COWLES FOUNDATION FOR RESEARCH IN ECONOMICS YALE UNIVERSITY

HAC ESTIMATION BY AUTOMATED REGRESSION By Peter C.B. Phillips July 004 COWLES FOUNDATION DISCUSSION PAPER NO. 470 COWLES FOUNDATION FOR RESEARCH IN ECONOMICS YALE UNIVERSITY Bx 088 New Have, Cecticut 0650-88 http://cwles.ec.yale.edu/

HAC Estimati by Autmated Regressi Peter C. B. Phillips Cwles Fudati, Yale Uiversity, Uiversity f Aucklad & Uiversity f Yrk April 0, 004 Abstract A simple regressi apprach t HAC ad LRV estimati is suggested. The methd explits the fact that the quatities f iterest relate t ly e pit f the spectrum the rigi. The ew estimatr is simply the explaied sum f squares i a liear regressi whse regressrs are a set f tred basis fuctis. Psitive defiiteess i the estimate is therefre autmatically efrced ad the techique ca be implemeted with stadard regressi packages. N kerel chice is eeded i practical implemetati but basis fuctis eed t be chse ad a smthig parameter crrespdig t the umber f basis fuctis eeds t be selected. A autmated apprach t makig this selecti based ptimizig the asympttic mea squared errr is derived. The limit thery f the ew estimatr shws that its prperties, icludig the cvergece rate, are cmparable t thse f cvetial HAC estimates cstructed frm quadratic kerels. ey wrds ad Phrases: Asympttic mea squared errr, autmati, bias, HAC estimati, lg ru variace, tred regressi, trigmetric plymial. JEL Classificati:C. Itrducti Attempts t rbustify iferece i ecmetrics have led t the systematic develpmet f techiques that take it accut ptetial hetergeeity ad autcrrelati i the data. Tw majr practical applicatis f this wrk ivlve HAC heterskedasticity ad autcrrelati csistet cvariace matrix estimati ad lg-ru variace LRV matrix estimati. All HAC ad LRV estimatrs that are cmmly used i ecmetric wrk are based kerel methds. These estimatrs My thaks g t Bruce Hase, Guid uersteieradtwrefereesfrcmmetsaearlier versi f the paper. NSF research supprt uder Grat N. SES 04-454 is ackwledged.

iherit their frm ad their asympttic prperties frm wrk i the earlier literature f spectral desity estimati, where kerel methds are agai dmiat. Autmated versis f these kerel methds have als bee develped. Autmati remves the eed fr discretiary badwidth chice i kerel estimati by implemetig data-determied badwidth selecti rules that are cmmly based asympttic mea squared errr frmulae. Badwidth selecti rules have bee built it sme ppular ecmetric sftware prgrams ad users may implemet them withut havig t make ay discretiary decisis. This cveiece has helped prmte the use f the methds i empirical research. Autmated techiques f this gere, just like the kerel methds which they are used, themselves belg t a lger pedigree f related wrk i statistics. The preset ctributi suggests a vel apprach t HAC/LRV estimati that des t ivlve the direct use f kerels. T the authr s kwledge, the apprach is ew ad has t bee suggested befre i ay earlier wrk i statistics r ecmetrics. Ulike cvetial prcedures the methd is t based kerel estimati, either by way f a lag kerel f weighted autcvariaces r by kerel smthig f the peridgram i the frequecy dmai. Hwever, we shall see that the apprach may be iterpreted as prducig a asympttic frm f kerel estimate. The idea is mtivated by the fact that the quatities f iterest i HAC/LRV estimati relate t ly e pit the spectrum ad that this pit the rigi refers t lg-ru behavir. This feature is explited by desigig a liear regressi f the variable f iterest a set f regressrs desiged t represet lg-ru behavir directly. The regressrs frm a set f tred basis fuctis. Ay set f basis fuctis may be used, but i the develpmet give here ad i empirical wrk it is geerally cveiet t use a rthrmal set f trigmetric plymials. Several examples are give. The ew HAC estimatr is simply part f the utput f this regressi ad is give by the explaied sum f squares i a liear regressi the tred basis. It ca be implemeted by stadard regressi packages. Psitive defiiteess i the estimate is autmatically efrced by its cstructi as a sum f squares ad this is s whatever the chice f basis fuctis. This prperty is imprtat, beig e f the mai ccers i Newey ad West 987 ad playig a sigificat rle i Adrews 99 regardig the selecti f suitable kerel fuctis. Curiusly, this is a example f a regressi that wuld cvetially be regarded i ecmetrics as misspecified r eve spurius because the regressrs are i fact irrelevat t the determiati f the depedet variable. Nevertheless, the cefficiets i this regressi prduce, up straightfrward rmalizati, a csistet HAC estimatr. The apprach has the advatage f the simple cveiece f least squares regressi ad kerel chice is eeded i its implemetati. Hwever, the tred basis fuctis eed t be chse ad a smthig parameter crrespdig t the umber f the tred fuctis actually used i the regressi als eeds t be selected. The smthig parameter chice ca be autmated based the behavir f the asympttic mea squared errr f the estimatr ad a rule fr such autmated implemetati is develped i the paper. As far as the chice f tred basis fuc-

tis is ccered, it is fte cveiet t use trigmetric fuctis ad it turs ut that the asympttic results are ivariat t the chice f basis withi the class f trigmetric plymials. The fact that csistet estimati is pssible usig apparetly irrelevat regressrs may appear smewhat magical. Hwever, prjectig a statiary time series t a space f treds, eve whe there is tred i the data, has the effect f islatig the lg-ru behavir i the time series ad this is what eables the direct regressi estimati f the lg-ru parameter. The idea has extesis ad may ther applicatis which are t discussed i the preset paper. Sme f these are csidered by the authr 004 i ther wrk.. Tred Regressi f Utreded Time Series The fllwig develpmet ccetrates the scalar case. Oly mir mdificatis are required t exted the results t the vectr case ad matrix HAC estimati. Accrdigly, let u t be a weakly depedet time series satisfyig u t C L ε t c j ε t j, j0 j a c j <, C 6 0,a>3, L j0 where ε t iid 0,σ ad E ε t v <, fr sme v>. Thetimeseriesu t is statiary with variace σ u P j0 c j σ, autcvariace fucti γ u h E u t u t+h σ P j0 c jc j+h, fiite v th abslute mmet E u t ν ³ P v j0 c j E εt v <, spectrum f u λ σ /π C e iλ, ad lg-ru variace ω πf u 0 σ C. The summability cditi i L esures that h h 3 γ u h <, which is helpful i sme techical derivatis belw ad meas that f u λ has ctiuus secd derivative f u P λ π σ h h γ u h e iλh. Allwace fr hetergeeity i ε t ad u t ca be made i the usual way with mir mdificatis t L c.f. Phillips ad Sl, 99. withut affectig the prcedures r the prperties discussed belw i a essetial way. Uder L, partial sums S t P t i u i satisfy the fuctial law e.g., Phillips ad Sl, 99 B : S b c P b c i u i B where bac sigifies the iteger part f a, is weak cvergece, ad B is Brwia mti with variace ω. Let { k } k be a cmplete rthrmal system i L [0, ]. Later, we will wrk with the explicit sequece 8 but it is sufficiet t assume that the fuctis k are twice ctiuusly differetiable [0, ]. We prpse a regressi f u t 3

a cllecti f determiistic regressrs { t k } k frmed by takig the first members f this rthrmal sequece evaluated ver t,..,.write this regressi i the frm u t b bk k t +e t : b 0 t + e t, t,...,; 3 k where t t,..., t 0. I bservati frmat 3 ca be writte as u Φ b b + e, with b Φ 0 Φ Φ 0 u. Let P Φ Φ 0 Φ Φ 0 ad cstruct the estimate µ Φ 0 u. 4 ˆω µ u 0 Φ u0 P u µ Φ 0 Φ As shw i Lemma A i the Appedix, P t t 0 t I + O ad P t t 0 t I +O. Stadard fuctial limit argumets ad Wieer itegrati reveal that fr fixed as we have / Z t u t d r db r :ξ d N 0,ω I. t It fllws immediately that 0 u0 P u d ξ0 ξ d ω χ, 5 where χ is chi-squared with degrees f freedm. Fr fixed, the asympttic mea ad variace f 5 are µ µ E u0 P u ω +, Var u0 P u ω4 +, 6 as. These results mtivate the lg-ru variace estimatr ˆω u0 P u u0 P u P, where u P P u, 7 which, i view f 6, ca be expected t be csistet fr ω whe as. The estimate ˆω issimplythesamplevariacefu P, the data prjected t the space spaed by the regressrs Φ. Thus, ˆω is that part f the sample variace f u t explaied by the regressi f u t t a determiistic tred basis. This explaied sum f squares may be regarded as ather way f thikig abut a lg-ru variace - the ctributi t the variati f u t that cmes frm lg-ru r tred-like behavir i the series. Thus, there wuld seem t be a strg heuristic mtivati fr csiderig estimates like 7. Nte that ˆω is a egative defiite quadratic frm i the data, whse matrix is the prjecti P. Thus, ˆω belgs t the geeral class f quadratic estimatrs f 4

ω. Geeral quadratic estimatrs were csidered i the early spectral aalysis literature but have received little subsequet atteti relative t kerel estimates. Oe reas is that, fr every such quadratic estimatr, e ca fid a crrespdig lag kerel estimatr with smaller mea squared errr Greader ad Rseblatt, 957, p.9. Iterestigly, as we will shw belw, ˆω turs ut itself t be asympttically equivalet t a lag kerel estimatr ad t have ice asympttic prperties aalgus t thse f quadratic kerel estimates. Thus, there is eed t adapt ˆω it kerel frm ad, f curse, ˆω is psitive by cstructi. T develp a csistet estimati techique, we eed t allw fr the umber f regressrs t pass t ifiity with the sample size i such a way that the regressi 3 remais feasible. Accrdigly, we impse the fllwig rate cditi + 0, R which requires t g t ifiity faster tha but slwer tha. T establish a cetral limit therem fr ˆω, we eed the further cditi 4/5, which ctrls the expasi rate s that there is bias i the limit. Fr a explicit limit thery, icludig a explicit expressi fr the limitig bias f ˆω, it is cveiet t use the rthrmal sequece k r µ si k ¾ πr, k,,... 8 The fuctis 8 are the eigevectrs f the cvariace kerel f Brwia mti c.f., Phillips, 998 ad frm a rthrmal system fr L [0, ]. Of curse, ther rthrmal sequeces ca be used. Hwever, it turs ut that the asympttic results give belw are ivariat t the chice f the rthrmal sequece withi the class f trigmetric plymials because the estimates are asympttically equivalet t the same lag kerel estimatr. This pit is discussed belw i secti 4. Uder this set-up, we give frmulae fr the limitig bias, variace ad mea squared errr ad a limit distributi thery fr ˆω. The details f the prfs are differet frm thse f the cvetial literature HAC estimati ad are f sme idepedet iterest, s they are prvided here. But the fial results ed up beig qualitatively similar, as the fllwig result shws. Therem Uder cditis L ad R a lim E ˆω ω P π 6 h h γ u h :D; b If 4/5, the ˆω ω N 0, ω 4 ; c If 5 / 4, thelim 4 E ˆω ω D +ω 4. Part a shws that ˆω hasbiasfrder / f the frm E ˆω ω + D [ + ], where D π h γ 6 u h : π 6 ω. 5 h

Frm b, the variace f ˆω is f O. S, icreases i the umber f regressrs icrease bias ad reduce variace. The situati is aalgus t badwidth chice i kerel estimati. Themeasquarederrrfˆω has the frm MSE ˆω Bias + Var 4 4 D + ω4. Optimizati f this quatity with respect t leads t the first rder cditi 4 3 D ω4 0, which gives the fllwig frmula fr the ptimal value f 4 4 5 ω 4 4D 5. 9 Of curse, this is aalgus t cvetial MSE ptimizati frmulae fr badwidth chice i kerel estimati e.g. Greader ad Rseblatt, 957. Frmula 9 ca be used t implemet a data-determied chice f i a cvetial way. Oe apprach is t use parametric estimates f D ad ω 4 i 9 as, fr example, i Newey ad West 994. The mst cmm ad cveiet methd i practice is a simple plug-i estimatr based the use f a parametric mdel fr prelimiary estimati f ω 4 ad D i 9. I the case f a first rder autregressi with fitted cefficiet â ad errr variace s, the stadard frmulae give ˆω s / â ad ˆD π 6 âs / â 4. Sme mdificatis t these frmulae may be desirable i cases where â is clse t uity. I the ctext f prewhiteig, fr example, Adrews ad Maha 99 prpsed a 0.97 rule i which â be replaced by 0.97 wheever â exceeds this value. It is kw that this particular rule seriusly iterferes with pwer i sme cases, especially statiarity testig c.f. Lee, 996. A alterative budary restricti that seems t imprve the size ad pwer prperties f prcedures based HAC estimates is the sample-size-depedet rule give i Sul, Phillips ad Chi 003, where â is replaced by / wheever it exceeds that value. 3. Asympttic Frm as a erel Estimatr Lemma Bc i the Appedix shws that the rthgal sequece k r i 8 satisfies the fllwig summati frmula πt s si πt+s k t k s si si k πt s si πt+s. 0 6

Usig this frmula ad 7, 3 ad 36 frm the prf f part b f the Therem, we fid that the HAC estimate ˆω has the fllwig asympttic frm ˆω si ª Ã πh si h + πh ª! u t u t+h + O p + lg + t,t+h µ Ã! h k ˆγ u h+o p + lg +, h + where µ h k si ª πh si ª ¾ πh cs πh si πh ª si ª πh may be regarded as a lag kerel fucti ad ˆγ u h P t,t+h u tu t+h is the sample autcvariace. The dmiat term i has the usual frm f a kerel estimate f ω ad is depedet, which serves the rle f a smthig parameter. Thus, ˆω behaves asympttically like a kerel estimate. Let M. Fr h/ smallwecawritethelagkerelk h as a fucti f h/m i apprximate frm as fllws k M µ h M with which we may assciate the fucti si ª πh M si πh M ª si πh ª M, πh M si πx k x πx, which is the lag kerel fr the Daiell estimate e.g., Priestley, 98, p. 44. This lag kerel is a smthed peridgram estimate with a rectagular spectral widw. Here, π/m is the width f the frequecy bad ver which the peridgram is beig smthed. The regressi based estimate ˆω is therefre clsely related t this well kw kerel estimate f ω ad has the same bias, variace ad mea squared errr as the Daiell estimate. Evidetly, therefre, the asympttic mea squared errr f ˆω is dmiated by that f the Bartlett - Priestley quadratic spectral widw Priestley, 98; Adrews, 99. 4. The Effect f Differet Tred Bases We may chse t use ther sequeces f rthgal fuctis [0, ] i the regressi 3 ad it is iterestig t explre the effects f such alterate chices the asympttic kerel frm f estimate ˆω. I the fllwig discussi, we cfie ur atteti t sequeces f trigmetric plymials. Parts a ad b f Lemma C i the Appedix give the fllwig summati frmulae fr the rthrmal sie ad csie trigmetric plymials k r 7

si{kπr} ad k r cs{kπr}, respectively, ¾ k t k s si πt s si k πt s si si πt+s πt+s ¾. 3 As i the calculati leadig t, the secd cmpet f 3 turs ut t be f smaller rder as, ad hlds with ¾ µ h k si πh si ª si ª πh cs πh ª πh si πh ª cs πh ª si ª πh µ si πh ª + O, which is asympttically equivalet t the lag kerel. Thus, these differet rthrmal trigmetric sequeces all lead t HAC estimates that are asympttically equivalet. If the cmplex rthrmal sequece k r e πikr is used, the the regressi cefficiets i 3 are themselves cmplex ad have the frm b Φ Φ Φ u, where Φ is the matrix f bservatis f the regressrs { t k : k,...,} ad sigifies cmplex cjugati ad traspsiti. I place f 3 we ca write u t k b bk e πikt + e t, 4 whichmayberegardedasafitted empirical versi f the Cramér represetati f the statiary prcess u t, which was ticed earlier i Phillips 996, Remark 5.a. Nte that λ k πk 0 fr all k,.., sice 0. Thus, the regressi 4 fcuses atteti the zer frequecy r lg-ru cmpet f the Cramér represetati f u t. The matrix Φ satisfies Φ Φ I ad is a scaled uitary matrix. S the k th elemet f b is simply the stadardized discrete Furier trasfrm, P t e πikt u t, f u t, which is well-kw t satisfy a cetral limit therem e.g. Haa, 970, p. 4 up rescalig. Additially, as shw i Phillips 999, therem 3., the asympttically ifiite cllecti f such elemets have the fllwig limit as ad are asympttically idepedet prvided but t t fast relative t. I particular ζ k Z e πikt ut e πikr db r, k,,... t 0 As is easily see, the limit variates ζ k R 0 eπikr db r are idepedet cmplex Gaussia N c 0,ω. Lettig P Φ Φ Φ Φ, we the have ˆω u0 P u b b b b 8 k ζ k p E ³ ζ k ω. 5

As such, the estimate ˆω may be iterpreted as the sample variace f the empirical estimates btaied frm the fitted regressi 4 f the rthgal prcess at the zer frequecy that appears i the Cramér represetati f u t. Observe that ζ k is πi u λ k, where I u λ is the peridgram f u t ad λ k πk are the fudametal frequecies. Thus we ca write ˆω π I u λ k, 6 which crrespds t a smthed peridgram estimate f the spectrum at the zer frequecy give that 0. The spectral widw i the estimate ˆω is clearly rectagular, just as that f the Daiell widw e.g., Haa, 970, p. 79. Agai the results are asympttically equivalet t thse f the estimate 7 based the siusidal sequece 8. Of curse, this set-up easily permits the use ther spectral widws. Let W diag{w λ,...,w λ } be a diagal matrix prescribig a particular weightig sequece based the widw fucti W λ. The the HAC estimate k ˆω W b b W b b π W λ I u λ k, has the usual frm f a smthed peridgram estimate with spectral widw W λ. I this way, the preset apprach accmmdates all cvetial kerel-based HAC estimates. 5. Discussi The estimate ˆω is straightfrward t cmpute, beig e f the utputs f a liear regressi ad it has a simple heuristic mtivati. The fact that regressi tred prduces this csistet estimate idicates that tred cefficiets carry ifrmati abut the lg-ru features f the data, eve thugh the true tred cefficiets i this regressi f a statiary time series are zer. I fact, as discussed i the preceedig secti, the estimate ˆω may be iterpreted as the sample variace f the cefficiets i a fitted regressi that is a empirical versi f the lg-ru part f the Cramér represetati f a statiary prcess. Simulatis t reprted here idicate that ˆω perfrms as well as curret idustry-stadard methds usig quadratic kerels, autmated badwidth selecti ad prewhiteig Adrews, 99; Adrews ad Maha, 99; De Haa, W.J., ad A. Levi, 997; Lee ad Phillips, 993; Newey ad West, 994. This may be expected give the asympttic relatiship betwee ˆω ad the Daiell kerel estimate. The regressi apprach develped here may be exteded t estimate the spectrum f u t at pits ther tha the rigi, althugh we d t pursue that pssibility here. k 9

6. Additial Lemmas ad Prfs 6. Lemma A Uder R, P t t 0 t I +O, ad P t t 0 t I + O, as. 6. Prf We first prvide a direct calculati whe the elemets f t are give by the trigmetric fuctis 8. I this case, the diagal elemets f P t t 0 t are t k µ t cs k πt ª µ si k ¾ πt t t cs k πt ¾ Re t t Re e ik π eik π e ik + 4 π Re πt ik e e ik π e ik π + 4 Re e ik π e ik π +4 cs k π ª cs k π ª +, ad the ff-diagals are µ µ t t k µ si k ¾ µ πt si ¾ πt t t cs k πt ¾ cs k + πt ¾ t Re πt πt ik e ik+ e t t Re e ik π eik π e ik π eik+ π eik+ π e ik+ π k dd, k + eve k eve, k+ dd. 0 k, k + bth dd r eve It fllws that P t t 0 t I + O ad P t t 0 t I + O. 0

I the geeral case, if k s is twice ctiuusly differetiable [0, ] by Euler summati we have µ t k Z µ t k dt + µ ³ k + k t ¾ + Z t [t] ¾ µ µ t t dt k 0 k Z µ µ k s ds + O +O, / sice t k ad 0 t k are uifrmly buded [0, ]. Similarly, P t t t k O uifrmly fr k 6. 6.3 Lemma B Fr k r si k πr ª, we have: a k t cs k b P k k t si πt si πt c P k k t k s si d σ4 C 4 P h si { πh si { } πh ; πt πt s si πt s ¾ si k π ª ; si si πt+s πt+s } σ4 C 4 [ + ], where is the differecig peratr k t k t k t ;. 6.4 Prf f Lemma B Fr part a µ t k µ si k πt " µ cs k ¾ µ si π t ¾ π t k µ si k π Fr part b k t µ si k ¾ πt Im e ik πt k k k πt Im e i πt ei Im e i πt si πt e i πt si πt si πt si πt. ¾ #.

Fr part c k t k s k Re e πt s i k k Re Re i i µ si k µ cs k k e he ik πt s ¾ µ πt si k π t s πt s i e πt s i e e πt s i ³e πt s i e πt s ³ e si Re e πt s i si πt s i e πt s πt s ¾ si πt s π t s cs si si πt s si si si πt s i πt s si e πt+s i πt s πt+s πt+s cs. Fr part d, we use Fejér s itegral givig σ 4 C 4 si ª πh si ª σ4 C 4 πh as required. 6.5 Lemma C h a If k r si{kπr}, k t k s si k ¾ πs ¾ µ cs k π t + s i e ik πt+s i e πt+s i si πt+s Z 0 πt+s πt+s i e πt+s i i π t + s i e πt+s i ¾ ³e πt+s i e πt+s i ³e πt+s i e πt+s ¾ si πt+s si πt+s si ª πr si dr [ + ] πrª σ4 C 4 [ + ] ω [ + ], πt s si πt s ¾ si si πt+s πt+s ¾ ; i

b If k r cs{kπr}, k t k s si k 6.6 Prf f Lemma C Fr part a k t k s k k k Re Re πt s si πt s si k πt ¾ si cs k k ¾ + k πs ¾ π t s cs k i πt s ik he e πt+s ik e πt s i e πt s i e πt s i e πt s i Re e πt+s i e πt s i e πt+s i ³e πt s i ³ e i πt s Re cs πt s +cs πt s +cs πt s cs cs πt s +cs πt+s +cs πt+s cs cs πt+s ¾ si πt s + si πt s cs πt s ¾ + si πt+s si πt+s cs πt+s si si πt+s πt+s ¾ π t + s e πt+s i e πt+s ¾ ; i e πt+s i ³e πt+s i ³ e πt+s cs πt s πt+s πt+s i 3

si si si si ¾ πt s si πt s πt s cs ¾ πt s si si πt s ¾ πt s πt s πt s si si si si πt+s πt+s ¾. ¾ πt+s si πt+s πt+s cs πt+s ¾ si si πt+s πt+s Fr part b k t k s k k k cs cs k k πt ¾ cs k πs ¾ π t s +cs k ¾ π t + s, which differs frm part a ly i the sig f the secd term. The stated result therefre fllws by the same calculatis. 6.7 Prf f Therem Part a E ˆω I view f Lemma A µ Φ 0 tr Φ tr h + k h + h + t,t+h t,t+h t 0 t+h γ u h t,t+h µ t 0 t+h γ u h +O k µ t µ t + h k γ u h +O µ, 4

ad, sice ω P h γ u h, E ˆω ω h + h h + + µ a k γ u h fr a>3 because γ u h h t,t+h k µ t µ t + h k γ u h +O µ µ µ t t + h µ k k γ u h +O k t,t+h, 7 h γ u h a h µ h a γ u h a, by cditi L. Als fr ay psitive iteger L <we have µ µ t t + h k k γ u h h + k t,t+h L µ µ t t + h k k γ u h h L k t,t+h + µ µ t t + h k k γ u h L < h < k t,t+h +O +O µ µ +O ad sice the elemets t k are buded uifrmly i t we have µ µ t t + h k k γ u h L < h < k t,t+h C γ u h C µ L 3 h 3 γ u h L 3. L < h < L < h < µ, 8 Nw chse L such that L 3/ + L 9 5

as. The, L 3 L < h < k ³ ad t,t+h k µ t µ t + h k Fr the first term f 8 we te that µ µ t t + h k k t,t+h µ t k + µ t k t,t+h t,t+h + µ µ t t + h k k t,t+h µ γ u h µ t + h k µ t k + O uifrmly i h L. It therefre fllws frm 7 - that µ t k. 0 µ, E ˆω ω L h L k µ + + O µ t,t+h µ µ t + h t k k γ u h µ t k. Next csider t,t+h µ t + h k µ t k µ si k µ t k ¾ µ πt si k ¾ µ π t + h si k ¾ πt t,t+h µ cs k ¾ πh cs k π t + h t,t+h cs k πt ¾¾ t,t+h µ h µ cs k ¾ πh " cs k π t + h cs k πt ¾ # 3 t,t+h Takig the first term f 3, averagig ver k, ad usig the fact that h L ad 6

L satisfies 9 s that L k k, we get µ cs k ¾ πh µ k µ πh µ h + π h 6 [ + ]. 4 Fr h 0 we ca write the secd term f 3 as " cs k π t + h cs k πt ¾ # t,t+h Z h/ cs / Z h/ / µ k π r + h cs {k πr} dr " ª si k π r + h k π si {k πr} k π h/ / # h/ / ¾ dr +O +O +O µ +O µ µ µ µ si k π h ¾ µ ¾ µ h + si k π +O k π µ si k π h ¾ µ ¾ µ si k π +O k π µ si k π h ¾ µ si k π h ¾ µ +O k π µ ¾ µ ¾ µ h + si k π si k π +O k π µ cs k π 3h ¾ si k π h ¾ µ +O k π 4 4 µ ¾ µ ¾ µ h +4 h cs k π si k π +O k π 4 4 µ cs k π 3h ¾ h 4 4 h ¾ µ +O, 5 4 7

fr h L ad L satisfyig 9. Averagig ver k we fid µ cs k π 3h ¾ h 4 4 h 4 k cs π +k 3h ¾ h 4 4 h 4 k cs k 3h ¾ h 4 4 h 4 k µ 3h k π h 4 4 h 4 k 4 π 9h µ h 6 4 + h 3 µ h µ L 3. 6 uifrmly i 0 h L. Withut gig thrugh the calculati, the same rate result hlds whe L h<0. We deduce frm 3, 4, 5 ad 6 that Thus, k µ h t,t+h E ˆω ω L h L k µ µ + + O π L µ 6 h π 6 h L h k µ t µ t + h k π h µ h 6 + t,t+h k µ t h γ u h+ h γ u h[+ ], ³ sice fr satisfyig R. Thus as stated. ³ lim E ˆω ω π 6 8. k µ t + h µ µ t k + O h µ t k γ u h µ h γ u h,

Part b Frm Lemma A ˆω u0 P u µ u 0 µ Φ Φ 0 Φ µ Φ 0 u µ u 0 µ Φ Φ 0 µ u +O k t k s µ u tu s +O k t,s k h + t,t+h k t k t + h u tu t+h +O Usig the device i Phillips ad Sl 99 we have the decmpsiti u t C ε t + ε t ε t, fr ε t c j ε t j, c j j0 µ. 7 c s, 8 where P j0 c j < uder L. The, 8, partial summati ad Lemma Ba yield t k t u t C C C + 3 C C t j+ k t ε t C k t ε t t k t ε t k ε k t ε t t t " t t t k t ε t k ε ε t cs µ k π t t # k t µ µ k ε t + O p + O p k t µ ε t + O p +, µ k π si k ª π k π 9

uifrmly i k. Thus, ˆω k k k +O p à C C k t t u +O t k t µ ε t + O p t k t t ε +O t t! k t ε t k Frm Lemma Bb, P k k t si πt " # Var k t ε t t by Lemma Bd. Hece, k si πt µ +O µ µ µ + O p. 9, ad s σ si 4 πt si πt t σ si πt si πt t à Z si! πr O 0 si πr dr O, " # k t ε t O p, 30 t ad it fllws frm 9 ad 30 that ˆω k k C k t t ε +O t +O p à + µ!. 3 Thus, ˆω ω C k à k t ε t σ + O p + t 5! 3 0

Write C C k k k t ε t σ t "Ã! k t ε t σ + t # k t k s ε tε s. 33 t>s I view f Lemma A, we have C C C k k k O p à + k t ε t σ t k t ε t σ + t Ã! k t σ k t Ã! ε t σ + O t!. 34 Thus, fr 4/5 we have frm 3-34 ˆω ω C k C C t k t k s ε tε s + p t>s k t k s ε t ε s + p t>s C h + C k si si si ª πh si ª t>s si t>s πt s πh si πt s πt+s πt+s si si t,t+h πt+s πt+s ε t ε t+h ε tε s + p ε t ε s + p. 35 We shw that the secd term i the fial expressi 35 is egligible. Nte that C si πt+s ε t ε s t>s si πt+s

has mea zer ad variace Hece ad s σ 4 t si πt+s t s si πt+s à σ 4 Z Z! r si {π r + p} O si dpdr π r + pª à σ 4 Z Z! r O si π r + pªdpdr Z " O σ4 cs # π r r + pª π si π dr r + pª à σ 4 Z O cs π r ª π si ª π r + cs ª! π r + π si ª π r + dr µ lg O. C si t>s si πt+s πt+s ˆω ω C h Next te frm Lemma Bd that h σ 4 C 4 h ε t ε s O p Ãr lg si ª πh ª si πh si ª πh si πh t,t+h ª σ 4 C 4 [ + ].! p, 36 ε t ε t+h + p. Fially, usig a martigale cetral limit argumet alg the same lies as that i Phillips, Su ad Ji 003, we may establish that C si ª πh si πh ª ε t ε t+h d N ³0, 4 σ 4 C N 0, ω 4, t,t+h givig the stated CLT fr ˆω ω.

Part c Part a shws that the bias f ˆω is give by E ˆω ω D [ + ], where D π 6 h h γ u h, while argumets as i part b shw the variace f ˆω t be It fllws that Var ˆω ª ω 4 [ + ]. MSE ˆω µ E ˆω ω µ ω 4 D + [ + ] ad, if 5 / 4, we get ³ 4 ³ lim MSE 4 ˆω lim E ˆω ω D +ω 4, as stated. 7. Refereces Adrews, D. W.. 99, Heterskedasticity ad autcrrelati csistet cvariace matrix estimati, Ecmetrica 59, 87 858. Adrews, D. W.., ad J.C. Maha 99, A imprved heterskedasticity ad autcrrelati csistet cvariace matrix estimatr, Ecmetrica 60, 953-966. De Haa, W.J., ad A. Levi 997, A practitier s guide t rbust cvariace matrix estimati, i Hadbk f Statistics 5, G.S. Maddala ad C.R. Ra, eds., Elsevier Amsterdam, pp.99-34. Greader, U. ad M. Rseblatt 957. Statistical Aalysis f Statiary Time Series. New Yrk: Wiley. Haa, E. J. 970. Multiple Time Series. New Yrk: Jh Wiley ad Ss. Lee, J. S. 996 O the pwer f statiary tests usig ptimal badwidth estimates, Ecmics Letters, 5, 3-37. Lee, C. C. ad P. C. B. Phillips 994, A ARMA prewhiteed lg-ru variace estimatr, Mauscript, Yale Uiversity. Newey, W.. ad. D. West 987. A simple psitive semi-defiite heterskedasticity ad autcrrelati csistet cvariace matrix, Ecmetrica 703 708. 3

Newey, W.. ad. D. West 994. Autmatic lag selecti i cvariace matrix estimati, Review f Ecmic Studies, 6, 63-653. Phillips, P.C.B. 996, Spurius Regressi Umasked Cwles Fudati Discussi Paper, #35. Phillips, P. C. B. 998. New Tls fr Uderstadig Spurius Regressis. Ecmetrica, 66, 99-36. Phillips, P. C. B. 999: Uit Rt Lg Peridgram Regressi, Yale Uiversity, Cwles Fudati Paper #44. Phillips, P. C. B. 004. Challeges f tredig time series ecmetrics, mime, Yale Uiversity. Phillips, P. C. B. ad V. Sl 99. Asympttics fr liear prcesses, Aals f Statistics 0, 97 00. Phillips, P. C. B., Y. Su ad S. Ji 003: Csistet HAC Estimati ad Rbust Regressi Testig Usig Sharp Origi erels with N Trucati, Cwles Fudati Discussi Paper N. 407 Priestley, M. B. 98. Spectral Aalysis ad Time Series, VlumeI. NewYrk: Academic Press. Sul, D., P. C. B. Phillips ad C-Y. Chi 003. Prewhiteig bias i HAC estimati, mimegraphed, Yale Uiversity. 4