for ; ; :::; as our daa windows or basis funcions. Similar o he Hanning window ( cos ) ; he above funcions have small side lobes and heir Fourier ransforms decay o zero raidly. As a resul, he associaed LV esimaor has a small bias due o secral leaage (Priesley, 98,. 563). his is an esecially desirable feaure for hyohesis esing where bias reducion is more imoran han he oin esimaion of he LV. Wih he above cosine basis funcions, is iid N(; I n ). As a resul, P is a Wishar disribuion W n (I n ; ): So ^ converges o a scaled Wishar disribuion. In he scalar case, he limiing disribuion reduces o he scaled chi-square disribuion. In general, for any conforming consan vecor z; z ^zz z converges in disribuion o : his resul can be used o es hyoheses regarding. he resuling es may have beer size roeries. See PSJ (6, 7) and Hashimzade and Vogelsang (7) for he same oin based on convenional ernel esimaors. We do no ursue his exension here as our main focus is on he inference for : 3.3 Large- Asymoics While he xed- asymoics may caure he randomness of ^ very well, i does no re ec he usual nonarameric bias or Parzen bias of ^: In his secion, we consider he asymoic roeries of ^ when boh and go o in niy such ha! : heorem Le Assumion hold. As! such ha! ; we have (a) E ^ B + o + O : (b) var vec(^) ( ) (I n + nn) ( + o ()) + O where B 3 X h h u (h) ; u (h) Eu u h ; nn is he n n commuaion marix, and I n is he n n ideniy marix. heorem exends heorem of Phillis (5), which is alicable only o scalar ime series wih nown mean. he bias erm here is di eren from ha given in heorem (i) in Phillis (5). his is because he basis funcions we used are a subse of he basis 8

11 funcions in Phillis (5). he advanage of droing cos ( )r; ; ; ::: is ha he esimaion uncerainy of does no a ec he bias and variance calculaion in large samles. More seci cally, we show in he roof ha ^ is asymoically equivalen o ~ " X X # " u X s s u s # ; () an esimaor ha is based on he rue bu unnown error erm u : his resul is in shar conras o exising resuls in he HAC esimaion lieraure. For convenional ernel HAC esimaors, he esimaion uncerainy in model arameers gives rise o a higher order bias erm, which is yically he same order of magniude as he asymoic variance. he higher order bias is no caured in he rs-order convenional asymoic heory, alhough i is re eced in he nonsandard xed-b asymoics. See for examle SPJ. We have hus rovided a novel way o eliminae he e ec of he esimaion uncerainy of he model arameers on he LV esimaion. We noe in assing ha he esimaion uncerainy may also be eliminaed using recursive OLS residuals. heorem (b) characerizes he asymoic behavior of he exac variance. his resul is di eren from heorem (ii) Phillis (5) as he laer rovides only he variance of he limiing disribuion of ^: In erms of momen calculaions, our resuls are sronger han hose in Phillis (5). Le (^; W ) Evec(^ ) W vec(^ ) be he mean squared error of vec(^) wih weighing marix W: I follows from heorem ha, u o smaller order erms: (^; W ) r hw Evec(^ )vec(^ ) i So he oimal is given by vec(b) W vec(b) r [W ( ) (I n + nn)] : r [W ( ) (In + nn )] 5 4vec(B) 45 : () W vec(b) his aroach o oimal choice is he same as ha for bandwidh choice in ernel LV esimaors. See, for examle, Andrews (99). 4 Auocorrelaion obus Inference for rend Parameers he hyoheses of ineres in his aer are H : r agains H : 6 r; where is a n marix and r is a vecor. he usual Wald saisic F ;OLS for esing H agains H is given by F ;OLS h i 3 (^ OLS ) ^ h i 3 (^ OLS ) : 9

12 When ; we can consruc he usual -saisic ;OLS 3 (^ OLS ) ^ : 4. Fixed- Asymoics Under he xed- asymoics and he null hyohesis Z F ;OLS! d r ( X Z Z ~ (r)dw n (r) dw n (r) ) ~ (s)dwn(s) Z r dw n (r) : I urns ou he scaling facor in he asymoic disribuion of ^ cancels ou wih ha in he asymoic disribuion of 3 (^ OLS ): o see his, we reresen he disribuion n W n (r) by W (r) for some marix and -dimensional Brownian moion W (r): hen for a xed ; we have Z F ;OLS! d where Z r r Z r ( dw (r) X Z dw (r) : Z ) ~ (r) dw (r) ~ (s) dw (s) X! ; Z dw (r) and ~ (r) dw (r): So he limiing disribuion of F ;OLS does no deend on and is ivoal. Since Z Z Z cov r dw (r); ~ (r) dw (r) r ~ (r) dr for all ; and are indeenden as boh are normal random variables. In addiion, s iidn(; I ) and P is a Wishar disribuion W (I ; ): Hence he limiing disribuion of F ;OLS is Hoelling s -square disribuion (Hoelling (93)): Since for ; we have F ;OLS! d (; ): + (; ) s F ; + ; ( + ) F ;OLS! d F ; + : + ( + );

13 where and + denoe indeenden random variables. When ; he above resul reduces o! d : ha is, he -saisic converges o he -disribuion wih degrees of freedom. hese xed- asymoic resuls can also be roved direcly using sandard echniques from mulivariae saisical analysis. We have herefore shown ha under he xed- asymoics, he scaled Wald saisic converges wealy o he F disribuion wih degrees of freedom (; + ) and he - saisic converges o he -disribuion wih degrees of freedom : hese resuls are very handy as criical values from he F disribuion or he disribuion can be easily obained from saisical ables or sandard economerics acages. Under he local alernaive hyohesis, H : r + c where c ~c () for some vecor ~c; we have, for ; ( + ) ( + ) F ;OLS! d X ( + ~c)! ( + ~c) : F ; + ; a noncenral F disribuion wih degrees of freedom (; + ) and noncenraliy arameer (~c) ~c c c c c: his resul follows from Proosiion 8. in Bilodeau and Brenner (999) where he noaion F c is he canonical F disribuion (Bilodeau and Brenner, 999, age 4). Similarly, he -saisic converges o he noncenral disribuion wih degrees of freedom and noncenraliy arameer c ( ) ~c: he local alernaive ower deends on c only hrough he noncenraliy arameer ~c ; he squared lengh of vecor ~c: he direcion of ~c does no maer. Hence, for he rs order asymoics given here, i is innocuous o assume ha ~c is uniformly disribued on he shere S () fx : x g. I urns ou ha his assumion grealy simli es he develomen of higher order exansions in laer secions. 4. Large- Asymoics When! such ha! ; he LV esimaor ^ is consisen. As a consequence F ;OLS! under H and F ;OLS! under H : When ; he above resul reduces o ;OLS! N(; ) under H and ;OLS! N (; ) under H : o comare he xed- asymoics wih he large- asymoics, we evaluae he di erence in heir quaniles. Le F; + be he quanile of he F ; + disribuion and F; be he quanile of he F ; disribuion. In oher words, F; is he quanile of he disribuion. By de niion and wih a

14 sligh abuse of noaion, we have, as! ; + P F ; + < F; + P < F; + ( + )! + EG F; + ( + ) " G F; + G F; E " + G F; E F ; + F ; ( + ) ( + ) G F; + G F; F ; + F; + G F ; F ; + o F ; + F ; F ; F;# + o + o! # : (3) herefore Bu hence F ; + G G G G + o + ; F; o : as! : (4) herefore he criical values from he F-disribuion are larger han hose from he - disribuion, re ecing he randomness in he denominaor of he Wald saisic. U o he order o(); he correcion erm + () increases wih and decreases wih : So when is small or is large, he di erence beween he F and aroximaions may be large. 5 High Order Exansion of he Finie Samle Disribuion In his secion, we consider a high order exansion of he Wald saisic in order o design a esing-oimal rocedure o selec : We mae he simli caion assumion ha u is normal, which faciliaes he derivaions. he assumion could be relaxed bu a he cos of much greaer comlexiy, see for examle, Sun and Phillis (9). Le V var(vec(u)), hen he GLS esimaor of sais es vec ^GLS (I n X) V (I n X) (In X) V vec (u) : Similarly, he OLS esimaor sais es vec ^OLS (I n X) (I n X) (In X) vec (u) :

15 So where vec ^OLS vec ^GLS + ; ; and more exlicily n (In X) (I n X) (In X) (In X) V (I n X) (In X) V o vec (u) : I follows from he asymoic equivalence of ^ OLS and ^ GLS ha Ec c O( ) for any vecor c: See Grenander and osenbla (957). I is easy o show ha h E vec ^GLS i : Hence, ^ GLS and are indeenden. In addiion, h ^u ni n I n X X X io X vec (u) ; and hus Evec ^GLS ^u (I n X) V (I n X) (In X) V V ni n h I n X X X X io : So ^ GLS is indeenden of boh and ^: Le F ;GLS be he Wald saisic based on he GLS esimaor: F ;GLS 3 (^ GLS ) ^ 3 (^ GLS ): Using he asymoic equivalence of he OLS and GLS esimaors and he above wo indeendence condiions, we can rove he following Lemma. Lemma 3 Le Assumion hold and assume ha " s iidn(; ): hen for ; (a) P ( +) F ;GLS < z EG z + + O ; (b) P ( +) F ;OLS < z P ( +) F ;GLS < z + O ; where G is he CDF of a random variable wih degrees of freedom, e ^ e ; e h and ;GLS var ^GLS 3 ( ;GLS ) 3 (^ GLS ) ( ;GLS ) 3 ; (^ GLS ) i : Lemma 3 shows ha he esimaion uncerainy of ^ a ecs he disribuion of he Wald saisic only hrough : aing a aylor exansion, we have + L + Q + o + + O ; 3

16 where L is linear in ^ and Q is quadraic in ^ : he exac exressions for L and Q are no imoran here bu are given in he roof of heorem 4. Plugging his sochasic exansion ino Lemma 3, we obain a higher order exansion of he nie samle disribuion of F ;OLS for he case where! such ha!. heorem 4 Le Assumion hold and assume ha " s iidn(; ). If! such ha!, hen ( + ) P F ;OLS < z G (z) + G (z) zb + G (z) z + o + o + O (5) where B B (; B; ) r n(b ) ( ) o : he rs erm in (5) comes from he sandard chi-square aroximaion of he Wald saisic. he second erm caures he nonarameric bias of he LV esimaor while he hird erm re ecs he variance of he LV esimaor. he resul is analogous o hose obained by SPJ for Gaussian locaion models and Sun and Phillis (SP, 9) for general linear GMM models wih saionary daa. However, here is an imoran di erence. For convenional ernel esimaors as used in SPJ and SP, he asymoics exansion conains a erm ha re ecs he bias due o he esimaion error of he model arameers. Such a erm does no aear here because he basis funcions we emloy are asymoically orhogonal o he regressors. o undersand he relaionshi beween he xed- and large- asymoics, we develo an exansion of he limiing F ; + disribuion as in (3): P (F ; + < z) G (z) + G (z) z + o ; as! : Comaring his wih heorem 4, we nd ha he xed- asymoics caures one of he higher order erms in he high order exansion of he large asymoics. Plugging z F; + ino he above equaion yields: G F; + + G F; + F ; + + o : his imlies ha ( + ) P F ;OLS < F; + + G F; + F ; +B +o + o + O : (6) herefore, use of criical value F ; + removes he variance erm G (z) z in he higher order exansion. he size disorion is hen of order O : In conras, if he 4

17 criical value from he convenional disribuion is used, he size disorion is of order O + O () : So when 3! ; using criical value F; + should lead o size imrovemens. We have hus shown ha criical values from he xed- asymoics is second order correc under he large- asymoics. he xed- asymoic disribuion of F ;OLS is ( + ) F ; + while is rs-order large- asymoic disribuion is : When is xed, he wo disribuions are di eren. Hence, he large- asymoic aroximaion is no even rs-order valid under he xed- asymoics. heorem 4 gives an exansion of he disribuion of ( + ) F ;OLS : he facor ( + ) is a nie samle correcion facor. Wihou his correcion, we can show ha, u o smaller order erms P F ;OLS < G + G B G ( ) + G : Comaring his wih (5), we nd ha he above exansion has an addiional erm G ( ) : For any given criical value ; his erm is negaive and grows wih ; he number of resricions in he hyohesis. As a resul, he error in rejecion robabiliy or he error in coverage robabiliy ends o be larger for larger : his exlains why convenional con dence regions end o have large under-coverage when he dimension of he roblem is high. In he res of he aer, we use he nie samle correced Wald saisic F ;OLS ( + ) F ;OLS and emloy criical value F; + o erform our es. For convenience, we refer o F;OLS as he F saisic and he es as he F es. F;OLS can be viewed as he sandard Wald saisic bu using he following esimaor for : \ + X ^ : So he nie samle correcion facor ( + ) can be viewed as a degree-of-freedom adjusmen. he following heorem gives he ye I and ye II errors of he F es. heorem 5 Le Assumion hold and assume ha " s iidn(; ). If! such ha!, hen (a) he ye I error of he F es is P F;OLS > F; + B G + o + o + O : (7) (b) Under he local alernaive H : r + ( ) ~c where ~c is uniformly disribued on he shere S () fx : x g; he ye II error of he F 5

18 es is P F;OLS < F; +jh G ; B + + Q ; + o + o + O G ; ; (8) where G ; () and G ; () are he CDF and df of he noncenral disribuion wih degrees of freedom and noncenraliy arameer and Q ; (z) G ; (z) G (z) G (z) G ; (z) z G (+); (z) : heorem 5(a) follows from heorem 4. he uniformiy of ~c on a shere enables us o use a similar argumen o rove heorem 5(b). A ey oin in he roof of heorem 4 is ha e is uniformly disribued on he uni shere S () ; which follows from he roaion invariance of he mulivariae sandard normal disribuion. he uniformiy of ~c ensures he same roery holds for he corresonding saisic e ( ;GLS ) 3 (^ GLS ) + ~c ( ;GLS ) 3 (^ GLS ) + ~c under he local alernaive hyohesis. he quaniy Q ; re ecs he di erence in curvaures of he wo CDF funcions G (z) and G ; (z) a he oin z : When we use he second order correc criical value F; +, he variance erm is removed under he null. However, due o he di erence in curvaures, he variance erm remains under he local alernaive hyohesis. he O() erm in heorem 5(b) caures his e ec. Since Q ; (z) > for all z > ; his erm increases monoonically wih. According o his erm, he value of should be chosen as large as ossible. his is no surrising. In order o imrove he ower of he F es, we should minimize he randomness of he LV esimaor, which calls for a large value. However, a large value may roduce large bias, which may lead o ower loss or size disorion. In he nex secion, we show ha here is an ooruniy o selec o rade o he bias e ec and variance e ec on he size and ower roeries. 6 Oimal Smoohing Parameer Selecion In his secion, we rovide a novel aroach o smoohing arameer selecion ha is mos suiable for semiarameric esing. 6. Oimal Formula In view of he asymoic exansion in (7) and ignoring he higher order erms, we can aroximae he ye I error of he F es by e I B G : 6

19 Similarly, from (8), he ye II error of he F es can be aroximaed by e II G ; B + G ; + G (+); : We choose o minimize he aroximae ye II error while conrolling for he aroximae ye I error. More seci cally, we solve min e II ; s:: e I where is a consan greaer han. Ideally, he ye I error is less han or equal o he nominal ye I error : In nie samles, here are always some aroximaion error and we allow for some discreancy by inroducing he olerance facor : For examle, when 5% and :; we aim o conrol he ye I error such ha i is no greaer han 6%. We may allow o deend on he samle size : For a larger samle size, we may require o ae smaller values. Noe ha boh he ye I and ye II errors deend on he asymoic bias of he esimaor ^ hrough B; he relaive bias of esimaing he variance of 3 (^ OLS ): Our esing-oriened crierion is in shar conras wih he crierion, which deends on a quadraic form of he asymoic bias of ^. In large samles, he quadraic form is of smaller order han he bias iself. So for esing roblems, i is more imoran o reduce he bias of he LV esimaor as comared o he oin esimaion of he LV marix. In addiion, he quadraic form is invarian o sign of B: he -oimal is he same for B and B: In conras, for he esing-oimal ; he sign of B (hence ha of B) is of vial imorance as shown below. he soluion o he minimizaion roblem deends on he sign of B: When B > ; he consrain e I is no binding and we have he unconsrained minimizaion roblem: min e II : he oimal is G! 3 (+); o 4BG 3 : (9) ; When B < ; he consrain e I may be binding and we have o use he uhn-ucer heorem o search for he oimum. Le be he Lagrange mulilier, and de ne L(; ) G ; B + + G ; + B G : G (+); () I is easy o show ha a he oimal ; he consrain e I is indeed binding and > : Hence, he oimal is o! ( ) B ; () G and he corresonding Lagrange mulilier is o G ; B G (+); + 3 G 4 [( ) ] 3 : G 7

20 Formulae (9) and () can be wrien collecively as o 4 4B h G ; G (+); o G 3 i ; where 8 < o : G ; ( ) G ( ) ; if B > + j Bj G (+); ( )[ ] 3 [G ( )] ; if B < 4[( )] 3 () he funcion L(; ) is a weighed sum of he ye I and ye II errors wih weigh given by he oimal Lagrange mulilier. When he size disorion is execed o be negaive, he oimal Lagrange mulilier is zero and we assign all weigh o he ye II error. In his case, he exansion rae of he oimal is O 3 : When he size disorion is execed o be osiive, he Lagrange mulilier is osiive. In his case, he loss funcion is a genuine weighed sum of ye I and ye II errors. he oimal has an exansion rae ha increases wih he olerance on he ye I error. When he ermied olerance is very low so ha s ; he oimal is bounded. he xed- rule can be inerreed as assigning increasingly more weigh o he ye I error as he samle size increases. On he oher hand, when he ermied olerance is high so ha O(); he oimal has an exansion rae of O( ); which is faser han he -oimal exansion rae. All else being equal, he oimal decreases wih B : his is execed, as he asymoic bias of ^ increases wih boh and B : When B is large, we should choose a small o o se he bias e ec. he formula for o deends on he noncenraliy arameer : For racical imlemenaion, we sugges choosing such ha he rs order ower of he es, as measured by G ; ; is 75%. ha is, we solve G; 75% for a given and a given signi cance level : As usual, we consider 5% and %: he value of can be easily comued using sandard saisical rograms. Since is an ineger greaer han or equal o ; in racice, we ae max(d o e ; ) as he value, where de is he ceiling funcion. o sum u, when he size disorion is execed o be negaive, he exansion rae of he oimal is O 3 : When he size disorion is execed o be osiive, he oimal has an exansion rae ha increases wih he olerance on he ye I error. he exansion can range from O() when he ermied olerance is very low o O( ) when he ermied olerance is very high. 6. Daa Driven Imlemenaion he oimal in () deends on he daa generaing rocess only hrough he arameer B: We can herefore wrie o o ( B): he unnown arameer B can be esimaed by a sandard lug-in rocedure based on a simle arameric model lie VA (e.g. Andrews (99)). More seci cally, he lug-in rocedure involves he following ses. Firs, we esimae he model using he OLS esimaor and comue he residuals f^u g : Second, we secify a mulivariae aroximaing arameric model and he model o f^u g by he sandard OLS mehod. hird, we rea he ed model as if i were he rue model for 8

21 he rocess fu g and comue B as a funcion of he arameers of he arameric model. Plugging he esimae B ino () gives he auomaic bandwidh ^: Suose we use a VA() as he aroximaing arameric model for u : Le ^A be he esimaed arameer marix and ^ be he esimaed innovaion covariance marix, hen he lug-in esimaes of and B are ^ (I n ^A) ^(In ^A ) ; (3) ^B 3 (I n ^A) 3 ^A^ + ^A ^ ^A + ^A ^ 6 ^A^ ^A +^( ^A ) + ^A^( ^A ) + ^ ^A (I ^A n ) 3 : (4) For he lug-in esimaes under a general VA() model, we refer o Andrews (99) for he corresonding formulae. Given he lug-in esimaes of and B; he daa-driven auomaic bandwidh can be comued as l ^ o max ^o ( B(; m ^B; ^)) ; : (5) 7 Simulaion Evidence his secion rovides some simulaion evidence on he nie samle erformance of he F es based on he lug-in rocedure ha minimizes he ye II error while conrolling for he ye I error. As in Vogelsang and Franses (5), we se n 6. he error follows eiher a VA() or VMA() rocess: u Au + " u A" + " where A I n ; " (v + f ; v + f ; :::; v n + f ) + and (v ; f ) is a Gaussian mulivariae whie noise rocess wih uni variance. Under his seci caion, he six ime series all follow he same VA() or VMA() rocess wih " s iidn(; ) for + I n + + J n; where J n is a marix of ones. he arameer deermines he degree of cross-deendence among he ime series considered. When ; he six series are uncorrelaed wih each oher. When ; he six series have he same air wise correlaion coe cien.5. he variance-covariance marix of u is normalized so ha he variance of each series u i is equal o one for all values of jj < : For he VA() rocess, (I n A) (I n A ) : For he VMA() rocess (I n + A )(I n + A ) : For he model arameers, we ae ; :5; :5; :75 and se and : We se he inerces and sloes o zero as he ess we consider are invarian o hose arameers. For each es, we consider wo signi cance levels 5% and %; wo di eren choices of he olerance arameer: : and :; and wo di eren samle sizes 3; 5: 9

22 able : ye I error of di eren ess for VA() error wih 3; : and Hybrid Hybrid : : : : : : As in Vogelsang and Franses (5), we consider he following null hyoheses: H : ; H : ; H 3 : 3 ; H 4 : ::: 6 ; where ; ; 3; 6; resecively. he corresonding marix is he rs rows of he ideniy marix I 6 : o exlore he nie samle size of he ess, we generae daa under hese null hyoheses. o comare he ower of he ess, we generae daa under he local alernaive hyohesis H : We examine he nie samle erformance of hree di eren esing mehods. he rs one is he new F es, which is based on he modi ed Wald saisic and esing-oimal and uses he F-disribuion as he reference disribuion. he second one is he convenional Wald es, which is based on he unmodi ed Wald saisic and -oimal and uses he disribuion as he reference disribuion. he las one is he es roosed by Vogelsang and Franses (5), which is based on he Barle ernel LV esimaor wih bandwidh equal o he samle size and uses he nonsandard asymoic heory. he hree mehods are referred as,, and resecively in he ables and gures below. able gives he emirical ye I error of he hree esing mehods for he VA() error wih samle size 3, olerance arameer :; and : he able also includes a hybrid rocedure ha emloys he -oimal and criical values from he F-disribuion. he only di erence beween he convenional mehod and he hybrid mehod lies in he criical values used. More seci cally, le ^ mse be he lug-in esimae of he -oimal given in () and ^F ;OLS ( ^ mse ) be he associaed Wald saisic. he hybrid mehod rejecs he null if ^F ;OLS ( ^ mse ) is larger han he criical value ( ^ mse )( ^ mse + ) F ; ^ where F mse + ; ^ is he -level criical value from mse + he F disribuion F ; ^mse +. In conras, he convenional mehod uses criical values from he disribuion. he signi cance level is 5%, which is also he nominal ye I error. Several aerns emerge. Firs, as i is clear from he able, he convenional mehod has a large size disorion. he size disorion increases wih boh he error deendence and he number of resricions being joinly esed. his resul is consisen wih our heoreical

23 analysis. he size disorion can be very severe. For examle, when :75 and 6, he emirical ye I error of he es is.538, which is far from.5, he nominal ye I error. Using he F criical values eliminaes he disorion o a grea exen. his is esecially rue when he size disorion is large. Inuiively, larger size disorion occurs when is smaller so ha he LV esimaor has a larger variaion. his is he scenario where he di erence beween he F criical values and criical values is larger. Second, he size disorion of he new mehod and he mehod is subsanially smaller han he convenional mehod. his is because boh ess emloy asymoic aroximaions ha caure he esimaion uncerainy of he LV esimaor. he smaller size disorion of he new mehod is also consisen wih ha of he hybrid mehod as boh are based on F -aroximaions. hird, comared wih he mehod, he new mehod has similar size disorion. Since he bandwidh is se equal o he samle size, he mehod is designed o achieve he smalles ossible size disorion. Given his observaion, we can conclude ha he new mehod succeeds in conrolling he ye I error. Due o he aroximaion error, he bound we imose on he aroximae ye I error does no fully conrol he emirical ye I error. his is demonsraed in able. he qualiy of aroximaion deends on he ersisence of he ime series. When he ime series is highly ersisen, he rs order asymoic bias of he LV esimaor may no aroximae he nie samle bias very well. As a resul, he aroximae ye I error, which is based on he rs order asymoic bias, may no fully caure he emirical ye I error. So i is imoran o ee in mind ha he emirical ye I error may sill be larger han he nominal ye I error even if we exer some conrol over he aroximae ye I error. Figures -4 resen he nie samle ower under he VA() error for di eren values of : We comue he ower using he 5% emirical nie samle criical values obained from he null disribuion. So he nie samle ower is size-adjused and ower comarisons are meaningful. he arameer con guraion is he same as hose for able exce he DGP is generaed under he local alernaives. wo observaions can be drawn from hese gures. Firs, he new es has higher ower han he es in mos cases exce when he error deendence is very high and he number of resricions being joinly esed is large. When he error deendence is low, he seleced value is relaively large and he variance of he associaed LV esimaor is small. In conras, he LV esimaor used in he es is inconsisen and is herefore execed o have a large variance. As a resul, he new es is more owerful han he es. On he oher hand, when he error deendence is high, he seleced values are small. In his case, boh he es and he new es emloy a LV esimaor wih large variance. he es can be more owerful in his scenario. Second, he new es is as owerful as he convenional Wald es. his resul is encouraging, as he size accuracy of a es is ofen achieved a he cos of sizable ower loss. An examle is he es. While i has more accurae size han he corresonding Wald es based on he Barle ernel LV esimaor, i is also less owerful. See Vogelsang and Franses (5) for deails. o shed furher ligh on he size and ower roeries of he new es and he corresonding Wald es under he convenional asymoics, we resen he mean and median of he seleced values for samle size 3 in able. he (samle) mean and median are comued over simulaion relicaions. I is clear ha for boh he esing-oriened crierion and he crierion, he mean and median of he seleced

24 able : he seleced values based on he esing-oriened crierion and he crierion for VA() error wih 3; :, and ^ es, ^es, ^es, 3 ^es, 6 ^mse Mean Median Mean Median Mean Median Mean Median Mean Median : : : able 3: ye I error for various ess based on VMA() error wih ; : Hybrid Hybrid : : : : : : value increase wih he error deendence. While he samle mean and median of he oimal are abou he same, he samle mean of he esing-oimal is less han is samle median, imlying ha he esing-oimal has relaively few high values. When he number of consrains is small, e.g. ; ; 3; he esing-oimal is smaller han he -oimal. his exlains why he ye I error of he new es is smaller han or abou he same as ha of he hybrid es. When he number of consrains is 6, he esing-oimal is larger, which exlains he higher size-adjused ower of he new es as comared o he hybrid es. When he samle size increases o 5, he esing-oimal becomes smaller han he -oimal for all values of considered. In his case, he new es has smaller size disorion han he hybrid es for all arameer con guraions considered bu is also slighly less owerful. able 3 resens he simulaed ye I errors for he VMA() error rocess. he qualiaive observaions for he VA() error remain valid. In fac, hese qualiaive observaions hold for oher arameer con guraions such as di eren samle sizes and di eren values of : All else being equal, he size disorion of he new mehod for : is slighly larger han ha for :: his is execed as we have a higher olerance for he ye I error when he value of is larger. Figures 5-8 resen he ower curves under he VMA() error. he gures reinforce and srenghen he observaions for he VA() error. I is clear now ha he new es is more owerful han he es and is as owerful as he convenional Wald es based on he -oimal and aroximaion. his is rue for all arameer combinaions considered.

25 In simulaions no reored here, we have considered VA() and VMA() errors wih negaive values of and hyoheses of he form ::: j for some j : For some of hese con guraions, B > : egardless of he sign of B; he new F es is ofen as owerful as, albei someimes slighly less owerful han, he convenional Wald es we consider here. On he oher hand, he new F es is much more accurae in size han he Wald es. In erms of he ye I error and size-adjused ower, he new F es dominaes he es in an overall sense. Comared o he hybrid es, he new F es achieves a smaller ye I error for a medium samle size a he cos of very small ower loss. 8 Conclusion he aer rooses a novel aroach o mulivariae rend inference in he resence of nonarameric auocorrelaion. he inference rocedure is based on a series ye LV esimaor. Comared o he convenional ernel ye LV esimaors, he series LV esimaor enjoys wo advanages. Firs, i is asymoically invarian o he inerce and rend arameers. his roery releases us from worrying abou he esimaion uncerainy of hose arameers. Second, he associaed (modi ed) Wald saisic converges o a sandard disribuion regardless of he asymoic seci caion of he smoohing arameer. his roery releases raciioners from he comuaion burden of simulaing nonsandard criical values. As a rimary conribuion of his aer, we roose a new mehod o selec he smoohing arameer in he series LV esimaor. he oimal smoohing arameer is seleced o minimize he ye II error hence maximize he ower of he es while conrolling for he ye I error. his new selecion crierion is fundamenally di eren from he crierion for he oin esimaion of he LV. Deending on he ermied olerance on he ye I error, he exansion rae of he esing-oimal smoohing arameer may be larger or smaller han he -oimal smoohing arameer. he xed smoohing arameer rule can be inerreed as exering increasingly igh conrol on he ye I error. Mone Carlo exerimens show ha he size of he new esing rocedure is as accurae as he nonsandard es of Vogelsang and Franses (5) wih bandwidh equal o he samle size. I is also as owerful as he convenional Wald es ha is based on he series LV esimaor and uses he -oimal smoohing arameer. he idea of esing-oimal smoohing arameer choice can be exended o usual ernel HAC esimaor. Sun () considers ernel HAC esimaion in a general GMM framewor and develos a esing-oimal rocedure for smoohing arameer choice. he mehod in Sun () can be adoed for rend esimaion and inference, leading o a esing-oriened bandwidh choice for he ye es. 3

26 4

27 (a) ρ (b) ρ δ δ (c) ρ.5 (d) ρ δ δ Figure : Size-adjused of Di eren esing Procedures for VA() Error wih 3; : and : (a) ρ (b) ρ δ δ (c) ρ.5 (d) ρ δ δ Figure : Size-adjused of Di eren esing Procedures for VA() Error wih 3; : and : 5

28 (a) ρ (b) ρ δ δ (c) ρ.5 (d) ρ δ δ Figure 3: Size-adjused of Di eren esing Procedures for VA() Error wih 3; : and 3: (a) ρ (b) ρ δ δ (c) ρ.5 (d) ρ δ δ Figure 4: Size-adjused of Di eren esing Procedures for VA() Error wih 3; : and 6: 6

29 (a) ρ (b) ρ δ δ (c) ρ.5 (d) ρ δ δ Figure 5: Size-adjused of Di eren esing Procedures for VMA() Error wih 3; : and : (a) ρ (b) ρ δ δ (c) ρ.5 (d) ρ δ δ Figure 6: Size-adjused of Di eren esing Procedures for VMA() Error wih 3; : and : 7

30 (a) ρ (b) ρ δ δ (c) ρ.5 (d) ρ δ δ Figure 7: Size-adjused of Di eren esing Procedures for VMA() Error wih 3; : and 3: (a) ρ (b) ρ δ δ (c) ρ.5 (d) ρ δ δ Figure 8: Size-adjused of Di eren esing Procedures for VMA() Error wih 3; : and 6: 8