Bandwidth Choice for Bias Estimators in Dynamic Nonlinear Panel Models

Bandwidth Choicefor Bias Estimators inDynamic Nonlinear Panel ModelsJinyong Hahn Guido KuersteinerUCLA UC DavisPreliminary and incompleteJanuary,2007Acknowledgment:The…rst author gratefully acknowledges…nancial support from NSF Grant SES-0313651. The second author gratefully acknowledges Financial support from NSF Grant SES0095132.AbstractThis paper considers bandwidth selection for spectral density estimators based on panel data sets.The spectral densities of greatest interest in this paper are the ones that appear in the bias expression for…xed e¤ects estimators in nonlinear dynamic panel models obtained by Hahn and Kuersteiner.The bias estimation problem is di¤erent from the usual HAC estimation problem because the need for positive de…niteness does not arise.As a consequence,the usual justi…cation for kernel smoothing of spectral estimators does not apply to this case.However,without kernel smoothing the bandwidth selection problem is signi…cantly more di¢cult because in this case not only the usual proportionality constants are data-dependent but also the optimal rate at which the bandwidth parameter grows with sample size.In this paper an in…nite order VAR model is used to obtain an estimate of the approximate mean squared error of the spectral estimator.It is shown that selecting the bandwidth parameter based on the estimated mean squared error criterion is asymptotically equivalent to the optimal infeasible bandwidth choice.Monte Carlo simulations show that truncated spectral estimates signi…cantly outperform kernel weighted estimates in terms of their e¤ectiveness in reducing bias in the panel application.1IntroductionHahn and Kuersteiner(2004)analyze the bias properties of general nonlinear panel models with…xed e¤ects. Maximum likelihood estimators of these models su¤er from what is know as the incidental parameter problem especially in short panel data sets with a limited amount of time series observations.One typically…nds that conventional cross-sectional asymptotic approximations where n tends to in…nity while T is…xed,lead to inconsistent parameter estimators.They adopt an alternative asymptotic approximation where n and T tend to in…nity at the same rate.Under these alternative asymptotics it can be shown that the limiting distribution of the maximum likelihood estimator is Gaussian but is not centered at zero.The non-centrality parameter is viewed as an approximation to the…nite sample bias and show that it is a function of the spectral density at zero frequency of a certain non-linear transformation of the data.Estimators of the bias can be used to bias correct the ML estimator but practical implementation of this procedure is complicated by the need to select a bandwidth parameter to estimate the spectrum at zero frequency.This paper discusses estimators of the bias term.The approximate mean squared error(MSE)of the bias estimator is derived and used to select the bandwidth in an automated way.Since the higher order MSE of the bias corrected maximum likelihood estimator depends on the bandwidth in the same way as the higher order MSE of the bias estimator itself,minimizing the MSE criterion used in this paper is also minimizing the higher order MSE of the bias corrected maximum likelihood estimator.The procedure thus is optimal not only in terms of an intermediate criterion but also in terms of the properties of the…nal stage estimator of interest.As is well known from the literature of heteroskedasticity and autocorrelation consistent(HAC)covariance matrix estimation,the MSE depends on nuisance parameters that are themselves functions of the temporal dependence structure of the data.An autoregressive approximations to the spectral density of suitable transformations of the observed data is proposed as an automatic way of estimating the nuisance parameters.In the literature on HAC estimation,such as Newey and West(1987,1994)and Andrews(1991)and Andrews and Monahan(1992)the use of kernel weighting functions for spectral density estimation at the zero frequency is motivated and justi…ed by the need to produce positive de…nite estimators of the spectral densities for all sample sizes.The bandwidth selection problem is then a relatively simple one,because the rate at which the bandwidth expands with the sample size is usually known and depends on the kernel used.In the case of bias correction there is no need to enforce positivity and thus the justi…cation of using kernel weighted spectral estimators is less clear.It is therefore proposed here to use a truncated estimate of the spectral density matrix at frequency zero.This has the advantage of faster rates of convergence of the higher order MSE for the bias estimator.However,using a truncated version of the spectral density matrix comes at the cost of a more di¢cult bandwidth selection problem.To the best of my knowledge this selection problem has not been solved in the time series literature,let allone for nonlinear dynamic panel models.A higher order approximation to the MSE of a certain spectral matrix at frequency zero is used to obtain a criterion function that depends on the bandwidth.The bandwidth is then chosen optimally by minimizing this approximate MSE.In order to estimate the unknown higher order MSE expression,a parametric VAR(h)model is used where the parameter h is tending to in…nity with the sample size.This procedure is shown to produce bandwidth choices that are asymptotically of the same order as the infeasible optimal bandwidth obtained from minimizing the unknown approximate MSE.2Bias EstimatorsSuppose that we are given a panel data model with a common parameter of interest 0and individual speci…c …xed e¤ects i 0,i =1;:::;n .As in Hahn and Kuersteiner (2004)a maximization estimator is de…ned by (1)b ;b 1;:::;b n =argmax ; 1;:::; n n Xi =1T X t =1(x it ; ; i )for some criterion function ( )that does not depend on T .Assume that is a sensible function in the sensethat,if n is …xed,and T !1,the estimator b ;b 1;:::;b n is consistent for ( 0; 10;:::; n 0).For simplicity of notation,assume dim ( i )=1.Hahn and Kuersteiner (2004)show that when n /T ! ,where 0< <1thenpnT b 0 !N p ;I 1 (I 0)1where for U i (x it ; ; i )@ (x it ; ; i )@i 0@ (x it ; ; i )@ i;V i (x it ; ; i )@ (x it ; ; i )@ i;the spectral quantitiesf V UiP 1l = 1Cov V it ;U i it l;f V V i P 1l = 1Cov (V it ;V it l )are de…ned and'V Ulim n 1P ni =1E V i it 1f V U i ;'V V 12lim n 1P ni =1 E V i it 2E U i i it f V V i ; 'V U'V V ;I lim n 1P n i =1I i ; I 1 :The bias can be expressed aslim n !1 1n n X i =1I i ! 11n nX i =1264f V Ui E h @V i (x it ; ; i )@ i i E U i i i (x it ; ; i ) f V V i 2 E h @V i (x it ; ; i )@ii 2375:It is more convenient to express as a weighted average of elements of a spectral density matrix at frequency zero.For this purpose de…ne(2)k it =k (x it ; 0; i 0)="V i (x it ; 0; i 0)Ui i (x it ; 0; i 0)#;^k it =k x it ;^ ;^ iand (3)a i;1= E U i i i (x it ; 0; i 0)2 E h @V i (x it ; 0; i 0)@ii 2;a i;2=1E h @V i (x it ; 0; i 0)@ i iwhere a i;1is a d 1 1vector where d =dim +1and a i;2is a scalar.Then de…ne the 2(d 1) d matrix~A i =2666664a i;10d 1 1 0d 1 10a i;2......... (00)a i;23777775and let (4)A i = I 1[I d 1;I d 1]~Aiwhere I d 1is the d 1dimensional identity matrix.Now de…ne kk i;j =E k it k 0it j ;f kk i =P 1l = 1 kki;j andf kk =lim n !1n 1P ni =1A i f kk i :The bias can then be expressed asf kk BwhereB ="10d 1 1#:Estimation of is done by replacing population moments with sample moments.In this formulation,the estimators b Ii ;b E h @V i (x it ; ; i )@ ii ,b E U i i i(x it ; ; i )and b E h @V i (x it ; ; i )@ ii are simple sample averages over t that do not need any truncation parameters.They will be ignored in the following discussion which is mainly concernedwith estimates of the spectral densities b fV U iand b f V V i:These estimators require truncation parameters because their population counterparts depend on in…nitely many terms.The estimator for the bias is therefore de…ned as(5)^ m ^f kk mB where^f kk m =1n nX i =1A i Pj j j m ^ kk i;j :and^ kk i;j=1Tmin(T;T +j )Xt =max(1;j )^k it ^k 0it j ;^ kk i; j=^ kk i;j:The bandwidth parameter m is chosen by minimizing the approximate mean squared error of ^ m:For this pur-pose de…ne kit (x it ; ; i )=@k (x it ; ; i )=@ with k 0;it =k it (x it ; 0; i 0)and ;v it = (E [V i ]) 1@ (x it ; 0; i 0)@:Then,it is shown in the Appendix that the largest order terms of ^ mdepending on m have an approximate MSE given by T 0(m )+mT (T 1+T 2) B 2whereT 0(m )=1nP n i =1A i P j j j >m kk i;j ;(6)T 1=1n P n i =1A i E k 0;it P 1u = 1E [ ;v i 1k i 1 u ]0 + P 1u = 1E [ ;vi 1k i 1 u ]E k 0;it0 and (7)T 2=1n P ni =1A i P 1u = 1E ;v i 1 ;v i 1 u E k 0;it E k 0;it0 De…ne m as the choice of m that minizes the approximate mean squared errorQ (m )= T 0(m )+2m T(T 1+T 2) B 2of ~ m;such that m solves m =arg min m 2f 0;1;:::; m gQ (m )where m is a prespeci…ed upper limit that does not depend on the data.3Nuisance Parameter EstimationIn order to obtain a feasible bandwidth parameter^m the criterion function Q(m)which depends on unknown nuisance parameters needs to be replaced with an empirical counterpart^Q(m):This can be easily achieved for the components T1and T2which can be replaced by the sample averages^T 1=1n P n i=1^A iT 1P T t=1^k it P m u= m T 1P min(T;T+u)t=max(1;u+1)^ ;vit^kit u 0(8)+ P m u= m T 1P min(T;T+u)t=max(1;u+1)^ ;vit^kit u T 1P T t=1^k it 0^T 2=1n P n i=1^A i T 1P T t=1^k it T 1P T t=1^k it 0(9)P m u= m T 1P min(T;T+u)t=max(1;u+1)h^ ;v it^ ;v it ioLemma1Assume that Conditions1-7hold and m=T!0as m;T!1:Let^T1and^T2be de…ned in8and9.Then^T1 T1=O p T 1=2^T2 T2=O p T 1=2 :Proof.The result follows from Lemmas3-19.For^T1and^T2the same arguments can be applied by notingthat the function ;vi1has the same properties as k it:The term T0(m)poses a more di¢cult estimation problem.Because in many panel data sets of interest the time dimension is too short to estimate the function T0(m)directly we propose to use a restriced VAR.In other words we approximate the spectral density matrix of k it with a…nite order vector autoregression of order h,or VAR(h).The approximate model with VAR coe¢cient matrices i1;h;::: ih;h is given by(10)^k it=^ ik;h+^ i1;h^k it 1+:::+^ ih;h^k it h+^v it;h:Let^U it;h= ^k0it;:::;^k0it h+1 0and de…ne^ i1;h=(T h) 1P T 1t=h^U it;h^k0it+1and^ i;h=(T h) 1P T 1t=h^U it;h^U0it;h: The estimated error covariance matrix is^ vi;h=(T h) 1P T t=h+1^v it;h^v0it;h where^v it;h=^k it ^ i1;h^k it 1 ::: ^ ih;h^k it h with coe¢cients^ i(h)0= ^ 0i1;h;:::;^ 0ih;h =^ 0i1;h^ 1i;h:The population analoge of^ i;h is the in…nite dimensional matrix i=( 0i1; 0i2;:::)0de…ned as 0i= 0i1;1 1i;1 where 0i1;1= kk i1;:::; kk ij;::: and i;1is an in…nite dimensional matrix with(m,n)th block element equal tokki(n m):If U it;1=(k it;k it 1;::::)then de…ne vi;1=Var(k it 0i U it;1):In order to obtain an approximationto the autocovariance function kk i;j de…ne matricesH i(h)=2666664i1;h i2;h ih;hI d0 0......I d03777775;H i(1)=2666664i1 i2 ihI0 00I0.........3777775where H i(h)is of dimension dh dh and H i(1)is in…nite dimensional.The autocovariance function kk i;j has the representation(11) kk i;j=1P l=0E01H l+j i(1)E1 vi;1E01H l i(1)E1where E1=(I d;0;:::)0:Also,let E h=(I d;0:::;0)0be a hd h matrix.Then an approximation to the autocovariance function kk ij is obtained as(12)^ i;h(j)=k maxP l=0E0h^H l+j i(h)E h^ vi;h E0h^H l i(h)E hwhere^H li(h)is de…ned in the same way as H li(h)except that the parameters ij;h have been replaced by^ ij;h:The estimator for the function T0(m)is based on the estimated autocovariance matrices implied by the VAR approximation and is formulated as^T 0(m)=1n P n i=1^A i k max P j=m+1^ i;h(j)!:The criterion function for selecting m is then formed as before^T(m)=^T0(m)+2mT ^T1+^T2 :Then an estimate of the approximate MSE of^ based on a VAR(h)approximation to the spectral density matrix of k it is B0^T(m)0^T(m)B:Theorem1Assume Conditions1,2,3,4,5,6and7hold.Also assume that for q (d+1)=2+2;some 0< <(100q+120) 1;assume that h;k max!1such that T =h!0;T =k max!0;h=o T1=5 and k max=o(T1=5 ).Then,for a sequence m0such that m0!1as T!1and uniformly in m for m<m0, P l=k max kk i;l =P1l=m kk i;l !0and P l=h k il k=P1l=m kk i;l !0as T!1it follows that^T(m) T(m)k T(m)k=o p(1):Corollary1For^Q(m)=B0^T(m)0^T(m)B it follows that ^Q(m) Q(m) =Q(m)o p(1): As is shown in Hannan and Deislter(1988,p.333)the result of Corollary1is enough to establish that ^m =m !p1where^m minimizes^Q(m):Hannan and Deistler also show that for models with exponentially decaying auto-covariance functions the optimal m does not depend on the constants T1and T2:It is therefore possible to specify two versions of the bandwidth selection rule.The…rst version of^m is de…ned as^m1such that ^m 1=arg minmB0^T(m)0^T(m)B:A modi…ed bandwidth selection rule2is de…ned as minimizing the following simpli…ed criterion function^m2=arg minmB0 ^T0(m)+m T1d 110d 0 ^T0(m)+m T1d 110d Bwhere1d 1is a d 1 1matrix consisting of all elements equal to one.Note that10d B=1such that the biascontribution to^ m that is increasing in m takes on the form mT1d 1;ie.all elements of are a¤ected in the same way.4ConclusionsThe problem of estimating the bias in a nonlinear dynamic panel model resulting form an incidental parameter is investigated.As was shown in earlier work in Hahn and Kuersteiner(2004),this bias depends on a spectral density matrix at frequency zero.It is argued that this spectral density matrix should be estimated by truncating lags,rather than kernel smoothing.A higher order analysis of the approximate MSE for the spectral density estimator leads to a criterion for selecting the bandwidth or lag truncation parameter.It is shown that a VAR approximation can be use to estimate the MSE criterion.The main result of this paper is to establish,that the truncation lag selected based on a feasible MSE criterion function asymptotically is equivalent to the truncation parameter selected using the infeasible criterion function.AppendixA Regularity ConditionsWe assume the following:Condition1For each >0,inf i h G(i)( 0; i0) sup f( ; ):j( ; ) ( 0; i0)j> g G(i)( ; )i>0,where b G(i)( ; i) T 1P T t=1(x it; ; i)and G(i)( ; i) E[(x it; ; i)].Condition2n;T!1such that nT! ,where0< <1.Condition3(i)For each i,f x it;t=1;2;:::g is a stationary mixing sequence;(ii)f x it;t=1;2;:::g are in-dependent across i;(iii)sup i j i(m)j Ca m for some a such that0<a<1and some C>0,where A i t (x it;x it 1;x it 2;:::),B i t (x it;x it+1;x it+2;:::),and i(m) sup t sup A2A i t;B2B i t+m j P(A\B) P(A)P(B)j. Condition4Let(x it; )be a function indexed by the parameter =( ; )2int ,where is a compact, convex subset of R p,p dim( )=d,and d 1=dim( ).Let =( 1;:::; k)be a vector of non-negative integers v i;j v j=P k j=1v j and D v(x it; )=@j j(x it; ) (@ v11:::@ k k).There exists a function M(x it)such that j D v(x it; 1) D v(x it; 2)j M(x it)k 1 2k for all 1; 22 and j v j 5.The function M(x it) satis…es sup 2 k D v(x it; )k M(x it)and sup i E h j M(x it)j10q+12+ i<1for some integer q p=2+2and for some >0.Condition5Let iT denote the smallest eigenvalue of iT=Var T 1=2P T t=1U i(x it; ; i) .We assume that inf i inf T iT>0.Condition6inf i j E[@V i(x it; 0; i0)/@ i]j>0.Condition7Let i1 ::: ik ::: id 1be the eigenvalues of I i in ascending order.Assume that (i)0<inf i i1 sup i id 1<1;(ii)lim n!1n 1P n i=1I i exists;(iii)letting I lim n!1n 1P n i=1I i,we assume that I is positive de…nite.Condition8Let the Wold representation of k it be given by k it=P1j=0B ij v it j where v it is a white noise sequence.Then B i(z)=P1j=0B ij z j has no roots for j z j 1where z2C. Condition9Uniformly in i it follows that P1j k j>m P1j j j>k kk i;j = P1j j j>m kk i;j !c1as m!1for some c12(0;1):Moreover,uniformly in m;1n P n i=1k A i k P1j=m kk i;j =k T0(m)k<1as n!1: Remark1The…rst part of the last condition seems to be a mild regularity condition in light of Condition3 and the fact that kk i;j ca j for some a 2(0;1)because of the mixing inequality.The secon part of the condition holds for any kk i that can be represented as an ARMA process.B Proofs and Auxiliary Results for MSE CalculationLet r 1=max(1;l )and r 2=min(T;T +l ):Also let r 1;j =max(1;l j )and r 2;j =min(T;T +l j ):The followingLemma establishes an approximation to ^fkk m=1nP n i =1^A i P j j j m^ kk i;j:Lemma 2Assume that Conditions 1-7hold and m=T !0as m;T !1:De…ne Let ^Sm =T 0(m )+T 4+T 8+T 14.Let ^mbe as de…ned in 5.Then ^ m=^S m B +o p m TandE ^S m B 2= T 0(m )+2m T (T 1+T 2) B 2+o m T2 :Proof.Follows from Lemmas 3-20.To prove Lemmas 3-20the following expansion for ^f kk mis obtained.Let K i;m =1TP ml = mP r 2t =r 1k it k 0it land de…nekit (x it ; ; i )=@k (x it ; ; i )=@ 0;kit (x it ; ; i )=@k (x it ; ; i )=@ ;k it (x it ; ; i )=@2k (x it ; ; i )=(@ )2;kit (x it ; ; i )=@2k (x it ; ; i )= @ @ 0 ;kit (x it ; ; i )=@ vec @k (x it ; ; i )=@ 0 =@ 0as well ask 0;it =kit (x it ; 0; i 0);k 0;it =k it (x it ; 0; i 0);k it =k it x it ;~ ;~ ik it =k it x it ;~ ;~ i k it=k it x it ;~ ;~ iwith ~ ;~ i such that ~ 0 b 0;k ~ i 0k k ^ i I 0k .Then use the multivariate mean value theorem to obtain the second order expansionb k it k it =k 0;it b 0 +k 0;it (b i i 0)+12^ 0 0 I k it ^ 0 +12k it (b i i 0)2+k it b 0 (b i i 0):We considervec^f kk f kk =J Xj =1T jwhere T 1=vec1n P n i =1A i E (K i;m ) f kk(13)T 2=vec 1n P ni =1A i (K im E (K i;m ))T 3=1n P n i =11T P m l = m P r 20t =r 1(I A i ) k it l k 0;it bT 4=1n P n i =1b i (0)T 3=2P ml = m P r 2t =r 1(I A i ) k it l k 0;it (14)T 5=1P n i =1b i (0)P ml = m P r 2t =r 1(I A i ) k it l k 0;it T 6=1n P n i =1b i (~")T 5=2P m l = m P r 2t =r 1(I A i ) k it l k 0;it T 7=1n P n i =11T P ml = m P r 2t =r 1(I A i ) k 0;it l k it bT 8=1n P n i =1(I A i )b i (0)TP ml = m P r 2t =r 1h k 0;it l k iti T 9=1n P n i =1(I A i )b i (0)T 2P ml = m P r 2t =r 1h k 0;it l k iti T 10=1n P n i =1(I A i )b i (~")T 5=2P m l = m P r 2t =r 1h k 0;it l k it i :T 11=1n P ni =11T P m l = mP r 2t =r 1(I A i )k 0;it lk0;itb b0 T 12=1n P n i =11T P ml = m P r 2t =r 1(I A i )k 0;it lk 0;it (bi i 0) b T 13=1n P n i =11T P m l = m P r 2t =r 1(I A i ) k 0;it l k 0;itb (b i i 0)T 14=1n P n i =11T P m l = m P r 2t =r 1(I A i )(bi i 0)2vec k 0;it k 00;it l (15)T 15=121n P n i =11T P m l = m P r 2t =r 1(I A i )(k it l I )vec ^ 0 0 I k it ^ 0T 16=1P n i =11P m l = m P r 2t =r 1(I A i ) k it l k it b 0 (b i i 0)T 17=1n P n i =1b i (0)T 3=2P m l = m P r 2t =r 1(I A i ) k it l k itb 0 T 18=121n P n i =1b i (0)TP ml = m P r 2t =r 1(I A i )(k it l k it )(b i i 0)T 19=1n P n i =1b i (~ )T 2P m l = m P r 2t =r 1(I A i ) k it l k itb 0 T 20=121n P n i =1b i (~ )T 2P m l = m P r 2t =r 1(I A i ) k it l k 0;it (b i i 0)T 21=121n P n i =11T P m l = m P r 2t =r 1(I A i ) ^ 0 0 I k it k it bT 22=121n P n i =11T P m l = m P r 2t =r 1(I A i ) k it k it b (b i i 0)T 23=1n P n i =1(I A i )b i (0)TP m l = m P r 2t =r 1 k it k itb 0 T 24=1n P n i =1(I A i )b i (0)T3=2P ml = m P r 2t =r 1(k it k it )(b i i 0)T 25=1n P n i =1(I A i )b i (~ )T 2P m l = m P r 2t =r 1 k 0;it l k itb 0 T 26=1n P n i =1(I A i )b i (~ )T 2P m l = m P r 2t =r 1 k 0;it l k it (b i i 0):For the last set of terms de…nek it ^ ;^ i =12 ^ 0 0 I k it ^ 0 +12k it (b i i 0)2+k it b 0(b i i 0):ThenT 27=1n P ni =11T P m l = m P r 2t =r 1(I A i )k0;it lkit^ ;^ i h b I iT 28=1n P n i =11T P m l = m P r 2t =r 1(I A i ) k it^ ;^ i k0;it l h I bi T 30=1n P n i =11T P m l = m P r 2t =r 1(I A i ) k it ^ ;^ i k it ^ ;^ iT 31=1n P n i =11T P m l = m P r 2t =r 1(I A i ) k 0;it l k it ^ ;^ i bT 32=1n P n i =11T P m l = m P r 2t =r 1(I A i ) k it ^ ;^ i k0;it(b i i 0)T 33=1n P n i =11T P m l = m P r 2t =r 1(I A i ) k 0;it l k it ^ ;^ i (b i i 0)T 34=1n P n i =11T P m l = mP r 2t =r 1(I A i ) k it ^ ;^ i k 0;it bT 35=1n P n i =11T P m l = m P r 2t =r 1(I A i )(b i i 0)vec k 0;it k it ^ ;^ i 0 T 36=1n P n i =11TP ml = m P r 2t =r 1(IA i )(b i i 0)vec k it ^ ;^ i 0k00;it l:The following Lemmas establish the mean and variances of T j ,j =1;::;36:Lemma 3T 1=vec((T 0(m ))+O (T 1)where (16)T 0(m )=1n P n i =1A i P j l j >m E k it k 0it land T 0(m )=O a 2+ m:Proof.ET 1=vec1nP n i =1A i E (K i;m ) f kk=vec 1n P n i =1A i P m l = m r 2 r 1T1 E k it k 0it l P j l j >m E k it k 0it l=O (T 1)+O a 2+ m;where the last line follows from Condition 3andk Var(T 1)k = E vec 1n P n i =1A i (K i;m E (K i;m )) vec 1n P n i =1A i (K i;m E (K i;m )) 0= n 11n P n i =1(I A i )Var (vec K i;m )(I A 0i ) n 11nP n i =1k A i k 2k Var (vec K i;m )k =O (n 1):since Var (vec K i;m )is uniformly bounded in i and m .Lemma 4E (T 2)=0and Var(T 2)=O n 1:Proof.See Lemma (3).Lemma 5Assume that Condition 2holds and m=T !0as m;T !1:Then T 3=O p (n 1=2T 1=2)+O p (n 1=2T 1m )and T 7=O p (n 1=2T 1=2)+O p (n 1=2T 1m ):Proof.For T 3use the expansionp nT ^ 0 =p n (0)+12r n T(0)+o p (1)obtained by Hahn and Kuersteiner (2004)such that1n P n i =11T P m l = m P r 20t =r 1(I A i ) k it l k 0;it b=1n 3=2P n i =11T 3=2P m l = m P r 2t =r 1(I A i ) k it l k0;itp n (0)+1n P n i =11TP m l = m P r 2t =r 112(I A i ) k it l k0;itrn T (0)+o p (1)1n 3=2P n i =11T3=2P m l = m P r 2t =r 1(I A i ) k it l k0;itwhere the last term is o p (n 1=2m=T )and thus is neglected.Then for the …rst term de…ne k 0;it =k 0;it E k 0;itsuch thatE1n 3=2P n i =11T 3=2P m l = m P r 2t =r 1(I A i ) k it l k 0;it p n (0) 1n 3=2P n i =1 E p n (0) 2 1=2 E 1T 3=2P m l = m P r 2t =r 1 k it l k 0;it 2!1=2k I A 0i k =O (n 1=2T 1=2):The last equality follows fromE p n (0) 2=tr1n P n i =1I i 1 1nT P n i 1;i 2=1P T t 1;t 2=1E U i 1t 1U 0i 2t 21n P n i =1I i 1=O (1)because of Conditions (3)and (4)andE 1T 3=2P m l = m P r 2t =r 1 k it lk 0;it 2=1T 3P m l 1;l 2= m P r 2t 1;t 2=r 1tr E k it 1 l 1k 0it 2 l 2 k 0;it 1 k 00;it 21T 3P m l 1;l2= m P r 2t 1;t 2=r 1 tr E k it 1 l 1k 0it 2 l 2 E k 0;it 1 k 00;it 2 +tr E k it 1 l 1 k 0;it 1 E k 0it 2 l 2 k 00;it 2 +tr E k 0it 2 l 2 k 0;it 1 E k it 1 l 1k 00;it 2+tr K 4 k it 1 l 1k 0it 2 l 2k 0;it 1 k 00;it 2 where K 4 k it 1 l 1k 0it 2 l 2k 0;it 1 k 00;it 2 is the matrix that has as its elements the forth order comulants of the elements of the matrix k it 1l 1k 0it 2 l 2k 0;it 1 k 00;it 2:The largest order term in the above expression is1T 3P m l 1;l 2= m P r 2t 1;t 2=r 1tr E k it 1 l 1 k 0;it 1 E k 0it 2 l 2 k 00;it 2 0=O (T 1)where the bound holds uniformly in i:Next consider1n 3=2P n i =11T 3=2P ml = mP r 2t =r 112(I A i ) k it l k 0;itrn T(0)=1n 3=2P ni =11T 3=2P m l = mP r 2t =r 112(I A i ) k it l k 0;itr n T 1n nX i =1I i ! 1 +o p (1)1n 3=2P ni =11T 3=2P ml = mP r 2t =r 112(I A i ) k it l k 0;it=O p (n 1=2T 1=2)by the same arguments as before.NextE 1n 3=2P n i =11T 3=2P m l = m P r 2t =r 1(I A i ) k it l E k 0;it p n (0)E p n (0) 2 1=21n 3=2P n i =1 E 1T3=2P m l = m P r 2t =r 1 k it l E k 0;it 2!1=2k I A 0i k =O (n 1=2T 1m )whereE1T 3=2P m l = m P r 2t =r 1 k it l E k 0;it 2=O (T 2m 2)by an argument analoguous to the proof of Lemma 20.The results can be shown for terms involving (0):This shows that1n P n i =11TP ml = m P r 20t =r 1(I A i ) k it l E k 0;itb =O p (n 1=2T 1m ):The analysis for T 7proceeds in the same way by noting that E 1T 3=2P m l = m P r 2t =r 1 E k 0;it l k it 2=O (T 2m 2)as before.Lemma 6De…ne(17) ;v it = (E [Vi ])1@ (x it ; 0; i 0)@:ThenE [T 4]=2m T 1n P n i =1(I A i ) P 1u = 1E [ ;vit k it u ] E k 0;i 1 +o (m T)=O (m=T )and Var(T 4)=O (n 1T 2m 2):Similarly,E [T 8]=2m T 1n P n i =1(I A i ) Ek 0;i 1 P 1u = 1E [ ;vit k it u ] +o m T =Om Tand Var(T 8)=O (n 1T 2m 2):e b i(0)= (E[V i ])11p TP Tt =1@ (x it ; 0; i 0)@+E V i (0)where(0)= 1n P ni =1I i11n pTP ni =1P Tt =1U itand de…ne(18) k it =k it Ek it and(19)Ujs= (E [V i ]) 1E V i1n P ni =1I i1U jsDe…ne T 4;1=1n P n i =11T 2P m l = m P r 2t =r 1P T s =1(I A i ) ;v is k it l k 0;it(20)T 4;2=1n 3=2P n i =11T 2P m l = m P r 2t =r 1P n j =1P T s =1(I A i )Ujs k it l k 0;it :(21)Then,for ET 4;1consider 1n P n i =11T 2P m l = m P r 2t =r 1P T s =1(I A i )E ;v is k it l k 0;it(22)=1n P n i =11T2P m l = m P r 2t =r 1P T s =1(I A i )cum ;v is ;k it l ; k it =O (T 2m )where cum ;v is;k it l ; k it is the matrix of third order cross-cumulants of ;v is ;k it l and k it and the relationship E ;v is k it l k it =cum ;v is ;k it l ; k itfollows from Shiryaev (1996,p.293).By the same argument as inAndrews (1991,Lemma 1)it follows that P 1t = 1P 1s = 1cum ;v is ;k it l ; kit<1uniformly in l such that the result follows.Next consider 1n P n i =11T2P m l = m P r 2t =r 1P T s =1(I A i )(E [ ;v is k it l ] E [kit ])(23)=2m T 1n P n i =1(I A i ) P 1u = 1E [ ;v it k it u ] E [k it ] +o (m T)=O (mT 1)by the mixing properties which imply that P 1u = 1k E [ ;v it k it u ]k <1:De…ne C is;t l;t = ;v is k it l k0;it and consider Var (T 4)whereVar (T 4;1)=1n 2T 4P n i =1P ml 1;l 2= m P r 2t 1;t 2=r 1P T s 1;s 2=1(I A i ) E(C is 1;t 1 l 1;t 1 E [C is 1;t 1 l 1;t 1])(C is 2;t 2 l 2;t 2 E [C is 2;t 2 l 2;t 2])0(I A 0i )=1n P n i =11TP m l 1;l 2= m P r 2t 1;t 2=r 1P T s 1;s 2=1(I A i )E C is 1;t 1 l 1;t 1C 0is 2;t 2 l 2;t 2 (I A 0i ) 1n P n i =11TP m l 1;l 2= m P r 2t 1;t 2=r 1P T s 1;s 2=1(I A i )E [C is 1;t 1 l 1;t 1]E [C is 2;t 2 l 2;t 2]0(I A 0i ):The second term in the last display is O (n 1T 4m 2)by 22.The matrix E h C is 1;t 1 l 1;t 1C 0is 2;t 2 l 2;t 2iis a matrix of sixth order cross moments of the elements in ;v is ;k it l and k0;it :We use the index (a;b;c;d;e;f )to denote indiviual elements of the random vectors ;v is 1;k it 1 l 1,k 0;it 1; ;v is 2;k it 2 l 2and k 0;it 2where ;v a;is 1is the a -thelement of ;v is 1and so forth.For any combination of elements (a;b;c;d;e;f )consider the index I f 1;2;:::;6g and denote byE (a;b;c;d;e;f )[I ]=E h f 12I g ;v a;is 1+f 1=2I g (f 22I g k b;it 1 l 1+f 2=2I g ) ::: f 62I g kf;0;it 2+f 6=2I gi with a similar de…nition holding for the cumulant cum (a;b;c;d;e;f )[I ]:Each element of EhC is 1;t 1 l 1;t 1C 0is2;t 2 l 2;t 2iis charactarized by a particular value for (a;b;c;d;e;f ):From Shiryaev (1996,p.292)it follows that for a typical element E (a;b;c;d;e;f )[f 1;::;6g ]of E h C is 1;t 1 l 1;t 1C 0is2;t 2 l 2;t 2iE (a;b;c;d;e;f )[f 1;::;6g ]=X[q r I r =f 1;2;:::;6gQ qr =1cum (a;b;c;d;e;f )[I r ]where the sum is over all possible decompositions of f 1;2;:::;6g and 1 q 6:All terms involving cumulants of order higher than two are smaller when averaged across time periods because of the mixing properties.The largest terms of E (a;b;c;d;e;f )[f 1;::;6g ]are covariance terms of the form1P m l 1;l 2= m P r 2t 1;t 2=r 1P T s 1;s 2=1Cov ;v a;is 1;k b;it 1 l 1Cov ;v d;is 2;k e;it 2 l 2 E kc;0;it 1 E h k f;0;it 2i =O T 2m 2 which shows that Var(T 4;1)=O (n 1T 2m 2):Next consider T 4;2where1n P n i =1 (E [V i ]) 1E V i (0)T 3=2P m l = m P r 2t =r 1(I A i ) k it l k 0;it p n (0) sup i(E [V i ]) 1E V i 1n 3=2P n i =1k (I A i )k 1T 3=2P m l = m P r 2t =r 1 k it l k 0;it :。