Modi ed Local Whittle Estimator for Long Memory Processes in the Presence of Low Frequency (and Other) Contaminations

Transcription

1 Modi ed Local Whittle Estiator for Long Meory Processes in the Presence of Low Frequency (and Other Containations Jie Hou y Boston University Pierre Perron z Boston University March 5, 203; Revised: January 22, 204 Abstract We roose a odi ed local-whittle estiator of the eory araeter of a long eory tie series rocess which has good roerties under an alost colete collection of containation rocesses that have been discussed in the literature, ostly searately. hese containations include rocesses whose sectral density functions doinate at low frequencies such as rando level shifts, deterinistic level shifts and deterinistic trends. We show that our odi ed estiator has the usual asytotic distribution alicable for the standard local Whittle estiator in the absence of such containations. We also show how the estiator can be odi ed to further account for additive noise and that our odi cation for low frequency containation reduces the bias due to short-eory dynaics. hrough extensive siulations, we show that the roosed estiator rovides substantial e ciency gains coared to existing seiaraetric estiators in the resence of containations, with little loss of e ciency when these are absent. JEL Classi cation Nubers: C22, C3, C4. Keywords: long eory rocess, rando level shifts, short eory dynaics, additive noise, local-whittle estiators. We are esecially grateful to Zhongjun Qu for helful coents, as well as two referees. y Deartent of Econoics, Boston University, 270 Bay State Rd., Boston, MA, 0225 (tyod@bu.edu. z Deartent of Econoics, Boston University, 270 Bay State Rd., Boston, MA, 0225 (erron@bu.edu.

2 Introduction Processes that are ersistent in the sense that the serial correlation between distant observations decay hyerbolically are called long eory rocesses. hey have found extensive use in caturing the behavior of any observed series since their introduction by Hurst (95. A long eory rocess is also characterized in the frequency doain by a sectral density function roortional to 2d as the frequency aroaches zero at a rate dictated by the the eory araeter d. In ters of araetric odeling, Granger and Joyeux (980 and Hosing (98 introduced the fractionally integrated ARF IM A(; d; q odel, a long-eory generalization of the short-eory ARM A(; q rocess. he estiators of the eory araeter are divided into araetric and sei-araetric ones. he theory of araetric estiators was develoed by Fox and aqqu (986 and Dahlhaus (989, aong others. Seiaraetric estiators of the eory araeter have becoe oular since they do not require nowing the seci c for of the short eory structure. hey are based on the eriodogras of the series, and can be categorized into two tyes: the log-eriodogra (LP estiator rst roosed by Gewee and Porter-Huda (983 and the local-whittle (LW estiator which is credited to Kunsch (987. he LP estiator is ain to OLS and the LW estiator to the MLE in the frequency doain. Robinson (995a,b analyzed the asytotic roerties of these two tyes of estiators. He showed that they are asytotically noral, have the sae convergence rate and that the asytotic variance of the LW estiator is saller than that of the LP estiator. here are, however, so-called containations that have an e ect on the bias and e ciency of these sei-araetric estiators, either in nite sales or even asytotically. Much of the literature so far has focused on roviding ethods to itigate the e ect of additive noise and/or short-eory dynaics, which have only a nite sale e ect. In the case of additive noise or so-called erturbed fractional rocesses, although both the LW and LP estiators reserve consistency and asytotic norality, as shown by Deo and Hurvich (200 and Arteche (2004, they can be severely biased. Hurvich and Ray (2003, Hurvich et al. (2005 and Arteche (2006, aong others, have roosed estiators that can reduce the e ect of noise by introducing an additive constant or olynoial ter in the sectral density function. hese ethods are all based on local Whittle estiators, given their exibility in accoodating ore structures in the seci ed data-generating rocess. he estiators are also strongly biased when substantial short-eory dynaics are resent. Aong others, Andrews and Sun (2004 considered an adative local olynoial Whittle estiator. By

3 substituting a olynoial structure for the constant ter used to aroxiate the behavior of the short eory coonent near frequency zero in the local Whittle estiator, they showed that their estiator has considerable e ciency gains coared to classic LW and LP estiators under the resence of short eory dynaics. Recently, Frederisen et al. (202 cobined the two ethods and roosed estiators that can siultaneously reduce the bias and ean squared error caused by short eory dynaics and noise erturbation. here are other low frequency containations (denoted as LFC that can have a ore serious e ect causing outright inconsistent estiates. hey ay be iortant enough to induce researchers to istaenly conclude that a short eory rocess with low frequency containations is actually a long eory rocess. Such an e ect is often called surious long eory. hese low frequency containations include, but are not con ned to, rando level shifts, deterinistic level shifts and deterinistic trends. A short-eory rocess containated by those coonents will exhibit hyerbolically decaying autocorrelations as well as a ole in its sectral density function at frequency zero, which are characteristics of a long eory rocess. Aong others, Diebold and Inoue (200, Granger and Hyung (2004, Miosch and St¼aric¼a (2004 and Perron and Qu (200 rovide theoretical exlanations for and siulation evidence of this surious long eory e ect. It has also been argued that odels incororating a short eory rocess with such low frequency containations rovide a better in-sale t and, in articular, forecast better coared to odels assuing a ure long eory rocess. Various studies reorted evidence that these fors of data containations are in fact very liely resent in the volatility of asset rices and considerably weaens the evidence of ure long-eory; see, e.g., Granger and Hyung (2004, Miosch and St¼aric¼a (2004, St¼aric¼a and Granger (2005, Perron and Qu (200, Lu and Perron (200, Qu and Perron (203, Varnesov and Perron (203 and u and Perron (203. Recent wor by Dolado et al. (2005, Ohanissian et al. (2008, Perron and Qu (200 and Qu (20 roosed tests in both the tie and frequency doain with varying degrees of success. Many have argued that the long-eory roerties of any econoic tie series are indeed surious. hese tests focus on distinguishing between a short eory rocess a ected by low frequency containations fro a true long eory rocess. So they do not o er ethods to estiate the eory araeter in the resence of low frequency containations when the true signal ay be of long or short eory. Recently, attention focused on roviding odi ed LP or LW estiators to account for low frequency containations. McClosey and Perron (203 roosed tried LP estiators that have desirable asytotic and nite sale roerties in the resence of low 2

4 frequency containations. Using a siilar triing technique, McClosey and Hill (203 roosed tried frequency doain quasi axiu lielihood estiator estiators for short-eory tie series odels (e.g., ARMA, GARCH and stochastic volatility odels that ay be containated by low frequency oveents. McClosey (203 considered a tried frequency doain quasi axiu lielihood estiator that can be used to consistently estiate the araeters of a long-eory stochastic volatility odel in the resence of low frequency containation assuing the signal to be an ARF IMA(; d; q rocess. Iacone (200 considered tried LW estiators. We roose odi ed LW estiators that wor under all inds of containations: low frequency, additive noise and short eory dynaics. Our ehasis is on accounting for low frequency containations and we show how to further odify the estiator to account for the other tyes. It adots techniques used in Andrews and Sun (2004, Hurvich et al. (2005 and Frederisen et al. (202 to introduce additive ters in the frequency doain quasi axiu lielihood function to cature the e ect of the low frequency containations, based on results of Perron and Qu (200 and McClosey and Perron (203 showing the sectral density function of low frequency containations to be of order O ( 2 near frequency zero. o account for additive noise, we follow Hurvich et al. (2005. Interestingly, our odi cation for low frequency containations also reduces the nite sale bias induced by short-eory dynaics, so that no further odi cation is necessary for this case. Our odi ed estiators have the following advantages: being seiaraetric, they do not require nowing the structure of the short eory rocess; they do not require triing so all data is used; unlie the tried LP estiator, they do not require the underlying rocess to be Gaussian; they have the sae asytotic variance as the standard LW estiator when no containation is resent; without low frequency containations, they are asytotically equivalent to the standard LW estiator that does not account for low frequency containations so that no e ciency loss is incurred by incororating our odi cations; they can easily be extended to a full araetric case. When low frequency containations are resent, it has, in ost cases, the sallest bias and ean-squared error aongst all existing estiators designed to control for low frequency containations, whether or not other tyes of containations are resent. o our nowledge, our contribution is the rst to rovide an estiator with good roerties under all reviously considered containations: low frequency, additive noise and short-eory dynaics. he structure of the aer is as follows. Section 2 resents the odel and soe reliinary results. Section 3 otivates and introduces our odi ed LW estiator that accounts 3

5 for ossible low frequency containations. Section 4 resents results about the consistency and liit distribution. Section 5 discusses how to extend the estiator to account for additive noise and short-eory dynaics. Section 6 resents the results of siulations to assess the nite sale roerties under a variety of ossible scenarios. Section 7 rovides brief concluding rears. All technical derivations are collected in a atheatical aendix. he following notation is used throughout:! stands d for convergence in distribution,! for convergence in robability,! for the liit as! (unless otherwise stated, a g b denotes the axiu of a and b, x y eans that x=y!. 2 he odel and reliinary results We start with soe basic de nitions of a long eory rocess. Let fy t g t= tie series with sectral density function f y ( at frequency given by be a stationary f y ( = G( 2d as! 0 + ( with G( a slowly varying function as! 0 + (i.e., for any real t, G(t=G(! as! 0 +. When d > 0, y t is a long-eory rocess with a sectral density function that increases for frequencies that get close to zero. he rate of divergence to in nity deends on the araeter d. Under soe general conditions, this low-frequency de nition is equivalent to the following long-lag autocorrelation de nition (Beran, 994. Let y ( be the autocorrelation function of y t. If y ( = c( 2d as!, with c( a slowly varying function as!, the rocess is said to have long eory. For 0 < d < =2, this ilies that the autocorrelations decreases to zero at a slow hyerbolic rate which deends on the araeter d, in contrast to the fast geoetric rate of decay that alies to a short-eory rocess. Exales of long-eory rocesses include the oular class of fractionally integrated autoregressive oving average odels, though in what follows we shall reain agnostic about the nature of the short-eory coonent iosing only high level assutions. When d = 0, y t is a short-eory rocess. he Data Generating Process (DGP considered is one where the series of interest, z t, is a long or short-eory rocess lus soe low frequency containation, viz., z t = c + y t + u t (2 where y t is a rocess with eory araeter d 2 [0; =2 and c a constant. Note that the value d = 0 is allowed so the DGP includes a short-eory rocess containated by 4

6 soe low frequency coonent. he rocess u t is the low frequency containation which will be de ned below. We suose that a sale of size is available. We de ne the eriodogras of the rocesses fz t ; y t ; u t g to be, for soe frequency ordinate, I z; = I = I z (, I y; = I y ( and I u; = I u ( where I w ( = (2 j t=w t e it j 2 for w = z; y; u, and their sectral density functions by f z; = f = f z (, f y; = f y ( and f u; = f u (. Seiaraeteric frequency doain estiators for non-containated fractional rocesses are all based on the local aroxiation ( and are robust to the nature of the short eory dynaics since they only use inforation fro eriodogra ordinates near the origin. he local Whittle (LW estiation ethod of Kunsch (987 and Robinson (995a has becoe oular because of its lielihood interretation, nice asytotic roerties (saller asytotic variance coared to log-eriodogra estiators, ild assutions (e.g., no need for a norality assution and ost iortantly in our case, the ossibility to easily odify it to accoodate the resence of containations. It is de ned as the iniizer of the (negative local Whittle lielihood function in the frequency doain Q(G 0 ; d = j= [log(g 0 2d j + I z ( j =(G 0 2d j ] where G 0 = G(0, = ( is the bandwidth which goes to in nity as! but at a slower rate than, j = 2j= are the Fourier frequencies. Concentrating with resect to G 0, the estiator of d is ^d LW = arg in d [log ^G 0 (d 2d P j= log j], where ^G 0 (d = P j= 2d j I z ( j. he tyes of rocesses considered for the low frequency containation (LFC coonent u t are laid out in the following de nition. De nition he low frequency containation coonent u t is generated by one of the following rocesses. Rando level shifts (RLS: u t = t= ;t where ;t = ;t t with t i:i:d: N(0; 2 and ;t i:i:d: Bernoulli(=; for soe 0. he coonents ;t, t are utually indeendent. 2 Deterinistic level shifts: u t = B i=c i ( i < t i where B is the ( xed nuber of regies (B is the nuber of breas, 0 < jc i j <, ( is the indicator function, 0 = 0 < < ::: < B < B = and i =! i with 0 <. < ::: < B <. 3 Deterinistic trends: u t = h(t= where h( is a deterinistic nonconstant function on [0; ] that is either Lischitz continuous or onotone with h( = 0. 4 Fractional trends: u t = O((t + =2, u 0 = 0; ju t+ u t j = O(ju t j=t where 2 ( =2; =2. his includes all cases for which h( is onotonic and bounded because we can sily substract h( fro h( and add h( to c in (2 to have the sae DGP. 5

7 Note that the robability of a level shift in the RLS odel is sale size deendent. If this were not the case, u t would have roerties siilar to that of a rando wal. A de ning characteristic of the RLS odel is that the average nuber of level shifts reains constant as the sale size grows. Note that can be zero so that the assution nests the no level shift or no containation case as well. Perron and Qu (200 considered the asytotic roerties of the eriodogra of this tye of rocess containating a short eory rocess and showed that, for any = ; :::; [=2], ( 2 = E(I u;! ( 2 =(4 3 as!. Miosch and St¼aric¼a (2004 considered the asytotic roerties of the eriodogra for a deterinistic level shift coonent when B = 2 (one level shift, with the addition of a short-eory coonent, and showed that E(I u; = O(= 2. Kunsch (987, Lea 2 considered the asytotic roerties of the eriodogra of a short-eory rocess containated by a bounded onotone trend. Qu (20, Lea extended Kunsch s results to the Lischitz continuous case and showed that E(I u; = O(= 2. Iacone (200 discussed the order of the eriodogra of in the case of a fractional trend and showed that E(I u; = O (= 2. he coon feature of these containating rocesses is that the ean of their eriodogra near frequency zero is of order O(= 2, or equivalently of order O( 2 since = 2= (note that the O ter could be o since it is ossible that E[(I u; =(= 2 ]! 0, a case we shall discuss further later. Processes with such LFC as additive coonents are non-stationary so they do not have the traditionally de ned sectral density function. Following coon ractice in such cases, we de ne their sectral density function to be the exectation of their eriodogra. Since the sectral density function of a long eory rocess near frequency zero is of order O( 2d, in general the sectral density function of such containating coonents doinates that of a long eory rocess at low frequencies and vice-versa at high frequencies. Note that in the reresentation (, when the rocess is containated by such LFC, we have G u G u (0 = li! ( 2 = E(I u;. Rear De nition could be relaced by the condition E(I u; = G u ((= 2 ( + O( with G u ( B where B is a xed bounded ositive constant. Hence, E(I u; =(= 2 need not converge to a constant, it only needs to be bounded as!. All LFC in De nition satisfy this roerty and all results to be resented reain valid under this general condition. Unlie short eory dynaics or containating noise, which cause only nite sale biases to the eory araeter estiator, the bias caused by LFC usually reains asytotically. o see when this alies, let A = ( 2 = E(I u;, then one can show 6

8 that 2d I O ( = 2d I y; + A O ( 2d = 2 2d P. So the bias introduced by LFC is of order 2d = (A = 2 2d. he following de nition will be useful. De nition 2 A LFC is said to be non-degenerate if li! f( 2 = E(I u; g > 0 for every. Otherwise it is said to be degenerate. An exale of a non-degenerate LFC is a RLS odel, in which case li! ( 2 = E(I u; = ( 2 =(4 3. An exale of a degenerate LFC is a onotone deterinistic trend. he bias caused by a non-degenerate LFC reains asytotically while the bias caused by a degenerate LFC can either reain or vanish asytotically, with the degree of the (otentially asytotic or nite sale bias deending on d and the bandwidth. 3 he odi ed local Whittle estiator Let the Fourier transfor of the rocess z t be h z ( j = (2 =2 ( P t= z je it j so that f z ( = E(I z ( = E(h z ( h z (, where denotes the colex conjugate value. One ay then de ne the frequency doain seudo Quasi Maxiu Lielihood Function (QMLF for h z ( as ' = log(f z ( + I z ( =f z (. When there is no containation in the data, f z ( reduces to f y ( and the standard LW estiator is the iniizer of the seudo-qmlf. With low frequency containation given by u t, a roble is how to construct a useful aroxiation to f z ( in such cases. Because the eriodogra of u t is of order O ( 2, a sensible strategy is to add a ter (G u= 2 to the sectral density function of y t to control for the low frequency containation. Accordingly, we consider the seudo sectral density function f, f z ( = G 0 2d + G u 2 =. Let = (G u=g 0 be the signal to noise ratio, the seudo sectral density function of the observed rocess is then: f, f z ( = G 0 2d +G u 2 = = G 0( 2d +(G u =G 0 2 = = G 0( 2d + 2 = = G 0g where g = ( 2d + 2 =. Rear 2 f is the seudo sectral density function in the sense that it is not the true sectral density function of the data, but an arti cial construct aied at roviding a good aroxiation to the behavior of the generalized sectral density function (i.e., the exectation of the eriodogra and an extended LW tye estiator with desirable roerties. his seudo sectral density function can then be used to aroxiate E(I z; and the seudo frequency doain QMLF is '(G; d; = P = ' (G; d;. Using the sae 7

9 technique as in Robinson (995a, we can concentrate G out of the QMLF using ^G = P = (I =g. Hence, the local Whittle (frequency doain QMLE estiator alicable under LFC, denoted as the LWLFC estiator, is ( ^d ; ^ = arg in (d; J (d;, where J (d; = log( = (I =g + log(g Rear 3 he coonent is an auxiliary variable in the sense that it is not a araeter of riary interest but is introduced as a tool used to control the in uence of the containations at low frequencies. Intuitively, is the aroriate signal to noise ratio to use as it easures the average of the relative agnitude of the containations across all frequencies. For the case of RLS containation, we have an exression for in ters of the araeters of the odel, given by = (G u =G 0 (2 2 = 2 "; see Perron and Qu (200. Rear 4 he ethod can be extended to the case with a araetric seci cation for the long-eory rocess. For exale, if y t is assued to follow the ARF IMA(; d; q rocess ( L d y t = ey t, where A(Ley t = B(L" t and " t i:i:d: N(0; 2 ", then we sily relace G 0 2d by 2 "(jb(e i j 2 =ja(e i j 2 [2j e i j 2d ] in the objective function J (d;. 4 Asytotic roerties We start by introducing the assutions required to obtain the consistency result for the LWLFC estiator. Many are the sae as in Robinson (995a, but soe are added or odi ed to accoodate the LFC coonents. Henceforth, we shall denote the true value of the long-eory araeter by d 0 and the true value of the signal-to-noise ratio by 0. Assution A. As! 0 +, f y ( G 0 2d 0 where G 0 2 (0; and d 0 2 [0; =2. Assution A2. =d = O(. For in a neighborhood of 0, f y ( is di erentiable and d log(f y ( Assution A3. y t is stationary and adits an in nite MA reresentation: y t E(y t = P j=0 j" t j with P j=0 2 j < where f" t g is a artingale di erence sequence with E(" t jf t = 0, E(" 2 t jf t = 2 ", E(" 3 t jf t = 3, and E(" 4 t = 4 where F t is the - eld generated by f" s ; s tg. Also, there exists a rando variable " such that E(" 2 < and for all > 0 and soe K > 0, P (j" t j > KP (j"j >. Rear 5 We require " t to have nite fourth oent even to establish consistency to invoe a strong law of large nubers for P = (I =g (d; and show that the convergence of the eory araeter estiate does not deend on the signal to noise ratio. 8 =

10 Assution A4. As!, ( (d2 0 3d 0+9=4 g(=2 = + =! 0. Rear 6 he requireent on the bandwidth to establish consistency dearts fro Robinson (995a who only requires that (= + (=! 0. his is due to the need to suress the iact of (I =g at low frequencies, < [( 2d 0=(2 d 0 ], in which case the eriodogra of the LFC doinates that of the long eory rocess. With the addition of the ter (= 2 in the QMLF, we can then bound ji =g j. However, to control the e ect of fi =g g at high frequencies where the eriodogra of the long eory rocess doinates that of the LFC, we need a larger bandwidth to suress the cuulative iact fro the low frequencies. he closer is d 0 to 0, the higher is the required bandwidth because the containation will then doinate at higher frequencies. he quantity ( (d 2 0 3d 0 + 9=4 g (=2 achieves its axiu value 5=9 when d 0 = 0. Hence, in ractice with an unnown eory araeter d 0, we need to choose a bandwidth of order greater than 5=9. Assution A5. u t is one of the LFC as stated in De nition. It will be useful to rst establish a liit result ertaining to the estiate ^ of the signal to noise ratio. his will be used in the roof of the consistency of ^d. Lea Under A-A5, if a non-degenerate LFC is resent, ^ is bounded above by zero. We now consider the consistency result and a reliinary bound on the convergence rate that will be used to establish the liit distribution of our estiator. heore Under A-A5: a ^d! d0 as! ; b j ^d d 0 j = o ((log( 3. Note that this result does not require ^ to be a consistent estiate, all that is required is that if LFC coonents are resent the robability liit of the estiate of ^ is bounded above by zero, which is guaranteed by Lea. his ilies that with robability arbitrarily close to one, ^ will be in a the set (0; and we can consider analyzing the liit of ^d for any value or sequences of in the set (0;. Before roceeding further, we need to discuss a roerty of the estiate of the signal-tonoise ratio ^ when there is no LFC resent. his, in conjunction with Lea, will allow us to derive the liit distribution of ^d for both cases with and without LFC. he required result is stated in the next lea, which is of indeendent interest. Lea 2 Suose no LFC is resent and that A-A4 hold, then, as! : ^ = O ( ( 2d 0=(2 2d 0! 0. 9

11 o rove the asytotic norality of ^d, further assutions are needed, soe of which are strengthened versions of Assutions A-A3. Assution A6. G 0 2 (0; and d 0 2 [0; =2. For soe 2 (0; 2], f y ( G 0 2d 0 ( + O( as! 0 +, where Assution A7. In a neighborhood of the origin, f y ( is di erentiable and df y (=d = O(f y (= as! 0 +. Assution A8. As!, (log 2! 0. he following theore states the asytotic distribution of the estiate ^d. heore 2 Under A-A8: =2 ( ^d d 0 d! N(0; =4 as!. Note that the asytotic variance of our estiator is the sae as that of the standard LW estiator of Robinson (995a alicable with no LFC. he intuitive reason is that, asytotically, the additional ter G u ( 2 = controls the e ect of LFC on the sectral density function well enough so that no e ciency loss ensues. When the agnitude of the LFC is wea, the asytotic distribution of heore 2 rovides a good aroxiation to the nite sale distribution. However, when the agnitude of the LFC is substantial, 2 =2 ( ^d d 0 does converge to a noral distribution raidly as increases (even with as sall as 52 but the aroach to a standard noral ay be slow, i.e., the ean and variance of 2 =2 ( ^d d 0 ay converge slowly to 0 and, resectively. Soe aroxiate forulas to coute the nite sale bias and variance of 2 =2 ( ^d d 0 have been found in unreorted siulations and they rovide good aroxiations. Unfortunately, they all deend on 0, the signal to noise ratio which cannot be identi ed when it is greater than zero, rendering the corrections not alicable in ractice. An iortant avenue of further research is to obtain a nite-sale scaling factor, say S, to relace in order to obtain good nite sale coverage rates for the LWLFC estiate. A conjecture is that S should be a decreasing function of 0 to re ect the iact of LFC on the variance of the eory araeter estiate. But since ^ is not a consistent estiator of 0, it is unliely that one can nd a good alicable forula. his roble about the coverage rate is not unique to our ethod, and alies to all existing ethods to estiate the eory araeter under soe containation. Alternative scaling factors have been roosed. For the log-eriodogra estiator, Gewee and Porter-Huda (983 suggested using the scaling factor S(l; =2, where S(l; = P j= (log j ( l + P =l log 2 for soe lower triing l, and its use was also discussed by Deo and Hurvich (200. For local Whittle-tye estiators, it was used by Hurvich et al. (2005 and Iacone (200. 0

12 5 Extension to the case of additive noise and short eory dynaics An advantage of LW-tye estiators is that, since they use the QMLF in the frequency doain, they can easily be odi ed to accoodate ore tyes of structures in the DGP, without the need to tri soe of the low frequencies. We consider two extensions to account for additive noise and short-eory dynaics. hese eleents do not cause an asytotic bias and, hence, the odi cations are aied solely at iroving the nite sale erforance. Consider rst the case where both LFC and additive noise are to be accounted for. o be recise, instead of (2, the DGP is now z t = c + y t + u t + w t, where, following Assution (H2 in Hurvich et, al (2005, the additive noise w t is a zero ean white noise with variance 2 w, such that for each s 6= t, E[w s " t ] = 0 and for each t, E[w t " t ] = w w, where " t is as de ned in A3 and w is the correlation between w t and " t, assued to be constant. Also, w t is indeendent of the LFC u t. Following Hurvich et al. (2005, we add a constant ter into the sectral density function, so that the odi ed seudo sectral density function is: f, f z ( = G 0 2d + G w + G u ( 2 = = G 0( 2d + (G w =G 0 + (G u =G 0 ( 2 = = G 0 ( 2d + w + ( u = 2 = G 0g (3 where, with a slight abuse of notation relabeling u = G u =G 0, g = ( 2d + w + ( u = 2 and the (aroxiate frequency doain QMLF is '(G; d; = P = ' (G; d; with = ( w ; u 0. Concentrating G out of the QMLF, the estiate of G is ^G = P = (I =g and the local Whittle QMLE estiator under noise erturbations and low frequency containations, denoted as the LWPLFC estiator, is ( ^d ; ^ = arg in (d; J (d;, where J (d; = log( = I g + log(g = For reasons discussed by Hurvich, et. al. (2005, the LWPLFC aroach is exected to wor when d 0 is not too close to zero. When d 0 = 0, the rocess is short-eory. We then have a cobination of two additive short-eory rocesses which cannot be identi ed searately. For the case of short eory dynaics lus LFC, we could follow the aroach of Andrews and Sun (2004 who add a olynoial structure into G 0, i.e., relace G 0 in (3 by G 0 ex( r ( j ; where r ( j ; = r s= s 2s j and = ( ; :::; r. However, unreorted siulations with r =, showed that doing so did not o er any gain in erforance over our LWLFC estiator with a saller value of the bandwidth (see the siulations in Section 6. his feature can be exlained as follows. Fro siulations to be reorted in the next

13 section, under strong short eory dynaics and RLS, the LWLFC estiator constructed with a large bandwidth has substantial bias but very sall variance, so that the overall MSE is alost entirely due to the bias. When a olynoial coonent is added, the uward bias is reduced but the variance is increased considerably so that the overall MSE is alost the sae or larger than that of the LWLFC estiator. With no RLS, the increased variance is saller so that the MSE is indeed reduced as reorted by Andrews and Sun (2004. At the root of the issue is the fact that both RLS and short eory dynaics cause uward biases in the estiate of the eory araeter. Hence, there is a confounding e ect so that the QMLF is at with resect to the correction factors for short eory dynaics and LFC. In unreorted siulations with both RLS and short eory dynaics, it was often found that either the coe cient to correct for short eory dynaics or the coe cient to account for LFC was very close to zero, desite having the true value of both coe cients greater than zero. As will be reorted in the siulations, the best way to account for short eory dynaics and RLS is to use the LWLFC estiator with a sall bandwidth. When both additive noise and short-eory dynaics are to be accounted for, three aroaches are ossible. One is to use the LWLFC estiator with a sall bandwidth, another is to use the LWPLFC with a large bandwidth, or we could follow the aroach of Frederisen et al. (202 who add olynoials and a constant as additive ters in the QMLF. One drawbac of the latter aroach is that the increase in the nuber of araeters can induce an iortant increase in variance resulting in increased ean-squared error. 6 Finite sale roerties he Data Generating Process (DGP used for the siulations is z t = y t +u t +w t, where y t is an ARF IMA(; d; 0 rocess given by ( L( L d y t = e t with e t i:i:d: N(0;, u t is a RLS rocess as described in De nition with 2 =, and w t i:i:d: N(0; 2 w is the additive noise coonent. he values used are: d = 0; 0:2; 0:45; = 0:0; 0:3; 0:6 and = 0; 5; 0; 20. he sale sizes are = 256, 52, 024, 2048 and 4096 in order to use of the fast Fourier transfor algorith with the whole data set. he estiate ^d is allowed to tae values in the set [ 0:99; 0:99] when evaluating the axiizers of the objective function. he value of the bandwidth is set to = for = 0:6; 0:7; 0:8, the choice being dictated by the fact that ust be larger than 5=9. hroughout, 500 relications are used. hese seci cations were also used by McClosey and Perron (203 so that we can ae direct coarisons of the relative erforance of our estiators with theirs (the sale sizes they used are 000 and 2000 but the inor di erences in should not be of concern given the rather large 2

14 di erences in erforance. he tried LP estiator of McClosey and Perron (203 deends on a lower triing and uer bandwidth, while ours deend on a bandwidth. We evaluate bias and Root Mean Squared Errors (RMSE. When aing coarisons, we do so using the values of the bandwidth (and triing for the LP estiator that gives the best RMSE for each of the statistics. We focus on rando level shifts as the containating coonent as this is arguably the ost relevant in ractice. he results are resented in ables -3 for the cases with only RLS and RLS lus short-eory dynaics, for which we focus on the LWLFC estiator. able 4 resents the results for the case of RLS lus additive noise, while able 5 resents results when all three tyes of containations are resent, in which cases we consider both the LWLFC and LWPLFC estiators. We do not ae a direct coarison with the tried LW estiator of Iacone (200. McClosey and Perron (203 erfored a coarison between the tried LP and LW estiators. hey concluded that the tried LP has generally saller bias and the tried LW generally lower variance and concluded that the overall erforance in the resence of RLS was coarable. 6. he case with only RLS he results for the case with only RLS are resented in the rst anels of ables -3 corresonding to the case = 0. Note rst that the best results in ters of RMSE are obtained with a large bandwidth using = 0:8, though biases are slightly saller with a saller bandwidth. Second, the results show that our estiator erfors better than McClosey and Perron s (203 tried LP estiator. When d 0 = 0, there is a 30-60% reduction in RMSE, when d 0 = 0:2 the reduction is in the range 30-40% while when d 0 = 0:45 it is in the range 5-20%. Hence, overall, the LWLFC estiator with a large bandwidth = 0:8, shows saller bias and RMSE than alternative estiators. When the rocess is uncontainated ( = 0, the bias and RMSE of our estiator is sall and close to that of the original LW estiator, so that very little e ciency loss is incurred when no containation is resent. 6.2 he case with RLS and short-run dynaics We now consider the case with both RLS and short-run dynaics (resented in ables -3 for non-zero values of. In this case the best results for the LWLFC estiator are obtained with a sall bandwidth, using = 0:6, and ore so as the agnitude of increases. Coared to the tried LP estiator, the reduction in RMSE is very substantial esecially for larger values of. For exale, with no RLS the reduction is around 65% when d = 0 and = 0:6, while it is around 40% when d = 0:45 and = 0:6. he LWLFC is able to reduce 3

15 bias and variance when both RLS and short-run dynaics are resent, even though it is designed to account only for LFC containation. As discussed in Section 5, the aroach of Andrews and Sun (2004 which adds a olynoial structure into G 0 does not o er additional iroveent. As stated in the above discussion, the results show that the LWLFC estiator has indeed very sall variance when both RLS and short-run dynaics are resent. 6.3 he case with RLS and additive noise he results for the case with RLS and additive noise are resented in able 4 for the LWLFC (which accounts only for LFC and LWPLFC estiators (which accounts for both. he variance of the noise is set to a large value 2 w = 4. he results show that the LWPLFC estiator has very sall biases irresective of the choice of the bandwidth. he biases are indeed orders of agnitude saller than those of the tried LP estiator which is severely a ected by noise. he sueriority of our estiator also holds when judged by the relative RMSE. According to the RMSE, the estiator erfors best with a high bandwidth ( = 0:8. he LWLFC estiator shows higher bias (though still uch saller than that of the tried LP but its variance is saller. In three out of the four cases analyzed (the excetion being d = 0:2 and = 20 the reduction in variance is not big enough so that the LWPLFC estiator has overall a saller RMSE when using a large bandwidth. As exected, the erforance of the LWPLFC iroves as d increases, for reasons exlained in Section he case with all three tyes of containations able 5 resents results with all three tyes of containations. We consider strong shorteory dynaics ( = 0:6 and a ediu value for the average nuber of level shifts ( = 0. For the additive noise, we use 2 w = ; 4; and we set d = 0:2; 0:45. he results show that both the LWLFC and LWPLFC erfor well. In general, the LWPLFC has better erforance when a large bandwidth is used, while the LWLFC is better with a sall bandwidth. For a large value of d 0 (0:45, the LWPLFC erfors slightly better than the LWLFC under the otial bandwidth alicable to each. When d 0 is sall (d 0 = 0:2 the LWLFC has slightly better erforance. his accords with Hurvich, et. al. (2005 who showed that the asytotic variance of the LW estiator increases as d 0 decreases. Overall, the results show an advantage of using the LWPLFC with a large bandwidth. Fro unreorted siulations, the erforance of the LWLFC and LWPLFC deteriorates as aroaches or with a oving-average araeter close to -, with or without noise. his is a roble coon to ost, if not all, versions of LW or LP estiators, tried or not. 4

16 6.5 Overall suary and recoendations he results showed that our estiators have good nite sale roerties and o er iroved ethods of inference coared to what is available in the literature. As with all existing seiaraetric estiators of this tye, the results can be sensitive to the choice of the bandwidth. In our case, a large bandwidth (e.g., = 0:8 is referable in ost cases. One excetion is when there is a strongly ositively correlated short-eory coonent, in which case a saller bandwidth ( = 0:6 is desirable. As of yet, there is no fully develoed ethod to choose the bandwidth. But soe aroaches are ossible for the ractioner to assess what is the best bandwidth to use. One is to estiate a reliinary araetric LFC odel with an AR coonent for the noise. Uon obtaining a large estiate of the AR coe cient a saller banddwidth is dictaded and vice versa if the coe cient is sall. While soewhat ad hoc, it should rovide a useful guide. 7 Conclusions We roosed a local-whittle estiator of the eory araeter of a long eory tie series rocess which has good roerties under an alost colete collection of containation rocesses that have been discussed in the literature. he estiator has any advantages: no assution of Gaussianity is required unlie the tried log-eriodogra estiator; there is no triing involved so that all inforation fro the low frequency coonents are retained; when there is no LFC, its erforance is coarable to that of the standard LW estiator so that no asytotic e ciency loss is incurred, with the loss of e ciency in nite sale being sall as revealed by the siulations; with a roer choice of the bandwidth, the extended estiator has good nite sale roerties with short-run dynaics and/or additive noise; it is sei-araetric so that there is no need for a full seci cation of the underlying short-eory structure, though it can also be extended to cover a fully seci ed araetric structure for the long-eory coonent such as an ARFIMA rocess. It does, nevertheless, have soe drawbacs. First, the erforance of the estiator is sensitive to the choice of the bandwidth. An adative, data-deendant ethod to select the bandwidth is an iortant avenue for future research. Note, however, that all current seiaraetric estiators exhibit sensitivity to the bandwidth choice. Also, when the estiator is extended to account for noise, as in Hurvich et. al (2005, the RMSE is roortional to (=d 0 so that when the true araeter d 0 is close to zero the reduction in bias is o set by an increase in variance and a ossible increase in the overall RMSE. 5

17 Aendix We rst introduce three leas which show that to soe extent the seudo sectral density function controls the eriodogra of the rocess well, in the sense that the ratio ji =f j is bounded and the average of (I =f is o (. Lea A. Let A = (2 P =2 t= z t cos( t, B = (2 P =2 t= z t sin( t, so that I = (A 2 +(B 2, and de ne the vector = ((f =2 A ; (f =2 B ; (f j =2 A j ; (f j =2 B j 0. Let ( ; 2 ; 3 ; 4 denote the joint cuulant of the rando variables ; 2 ; 3 ; 4 with n ; n 2 ; n 3 ; n 4 nonnegative integers that su to n. hen under Assutions A-A5, for 0 > 0 and letting M 0 = 0 =(2 2 2d 0, for any sequences of ositive integers and j such that > j and =! 0, the following result holds for n > 2: ( n ; n 2 2 ; n 3 3 ; n 4 4 n=2 nd = O(( (n +n 3 ( d 0 j =( + M (n 2+n 4 ( d 0 0 2d 0 2 2d 0 (n +n 3 ( + M 0 2d0 j 2 2d 0 (n 2+n 4 =2 which is O( if j ( 2d 0=(2 2d 0 and o( if j > ( 2d 0=(2 2d 0. Siilarly, for n > 2, the n-th cuulant of e = (A =(f =2 ; B =(f =2 0 are O(( n=2 nd 0 = n( d0 =( + M 0 ( 2d 0 = 2 2d 0 n=2. When 0 = 0, M 0 = 0 and the result reduces to ( n ; n 2 2 ; n 3 3 ; n 4 4 = O( n=(2 nd 0 =[ (n +n 3 ( d 0 j (n 2+n 4 ( d 0 ] Proof: his lea is a direct consequence of Lea A.3 in McClosey and Perron (203, henceforth MP, and the de nition of the seudo sectral density function f. he di erence in the results is sily due to the fact that we use f = 2d 0 + ( 0 = 2, while MP use f = 2d 0. Hence, a di erent exression is obtained when 0 > 0. Lea A.2 Under A-A5, with I =!! and M 0 = 0 =(2 2 2d 0, for j < : (i E(I =f = + [O( log + O(= +2d 0 ]=[ + M 0 ( 2d 0 = 2 2d 0 ] (ii E((! 2 =f = O( log + O( 2d 0 = 2 2d 0 =[ + M 0 ( 2d 0 = 2 2d 0 ] (iii E(!! j = O( O( 2d 0 =( d 0 j d 0 log j + f f j ( + M0 ( 2d 0 = 2 2d 0( + M0 ( 2d 0 =j 2 2d 0 (iv E(!! j = O( O( 2d 0 =( d 0 j d 0 log j + f f j ( + M0 ( 2d 0 = 2 2d 0( + M0 ( 2d 0 =j 2 2d 0 Proof: For art (i, we have E(I u; =( 2 = O (. Hence, fro heore in MP, E( I f = E( I f y; f y; f = f y; f E( I f y; = f y; f E( I y; f y; + I u; f y; + 2I yu; f y; A-

18 = = For art (ii, 2d 0 2d 0 + ( 0 = 2 ( + O( log + ( 2 + M 0 2d 0 2 2d 0 3 2d0 + O( 2 2 2d 0 2d0 + M 0 ( 2d 0 = 2 2d 0 ( + M 0 + O(log + ( 2 2d O( +2d 0 = + O( log + O(= +2d 0 + M 0 ( 2d 0 = 2 2d 0 E( (! 2 = E( (! 2 f y; = f y; E( (! 2 = f f y; f f f y; For art (iii, = O( log + O( 2d0 = 2 2d0 + M 0 ( 2d 0 = 2 2d 0 + M 0 ( 2d 0 = 2 2d 0 O( log + 2d 0 2 2d 0 E(!! j!! j fy; f y;j = E( = f f j fy; f y;j f f j fy; f y;j!! j E( f f j fy; f y;j = ( + M 0 ( 2d 0 = 2 2d 0 ( + M 0 ( 2d 0 = 2 2d 0 =2 O( log j = O( log j + O( 2d0 = d0 j d0 [( + M 0 ( 2d 0 = 2 2d 0( + M0 ( 2d 0 =j 2 2d 0] =2 and the roof is entirely analogous for art (iv. + 2d 0 d 0 j d 0 Lea A.3 Under A-A5: if a = is bounded away fro zero or b there is no LFC in data, then: ji =f j is bounded, and 2 P = (I =f = o (. Proof: First, = ( I f For the rst ter, we have: = = ( I f I y; f y; + = ( I y; f y; = ( I f I y; f y; = whose rst coonent is such that, = ( I f I y; f y; + = ( I f I y; f y; = ( I f = = ( I y; f y; ( I y; f y; = = ( I y; f z; f u; f z; + I u; f z; + 2 I yu; f z; = I y; f y; + I u; f z; + 2 I yu; f z; A-2 = ( I u; f u; f z; ( I y; f y; ( f u; f z; + 2 I yu; f z;

19 Note that Ej I u; f z; f u; j = Ej( I u; f u; =( f z; f u; j = f u; f z; Ej I u; f u; Fro MP (Lea A.3 with n = n 2 = n 3 = n 4 = and = 2 = 3 = 4 = I u; =f y; : j Ej I u; f u; j Ej I u; f u; j + [E(j I u; f u; j 2 ] =2 + C So Ej = ( I u; f u; j f z; = j f u; f z; jej I u; f u; j C! 0 if =! 0. We also have Ej P = (I y; =f y; (f u; =f z; j! 0, since jf u; =f z; j <. Fro MP, Perron and Qu (200 and Qu (2008: I yu ( = O ( =2 (+d 0 and f = f z; = f y; + f u; = G 2d 0 + G u 2. Hence, and j I yu; f z; j s Ej 2 O ( =2 (+d 0 O ( 2d 0 + O ( 2 s O ( =2 d 0 + O ( =2 d 0 = if =! 0. Hence, I yu; f z; j 2 = Ej I yu; j < 2 O ( = O ( f z;! 0 < O ( Ej Ej = = ( I f I y; f y; j = Ej ( I u; f u; j + Ej f z; = = ( I u; f u; f z; ( I y; f y; ( I y; f y; ( f u; f z; j + Ej 2 ( f u; f z; + 2 I yu; f z; j = I yu; f z; j! 0 if =! 0. It is easy to show that Ej P = (I =f I y; =f y; j! 0, and the fact that Ej P = (I y;=f y; j! 0 follows fro Hurvich et. al. (2005. So Ej = ( I f j Ej = ( I f I y; f y; j + Ej = ( I f I y; f y; j + Ej = ( I y; f y; j! 0 if =! 0. Note that during the roof we also showed that ji =f j ji =f I y; =f y; j + ji y; =f y; j + is bounded. A-3

20 Proof of Lea : Let M = ^ =(2 2 2 ^d and M 0 = 0 =(2 2 2d 0. We analyze the artial derivative of the objective function with resect J ( ^d ; ^ = [ g ( ^d ; ^ 2 ( I g ( ^d ; ^ I (g ( ^d ; ^ 2 2 ] = = [ 2 I G 0 ( = G 0 g ( ^d ; ^ P j= (I j=g j ( ^d ; ^ ] 2 ^d + (^ = 2 = [ I G 0 ( f = P g (d 0 ; 0 j= (I j=g j ( ^d ; ^ g ( ^d ; ^ 2 ] 2 ^d + (^ = 2 = f 2 ( [ ( I ( 2d0 + ( 0 = 2 = 2 ^d + (^ = 2 f 2 ^d + (^ = 2 (A. I j n ( 2d0 j + ( 0 = 2 j ]g f j= j 2 ^d j + (^ = 2 j Using suation by arts, (A. becoes: 2 f ( [ ( I 2 ^d 2d 0 = 2 ^d + (^ = 2 ( + M 0( 2d 0 = 2 2d 0 f + M ( 2d = 2 2d I j n( 2 ^d 2d 0 j ( + M 0( 2d 0 = 2 2d 0 ]g f j + M ( 2 ^d = 2 2 ^d = ( j= 2 = ( 2 ^d + (^ = 2 n( 2 ^d j= [( + f n( j= j = = = I j f j 2 ^d 2d 0 j ( + M 0( 2d 0 = 2 2d 0 + M ( 2 ^d = 2 2 ^d ] 2 j 2 ^d j + (^ = 2 j = [ ( I f 2 ^d 2d 0 ( + M 0( 2d 0 = 2 2d 0 + M ( 2 ^d = 2 2 ^d ( 2 j+ 2 ^d j+ + (^ = 2 j+ ( ( I 2 ^d 2d 0 ( + M 0( 2d 0 = 2 2d 0 f + M ( 2 ^d = 2 2 ^d = 2 I f 2 ^d 2d 0 ( + M 0( 2d 0 = 2 2d 0 + M ( 2 ^d = 2 2 ^d ]g [ + (^ = 2 = I f 2 ^d 2d 0 ( + M 0( 2d 0 = 2 2d 0 + M ( 2 ^d = 2 2 ^d n( A-4 j= ] I j f j 2 ^d 2d 0 j ( + M 0( 2d 0 = 2 2d 0 + M ( 2 ^d = 2 2 ^d ]

21 [( + j= [j n( = 2 j 2 ^d j + (^ = 2 j j = ( 2 j+ 2 ^d j+ + (^ = 2 j+ ( I f 2 ^d 2d 0 ( + M 0( 2d 0 = 2 2d 0 + M ( 2 ^d = 2 2 ^d I f 2 ^d 2d 0 ( + M 0( 2d 0 = 2 2d 0 + M ( 2 ^d = 2 2 ^d ] Now, suose ^! 0 with (@=@J ( ^d ; ^ = 0. We de ne h [+M 0 ( 2d 0 = 2 2d 0 ]=[+ M ( 2 ^d = 2 2 ^d ]. We consider two cases. In the rst, suose M! 0 at a slow rate such that for soe sall ; we still have M ( 2 ^d = 2 2 ^d!. Let = inf fm ( 2 ^d = 2 2 ^d! 0g, then 8 < h : M 0 M ( 2( ^d d 0 + M ( 2 ^d= 2 2 ^d when + M 0 ( 2d 0 = 2 2d 0 when > Note that we ust have either (M 0 =M (= 2( ^d d 0 or M 0 ( 2d 0 = 2 2d 0 go to in nity for soe sall. Also, M 0 ( 2d 0 = 2 2d 0 = M 0 M ( 2( ^d d 0 (M ( 2( ^d d 0 2d0 2 2d 0 when >. Hence, 8 2 ^d 2d 0 ( + M 0( 2d 0 = 2 2d 0 < s + M ( 2 ^d = 2 2 ^d : Let = M 0 M ( 2( ^d d 0 (M 2 ^d 2 2 ^d = o ( M 0 M ( 2( ^d d 0 ] (M 0 =M when o (M 0 =M when > a j = I f 2 ^d 2d 0 ( + M 0( 2d 0 = 2 2d 0 + M ( 2 ^d = 2 2 ^d then we now that a j = O (M 0 =M when and a j = o (M 0 =M when >. So, fa j g is a ositive sequence whose rst few ters have higher order than the rest. So we have (j= P j = ( I f 2 ^d 2d 0 ( +M 0( 2d 0 = 2 2d 0 +M ( n( P I = f 2 ^d 2d 0 ( +M 0( 2d 0 = 2 2d 0 +M ( 2 ^d= 2 2 ^d C j < 0 A-5 2 ^d= 2 2 ^d (A.2

22 where C j is soe constant. Under the second case, M! 0 fast enough so that, for any, M ( 2 ^d = 2 2 ^d O (. For this case, h + M 0 ( 2d 0 = 2 2d 0 and 2 ^d 2d 0 ( + M 0( 2d 0 = 2 2d 0 + M ( 2 ^d = 2 2 ^d 2 ^d 2d 0 ( + M 0 ( 2d 0 = 2 2d 0 (= 2d 0 2 ^d + M 0 ( 2 ^d = 2 2 ^d If d 0 ^d, the last exression is decreasing in for all = ; :::; ; if d 0 < ^d, the rst is increasing in ; but always saller than, and the second is decreasing in and goes to in nity when is sall. Hence, (A.2 still holds. Since for large enough, 2 j ( 2 ^d j + (^ = 2 j j+( 2 2 ^d j+ + (^ = j+ 2 D j > 0 where D j is soe constant, we have shown that (@=@J ( ^d ; ^ < 0 if ^! 0, which is a contradiction. So ^ has to be bounded fro zero when 0 > 0. Proof of heore : First, we consider the case when LFC indeed exists in the true DGP. he roof for the case with no LFC will follow with trivial odi cations. Note that if LFC coonents are resent, the robability liit of the estiate ^ is bounded above zero, by Lea. his ilies that with robability arbitrarily close to one, ^ will be in the set (0; and, without loss of generality, we can consider analyzing the liit of ^d for any sequence or values of in the set (0;. Accordingly, we want to show that, with robability arbitrarily close to one for large and, if f g is a sequence bounded above fro zero and if f ^d g iniizes J (d; given f g, then for ^d such that j ^d d 0 j for any > 0, we have J ( ^d ; J (d 0 ; > 0, which delivers a contradiction showing that in the liit the iniizer of J (d; ust converge to d 0. Let G(d; = P = I =g, where g = ( 2d + ( = 2. We rst have: J ( ^d ; J (d 0 ; = [log G( ^d ; + [log G(d 0 ; + = = log( 2 ^d ( ^d log( 2 ^d ( + 2+2d 0 ] = log G( ^d ; log G(d 0 ; + G( = log ^d ; G 0 ( P = 2( ^d d 0 2( ^d d 0 + = = = log G(d 0; G 0 log( 2( ^d d 0 + log( log( + ( = 2+2 ^d + ( = 2+2d 0 A-6 ( + ( = 2+2 ^d + ( = 2+2d 0 = 2( ^d d 0

23 G( = log ^d ; G 0 ( P ^d = 2( d 0 log(2( ^d d 0 + 2( ^d d 0 [ (log G( = log ^d ; G 0 ( P 2( ^d d 0 [ = ^d = 2( d 0 log G(d 0; G 0 + log( 2( ^d d 0 (2( ^d d 0 + log ] + log (log ] + = log( + 2( ^d d 0 + 2( ^d d 0 = log( + ( = 2+2 ^d + ( = 2+2d 0 log G(d 0; + log( 2( ^d d 0 + G 0 = ( 2( ^d d 0 = log( + ( = 2+2 ^d (A.3 + ( = 2+2d 0 Note that for the last ter of (A.3, we have log(+2( ^d d 0 +2( ^d d 0 (=6( ^d d 0 2 (=6 2. Hence, if we can show that the other ve ters are o (, we can derive a contradiction. he third and fourth are o ( fro Robinson (995a. Proof that the rst ter of (A.3 is o (. o show that, it is equivalent to rove that G( ^d ; =[G 0 ( P ^d = 2( d 0 ] = o (. o that e ect, G( ^d ; G 0 ( P = 2( ^d d 0 = (G 0 ( = ( = ( = = = ( = 2( ^d d 0 2( ^d d 0 2( ^d d 0 f f f [ = = ( 2( ^d d 0 f = = I 2 ^d ( + ( = 2+2 ^d I ^d G 0 2d 0 ( + ( = 2+2 ^d 2( d0 G 0 = 2( ^d d 0 2( ^d d 0 [ I + ( 0 = 2+2d0 ]g f + ( = 2+2 ^d ( 2( ^d d 0 ( I + ( 0 = 2+2d0 f + ( = 2+2 ^d = ]g g 2( ^d d 0 g Fro Robinson (995a, P = (=2( ^d d 0 = o ( if ^d d 0 6= =2. When, I + ( = 2+2d0 f + ( = 2+2 ^d = ( I f + I f O(( 2d 0 : Fro Lea A.2, P = (I =f = o (, hence P = I f = = ( I f A-7 I y; f y; + = = I =f = + o (. Also, I y; f y;

24 he rst ter is o ( fro Lea A.3 and the second is o ( = o ( fro Hurvich et. al. (2005. Cobining these results, we have P = I =f = + o (. Now, = = + ( 2( ^d d 0 ( I + ( 0 = 2+2d0 f + ( = 2+2 ^d ( 2( ^d d 0 ( I + ( f 2( ^d d 0 O( ( 2 2d 0 I f = = = = ( 2( ^d d 0 ( I f = I y; f y; + ( 2 ^d O( 2 ( 2d 0 ( I f = ( 2( ^d d 0 ( I y; f y; (A.4 I y; f y; + = ( 2 ^d O( 2 ( 2d 0 I y; f y; We will show that all four ters of (A.4 are o ( by showing that the exectations of their absolute values are o (. For the rst ter, we have fro Lea A.3, E(j = ( 2( ^d d 0 ( I f I y; f y; j = ( 2( ^d d 0 Ej( I f I y; f y; j Fro Lea A., we now that Ej(I =f I y; =f y; j C(= d 0 C(= d 0, where C is soe constant not deending on and. We also have ( 2( ^d d 0 = = ( 2( ^d d 0 ( = 2( ^d d 0 = ( +2( ^d d 0 ( 2( ^d d 0 = O (! 0 = where the last equality is fro Robinson (995a, eq. (3.7. Hence, P = (=2( ^d d 0 =( + 2( ^d d 0, which shows that the rst ter is o (. For the second ter, Ej = ( 2( ^d d 0 ( I y; f y; j = Ej = ( 2( ^d d 0 ( I y; f y; = ( 2( ^d d 0 ( I y; f y; j Ej = ( 2( ^d d 0 ( I y; j + ( f y; +2( ^d For the third ter, E(j ( 2 ^d O( ( 2 2d 0 ( I f = d 0 Ej I y; f y; j = Ej A-8 = = ( 2( ^d d 0 ( I y; f y; j = o ( ( 2 ^d O( 2d 0 ( I f I y; f y; j