Nonlinear Log-Periodogram Regression for Perturbed Fractional Processes

Transcription

1 Nonlinear Log-Periodogra Regression for Perturbed Fractional Processes Yixiao Sun Departent of Econoics Yale University Peter C. B. Phillips Cowles Foundation for Research in Econoics Yale University First Draft: October 2000 This Version: October 200 The authors thank Feng Zhu for his careful reading of the first draft and helpful coents. Sun acknowledges the Cowles Foundation for support under a Carl Anderson fellowship. Phillips thanks the NSF for support under Grant No. SES Address correspondence to: Yixiao Sun, Departent of Econoics, Yale University, P.O. Box , New Haven, CT , USA; Tel.: ; Eail: yixiao.sun@yale.edu.

2 ABSTRACT This paper studies fractional processes that ay be perturbed by weakly dependent tie series. The odel for a perturbed fractional process has a coponents fraework in which there ay be coponents of both long and short eory. All coonly used estiates of the long eory paraeter such as log periodogra LP regression ay be used in a coponents odel where the data are affected by weakly dependent perturbations, but these estiates suffer fro serious downward bias. To circuvent this proble, the present paper proposes a new procedure that allows for the possible presence of additive perturbations in the data. The new estiator resebles the LP regression estiator but involves an additional nonlinear ter in the regression that takes account of possible perturbation effects in the data. Under soe soothness assuptions at the origin, the bias of the new estiator is shown to disappear at a faster rate than that of the LP estiator, while its asyptotic variance is inflated only by a ultiplicative constant. In consequence, the optial rate of convergence to zero of the asyptotic MSE of the new estiator is faster than that of the LP estiator. Soe siulation results deonstrate the viability and the bias-reducing feature of the new estiator relative to the LP estiator in finite saples. JEL Classification: C3; C4; C22; C5 Keywords: Asyptotic bias; Asyptotic norality; Bias reduction; Fractional coponents odel; Perturbed fractional process; Rate of convergence

3 Introduction Fractional processes have been gaining increasing popularity with epirical researchers in econoics and finance. In part, this is because fractional processes can capture fors of long run behavior in econoic variables that elude other odels, a feature that has proved particularly iportant in odelling inter-trade durations and the volatility of financial asset returns. In part also, fractional processes are attractive to epirical analysts because they allow for varying degrees of persistence, including a continuu of possibilities between weakly dependent and unit root processes. For a pure fractional process, short run dynaics and long run behavior are driven by the sae innovations. This ay be considered restrictive in that the innovations that drive long run behavior ay arise fro quite different sources and therefore differ fro those that deterine the short run fluctuations of a process. To accoodate this possibility, the odel we consider in the present paper allows for perturbations in a fractional process and has a coponents structure that introduces different sources and types of variation. Such odels provide a echanis for siultaneously capturing the effects of persistent and teporary shocks on the realized observations. They see particularly realistic in econoic and financial applications when there are any different sources of variation in the data and both long run behavior and short run fluctuations need to be odeled. Specifically, a perturbed fractional process z t is defined as a fractional process y t that is perturbed by a weakly dependent process u t as follows where µ is a constant and y t = L d 0 w t = z t = y t + µ + u t, t =, 2,..., n, k=0 Γd 0 + k Γd 0 Γk + w t k, 0 < d 0 < /2. 2 Here, y t is a pure fractional process and u t and w t are independent Gaussian processes with zero eans and continuous spectral densities f u λ and f w λ, respectively. We confine attention to the case where the eory paraeter d 0 0, 2 largely for technical reasons that will becoe apparent later. The case is certainly the ost relevant in epirical practice, at least for stationary series, but the restriction is an iportant one. To aintain generality in the short run coponents of z t we do not ipose specific functional fors on f u λ and f w λ. Instead, we allow the to belong to a faily that is characterized only by regularity conditions near the zero frequency. This forulation corresponds to the conventional seiparaetric approach to odelling long range dependence. By allowing for the presence of two separate stochastic coponents, the odel captures echaniss in which different factors ay coe into play in deterining long run and short run behaviors. Such echaniss ay be expected to occur in the generation of acroeconoic and financial data for several reasons. For exaple, tie series observations of acroeconoic processes often reflect short run copetitive forces as well as long run growth deterinants. Additionally, econoic

4 and financial tie series frequently arise fro processes of aggregation and involve errors of easureent, so that the presence of an additive, short eory disturbance is quite realistic. For instance, if the underlying volatility of stock returns follows a fractional process, then realized volatility ay follow a perturbed fractional process because the presence of a bid-ask bounce adds a short eory coponent to realized returns, with consequent effects on volatility. Soe epirical odels now in use are actually special cases of perturbed fractional processes. Aong these, the long eory stochastic volatility odel is growing in popularity for odelling the volatility of financial tie series see Breidt, Crato and De Lia, 998, and Deo and Hurvich, 999. This odel assues that log rt 2 = y t + µ + u t, where r t is the return, y t is an underlying fractional process and u t = iid0, σ 2, thereby coing within the fraework of. Another exaple is a rational expectation odel in which the ex ante variable follows a fractional process, so that the corresponding ex post variable follows with u t being a artingale difference sequence. Sun and Phillips 2000 used this fraework to odel the real rate of interest and inflation as perturbed fractional processes and found that this odel helped explain the epirical incopatibility of eory paraeter estiates of the coponents in the ex post Fisher identity. The study by Granger and Marol 997 provides a third exaple, addressing the frequently observed property of financial tie series that the autocorrelogra can be low but positive for any lags. Granger and Marol explained this phenoenon by considering tie series that consist of a long eory coponent cobined with a white noise coponent that has a uch larger variance, again coing within the fraework of. The ain obect in the present paper is to develop a suitable estiation procedure for the eory paraeter d 0 in. As we will show, existing procedures for estiating d 0 typically suffer fro serious downward bias in odels where there are additive perturbations like. The present paper therefore proposes a new procedure that allows for the possible presence of such perturbations in the data. The spectral density f z λ of z t can be written as f z λ = 2 sin λ 2 2d 0 f λ, where f λ = f w λ+2 sin λ 2 2d 0 f u λ is a continuous function over [0, π]. So, f z λ satisfies a power law around the origin of the for f z λ G 0 λ 2d 0 as λ 0+, for soe positive constant G 0. Therefore, we can estiate d 0 by using the linear log-periodogra LP regression introduced by Geweke and Porter-Hudak 983. Building on the earlier work of Künsch 986, Robinson 995a established the asyptotic norality of the LP estiator. Subsequently, Hurvich, Deo and Brodsky 998 hereafter HDB coputed the ean square error of the LP estiator and provided an MSE-optial rule for bandwidth selection. The LP estiator has undoubted appeal. It is easy to ipleent in practice and has been coonly eployed in applications. However, when the spectral density of u t doinates that of w t in a neighborhood of the origin, the estiator ay be biased downward substantially, especially in sall saples. One source of the bias is the error of approxiating the logarith of f λ by a constant in a shrinking neighborhood of the origin. This crude approxiation also restricts the rate of convergence. The rate of convergence of the LP estiator will be shown to be n 2d 0/4d 0 +, which 2

5 is quite slow, especially when d 0 is close to zero. To alleviate these probles, we take advantage of the structure of our odel and propose to estiate the logarith of f λ locally by c + βλ 2d 0. Our new estiator d is defined as the iniizer of the su of the squared regression errors in a regression of the for log I z = α 2d log λ + βλ 2d + error, =, 2,...,, 3 where I z = I z λ = n 2πn z t expitλ 2, λ = 2π n, 4 t=0 and is a positive integer saller than the saple size n. The new estiator can be seen as a way of utilizing paraetric inforation in a nonparaetric setting. We approxiate the unknown function locally by a nonlinear function instead of a constant. One otivation for the nonlinear LP regression estiator is the local nonlinear least square estiator in the nonparaetric literature. Linton and Gozalo 2000 found that the local nonlinear estiator had superior perforance copared to the usual kernel estiator when the local nonlinear paraeterization is close to the unknown function. Analogously, we expect the nonlinear log periodogra regression estiator to work well in the presence of perturbations, especially when the perturbations are relatively large. In this paper we investigate the asyptotic and finite saple properties of d. We deterine its asyptotic bias, variance, asyptotic ean squared error AMSE, and asyptotic norality, and we calculate the AMSE optial choice of bandwidth and its plug-in version. In the presence of the weakly dependent coponent, we find that the asyptotic bias of d is of order 4d 0 /n 4d 0, provided that f w and f u are boundedly differentiable around the origin, whereas that of the LP estiator d LP has the larger order 2d 0 /n 2d 0. The asyptotic variances of d and d LP are both of order. In consequence, the optial rate of convergence to zero of d is of order n 4d 0/8d 0 +, whereas that of d LP is of the larger order n 2d 0/4d 0 +. We find that d is asyptotically noral with ean zero, provided that 8d0+ /n 8d 0 0, whereas d LP is asyptotically noral only under the ore stringent condition 4d0+ /n 4d 0 0. Soe Monte Carlo siulations show that the asyptotic results of the paper iic the finite saple properties of the new estiator quite well. For the fractional coponent processes considered in the siulations, the new estiator d has a lower bias, a higher standard deviation, and a lower RMSE copared to the LP estiator d LP, as the asyptotic results suggest. The lower bias leads to better coverage probabilities for d over a wide range of than for d LP. On the other hand, the lower standard deviation of d LP leads to shorter confidence intervals than confidence intervals based on d. The paper by Andrews and Guggenberger 999 is ost related to our work. They considered the conventional fractional odel i.e., varu t = 0 and proposed to approxiate log f w λ by a constant plus a polynoial of even order. Andrews and Sun 2000 investigated the sae issue in the context of a local Whittle estiator. Other related papers include Henry and Robinson 996, Hurvich and Deo 999 3

6 and Henry 999. These papers consider approxiating log f λ by a ore sophisticated function than a constant for the purpose of obtaining a data-driven choice of. The present paper differs fro those papers in that a nonlinear approxiation is used in order to achieve bias reduction and to increase the rate of convergence in the estiation of d 0. Also, the nonlinear polynoial function used here depends on the eory paraeter d 0 whereas this is not so in the work ust entioned and the estiation procedure for d 0 utilizes this inforation. The rest of the paper is organized as follows. Section 2 forally defines the new estiator. Section 3 outlines the asyptotics of discrete Fourier transfors and log-periodogra ordinates, which are used extensively in later sections. Section 4 establishes consistency and derives asyptotic norality results for the new estiator. Asyptotic bias, asyptotic MSE, and bandwidth selection are also considered. Section 5 investigates the finite saple perforance of the new estiator by siulations. Proofs are collected in the Appendix. 2 Nonlinear Log Periodogra Regression This section otivates a new estiator that explicitly accounts for the additive perturbations in. Throughout, is taken as the data generating process and then Taking the logariths of 5 leads to f z λ = 2 sin λ 2 2d 0 f λ. 5 logf z λ = 2d 0 log λ + log f λ 2d 0 log2λ sin λ. 6 2 Replacing f z λ by periodogra ordinates I z λ evaluated at the fundaental frequencies λ, =, 2,..., yields logi z = c 0 2d 0 log λ + log f λ + U + Oλ 2, 7 where c 0 = is the Euler constant and U = log[i z λ /f z λ ] + c 0. By virtue of the continuity of f λ, we can approxiate log f λ by a constant over a shrinking neighborhood of the zero frequency. This otivates log-periodogra regression on the equation logi z = constant 2d log λ + error. 8 The LP estiator d LP is then given by the least squares estiator of d in this regression. If {U } = behave asyptotically like independent and identically distributed rando variables, then the LP estiator is a reasonable choice. In fact, under assuptions to be stated below, we establish that d LP d 0 Nb LP, π2 24 where b LP = O 2d 0 + /n 2d 0 and signifies asyptotically distributed. The asyptotic bias of d LP itself is therefore of order O 2d 0 /n 2d 0, which can be quite large. 4

7 To reduce the bias, we can approxiate log f λ by a siple nonlinear function of frequency under the following assuptions: Assuption : Either a σ u = varu t = 0 for all t, so f u λ 0, for λ [ π, π] or: b σ u 0 and f u λ is continuous on [ π, π], bounded above and away fro zero with bounded first derivative in a neighborhood of zero. Assuption 2: f w λ is continuous on [ π, π], bounded above and away fro zero. When σ u = 0, f w λ is three ties differentiable with bounded third derivative in a neighborhood of zero. When σ u 0, f w λ is differentiable with bounded derivative in a neighborhood of zero. Assuptions b and 2 are local soothness conditions and hold for any odels in current use, including ARMA odels. They allow us to develop a Taylor expansion of log f λ about λ = 0 with an error of the order of the first oitted ter. Specifically, when σ u = 0, When σ u 0, So, in either case log f λ log f λ = log f w 0 + Oλ 2. 9 = log f w λ + log[ + 2 sin λ f 2 2d 0 uλ f w λ ] { } = log f w λ + log + λ 2d 0 + Oλ 2 fu 0 f w 0 + Oλ2 = log f w 0 + f u0 f w 0 λ2d 0 + Oλ 4d 0. 0 log f λ = log f w 0 + f u0 f w 0 λ2d 0 + Oλ r where O holds uniforly over =, 2,..., and r = 4d 0 {σ u 0} + 2{σ u = 0}. Cobining 7 with produces the nonlinear LP regression odel: where logi z = 2d 0 log λ + α 0 + λ 2d 0 β 0 + U + ε, 2 α 0 = log f w 0 c 0, β 0 = f u 0/f w 0, and ε = log f λ log f w 0 β 0 λ 2d 0 2d 0 [log2 sin λ 2 log λ ]. 3 The new estiator is then defined as the iniizer of the su of squared regression errors in this odel, i.e. ˆα, d, β = arg in SSEα, d, β, 4 α,d,β 5

8 where SSEα, d, β = = Concentrating 5 with respect to α, we obtain [logi z α + 2d log λ λ 2d β] 2. 5 d, β = arg in Qd, β, 6 d D,β B with Qd, β = {log I z = +2dlog λ log I zk k= k= log λ k βλ 2d k= λ 2d k }2. 7 where B is a copact and convex set, D = [d, d 2 ] is a closed interval of adissible values for d 0 with 0 < d < d 2 < /2. Here d and d 2 can be chosen arbitrarily close to 0 and /2, respectively. We write θ = d, β, Θ = D B for convenience and assue the true value of θ lies in the interior of the adissible set. 3 Log-periodogra Asyptotics and Useful Leas To establish the asyptotic properties of the new estiator, we need to characterize the asyptotic behavior of the log-periodogra ordinates U = log[i z λ /f z λ ] + c 0. Define A z = n z t cos λ t and B z = n z t sin λ t, 8 2πn 2πn t=0 t=0 then U = ln A 2 z f z + B2 z f z + c 0, =,...,. 9 In view of the Gaussianity of A z and B z, we can evaluate the eans, variances, and covariances of U, if the asyptotic behavior of the vector A z /f /2 z, B z/f /2 z, A zk/f /2 zk, B zk/f /2 zk is known. The properties of this vector depend in turn on those of the discrete Fourier transfors of z t, defined as w λ = 2πn /2 n z te itλ. The asyptotic behavior of w λ is given in the following lea which is a variant of results given earlier by several other authors Robinson, 995a, HDB, 998, Andrews and Guggenberger, 999. Lea Let Assuptions and 2 hold. Then uniforly over and k, k <, /n 0, 6

9 a E [w λ w λ /f z λ ] = + O log, b E [w λ w λ /f z λ ] = O log, [ c E w λ w λ k / f z λ f z λ k /2] = O k log, d [ E w λ w λ k / f z λ f z λ k /2] = O k log. It follows directly fro Lea that for k <, EA 2 z/f z = 2 + Olog EA z B z /f z = O log, EBz/f 2 z = 2 + Olog,, EA z B zk /f z f zk /2 = O log. 20 k Using these results and following the sae line of derivation as in HDB 998, we can prove Lea 2 below. Since the four parts of this lea are proved in a siilar way to Leas 3, 5, 6 and 7 in HDB, the proofs are oitted here. Lea 2 Let Assuptions and 2 hold. Then a Cov U, U k = O log 2 /k 2, uniforly for log 2 k <, b li n sup EU 2 <, c E U = O log /, uniforly for log 2, d V ar U = π 2 /6 + O log /, uniforly for log 2. With the asyptotic behavior of U in hand, we can proceed to show that the noralized sus = c U are uniforly negligible under certain conditions on the coefficients c. Quantities of this for appear in the noralized Hessian atrix below. Lea 3 Let {c d, β} = be a sequence of functions such that, for soe p 0, and for soe q 0, sup c = Olog p uniforly for, 2 d,β Θ sup c c = O log q uniforly for. 22 d,β Θ Then sup d,β Θ c U = O p log axp,q. 23 = 7

10 We can ipose additional conditions to get a tighter bound. For exaple, if we also require that sup d,β Θ c = O, then sup d,β Θ = c U = Op logq, as is readily seen fro the proof of the lea. Further, the lea reains valid if we reove the sup operator fro both the conditions and the conclusion. The following lea assists in establishing the asyptotic norality of the nonlinear log-periodogra regression estiator. Lea 4 Let a kn = a k be a triangular array for which ax a k = o, k k=+ 0.5+δ a 2 k ρ, k=+ 0.5+δ a k p = O, 24 for all p, and 0 < δ < 0.5. Then, k=+ 0.5+δ a k U k d N 0, π2 6 ρ. 25 The proof of this lea is based on the ethod of oents and involves a careful exploration of the dependence structure of the discrete Fourier transfors. Robinson s arguent 995a, pp fors the basis of this developent and can be used here with soe inor odifications to account for differences in the odels. Details are oitted here and are available upon request. 4 Consistency, Asyptotic Norality and Bandwidth Choice We first establish asyptotic properties for the LP estiator in the context of the coponents odel. The following theore gives the liit theory and provides a benchark for later coparisons. Theore Let Assuptions and 2 hold. Let = n and 2r + n 2r K σ {σ u 0} + K 0 {σ u = 0} 26 as n, where r = 2d 0 {σ u 0}+2{σ u = 0} and K σ, K 0 > 0 are constants. Then where dlp d 0 Nb LP, π2, b LP = 2π 2d f 0 u0 d 0 f w 0 2d K σ{σ u 0} 2π2 9 f w0 f w 0 + d 0 K 0 {σ u = 0}

11 When σ u 0, the ratio 2r + /n 2r = 4d0+ /n 4d 0 K σ in 26. This delivers an upper bound of order On 4d 0/+4d 0 on the rate at which can increase with n and allows for larger choices of for larger values of d 0. Intuitively, as d 0 increases, the containation fro perturbations at frequencies away fro the origin becoes relatively saller and we can expect to be able to eploy a wider bandwidth in the regression. To eliinate the asyptotic bias b LP in 27 altogether, we use a narrower band and set = on 4d 0/+4d 0 in place of 26. Deo and Hurvich 999 established a siilar result under the assuption that u t is iid, but not necessarily Gaussian. Their assuption that 4d0+ log 2 /n 4d 0 = o is slightly stronger than the assuption ade here. When σ u 0, the liit distribution 27 involves the bias d 0 b LP = 2π 2d f 0 u0 f w 0 2d K σ < 0, 29 which is always negative, as one would expect, because of the effect of the short eory perturbations. Correspondingly, the doinating bias ter of d LP has the for b n,lp = 2π 2d f 0 u0 d 0 2d0 f w 0 2d < The agnitude of the bias obviously depends on the quantity f w 0/f u 0, which is the ratio of the long run variance of the short eory input of y t to that of the perturbation coponent u t. The ratio can be interpreted as a long run signal-noise ratio SNR, easuring the strength in the long run of the signal fro the y t inputs relative to the long run signal in the perturbations. The stronger the long run signal in the perturbations, the greater the downward bias and the ore difficult it becoes to estiate the eory paraeter accurately. One ight expect these effects to be exaggerated in sall saples where the capacity of the data to discriinate between long run and short run effects is reduced. When σ u = 0, the theore contains essentially the sae results proved in HDB. In this case, the doinating ter in the bias of d LP is given by b n,lp = 2π2 9 f w0 f w 0 + d 0 6 n 2d 0 2 n 2. 3 HDB showed that the doinating bias of d LP in the case of pure fractional process regression is given by the expression 2π2 f w0 9 f w 0 n The presence of the additional factor d 0 /6 in the second ter of our expression 3 arises fro the use of a slightly different regressor in the LP regression. In particular, we eploy 2 log λ as one of the regressors in 3, while HDB use 2 log2 sin λ /2. These regressors are norally considered to be asyptotically equivalent. However, while the use of 2 log λ rather than 2 log2 sin λ /2 has no effect on the asyptotic variance, it does affect the asyptotic bias. 9

12 We now investigate the asyptotic properties of the nonlinear log-periodogra regression estiator. Theore 2 Let Assuptions and 2 hold. a If + n 0 as, n, then d d 0 = o p. b If for soe arbitrary sall > 0, n + d d 0 = O p 0 n 2d and β β0 = o p. n4d 0 + 4d , as, n, then Theore 2 shows that d is consistent under ild conditions. All that is needed is that approaches infinity slower than the saple size n. As shown by HDB, triing out low frequencies is not necessary. This point is particularly iportant in the present case. In seeking to reduce containations fro the perturbations, the lowest frequency ordinates are the ost valuable in detecting the long eory effects. It is not straightforward to establish the consistency of β, because, as n, the obective function becoes flat as a function of β. The way we proceed is, in fact, to show first that d converges to d 0 at soe slower rate, ore precisely, d d 0 = O p n 2d 0. We prove this rate of convergence stepwise. We start by showing that d d 0 = o p n d/2 for 0 < d < d 0, using the fact that βλ 2d = O n 2d uniforly in d, β Θ. We can then deduce that d d 0 = o p n d0+. With this faster rate of convergence, we have better control over soe quantities and can obtain an even faster rate of convergence for d. Repeating this procedure leads to d d 0 = O p n 2d 0, as desired. The idea of the proof ay be applicable to other nonlinear estiation probles when the involved variables are integrated of different orders or have different stochastic orders. We prove the rate of convergence of d without using the consistency of β. This is unusual because in ost nonlinear estiation probles it is coon to prove the consistency of all paraeters first in order to establish rates of convergence. The approach is successful in the present case because when d is close to d 0, the regressor λ 2d evaporates as n and approaches zero approxiately at the rate of /n 2d 0. We proceed to show that under soewhat stronger conditions and if suitably noralized, d d 0, β β 0 is asyptotically noral. follows: The new assuption is as Assuption 3: n 4d 0+ / 4d for soe arbitrary sall > 0 and 2r+ /n 2r = O as, n, where r = 4d 0 {σ u 0} + 2{σ u = 0}. The two conditions in Assuption 3 are always copatible because r 4d 0 and is arbitrarily sall. The lower bound on the growth rate of ensures the consistency of d and β, which validates the use of the first order conditions. The upper bound on the growth rate of guarantees that the noralized gradient of Qd, β is O p, which is required for the asyptotic norality of d, β. 0

13 When σ u = 0, the upper bound becoes 5 /n 4 = O, which is the sae as the upper bound for asyptotic norality of the LP estiator for a pure fractional process. When σ u 0, the upper bound becoes 8d0+ /n 8d 0 = O, which is less stringent than the upper bound given in Theore. It therefore allows us to take larger than in conventional LP regression applied to the fractional coponents odel. In consequence, by an appropriate choice of, we have asyptotic norality for d with a faster rate of convergence than is possible in LP regression. However, for any 0 < d 0 < /2, the upper bound is ore stringent than = On 4/5, the upper bound for asyptotic norality of LP regression in a pure fractional process odel. Hence, the existence of the weakly dependent perturbations in requires the use of a narrower bandwidth than LP regression for a pure fractional process. Interestingly, as d 0 approaches /2, the upper bound becoes arbitrarily close to = On 4/5. We now proceed to establish asyptotic norality. The first order conditions for 6 are: S n d, β = 0, 33 where e d, β = log I z S n d, β = = x d, β x d, β x 2 d, β x 2 d, β x d, β = 2 log λ βλ 2d, x d, β = x 2 d, β = λ 2d, x 2 d, β = e d, β, 34 x k, k= x 2k, and 35 k= log I zk +2dlog λ k= Expanding S n d, β about S n d 0, β 0, we have log λ k k= βλ 2d k= βλ 2d k 0 = S n d 0, β 0 + H n d 0, β 0 d d 0, β β 0 + [H n H n d 0, β 0 ] d d 0, β β 0, 37 where H n is the Hessian atrix, H n is the Hessian evaluated at d, β, the ean values between d 0, β 0 and d, β. The eleents of the Hessian atrix are:. 36 H n, d, β = H n,2 d, β = H n,22 d, β = x x 2 β = = x x x 2 x 2 = x 2 x 2 2. = e log λ 2 2 λ 2d, = e log λ 2 λ 2d, 38

14 Define the diagonal atrix D n = diag, λ 2d 0. We show in the following lea that the noralized Hessian Dn H n d 0, β 0 Dn converges in probability to a 2 2 atrix defined by Ω = 4 4d 0 2d d 0 2d d 2 0 4d 0 +2d 0 + 2, 39 and the asyptotic bias of the noralized score Dn S n d 0, β 0 is b n, where and b n = {σ u 0} /2 λ 4d 0 b n + {σ u = 0} /2 λ 2 b 2n, 40 b n = f w0 2 8d 0 2fu0 2 b 2n = 4d d 2 0 2d 0 +4d 0 +6d 0 + f w0 f w 0 + d d d 0 +32d 0 +,. 4 Before stating the lea, we need the following notation. Let J n d, β be a 2 2 atrix whose i, -th eleent is J n,i = x ik d, β x i d, β x k d, β x d, β, 42 k= and let Θ n be a set defined by Θ n = {d, β : λ d 0 d d 0 < ε and β β 0 < ε}. 43 Lea 5 Let Assuptions -3 hold. We have a sup d,β Θn Dn H n d, β J n d, βdn = o p, b sup d,β Θn Dn [J n d, β J n d 0, β 0 ]Dn = o p, c Dn J n d 0, β 0 Dn Ω, d Dn S n d 0, β 0 + b n N0, π2 6 Ω. Theore 3 Let Assuptions, 2 and 3 hold, then D n d d0 β β 0 Ω b n N0, π2 6 Ω 44 where Ω = [ 6d 3 0 2d 6d d 0 6d d d d 0 + 4d 6d d ]. 2

15 Reark Fro the above theore, we deduce iediately that the asyptotic variance of d d 0 is π2 24 C d, where C d = + 4d 0+ >. Approxiating log f 4d 2 0 locally by a nonlinear function instead of a constant therefore inflates the usual asyptotic variance of the LP regression estiator in a pure fractional odel by the factor C d. This is to be expected, as adding ore variables in regression usually inflates variances. Reark 2 The asyptotic bias of d, β is equal to Dn Ω b n. Soe algebraic anipulations show that when σ u = 0, Dn Ω b n = 2π2 f w0 9 f w 0 + d 0 6 n 2, 45 d 0 2d 0 + d 0 2d d d 0 + d 2 0 2d 0+3 and when σ u 0, Dn Ω b n = 2π4d 0 fw0 2 fu0 2 n 4d 0 d 0 2d 0 + 4d d d d 0 +6d Reark 3 When σ u 0, according to 46 the asyptotic bias of d is of order 4d 0 /n 4d 0. In contrast, the asyptotic bias of the LP estiator is of order 2d 0 /n 2d 0, as shown above in 30. The asyptotic bias of the new estiator is therefore saller than that of the LP estiator by order 2d 0 /n 2d 0. When σ u = 0, the asyptotic bias of d is of the sae order as that of d LP, as seen fro 3 and 45. The relative agnitude depends on the value of d 0 and the curvature of f w λ at λ = 0. Reark 4 Note that β converges ore slowly by a rate of n 2d 0 than d. Heuristically, the excitation levels of the two regressors log λ and λ 2d 0 and thus their inforation content are different. More specifically, we have = log λ k= log λ k/ 2 = O whereas = λ2d 0 k= λ2d 0 k / 2 = Oλ 2d 0. Next, we consider issues of bandwidth choice in the case where σ u 0. Following the above rearks, the asyptotic ean-squared error AMSE of d is where AMSE d = K 2 n 8d 0 + π2 24 C d, 47 K = 2π 4d 0 β 2 d 0 2d d d Straightforward calculations yield the value of that iniizes AMSE d, viz. opt π 2 C d = [ 92d 0 K 2 /8d 0+ n 8d 0/8d 0 + ], 49 3

16 where [ ] denotes the integer part. In contrast, the AMSE of d LP is where AMSE d LP = K 2 LP n 4d 0 + So the AMSE-optial bandwidth for d LP is π2 24, 50 K LP = 2π 2d 0 β 0 d 0 2d opt LP = [ π 2 96d 0 KLP 2 /4d0+ n 4d 0/4d 0 + ]. 52 When = opt, the AMSE of d converges to zero at the rate of n 8d 0/8d 0 +. In contrast, when = opt LP, the AMSE of d LP converges to zero only at the rate of n 4d 0/4d 0 +. Thus, the optial AMSE of d converges to zero faster than that of d LP. Of course, the optial bandwidth 49 depends on the unknown quantities d 0 and β 0. Consistent estiates are readily available under Assuptions, 2 and 3. A data dependent choice of for the coputation of d can be obtained by plugging initial estiates of β 0 and d 0 into Siulations 5. Experiental Design This section investigates the finite saple perforance of the new estiator in coparison with conventional LP regression. The chosen data generating process is z t = L d 0 w t + u t, 53 where {w t : t =, 2,..., n} are iid N0,, {u t : t =, 2,..., n} are iid N0, σ 2 u and {w t } are independent of {u t }. We consider the following constellation of paraeter cobinations d 0 = 0.25, 0.45, 0.65, 0.85, and σ 2 u = 0, 4, 8, In view of the fact that the LP estiator is consistent for both stationary fractional processes d 0 < 0.5 and nonstationary fractional processes 0.5 d 0 < see Ki and Phillips, 999, we expect the new estiator to work well for nonstationary fractional coponent processes for this range of values of d 0 as well as for stationary fractional coponent processes over 0 < d 0 < 0.5. Hence it is of interest to include soe values of d 0 that fall in the nonstationary zone. The value of σ 2 u deterines the strength of the noise fro the perturbations. The long run SNR increases as σ 2 u decreases. When σ 2 u = 0, z t is a pure fractional process 4

17 with an infinite long run SNR. The inverse of the long run SNR, viz. f u 0/f w 0, takes the values 0, 4, 8, 6. These are close to the values in Deo and Hurvich 999. In their siulation study, the ratio f u 0/f w 0 takes the values 6.7 and We consider saple sizes n = 52 and Because n has the coposite for 2 k k integer for these choices, zero-padding is not a concern when we use the fast Fourier transfor to copute the periodogra. For each saple size and paraeter cobination, 2000 replications are perfored fro which we calculate the biases, standard deviations and root ean square errors of d and d LP, for different selections of the bandwidth. Then, for each paraeter cobination, we graph each of these quantities as functions of = 4, 5,..., n/2. The results are shown in panels a-c of Figs. 5. In addition, we copute the coverage probabilities, as functions of, of the noinal 90% confidence intervals that are obtained using the asyptotic norality results of Theores and 3. When constructing these confidence intervals, we estiate the variances of d and d LP using finite saple expressions rather than the liit expressions, because the forer yield better finite saple perforance for all paraeter cobinations and for both estiators. The variance of d is estiated by the, eleent of the inverse of the Hessian atrix, which is π 2 6 H 22,n d, β H d, βh 22 d, β H2 2 d, β, 55 whereas the variance of d LP is estiated by π 2 log λ log λ k 24 = k= We calculate the average lengths of the confidence intervals as functions of. The coverage probabilities and the average lengths are graphed against in panels d and e of Figs Results We report results only for the cases d 0 = 0.45 and d 0 = 0.85, since these are representative of the results found in the other two cases, d 0 = 0.25 and 0.65, respectively. Also, for each value of d 0, we discuss only the cases σ 2 u = 0 and σ 2 u = 8, as the results for the other values of σ 2 u were qualitatively siilar. We first discuss the results when d 0 = 0.45 and σ 2 u = 0. In this case, z t is a pure fractional process. Fig. a shows that the bias of d is positive and larger than that of d LP. The positive bias of d confors to our asyptotic results. Fro Reark 2, the asyptotic bias of d is π2 27 n 2 d 0 2d 0 + 2d 0 + 3, which is always positive for d 0 <. Fig. b shows that the variance of d is larger than that of d LP, as predicted by Theore 3. Coparing RMSE s in Fig. c, we see that the RMSE of d is larger than that of d LP. The inferior perforance of d in this case is not surprising since the LP estiator is designed for pure fractional processes, whereas 5

18 our estiator d allows for additional noise in the syste and is designed for perturbed fractional processes. However, it is encouraging that the LP estiator outperfors the new estiator only by a sall argin. Apparently, the cost of including the additional regressor, even when it is not needed, is sall. Next, we discuss the results when d 0 = 0.45 and σ 2 u = 8. Fig. 2a shows that the LP estiator d LP has a large downward bias in this case, whereas the new estiator d has a uch saller bias. Apparently, the bias-reducing feature of d established in the asyptotic theory is anifest in finite saples. Fig. 2b shows that the standard error of dlp is less than that of d for all values of, again consistent with the asyptotic results. For each estiator, the standard error declines at the approxiate rate / as increases, because is the effective saple size in the estiation of d 0. Fig. 2c shows that the RMSE of d is saller than that of d LP over a wide range of values. Fig. 2d shows that the coverage probability of d is fairly close to the noinal value of 0.9, provided that is not taken too large. In contrast, d LP has a true coverage probability close to 0.9 only for very sall values of. This is due to the large bias of d LP. However, the larger standard error of d leads to longer confidence intervals on average, and this is apparent in Fig. 2e. We now turn to the results when d 0 = 0.85 and σ 2 u = 0. Figure 3 shows that both d LP and d work reasonably well for nonstationary fractional processes /2 d 0 <. Copared with Fig., we observe that the difference in the standard errors of these two estiators becoes saller while the difference in the biases reains ore or less the sae. Although d LP is still a better estiator than d in this case, the advantage of d LP has clearly diinished with the increase in d 0. Figure 4 provides results for the case d 0 = 0.85 and σ 2 u = 8. Fig. 4a shows that the bias reduction fro using d is substantial. For exaple, when = 40, the bias of d LP is 0.8, while that of d is only The evidence therefore suggests that d is effective in reducing bias not only in stationary fractional coponent odels but also in nonstationary odels. Fig. 4b shows that the standard error of d is only slightly larger than that of d LP. The large bias reduction and sall variance inflation lead to a saller RMSE for d over a wide range of values, as shown in Fig. 4c. In addition, the coverage probability based on d LP decreases very rapidly as increases, whereas that based on d decreases uch ore slowly. In fact, the coverage probability based on d is close to 0.9 over a wide range of values. Fig. 4e shows that the superior perforance of the coverage probability of d coes at the expense of having longer confidence intervals on average than those based on d LP. The siulations also reveal that the bias of d LP is always negative when σ u > 0 and that the absolute value of the bias increases with σ 2 u, due to stronger containation fro the perturbations that this produces. In addition, d is ore effective in bias reduction for larger values of d 0. Intuitively, when d 0 is sall, the bias of d LP is sall no atter what value σ u ay take. For a large value of σ u, the perturbation coponent doinates the fractional coponent, so that d LP would be around 0. In this case, the bias of d LP is sall only because the true value of d 0 itself is sall. Also, for sall values of σ u, the bias fro containation is naturally going to be sall. Therefore, in both cases, the bias of d LP will be sall when d 0 is sall and 6

19 there is not uch scope for d to anifest its bias-reducing capacity. Finally, for the large saple size n = 2048, the qualitative coparisons ade and conclusions reached for the n = 52 saple size continue to apply. Fig. 5 presents the results for one particular specification d 0 = 0.85, σ 2 u = 8 which shows that d has a uch saller bias and a slightly larger variance than d LP. The RMSE of d is uch saller than that of d LP over a wide range of values. To su up, the siulations show that, for fractional coponent processes, the new estiator d has a lower bias, a higher standard deviation, and a lower RMSE in coparison to the LP estiator d LP, corroborating the asyptotic theory. The lower bias generally leads to iproved coverage probability in confidence intervals based on d over a wide range of. On the other hand, the lower standard deviation of d LP leads to shorter confidence intervals than those based on d. 6 Conclusion In epirical applications it has becoe custoary practice to investigate the order of integration of the variables in a odel when nonstationarity is suspected. This practice is now being extended to include analyses of the degree of persistence using fractional odels and estiates of long eory paraeters. Nonetheless, for any tie series, and particularly acroeconoic variables for which there is liited data, the actual degree of persistence in the data continues to be a controversial issue. The epirical resolution of this proble inevitably relies on our capacity to separate lowfrequency behavior fro high-frequency fluctuations and this is particularly difficult when short run fluctuations have high variance. Actual epirical results often depend critically on the discriinatory power of the statistical techniques being eployed to ipleent the separation. The odel used in the present paper provides soe assistance in this regard. It allows for an explicit coponents structure in which there are different sources and types of variation, thereby accoodating a separation of short and long eory coponents and allowing for fractional processes that are perturbed by weakly dependent effects. Copared to the conventional forulation of a pure fractional process like 2, perturbed fractional processes allow for ultiple sources of high-frequency variation and, in doing so, see to provide a richer setting for uncovering latent persistence in an observed tie series. In particular, the odel provides a echanis for siultaneously capturing the effects of persistent and teporary shocks and sees realistic in econoic and financial applications when there are any different sources of variation in the data. The new econoetric ethods we have introduced for estiating the fractional paraeter in such odels take account of the presence of additive disturbances, and help to achieve bias reduction and attain a faster rate of convergence. The asyptotic theory is easy to use and sees to work reasonably well in finite saples. The ethods of the paper can be extended in a nuber of directions. First, it is of interest to study the perforance of the ethods here under non-gaussian errors, as in Deo and Hurvich 999 for LP regression. Second, the nonlinear approxiation 7

20 approach can be used in cobination with other estiators, such as the local Whittle estiator Robinson 995b, which sees natural in the present context because the procedure already uses optiization ethods. In addition, the idea of using a nonlinear approxiation can be applied to nonstationary fractional coponent odels and used to adapt the ethods which have been suggested elsewhere e.g., Phillips, 999, Shiotsu and Phillips, 200 for estiating the eory paraeter in such odels to cases where there are fractional coponents. Appendix of Proofs Proof of Lea. A spectral density satisfying Assuptions and 2 also satisfies Assuptions and 2 of Robinson 995a. In consequence, the lea follows fro Theore 2 of Robinson 995a. Since we noralize the discrete Fourier transfor by the spectral density fz /2 λ instead of the power function Cg /2 λ d, 4.2 of Robinson 995a is always zero and the extra ter n inα,β in Robinson 995a does not arise in our case. Proof of Lea 3. Note that c U = = [log 2 ] = c U + =[log 2 ]+ c U F + F 2. A. But E sup d,β Θ F is less than E [log 2 ] = sup d,β Θ c U logp [log 2 ] = EU 2 /2 = Olog p+2 / A.2 by Lea 2b. Hence Let s r = sup F = O p log p+2 / = O p logp. d,β Θ r k=[log 2 ]+ U r, r = [log 2 ] +,..., and s [log 2 ] = 0. Then, fro Lea 2a, c and d, it follows that Es 2 r = = r k=[log 2 ]+ r k=[log 2 ]+ EU 2 k + 2 π2 [log 2 ]+ k< r 6 + k log k + 2 = Or + Or log 2 r/ log 2, EU U k r [log 2 +] k<<r Ok 2 log 2 A.3 A.4 A.5 8

21 which iplies s r = O p r /2. Using this result and partial suation, we have: sup F 2 sup c U d,β Θ d,β Θ =log 2 + = sup d,β Θ s c c + sup =log 2 d,β Θ s c + = O p /2 O log q + O p logp = logq =log 2 + =log 2 + O p /2 + O p logp = O p logq + O p logp = O p logaxp,q. A.6 Cobining A.3 with A.6, we get sup d,β Θ = c U = Op logaxp,q. Proof of Theore. When σ u = 0, the theore is essentially the sae as results already established in HDB. Only one odification is needed. HDB use 2 log2 sin λ /2 as one of the regressors while we eploy 2 log λ. The use of 2 log λ rather than 2 log2 sin λ /2 has no effect on the asyptotic variance, but it does affect the asyptotic bias. This is because the asyptotic bias coes fro the doinating ter in ε and this ter is different for different regressors. Using 2 log2 sin λ /2 as the regressor yields f ε = log f w λ log f w 0 = w0 2f w0 λ 2 + o. A.7 In contrast, using 2 log λ as the regressor yields ε = log f w λ log f w 0 2d 0 log2 sin λ = f w0 2f w0 + d 0 λ o. 2 log λ A.8 With this adustent, the arguents in HDB go through without further change. Now consider the case σ u 0. Rewrite the spectral density of z t as f z λ = λ 2d 0 gλ, A.9 where gλ = λ 2 sin λ/2 2d 0 f λ. Since gλ g0 = + O λ 2 f w 0 + λ 2d 0 f u 0 + Oλ 2 f w 0 = Oλ 2d 0 A.0 9

22 2d 0 f w0 2d n 2d 0 as λ 0+, gλ is sooth of order 2d 0. Cobining this with our assuption that and 4d0+ /n 4d 0 = O verifies Assuptions and 2 of Andrews and Guggenberger 999. Hence their Theore is valid with r = 0, s = 2d 0 and q = 2d 0. It is easy to show that the ter O q /n q in their theore is actually 2π 2d 0 fu0 d 0. Andrews and Guggenberger established asyptotic norality under their assuption 3 that 4d0+ /n 4d 0 = o. In fact, asyptotic norality holds under our assuption 4d0+ /n 4d 0 = O as long as an asyptotic bias of order O is allowed. Proof of Theore 2. Let V d, β = 2d d 0 log λ βλ 2d + β 0 λ 2d 0. Then we can decopose Qd, β into three parts as follows: = Qd, β = {U U + ε ε + V V } 2 A. = V V = U + ε V V + = U + ε Ū ε2, where the dependence on d, β has been suppressed for notational siplicity. Since = U + ε Ū ε2 is independent of d, β, we only need to consider the first two ters. Part a We prove part a by showing that 2 = U + ε V V = o p uniforly in d, β and = V V 2 converges uniforly to a function, which has a unique iniizer d 0. First, we show sup U V d,β Θ V = O p. A.2 = We proceed by verifying the conditions in Lea 3. The first condition holds because sup V d, β V d, β d,β Θ 2 sup d d 0 log λ d,β Θ log λ + 2 = sup = d,β Θ β λ 2d = λ 2d = 2 sup d d 0 log + O = Olog uniforly over. A.3 d,β Θ The second condition holds because sup V d, β V d, β d,β Θ 2 sup d d 0 log + 2 sup βλ 2d d,β Θ d,β Θ 2d = O for all, A.4 20

23 where the final line follows fro the fact that sup d,β Θ 2d = O. In addition, V d, β V d, β = 2 sup d d 0 log λ d,β Θ = 2 sup d d 0 log d,β Θ log λ + O = log + O = O. = A.5 Hence A.2 is satisfied. Next, we show that sup d,β Θ = ε V d, β V d, β = O p λ 4d 0. Under Assuptions and 2, ε = log f λ log f w 0 2 sin λ 2 2d 0 β 0 2d 0 [log2 sin λ 2 log λ ] = Oλ r = Oλ 4d 0, A.6 so we have, using A.4 and A.5 sup d,β Θ sup d,β Θ = λ 4d 0 Finally, ε V d, β V d, β = ε r V d, β V d, β = r= + sup d,β Θ ε V d, β V d, β = O p r 4d 0 + O pλ 4d 0 = O p λ 4d 0 = o p. A.7 = r= V V 2 = = 2d d 0 log = = 4d d log k + o k= log 2 = log k 2 + o k= = 4d d o, A.8 where o holds uniforly over d, β Θ. Here we have eployed = log2 k= log k 2 = + o. 2

24 Let D δ = {d : d d 0 > δ}. In view of A.2, A.7 and A.8, we have P d Dδ P Q d, β Qd 0, β 0 P in V d D δ,β B V U + ε V V 0 = = P in V d D δ,β B V 2 sup 2 U + ε V = d D δ,β B V = P in V d D δ,β β V 2 o = P in d D δ 4d d 0 2 o 0, A.9 which copletes the proof. Part b Copared with log λ, λ 2d 0 is negligible since d 0 > 0. Due to the difference in the orders of agnitude of the regressors, it is not straightforward to establish the consistency of β. In fact, we proceed by showing first that d converges to d 0 at soe preliinary rate and then go on to show that d d 0 = O p n 2d 0. We obtain this rate sequentially. First, we show that d d 0 = o p n d/2. Fro Q d, β Qd 0, β 0 0, we get 2 V d, β V 2 d, β = sup U + ε V d, β V d, β A.20 = d,β Θ 2 U + ε V d, β V d, β = = O p + O p λ 4d 0 = o p n 2d, A.2 where the last equality follows fro the assuptions that n 4d0+ / 4d0+ + = o and that d d > 0. But = V d, β V d, β 2 equals 2 2 d d 0 log log d + Oλ2 + Oλ 2d 0 = = = 4 d d o + Oλ 2d 0 + Oλ 2 d = 4 d d o + O n 2d. A.22 Therefore, 4 d d o + O p n 2d o p n 2d, A.23 22

25 which iplies that d d 0 is at ost O p n d. Thus d d 0 = o p n d/2. Second, we show that d d 0 = o p n d 0+. Since d d 0 = o p n d/2, we only need consider d D n = {d : d d 0 < ε n d/2 } for soe sall ε > 0. Approxiating sus by integrals and using the forulae: = k log = + o, k 0, A.24 k + 2 and = k log 2 = 2 + o, k 0, A.25 k + 3 we deduce that = V 2d, 2 V d, β is equal to 8d d β2 λ 4d 4d + + β2 0λ 4d0 4d d d 0βλ 2d 4d d 0β 0 λ 2d 0 2d + 2d o 2 2ββ 0λ 2d+2d 0 + o 2d d 0 + βλ2d 2d + 2d 0 + 2d + β 0λ 2d0 + o 2d 0 + [ ] 2 = 4d d dβλ 2d 2d + 4d + 2d 0 β 0 λ 2d0 2d 0 + 4d o A dd 0ββ 0 λ 2d+2d 0 2d + 2d 0 + 4d + 4d0 + 2d + 2d o = 4d d Oλ 4d 0 = 4d d oλ 2d 0+, A.27 where the O and o ters hold uniforly over d, β D n B. The last line follows because when d d 0 ε n d /2, λ 2d = λ 2d 0 λ 2d 2d 0 = λ 2d 0 exp2d 2d 0 log λ constλ 2d 0 expλ d /2 log λ = Oλ 2d 0. A.28 Using Q d, β Qd 0, β 0 0 again, we have V d, β V d, β 2 O p + O p λ 4d 0 = o p n 2d 0+, A.29 = where the equality follows fro the assuption n 4d 0+ / 4d 0+ + = o. Cobining A.27 and A.29, we get Hence d d 0 = o p n d d d oλ 2d 0+ o p n 2d 0+. A.30 23

26 Next, we show that d d 0 = o p n 3d 0+ /2. We first prove that = U V V = o p n 3d 0+ and = ε V V = o p uniforly in d, β D n B, where D n = {d : d d 0 < ε n d0+ }. Following the sae steps as in the proof of Lea 3, we copute the orders of V d, β V d, β, V d, β V d, β and V d, β V d, β as follows. First, sup d,β D n B V d, β V d, β is bounded by 2 sup d,β D n B d d 0 log λ = 2 sup d d 0 log + Oλ 2d 0 d,β Θ log λ + 2 = sup β λ 2d n B d,β D n 3d 0+ = λ 2d = O n d 0+ log uniforly over. A.3 Siilarly, sup d,β D n B V d, β V d, β is bounded by 2 sup d d 0 log λ d,β D n B = 2 sup = O d,β D n B = log λ + Oλ 2d 0 d d 0 log log + Oλ 2d 0 = n d 0+. A.32 Furtherore, sup d,β D n B V d, β V d, β is not greater than 2 sup d d 0 log d,β D n B + 2 sup βλ 2d n B 2d d,β D = O n d 0+ for all. A.33 Invoking the sae arguent as in the proof of Lea 3, we obtain sup U V d,β D n B V = O p = n d 0+ = o p n 3d 0+, and sup ε V d,β D V = o p n d 0+ +4d 0 = o p n 3d 0+,A.34 n B = as desired. In addition, it follows fro A.27 that when d D n, d o + oλ 3d 0+ = V V 2 = 4d. Applying the sae arguent as before, we get 4 d d o + oλ 3d 0+ o p n 3d 0+, A.35 24

27 and so d d 0 = o p n 3d0+ /2. Repeating the procedure again we obtain d d 0 = o p n 7d 0+ /4 if 7+ /4 < 2. Further iterations of this procedure lead to d d 0 = o p n 2 2 k +, k = 0,, 2, 3,... if 2 2 k + < 2. We stop the iteration if we obtain d d 0 = o p n 2 2 k 0 + for soe k 0 0 such that 2 2 k 0 + < 2 and 4 2 k In this case, we have sup U V V d,β Dn B = = O p n 2 2 k 0 + d 0 = o p n 4 2 k 0 + d 0 = o p n 4d 0, A.36 and sup d,β D n B ε V V = o p = n 4d 0, A.37 where D n = {d : d d 0 < ε n 2 2 k 0 + }. Applying the sae arguent as before, we deduce 4 d d o + Oλ 4d 0 o p n 4d 0. A.38 In consequence, d d 0 = O p n 2d 0. Now, since 2d + 2d d + 4d 0 + = 4d 2 8dd 0 + 4d 2 0 = 4d d 0 2 > 0, we deduce fro A.26 that 2 V V 2 2dβλ 2d 2d + 4d + 2d 0 β 0 λ 2d0 2d o. A.39 4d 0 + = In view of = V d, β V d, β 2 o p λ 4d 0, we obtain 2 2 d 2 d βλ 2d 2 d 0 β + 4 d 0 λ 2d 0 + 2d o o p λ 4d 0. 4d 0 + A.40 25

28 But = = = = 2d βλ 2 d 2d 2 d 0 β + 4 d 0 λ 2d 0 + 2d 0 + 4d d 2 d βλ 2 2 d + 4 d dβ 0 λ 2 d d + 4 d + dβ 0 λ 2 d d + 4 d dβ 0 λ 2d d + 4 d dβ 0 λ 2d 0 2d 2 d 0 β + 4 d 0 λ 2d 0 + 2d 0 + 4d d 2 dλ β β0 2 d + 4 d + 2 dβ 0 λ 2 d λ 2d d + 4 d d 2d 2 d d + 2d 0 + Oλ 2d 0 4d d 2 dλ 2 d β + 4 d β0 + O p λ 2 d log λ d d 0 + O p λ 4d d 2 dλ 2 d β + 4 d β0 + o p λ 3d 0, + A.4 where d is between d and d 0. So 2 d 2 dλ 2 d + 4 d + 2 β β0 + o p λ 3d 0 o p λ 4d 0. A.42 This iplies that 4 dλ 4 d d 0 2 d d β β 0 2 o p λ 4d 0, + A.43 fro which we deduce that β β 0 = o p. Proof of Lea 5 Part a The 2,2 eleent of sup θ Θn Dn H n d, β J n d, βdn is zero, so it suffices to consider the, and,2 eleents. Since sup θ Θn β I z + 2d log λ βλ 2d = α 0 + U + ε + d d 0 log λ 2 + β 0 λ 2d 0 βλ 2d, = e log λ 2 2 λ 2d, the, eleent, is bounded by L + L 2 + L 3 + L 4, 26