HAC ESTIMATION BY AUTOMATED REGRESSION PETER C. B. PHILLIPS COWLES FOUNDATION PAPER NO. 1158

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1 HAC ESTIMATION BY AUTOMATED REGRESSION BY PETER C. B. PHILLIPS COWLES FOUNDATION PAPER NO. 58 COWLES FOUNDATION FOR RESEARCH IN ECONOMICS YALE UNIVERSITY Box 088 New Have, Coecticut

2 Ecoometric Theory,, 005, 6 4+ Prited i the Uited States of America+ DOI: 0+070S HAC ESTIMATION BY AUTOMATED REGRESSION PETER C.B. PHILLIPS Cowles Foudatio, Yale Uiversity Uiversity of Aucklad ad Uiversity of York A simple regressio approach to HAC ad LRV estimatio is suggested+ The method exploits the fact that the quatities of iterest relate to oly oe poit of the spectrum ~the origi!+ The ew estimator is simply the explaied sum of squares i a liear regressio whose regressors are a set of tred basis fuctios+ Positive defiiteess i the estimate is therefore automatically eforced, ad the techique ca be implemeted with stadard regressio packages+ No kerel choice is eeded i practical implemetatio, but basis fuctios eed to be chose ad a smoothig parameter correspodig to the umber of basis fuctios eeds to be selected+ A automated approach to makig this selectio based o optimizig the asymptotic mea squared error is derived+ The limit theory of the ew estimator shows that its properties, icludig the covergece rate, are comparable to those of covetioal HAC estimates costructed from quadratic kerels+. INTRODUCTION Attempts to robustify iferece i ecoometrics have led to the systematic developmet of techiques that take ito accout potetial heterogeeity ad autocorrelatio i the data+ Two major practical applicatios of this work ivolve HAC ~heteroskedasticity ad autocorrelatio cosistet! covariace matrix estimatio ad log-ru variace ~LRV! matrix estimatio+ All HAC ad LRV estimators that are commoly used i ecoometric work are based o kerel methods+ These estimators iherit their form ad their asymptotic properties from work i the earlier literature of spectral desity estimatio, where kerel methods are agai domiat+ Automated versios of these kerel methods have also bee developed+ Automatio removes the eed for discretioary badwidth choice i kerel estimatio by implemetig data-determied badwidth selectio rules that are commoly based o asymptotic mea squared error formulas+ Badwidth selec- My thaks go to Bruce Hase, Guido uersteier, ad two referees for commets o a earlier versio of the paper+ NSF research support uder grat SES is ackowledged+ Address correspodece to Peter Phillips, Cowles Foudatio for Research i Ecoomics, Yale Uiversity, Box 0887, New Have, CT , USA; peter+phillips@yale+edu Cambridge Uiversity Press $+00

3 HAC ESTIMATION BY AUTOMATED REGRESSION 7 tio rules have bee built ito some popular ecoometric software programs, ad users may implemet them without havig to make ay discretioary decisios+ This coveiece has helped promote the use of the methods i empirical research+ Automated techiques of this gere, just like the kerel methods o which they are used, themselves belog to a loger pedigree of related work i statistics+ The preset cotributio suggests a ovel approach to HAC0LRV estimatio that does ot ivolve the direct use of kerels+ To the author s kowledge, the approach is ew ad has ot bee suggested i ay earlier work i statistics or ecoometrics+ Ulike covetioal procedures the method is ot based o kerel estimatio, either by way of a lag kerel of weighted autocovariaces or by kerel smoothig of the periodogram i the frequecy domai+ However, we shall see that the approach may be iterpreted as producig a asymptotic form of kerel estimate+ The idea is motivated by the fact that the quatities of iterest i HAC0LRV estimatio relate to oly oe poit o the spectrum ad that this poit ~the origi! refers to log-ru behavior+ This feature is exploited by desigig a liear regressio of the variable of iterest o a set of regressors desiged to represet log-ru behavior directly+ The regressors form a set of tred basis fuctios+ Ay set of basis fuctios may be used, but i the developmet give here ~ad i empirical work! it is geerally coveiet to use a orthoormal set of trigoometric polyomials+ Several examples are give+ The ew HAC estimator is simply part of the output of this regressio ad is give by the explaied sum of squares i a liear regressio o the tred basis+ It ca be implemeted by stadard regressio packages+ Positive defiiteess i the estimate is automatically eforced by its costructio as a sum of squares, ad this is so whatever the choice of basis fuctios+ This property is importat, beig oe of the mai cocers i Newey ad West ~987! ad playig a sigificat role i Adrews ~99! regardig the selectio of suitable kerel fuctios+ Curiously, this is a example of a regressio that would covetioally be regarded i ecoometrics as misspecified ~or eve spurious! because the regressors are i fact irrelevat to the determiatio of the depedet variable+ Nevertheless, the coefficiets i this regressio produce, upo straightforward ormalizatio, a cosistet HAC estimator+ The approach has the advatage of the simple coveiece of least squares regressio, ad o kerel choice is eeded i its implemetatio+ However, the tred basis fuctios eed to be chose, ad a smoothig parameter correspodig to the umber of the tred fuctios actually used i the regressio also eeds to be selected+ The smoothig parameter choice ca be automated based o the behavior of the asymptotic mea squared error of the estimator, ad a rule for such automated implemetatio is developed i the paper+ As far as the choice of tred basis fuctios is cocered, it is ofte coveiet to use trigoometric fuctios, ad it turs out that the asymptotic results are ivariat to the choice of basis withi the class of trigoometric polyomials+

4 8 PETER C.B. PHILLIPS The fact that cosistet estimatio is possible usig apparetly irrelevat regressors may appear somewhat magical+ However, projectig a statioary time series oto a space of treds, eve whe there is o tred i the data, has the effect of isolatig the log-ru behavior i the time series, ad this is what eables the direct regressio estimatio of the log-ru parameter+ The idea has extesios ad may other applicatios that are ot discussed i the preset paper+ Some of these are cosidered by the author ~004! i other work+. TREND REGRESSION OF UNTRENDED TIME SERIES The followig developmet cocetrates o the scalar case+ Oly mior modificatios are required to exted the results to the vector case ad matrix HAC estimatio+ Accordigly, let u t be a weakly depedet time series satisfyig ` u t C~L!«t j0 c j «tj, ` j a 6c j 6 `, C~! 0, a 3, L) j0 where «t iid~0, s! ad E~6«t 6 v! `, for some v + The time series u t is statioary with variace s u ` j0 c j s, autocovariace fuctio g u ~h! E~u t u th! s ` j0 c j c jh, fiite vth absolute momet E6u t 6 ~ ` j0 6c j 6! v E6«t 6 v `, spectrum f u ~l! ~s 0p!6C~e il!6, ad log-ru variace v pf u ~0! s C~! + The summability coditio i L esures that ` h` h 3 6g u ~h!6 `, ) which is helpful i some techical derivatios that follow ad meas that f u ~l! has cotiuous secod derivative f ~! u ~l! ~s ` 0p! h` h g u ~h!e ilh + Allowace for heterogeeity i «t ad u t ca be made i the usual way with mior modificatios to L ~cf+ Phillips ad Solo, 99! without affectig the procedures or the properties discussed subsequetly i a essetial way+ t Uder L, partial sums S t i u i satisfy the fuctioal law ~e+g+, Phillips ad Solo, 99! {{} u i B ~{! : S {{} M i M B~{!, ) where {a} sigifies the iteger part of a, is weak covergece, ad B~{! is Browia motio with variace v + Let $w k % ` k be a complete orthoormal system i Later, we will work with the explicit sequece ~8!, but it is sufficiet to assume that the fuctios w k are twice cotiuously differetiable + We propose a regres-

5 [ Z sio of u t o a collectio of determiistic regressors $w k ~t0!% k formed by takig the first members of this orthoormal sequece evaluated over t,+++,+ Write this regressio i the form u t k b k w k t e t : Z b ' w t e t, t,+++,, 3) where w t ~w ~t0!,+++,w ~t0!! ' + I observatio format ~3! ca be writte as u F bz e, with bz ~F ' F! F ' u+ Let P F ~F ' F! ' F ad costruct the estimate v u' P u u' F F ' F F ' u+ 4) M M As show i Lemma A i Sectio 6, ' t w t w t I O~0! ad ~ ' t w t w t! I O~0!+ Stadard fuctioal limit argumets ad Wieer itegratio reveal that for fixed as r ` we have t 0 w t u t r d w ~r! db~r! : j d N~0, v 0 I!+ It follows immediately that HAC ESTIMATION BY AUTOMATED REGRESSION 9 u' P u r d j ' j d v x, 5) where x is chi-squared with degrees of freedom+ For fixed, the asymptotic mea ad variace of ~5! are E u' P u v o~!, Var u' P u v4 o~!, 6) as r `+ These results motivate the log-ru variace estimator v[ u' P u u ' P u P, where u P P u, 7) which, i view of ~6!, ca be expected to be cosistet for v whe r ` as r `+ The estimate v[ is simply the sample variace of u P, the data projected oto the space spaed by the regressors F + Thus, v[ is that part of the sample variace of u t explaied by the regressio of u t oto a determiistic tred basis+ This explaied sum of squares may be regarded as aother way of thikig about a log-ru variace the cotributio to the variatio of u t that comes from log-ru ~or tredlike! behavior i the series+ Thus, there would seem to be a strog heuristic motivatio for cosiderig estimates like ~7!+

6 [ 0 PETER C.B. PHILLIPS Note that v[ is a oegative defiite quadratic form i the data, whose matrix is the projectio P + Thus, v[ belogs to the geeral class of quadratic estimators of v + Geeral quadratic estimators were cosidered i the early spectral aalysis literature but have received little subsequet attetio relative to kerel estimates+ Oe reaso is that, for every such quadratic estimator, oe ca fid a correspodig lag kerel estimator with smaller mea squared error ~Greader ad Roseblatt, 957, p+ 9!+ Iterestigly, as we will show later, v turs out itself to be asymptotically equivalet to a lag kerel estimator ad to have ice asymptotic properties aalogous to those of quadratic kerel estimates+ Thus, there is o eed to adapt v[ ito kerel form, ad, of course, v[ is positive by costructio+ To develop a cosistet estimatio techique, we eed to allow for the umber of regressors to pass to ifiity with the sample size i such a way that the regressio ~3! remais feasible+ Accordigly, we impose the followig rate coditio o : r 0, R) which requires to go to ifiity faster tha M but more slowly tha + To establish a cetral limit theorem for v[, we eed the further coditio o~ 405!, which cotrols the expasio rate so that there is o bias i the limit+ For a explicit limit theory, icludig a explicit expressio for the limitig bias of v[, it is coveiet to use the orthoormal sequece w k ~r! M sik pr, k,,++++ 8) The fuctios ~8! are the eigevectors of the covariace kerel of Browia motio ~cf+, Phillips, 998! ad form a orthoormal system for Of course, other orthoormal sequeces ca be used+ However, it turs out that the asymptotic results give subsequetly are ivariat to the choice of the orthoormal sequece withi the class of trigoometric polyomials because the estimates are asymptotically equivalet to the same lag kerel estimator+ This poit is discussed i Sectio 4+ Uder this setup, we give formulas for the limitig bias, variace, ad mea squared error ad a limit distributio theory for v[ + The details of the proofs are differet from those of the covetioal literature o HAC estimatio ad are of some idepedet iterest, so they are provided here+ But the fial results ed up beig qualitatively similar, as the followig result shows+ THEOREM + Uder coditios L ad R i) lim r`~0! E~ v[ v! ~p 06! h` h g u ~h! : D; ii) If o~ 405!, the M~ v[ v! N~0,v 4!; iii) If r, the lim r`~0! 4 E~ v[ v! D v 4. `

7 HAC ESTIMATION BY AUTOMATED REGRESSION Part ~i! shows that v[ has bias of order 0 of the form E~ v[! v ` D@ o~!#, where D p 6 h g u ~h! : p h` 6 v ~! + From ~ii!, the variace of v[ is of O~!+ So, icreases i the umber of regressors icrease bias ad reduce variace+ The situatio is aalogous to badwidth choice i kerel estimatio+ The mea squared error of v[ has the form MSE~ v[! Bias Var 4 D v4 4 + Optimizatio of this quatity with respect to leads to the first-order coditio ~ !D ~v 4 0! 0, which gives the followig formula for the optimal value of : v ) 4D Of course, this is aalogous to covetioal mea squared error optimizatio formulas for badwidth choice i kerel estimatio ~e+g+, Greader ad Roseblatt, 957!+ Formula ~9! ca be used to implemet a data-determied choice of i a covetioal way+ Oe approach is to use oparametric estimates of D ad v 4 i ~9! as, for example, i Newey ad West ~994!+ The most commo ad coveiet method i practice is a simple plug-i estimator based o the use of a parametric model for prelimiary estimatio of v 4 ad D i ~9!+ I the case of a first-order autoregressio with fitted coefficiet a[ ad error variace s, the stadard formulas give v[ s 0~ a! [ ad DZ ~p 06! as [ 0~ a! [ 4 + Some modificatios to these formulas may be desirable i cases where a[ is close to uity+ I the cotext of prewhiteig, for example, Adrews ad Moaha ~99! propose a 0+97 rule i which a[ is replaced by 0+97 wheever a[ exceeds this value+ It is kow that this particular rule seriously iterferes with power i some cases, especially statioarity testig ~cf+ Lee, 996!+ A alterative boudary restrictio that seems to improve the size ad power properties of procedures based o HAC estimates is the sample-size-depedet rule give i Sul, Phillips, ad Choi ~003!, where a[ is replaced by 0M wheever it exceeds that value+ 3. ASYMPTOTIC FORM AS A ERNEL ESTIMATOR Lemma B~iii! i Sectio 6 shows that the orthogoal sequece w k ~r! i ~8! satisfies the followig summatio formula: p~t s! p~t s! si si w k k w t k s si p~t s! si p~t s! + 0)

8 [ PETER C.B. PHILLIPS Usig this formula ad ~7!, ~3!, ad ~36! from the proof of part ~ii! of Theorem, we fid that the HAC estimate [ v has the followig asymptotic form: v where h si ph si O p M Mlog h k h ph t, th k h g[ u ~h! O p M si ph si ph cos u t u th Mlog, ) ph ph si si ph ) may be regarded as a lag kerel fuctio ad g[ u ~h! ~0!t, th u t u th is the sample autocovariace+ The domiat term i ~! has the usual form of a kerel estimate of v ad is depedet o, which serves the role of a smoothig parameter+ Thus, v[ behaves asymptotically like a kerel estimate+ Let M+ For h0 small we ca write the lag kerel k ~h0! as a fuctio of h0m i approximate form as follows: si ph si ph M M k M h M si ph M ; ph M with which we may associate the fuctio si px k~x! px, which is the lag kerel for the Daiell estimate ~e+g+, Priestley, 98, p+ 44!+ This lag kerel is a smoothed periodogram estimate with a rectagular spectral widow+ Here, p0m is the width of the frequecy bad over which the periodogram is beig smoothed+ The regressio-based estimate v[ is therefore closely related to this well-kow kerel estimate of v ad has the same bias, variace, ad mea squared error as the Daiell estimate+ Evidetly, therefore, the asymptotic mea squared error of v[ is domiated by that of the Bartlett Priestley quadratic spectral widow ~Priestley, 98; Adrews, 99!+,

9 Z Z 4. THE EFFECT OF DIFFERENT TREND BASES We may choose to use other sequeces of orthogoal fuctios i the regressio ~3!, ad it is iterestig to explore the effects of such alterate choices o the asymptotic kerel form of estimate v[ + I the followig discussio, we cofie our attetio to sequeces of trigoometric polyomials+ Parts ~i! ad ~ii! of Lemma C i Sectio 6 give the followig summatio formulas for the orthoormal sie ad cosie trigoometric polyomials w k ~r! M si$kpr% ad w k ~r! M cos$kpr%, respectively: si s! p~t w k k w t k si s! p~t s 7 + p~t s! p~t s! si si 3) As i the calculatio leadig to ~!, the secod compoet of ~3! turs out to be of smaller order as, r `, ad ~! holds with k si ph h si ph si ph cos ph cos ph si ph si ph O si ph, which is asymptotically equivalet to the lag kerel ~!+ Thus, these differet orthoormal trigoometric sequeces all lead to HAC estimates that are asymptotically equivalet+ If the complex orthoormal sequece w k ~r! e pikr is used, the the regressio coefficiets i ~3! are themselves complex ad have the form b ~F * F! ~F * u!, where F is the matrix of observatios of the regressors $w k ~t0! : k,+++,% ad * sigifies complex cojugatio ad traspositio+ I place of ~3! we ca write u t b k e ~pikt0! e t, 4) k HAC ESTIMATION BY AUTOMATED REGRESSION 3

10 [ Z 4 PETER C.B. PHILLIPS which may be regarded as a fitted empirical versio of the Cramér represetatio of the statioary process u t, which was oticed earlier i Phillips ~996, Remark 5+a!+ Note that l k pk0 r 0 for all k,+++, because 0 r 0+ Thus, the regressio ~4! focuses attetio o the zero frequecy ~or log-ru! compoet of the Cramér represetatio of u t + The matrix F satisfies F * F I ad is a scaled uitary matrix+ So the kth elemet of b is simply the ~stadardized! discrete Fourier trasform, ~0! t e pikt0 u t, of u t, which is well kow to satisfy a cetral limit theorem ~e+g+, Haa, 970, p+ 4! upo rescalig+ Additioally, as show i Phillips ~999, Theorem 3+!, the ~asymptotically ifiite! collectio of such elemets have the followig limit as r ` ad are asymptotically idepedet provided r ` but ot too fast relative to + I particular z k e pikt0 u t e M t 0 pikr db~r!, k,,++++ As is easily see, the limit variates z k * 0 e pikr db~r! are idepedet complex Gaussia N c ~0, v!+ Lettig P F ~F * F! F *, we the have v[ u' P u bz * bz 6z k 6 r p E~6z k 6! v + 5) k As such, the estimate v[ may be iterpreted as the sample variace of the empirical estimates ~obtaied from the fitted regressio ~4!! of the orthogoal process ~at the zero frequecy! that appears i the Cramér represetatio of u t + Observe that 6z k 6 is pi u ~l k!, where I u ~l! is the periodogram of u t ad l k pk0 are the fudametal frequecies+ Thus we ca write v p I u ~l k!, 6) k which correspods to a smoothed periodogram estimate of the spectrum at the zero frequecy ~give that 0 r 0!+ The spectral widow i the estimate v[ is clearly rectagular, just as that of the Daiell widow ~e+g+, Haa, 970, p+ 79!+ Agai the results are asymptotically equivalet to those of the estimate ~7! based o the siusoidal sequece ~8!+ Of course, this setup easily permits the use of other spectral widows+ Let W diag$w~l!,+++,w~l!% be a diagoal matrix prescribig a particular weightig sequece based o the widow fuctio W~l!+ The the HAC estimate v[ W bz * W bz p W~l!I u ~l k! k has the usual form of a smoothed periodogram estimate with spectral widow W~l!+ I this way, the preset approach accommodates all covetioal kerelbased HAC estimates+

11 HAC ESTIMATION BY AUTOMATED REGRESSION 5 5. DISCUSSION The estimate v[ is straightforward to compute, beig oe of the outputs of a liear regressio, ad it has a simple heuristic motivatio+ The fact that regressio o tred produces this cosistet estimate idicates that tred coefficiets carry iformatio about the log-ru features of the data, eve though the true tred coefficiets i this regressio of a statioary time series are zero+ I fact, as discussed i the precedig sectio, the estimate v[ may be iterpreted as the sample variace of the coefficiets i a fitted regressio that is a empirical versio of the log-ru part of the Cramér represetatio of a statioary process+ Simulatios ~ot reported here! idicate that v[ performs as well as curret idustry-stadard methods usig quadratic kerels, automated badwidth selectio, ad prewhiteig ~Adrews, 99; Adrews ad Moaha, 99; De Haa ad Levi, 997; Lee ad Phillips, 994; Newey ad West, 994!+ This may be expected give the asymptotic relatioship betwee v[ ad the Daiell kerel estimate+ The regressio approach developed here may be exteded to estimate the spectrum of u t at poits other tha the origi, although we do ot pursue that possibility here+ 6. ADDITIONAL LEMMAS AND PROOFS LEMMA A+ Uder R, ' t w t w t ' w t w t! I O~0!, asr`+ I O~0!, ad ~ t Proof+ We first provide a direct calculatio whe the elemets of w t are give by the trigoometric fuctios ~8!+ I this case, the diagoal elemets of ' t w t w t are cos~k! pt t w k t t t si k pt t pt cos ~k! Re Re e i~k!~p0! e i~k!p 4 Re e i~k!~p0! 4 e i~k!~p0! e i~k!~p0! 4 Re cos~k! p cos~k! p, i~k!~pt0! e t i~k!~p0! e 6e i~k!~p0! 6

12 6 PETER C.B. PHILLIPS ad the off-diagoals are t w k t w t t t sik pt si pt cos~k! pt cos~k! pt Re e i~k!~pt0! t Re e i~k!~p0! e i~k!p e i~k!~p0! i~k!~pt0! e t e i~k!~p0! e i~k!p e i~k!~p0! k odd, k eve k eve, k odd 0 k, k both odd or eve+ It follows that t ' w t w t I O~0! ad ~ ' t w t w t! I O~0!+ I the geeral case, if w k ~s! is twice cotiuously differetiable by Euler summatio we have t w k t w k t dt w k w k t w k ' w k t dt 0 w k ~s! ds O O, because w k ~t0! ad w ' k ~t0! are uiformly bouded Similarly, t w k ~t0!w ~t0! O~0! uiformly for k +

13 HAC ESTIMATION BY AUTOMATED REGRESSION 7 LEMMA B+ For w k ~r! M si$~k _!pr%, we have i) Dw k ~t0! M cos$~k _!~p~t _ 0!%si$~k _!~p0!%; ii) k w k ~t0! M@si ~pt0!0si~pt0!#; iii) k w k ~t0!w k ~s0! s!0%0si$ _ ~p~t s!0!%# s!0%0si$ _ ~p~t s!0!%#; iv) ~s 4 C~! 4 0! $ph0%0si $ _ ~ph0!%# s 4 C~! o~!#, where D is the differecig operator Dw k ~t0! w k ~t0! w k ~t 0!. Proof of Lemma B+ For part ~i! Dw k t Msik pt sik Mcosk p t k For part ~ii! w k t M k MIm sik pt i~k0!~pt0! e k MIme e i~pt0! i~pt0! e i~pt0! si pt MImei~pt0! si pt pt si M si pt + sik p~t! + p

14 8 PETER C.B. PHILLIPS k For part ~iii! w k t w k s k k Re sik pt sik ps cosk p~t s! k i~k0!~p~ts!0! e i~k0!~p~ts!0! # p~t s! Ree e ~p~ts!0!i ~0!~p~ts!0!i e ~p~ts!0!i e e ~p~ts!0!i ~0!~p~ts!0!i e ~p~ts!0!i i ~e ~p~ts!0!i e ~p~ts!0!i! Ree ~p~ts!0!i i ~e ~0!~p~ts!0!i e ~0!~p~ts!0!i! i ~e ~p~ts!0!i e ~p~ts!0!i!! e ~p~ts!0!i i ~e ~p~ts!0!i e ~0!~p~ts!0!i si p~t s! Ree ~p~ts!0!i si p~t s! si p~t s! e ~p~ts!0!i si p~t s! cos p~t s! si p~t s! si p~t s! cos p~t s! si p~t s! si p~t s! p~t s! p~t s! si si p~t s! si si p~t s! +

15 For part ~iv!, we use Fejér s itegral, givig ph s 4 C~! 4 si h si ph s 4 C~! 4 HAC ESTIMATION BY AUTOMATED REGRESSION 9 s 4 C~! 4 0 pr si si dr@ o~!# o~!# o~!#, as required+ LEMMA C+ i) If w k ~r! M si$kpr%, k w k t w k s ii) If w k ~r! M cos$kpr%, k w k t w k s si p~t s! p~t s! si si s! p~t ; p~t s! si si p~t s! p~t s! si si s! p~t + p~t s! si

16 30 PETER C.B. PHILLIPS k Proof of Lemma C+ For part ~i! w k t w k s k p~t cosk k sik pt sik ps s! cosk p~t s! Re k Ree e ~p~ts!0!i ~p~ts!0!i e ~p~ts!0!i e e ~p~ts!0!i ~p~ts!0!i e ~p~ts!0!i Re e ~p~ts!0!i e e ~p~ts!0!i ~p~ts!0!i ~p~ts!0!i ik~p~ts!0! e ik~p~ts!0! # Re~e ~p~ts!0!i!~ e ~p~ts!0!i! p~t s! cos ~e ~p~ts!0!i!~ e ~p~ts!0!i! p~t s! cos p~t s! p~t s! ~!p~t s! cos cos cos p~t s! cos p~t s! p~t s! ~!p~t s! cos cos cos p~t s! cos si si s! p~t p~t s! p~t s! cos si si s! p~t p~t s! p~t s! cos

17 si si s! p~t p~t s! p~t s! cos si si s! p~t p~t s! p~t s! cos si p~t si s! p~t s! si p~t s! si p~t s! HAC ESTIMATION BY AUTOMATED REGRESSION 3 si p~t s! si p~t s! si s! p~t p~t s! si si s! p~t + p~t s! si For part ~ii! k w k t w k s k p~t cosk k cosk pt cosk ps s! cosk p~t s!, which differs from part ~i! oly i the sig of the secod term+ The stated result therefore follows by the same calculatios+

18 3 PETER C.B. PHILLIPS E~ [ Proof of Theorem + Part i). I view of Lemma A v! tr F ' F tr h t, th k h h t, th t, th ad, because v ` h` g u ~h!, E~ v[ v! h k ` 6h6 g u ~h! h k t, th t, th ' w t w th ' w t w th g u ~h! g u ~h! O w k w t k t h g u ~h! O, w k k t w t h u~h! g O w k k t w t h u~h! g O o a, 7) for a 3 because ` ` g u ~h! 6h6 6h6 ` 6g u ~h!6 a 6h6 6h6 a 6g u ~h!6 o a, by coditio L+ Also for ay positive iteger L we have h k L hl k L 6h6 t, th t, th k w k t w k t h u~h! g O u~h! g O w k k t w t h w k t w k t h t, th g u ~h! O, 8)

19 ad because the elemets w k ~t0! are bouded uiformly i t we have L 6h6 k t, th w k k t w t h g u~h! C 6g u ~h!6 C 3 L 6h6 L h 3 6g u ~h!6 L o 6h6 L3 Now choose L such that HAC ESTIMATION BY AUTOMATED REGRESSION 33 + L 30 L o~! 9) as r `+ The, 0L 3 o~ 0! ad L 6h6 k t, th For the first term of ~8! we ote that t, th w k w t k t h w t, th k t t, th w k k t w t h g u~h! o w k w t k t h w k t + 0) w t, th k w t k t h w k t O, ) uiformly i 6h6 L + It therefore follows from ~7! ~! that E~ v[ v! L hl k o t, th w k k t w t h k w t g u~h! O + )

20 34 PETER C.B. PHILLIPS Next cosider t, th w k w t k t h w k t sik t, th pt sik p~t h! sik pt t, thcosk ph cos~k! t, th cos~k! pt 6h6 cosk ph t, thcos~k! pt h pt h cos~k! pt + 3) Takig the first term of ~3!, averagig over k, ad usig the fact that 6h6 L ad L satisfies ~9! so that L 0 o~!, we get cosk ph k k k ph o h p h o~!# + 4)

21 For h 0 we ca write the secod term of ~3! as pt h pt cos~k!! t, thcos~k 0 dr O cos$~k!pr% dr O h0 cos~k!pr h 0 h0 h h0!pr si~k ~k!p 0 si$~k!pr% h0 ~k!p 0!p ~k!psi~k h O O si~k!p h!p ~k!psi~k h O si~k!p O!p ~k!psi~k h si~k!p h O!p ~k!psi~k h si~k!p O!p ~k!pcos~k 4 3h HAC ESTIMATION BY AUTOMATED REGRESSION 35 si~k!p h 4 O!p ~k!pcos~k h 4 4 si~k!p h 4 O cos ~k!p 3h 4 h 4 4 h O, 5)

22 36 PETER C.B. PHILLIPS for 6h6 L ad L satisfyig ~9!+ Averagig over k we fid k cos~k!p 4 3h h 4 h 4 k cosp ~k! 3h 4 h 4 h 4 cos~k! 3h k 4 h 4 h 4 3h ~k! p k 4 4 p 9h 6 h 4 o h 3 3 h 4 h 4 o h o L 6) uiformly i 0 h L + Without goig through the calculatio, the same rate result holds whe L h 0+ We deduce from ~3! ~6! that k Thus, t, th w k t w k t h 6h6 p h E~ v[ v! L hl k o h 6 t, th o O L p 6 hl p ` 6 h` w k t + w k k t w t h k w t g u~h! 6h6 h g u ~h! o O h g u ~h!@ o~!# because 0 o~ 0! for satisfyig R+ Thus lim r` as stated+ E~ v[ v! p ` 6 h` h g u ~h!,

23 [ I I Part ii). From Lemma A F ' u v u' P u u' F F ' F M M u' F F ' u O M M w k w t k u s t u s O k k t, s h HAC ESTIMATION BY AUTOMATED REGRESSION 37 t, th w k t w k t h u t u th O + 7) Usig the device i Phillips ad Solo ~99! we have the decompositio u t C~!«t «I t «I t, for «I t ` j0 c j «tj, ` c j c s, 8) j where ` j0 6cI j 6 ` uder L+ The, ~8!, partial summatio, ad Lemma B~i! yield w M t k u t t C~! M C~! M C~! M 30 t w t k «t t C~! M w t k t «t w t k «t t w k ~! k p C~! M C~! M w t k D t «I t M w k~! «I «I M «I t M cosk p t sik p k p M t w t k «t t O pmo p k w t k «t t O p M Dw k t t «I

24 [ [ 38 PETER C.B. PHILLIPS uiformly i k + Thus, v k M t C~! k M t C~! k M t O p w k t t u O w k t «t O p O w k t t «O M t k w k t «t O p + 9) From Lemma B~ii!, k w k ~t0! M@si ~pt0!0si~pt0!#, ad so pt Var M w t k k t t si 4 «s t pt si pt s si t pt si O pr si 0 pr si dr by Lemma B~iv!+ Hece, MM t k O~!, w k t «t O p ~!, 30) ad it follows from ~9! ad ~30! that v C~! k M t w k t «t O O p M + 3)

25 Thus, M~ v[ v! C~! M Write C~! M k C~! M M t k O p k M t w k t «t s ) w k t «t s w t k t I view of Lemma A, we have C~! M k t C~! M C~! M k k O p M M w k t «t s t t «t s w ts k k t w s «t «s + w k t ~«t s! w t k t s w k t ~«t s! O M 33) M + 34) Thus, for o~ 405! we have from ~3! ~34! M~ v[ v! HAC ESTIMATION BY AUTOMATED REGRESSION 39 C~! M k ts w k t w k s «t «s o p ~! C~! M C~! M ts w k t w k s «t «s o p ~! ts k p~t s! p~t s! si si si p~t s! si p~t s! t «s o p~! «

26 40 PETER C.B. PHILLIPS C~! MM h C~! M si ph si ph p~t s! si ts si M t, th «t «th p~t s! «t «s o p~!+ 35) We show that the secod term i the fial expressio ~35! is egligible+ Note that p~t s! C~! si M ts si p~t s! «t «s has mea zero ad variace p~t s! si s 4 t t s Os 4 Os 4 si p~t s! r0 Os 4 0 p Os 4 0 O log + si $p~r p!% si dpdr p~r p! r0 p si dpdr p~r p! cos p r~0! ~r p! si p dr ~r p! 0 cos p r si p r p cos p r si p r dr

27 Hece C~! M ad so p~t s! si ts si p~t s! M~ v[ v! C~! MM h «t «s O p log o p ~!, 36) si ph si ph Next ote from Lemma B~iv! that ph s 4 C~! 4 si h si ph s o~!# + M «t «t, th th o p~!+ Fially, usig a martigale cetral limit argumet alog the same lies as that i Phillips, Su, ad Ji ~003!, we may establish that C~! MM h si si ph [ N~0,v 4!, HAC ESTIMATION BY AUTOMATED REGRESSION 4 ph M «t «t, th th r d N~0,s 4 C~! 4! givig the stated cetral limit theorem for M~ v[ v!+ Part iii). Part ~i! shows that the bias of v[ is give by E~ v[ v! ` D@ o~!#, where D p 6 h` whereas argumets as i part ~ii! show the variace of v[ to be h g u ~h!, Var $ v[ % o~!# +

28 4 PETER C.B. PHILLIPS It follows that MSE~ v[! E~ v[ v! ad, if r, we get lim r` as stated+ 4 REFERENCES MSE~ v[! lim r` 4 D o~!#, E~ v[ v! D v 4, Adrews, D+W++ ~99! Heteroskedasticity ad autocorrelatio cosistet covariace matrix estimatio+ Ecoometrica 59, Adrews, D+W++ &J+C+ Moaha ~99! A improved heteroskedasticity ad autocorrelatio cosistet covariace matrix estimator+ Ecoometrica 60, De Haa, W+J+ &A+ Levi ~997! A practitioer s guide to robust covariace matrix estimatio+ I G+S+ Maddala & C+R+ Rao ~eds+!, Hadbook of Statistics, vol+ 5, pp Elsevier+ Greader, U+ &M+ Roseblatt ~957! Statistical Aalysis of Statioary Time Series. Wiley+ Haa, E+J+ ~970! Multiple Time Series+ Wiley+ Lee, J+S+ ~996! O the power of statioary tests usig optimal badwidth estimates+ Ecoomics Letters 5, Lee, C+C+ &P+C+B+ Phillips ~994! A ARMA Prewhiteed Log-Ru Variace Estimator+ Mauscript, Yale Uiversity+ Newey, W++ &+D+ West ~987! A simple positive semi-defiite heteroskedasticity ad autocorrelatio cosistet covariace matrix+ Ecoometrica 55, Newey, W++ &+D+ West ~994! Automatic lag selectio i covariace matrix estimatio+ Review of Ecoomic Studies 6, Phillips, P+C+B+ ~996! Spurious Regressio Umasked+ Cowles Foudatio Discussio paper 35, Yale Uiversity+ Phillips, P+C+B+ ~998! New tools for uderstadig spurious regressios+ Ecoometrica 66, Phillips, P+C+B+ ~999! Uit Root Log Periodogram Regressio+ Cowles Foudatio paper 44, Yale Uiversity+ Phillips, P+C+B+ ~004! Challeges of Tredig Time Series Ecoometrics+ Mimeo, Yale Uiversity+ Phillips, P+C+B+ &V+ Solo ~99! Asymptotics for liear processes+ Aals of Statistics 0, Phillips, P+C+B+, Y+ Su, &S+ Ji ~003! Cosistet HAC Estimatio ad Robust Regressio Testig Usig Sharp Origi erels with No Trucatio+ Cowles Foudatio Discussio paper 407, Yale Uiversity+ Priestley, M+B+ ~98! Spectral Aalysis ad Time Series, vol+ I+ Academic Press+ Sul, D+, P+C+B+ Phillips, &C+-Y+ Choi ~003! Prewhiteig Bias i HAC Estimatio+ Mimeo, Yale Uiversity+