OPTIMAL BANDWIDTH SELECTION IN HETEROSKEDASTICITY-AUTOCORRELATION ROBUST TESTING. YIAIAO SUN, PETER C. B. PHILLIPS, and SAINAN JIN

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1 OPIMAL BANDWIDH SELECION IN HEEROSKEDASICIY-AUOCORRELAION ROBUS ESING BY YIAIAO SUN, PEER C. B. PHILLIPS, and SAINAN JIN COWLES FOUNDAION PAPER NO. COWLES FOUNDAION FOR RESEARCH IN ECONOMICS YALE UNIVERSIY Box 88 New Haven, Connecticut

2 Econometrica, Vol. 76, No. January, 8, OPIMAL BANDWIDH SELECION IN HEEROSKEDASICIY AUOCORRELAION ROBUS ESING YIXIAO SUN University of California, San Diego, La Jolla, CA 993, U.S.A. PEER C. B. PHILLIPS Cowles Foundation, Yale University, New Haven, C 65, U.S.A.; University of Auckland, Auckland, New Zealand; and University of York, York, U.K. SAINAN JIN Guanghua School of Management, Peking University, Beijing, 87, China he copyright to this Article is held y the Econometric Society. It may e downloaded, printed and reproduced only for educational or research purposes, including use in course packs. No downloading or copying may e done for any commercial purpose without the explicit permission of the Econometric Society. For such commercial purposes contact the Office of the Econometric Society contact information may e found at the wesite or in the ack cover of Econometrica. his statement must the included on all copies of this Article that are made availale electronically or in any other format.

3 Econometrica, Vol. 76, No. January, 8, OPIMAL BANDWIDH SELECION IN HEEROSKEDASICIY AUOCORRELAION ROBUS ESING BY YIXIAO SUN,PEER C. B. PHILLIPS, AND SAINAN JIN his paper considers studentized tests in time series regressions with nonparametrically autocorrelated errors. he studentization is ased on roust standard errors with truncation lag M for some constant ] and sample size It is shown that the nonstandard fixed- limit distriutions of such nonparametrically studentized tests provide more accurate approximations to the finite sample distriutions than the standard small- limit distriution. We further show that, for typical economic time series, the optimal andwidth that minimizes a weighted average of type I and type II errors is larger y an order of magnitude than the andwidth that minimizes the asymptotic mean squared error of the corresponding long-run variance estimator. A plug-in procedure for implementing this optimal andwidth is suggested and simulations not reported here confirm that the new plug-in procedure works well in finite samples. KEYWORDS: Asymptotic expansion, andwidth choice, kernel method, long-run variance, loss function, nonstandard asymptotics, roust standard error, type I and type II errors.. INRODUCION IN IME SERIES REGRESSIONS with autocorrelation of unknown form, the standard errors of regression coefficients are usually estimated nonparametrically y kernel-ased methods that involve some smoothing over the sample autocovariances. he underlying smoothing parameter may e defined as the ratio of the andwidth to the sample size and is an important tuning parameter that determines the size and power properties of the associated test. It therefore seems sensile that the choice of should take these properties into account. However, in conventional approaches e.g., Andrews 99; Newey and West 987, 994 and most practical software, the parameter is chosen to minimize the asymptotic mean squared error AMSE of the long-run variance LRV estimator. his approach follows what has long een standard practice in the context of spectral estimation Grenander and Rosenlatt 957 where the focus of attention is the spectrum or, in the present context, the LRV. Such a choice of the smoothing parameter is designed to e optimal in the AMSE sense for estimation of the relevant quantity here, the asymptotic standard error or LRV, ut is not necessarily est suited for hypothesis testing. An earlier version of the paper is availale as Sun, Phillips, and Jin 6. he present version is a sustantially shorter paper and readers are referred to the earlier version for further details, discussion, proofs, and some asymptotic expansion results of independent interest. he authors thank Whitney Newey, two anonymous referees, and many seminar participants for comments on earlier versions. Phillips acknowledges partial research support from the Kelly Foundation and the NSF under grant SES Jin acknowledges financial support from the NSFC grant

4 76 Y. SUN, P. C. B. PHILLIPS, AND S. JIN In contrast to the aove convention, the present paper develops a new approach to choosing the smoothing parameter. We focus on two-sided tests and consider choosing to optimize a loss function that involves a weighted average of the type I and type II errors, a criterion that addresses the central concerns of interest in hypothesis testing, alancing possile size distortion against possile power loss in the use of different andwidths. his new approach to automatic andwidth selection requires improved measurement of type I and type II errors, which are provided here y means of asymptotic expansions of oth the finite sample distriution of the test statistic and the nonstandard limit distriution. We first examine the asymptotic properties of the statistical test under different choices of Using a Gaussian location model, we show that the distriution of the conventionally constructed t statistic is closer to its limit distriution derived under the fixed- asymptotics than that derived under the small- asymptotics. More specifically, when is fixed, the error in the rejection proaility ERP of the nonstandard t-test is O, while that of the standard normal test is O he result is related to that of Jansson 4, who showed that the ERP of the nonstandard test ased on the Bartlett kernel with is Olog/. Our result strengthens and generalizes Jansson s result in two aspects. First, we show that the log factor can e dropped. Second, while Jansson s result applies only to the Bartlett kernel with, our result applies to more general kernels with oth and< On the other hand, when is decreasing with the sample size, the ERP of the nonstandard t-test is smaller than that of the standard normal test, although they are of the same order of magnitude. As a consequence, the nonstandard t-test has less size distortion than the standard normal test. Our analytical findings here support earlier simulation results in Kiefer, Vogelsang, and Bunzel,hereafter KVB, Kiefer and Vogelsang a,, hereafter KV, Gonçalves and Vogelsang 5, and our own work Phillips, Sun, and Jin 6, 7, hereafter PSJ. he theoretical development is ased on two high-order asymptotic expansions. he first is the expansion of the nonstandard limiting distriution around its limiting chi-squared form. he second is the expansion of the finite sample distriution of the t statistic under conventional joint limits where the sample size and simultaneously. hese higher-order expansions enale us to develop improved approximations to the type I and type II errors. For typical economic time series, the dominant term in the type I error increases as decreases while the dominant term in the type II error decreases as decreases. Since the desirale effects on these two types of errors generally work in opposing directions, there is an opportunity to trade off these effects. Accordingly, we construct a loss function criterion y taking a weighted average of these two types of errors and show how may e selected in such a way as to minimize the loss. Our approach gives an optimal that generally has a shrinking rate of at most O q/q+ and that can even e O for certain loss functions,

5 BANDWIDH SELECION IN ROBUS ESING 77 depending on the weights that are chosen. Note that the optimal that minimizes the asymptotic mean squared error of the corresponding LRV estimator is O q/q+ cf., Andrews 99. hus, optimal values of for LRV estimation are smaller as than those that are most suited for statistical testing. he fixed- rule is otained y attaching sustantially higher weight to the type I error in the construction of the loss function. his theory therefore provides some insight into the type of loss function for which there is a decision theoretic justification for the use of fixed- rules in econometric testing. o implement the optimal andwidth selection rule for two-sided tests in a location model, users may proceed via the following steps:. Specify the null hypothesis H : β β and an alternative hypothesis H : β β c >, where c may reflect a value of scientific interest or economic significance if such a value is availale. In the asence of such a value, we recommend that the user may set the default discrepancy parameter value δ inelow.. Specify the significance level α of the test and the associated two-sided standard normal critical value z α that satisfies z α α/ 3. Specify a weighted average loss function see 45 elow that captures the relative importance of the type I and II errors, and reflects the relative cost of false acceptance and false rejection. Normalize the weight for the type II error to e unity and let w e the weight for the type I error. 4. Estimate the model y ordinary least squares OLS and construct the residuals û t 5. Fit an AR model to the estimated residuals and compute ˆd ˆρ ˆρ ˆσ û t ˆρû t δ ˆρ c ˆσ t where ˆρ is the OLS estimator of the autoregressive AR coefficient. Set δ as a default value if the user is unsure aout the alternative hypothesis. 6. Specify the kernel function to e used in heteroskedasticity autocorrelation consistent HAC standard error estimation. Among the commonly used positive definite kernels, we recommend a suitale second-order kernel q such as the Parzen or quadratic spectral QS kernel. 7. Compute the automatic andwidth, qg ˆd[wG z α G δ z] /3 α cz K α δz /3 α ˆd[wG ˆ z α G δ z α ] >, log ˆd[wG z α G δ z] α

6 78 Y. SUN, P. C. B. PHILLIPS, AND S. JIN where 3 { 6 Parzen kernel, g 4 QS kernel, { 539 Parzen kernel, c QS kernel. G δ is the cumulative distriution function cdf of a noncentral chisquared variate with degree of freedom and noncentrality parameter δ and K δ is defined in equation Compute the HAC standard error using andwidth ˆM ˆ and construct the usual t statistic t ˆ 9. Let z α ˆ z α + k 3 ˆ + k4 ˆ,wherek 3 and k 4 are given in ale I. Reject the null hypothesis if t ˆ z α ˆ. he rest of the paper is organized as follows. Section reviews the firstorder limit theory for the t-test as with the parameter fixed and as with approaching zero. Section 3 develops an asymptotic expansion of the nonstandard distriution under the null and local alternative hypotheses as aout the usual central and noncentral chi-squared distriutions. Section 4 develops comparale expansions of the finite sample distriution of the statistic as and at the same time. Section 5 compares the accuracy of the nonstandard approximation with that of the standard normal approximation. Section 6 proposesa selection rulefor that is suitale for implementation in semiparametric testing. he last section provides some concluding discussion. Proofs and relevant technical results are availale in Sun, Phillips, and Jin 6, 7, hereafter SPJ.. HEEROSKEDASICIY AUOCORRELAION ROBUS INFERENCE hroughout the paper, wefocusoninference aout β in the location model, 4 y t β + u t t where u t is a zero mean process with a nonparametric autocorrelation structure. he nonstandard limiting distriution in this section and its asymptotic expansion in Section 3 apply to general regression models under certain conditions on the regressors; see KV a,, 5. On the other hand, the asymptotic expansion of the finite sample distriution in Section 4 applies only to the location model. Although the location model is of interest in its own right, this is a limitation of the paper and some possile extensions are discussed in Section 7. OLS estimation of β gives ˆβ Ȳ t y t and the scaled and centered estimation error is 5 ˆβ β S

7 BANDWIDH SELECION IN ROBUS ESING 79 where S t t τ u τ Let û τ y τ ˆβ e the demeaned time series and let Ŝt t τ ûτ e the corresponding partial sum process. he following condition is commonly used to facilitate the limit theory e.g., KVB and Jansson 4. ASSUMPION : S [r] satisfies the functional law 6 / S [r] ωw r r [ ] where ω is the long-run variance of u t and Wris standard Brownian motion. Under Assumption, 7 / Ŝ [r] ωv r r [ ] where V is a standard Brownian ridge process, and 8 ˆβ β ωw N ω which provides the usual asis for roust testing aout β It is standard empirical practice to estimate ω using kernel-ased nonparametric estimators that involve some smoothing and possily truncation of the autocovariances. When u t is stationary, the long-run variance of u t is 9 ω γ + γj j where γj Eu t u t j Correspondingly, heteroskedasticity autocorrelation consistent HAC estimates of ω typically have the form ˆω M ˆγj j + j k ˆγj M j û t+j û t for j, t t j+ û t+j û t for j<, involving the sample covariances ˆγj In, k is some kernel function and M is a andwidth parameter. Consistency of ˆω M requires M and M/ as e.g., Andrews 99, Newey and West 987, 994.

8 8 Y. SUN, P. C. B. PHILLIPS, AND S. JIN o test the null H : β β against H : β β the standard approach relies on a nonparametrically studentized t-ratio statistic of the form t ˆωM / ˆβ β / ˆωM which is asymptotically N. Useoft ˆωM is convenient empirically and therefore widespread in practice, in spite of well known prolems of size distortion in inference. o reduce size distortion, KVB and KV 5 proposed the use of kernelased estimators of ω in which M is set proportional to, that is, M for some ]. In this case, the estimator ˆω ecomes ˆω j + j k ˆγj and the associated t statistic is given y 3 t / ˆβ β / ˆω When the parameter is fixed as KV 5 showed that under Assumption ˆω ω Ξ where the limit Ξ is random and given y 4 Ξ k r sdv rdv s with k k /, and the t statistic has a nonstandard limit distriution. Under the null hypothesis, 5 t WΞ / whereas under the local alternative H : β β + c / 6 t δ + WΞ / where δ c/ω hus, the t statistic has a nonstandard limit distriution that arises from the random limit of the LRV estimate ˆω when is fixed as However, as decreases, the effect of this randomness diminishes, and when, the limit distriutions under the null and local alternative oth approach those of conventional regression tests with consistent LRV estimates. It is important to point out that in the nonstandard limit distriution, W is independent Ξ his is ecause W is uncorrelated with Vr for any r [ ], and oth W and Vrare Gaussian.

9 BANDWIDH SELECION IN ROBUS ESING 8 In related work, the present authors PSJ 6, 7 proposed using an estimator of ω of the form 7 ˆω ρ j + [ j k ] ρ ˆγj which involves setting M equal to and taking an aritrary power ρ of the traditional kernel. Statistical tests ased on ˆω and ρ ˆω share many of the same properties, which is explained y the fact that ρ and play similar roles in the construction of the estimates. he present paper focuses on ˆω and tests associated with this estimate. Ideas and methods comparale to those explored in the present paper may e pursued in the context of estimates such as ˆω and ρ are reported in other work PSJ 5a, EXPANSION OF HE NONSANDARD LIMI HEORY his section develops asymptotic expansions of the limit distriutions given in 5 and6 as the andwidth parameter hese expansions are taken aout the relevant central and noncentral chi-squared limit distriutions that apply when corresponding to the null and the local alternative hypotheses. he expansions are of some independent interest. For instance, they can e used to deliver correction terms to the limit distriutions under the null, therey providing a mechanism for adjusting the nominal critical values provided y the usual chi-squared distriution. he latter correspond to the critical values that would e used for tests ased on conventional consistent LRV estimates. he expansions and later developments in the paper make use of the following kernel conditions: ASSUMPION : i kx : R [ ] is symmetric, piecewise smooth with k and kxx dx <. ii he Parzen characteristic exponent defined y { } kx 8 q max q : q Z + g q lim < x x q is greater than or equal to. iii kx is positive semidefinite, that is, for any square integrale function fx, ks tfsftdsdt Assumption imposes only mild conditions on the kernel function. All the commonly used kernels satisfy i and ii. he assumption kxx dx < ensures the integrals that appear frequently in later developments are finite and validates use of the Riemann Leesgue lemma in proofs. he assumption

10 8 Y. SUN, P. C. B. PHILLIPS, AND S. JIN of positive semidefiniteness in iii ensures that the associated LRV estimator is nonnegative. Commonly used kernels that are positive semidefinite include the Bartlett kernel, the Parzen kernel, and the QS kernel, which are the main focus of the present paper. For the Bartlett kernel, the Parzen characteristic exponent is For the Parzen and QS kernels, the Parzen characteristic exponent is. z} e the nonstandard limiting distriution and let G λ G ; λ e the cdf of a noncentral χ λ variate with noncentrality parameter λ he following theorem estalishes the asymptotic expansion of F δ z around G δ z Let F δ z : P{ δ + WΞ / HEOREM : Let Assumption hold. hen 9 F δ z G δ z + p δ z + q δ z + o where the term o holds uniformly over z R + as, p δ z c G δ z z 4 c G δ z z q δ z G δ z z c 3 + G δ z z 4 c 4 c G δ z z 6 c c and c c 3 kx dx c kx x dx c 4 k x dx k x x dx As is apparent from the proof given in SPJ 7, the term c G δ z z 4 in p δ z arises from the randomness of Ξ whereas the term c G δ z z in p δ z arises from the asymptotic ias of Ξ Although Ξ converges to as we have varξ c + o and EΞ c + o as estalished in the proof. Ξ is not centered exactly at ecause the regression errors have to e estimated. he terms in q δ z are due to the first- and second-order iases of Ξ, the variance of Ξ, and their interactions. It follows from heorem that when δ F z Dz +[c D z z 4 c D z z ] + [ D z z c 3 + ] D z z 4 c 4 c D z z 6 c c + o

11 BANDWIDH SELECION IN ROBUS ESING 83 where D G is the cdf of χ distriution. For any α let z α R+ z α R+ such that Dz α and F α z α α hen, using a Cornish Fisher-type expansion, we can otain high-order corrected critical values. Before presenting the corollary elow, we introduce some terminology. We call the critical value that is correct to O the second-order corrected critical value and call the critical value correct to O the third-order corrected critical value. We use this convention throughout the rest of the paper. he following quantities appear in the corollary: k c + c z + α c z 4 α k c + 3 c c c + c c 4 + c + 3 c c c c 4 z 4 + α 6 c k 3 c + c z α + 4 c z 3 α k 4 8 c c c + 6 c + c c 4 z α + 4 c c c c + 8 c 4 z α z 6 α 6 c z 8 α z 3 α + 8 c z5 α 3 c z7 α COROLLARY : For asymptotic chi-squared and normal tests: i he second-order corrected critical values are 3 z α z α + k + o z α z α + k 3 + o ii he third-order corrected critical values are 4 z α z α + k + k + o z α z α + k 3 + k 4 + o where z α is the nominal critical value from the standard normal distriution. Our later developments require only the second-order corrected critical values. he third-order corrected critical values are given here ecause the second-order correction is not enough to deliver a good approximation to the exact critical values when the Parzen and QS kernels are employed and is large. For the Bartlett, Parzen, and QS kernels, we can compute c, c, c 3,andc 4 either analytically or numerically. ale I gives the high-order corrected critical values for these three kernels. hese higher order corrected critical values

12 84 Y. SUN, P. C. B. PHILLIPS, AND S. JIN ABLE I HIGH-ORDER CORRECED CRIICAL VALUES z α z α + k + k + o z α z α + k 3 + k 4 + o α 5% z α 96 α % z α 645 k k k 3 k 4 k k k 3 k 4 Bartlett Parzen QS provide excellent approximations to the exact ones when is smaller than.5 and reasonaly good approximations for other values of When δ and the second-order corrected critical values are used, we can use heorem to calculate the local asymptotic power, measured y P{ δ + WΞ / >z α } as in the following corollary. COROLLARY 3: Let Assumption hold. hen the local asymptotic power satisfies 5 P { } δ + WΞ / >z α Gδ z c α z 4 K α δz α + o as, where 6 K δ z δ / j j! j is positive for all z and δ e δ / z j / e z/ j Ɣj + / j+/ z According to Corollary 3, the local asymptotic test power, as measured y P{ δ + WΞ / >z α } decreases monotonically with at least when is small. For a given critical value, we can show that z 4K α δz α achieves its maximum around δ, implying that the power increase resulting from the choice of a small is greatest when the local alternative is in an intermediate neighorhood of the null hypothesis. For any given local alternative, the function is monotonically increasing in z α herefore, the power improvement due to the choice of a small increases with the confidence level α his is expected. When the confidence level is higher, the test is less powerful and the room for power improvement is greater. 4. EXPANSIONS OF HE FINIE SAMPLE DISRIBUION his section develops a finite sample expansion for the simple location model. his development, like that of Jansson 4, relies on Gaussianity,

13 BANDWIDH SELECION IN ROBUS ESING 85 which facilitates the derivations. he assumption could e relaxed y taking distriutions ased for example on Gram Charlier expansions, ut at the cost of much greater complexity see, for example, Phillips 98, Velasco and Roinson. he following assumption facilitates the development of the higher order expansion. ASSUMPION 3: u t is a mean zero covariance-stationary Gaussian process with h h γh < where γh Eu t u t h We develop an asymptotic expansion of P{ ˆβ β / ˆω z} for β β + c/ Depending on whether c is zero or not, this expansion can e used to approximate the size and power of the t-test. Since u t is in general autocorrelated, ˆβ and ˆω are statistically dependent, which makes it difficult to write down the finite sample distriution of the t statistic. o tackle this difficulty, we decompose ˆβ and ˆω into statistically independent components. Let u µ u y y y l, and Ω varu hen the generalized least squares estimator of β is β l Ω l l Ω y and 7 ˆβ β β β + l l l ũ where ũ I l l Ω l l Ω u which is statistically independent of β β Since ˆω can e written as a quadratic form in ũ, ˆω is also statistically independent of β β Next, it is easy to see that 8 ω : var ˆβ β l Ω l ω + O and it follows from Grenander and Rosenlatt 957 that 9 ω : var β β l Ω l ω + O herefore / l ũ N O. Comining this result with the independence of β and ũ, ˆω we have 3 P{ ˆβ β / ˆω z} P{ β β/ ω + c/ ω z ˆω / ω / l ũ/ ω } E z ˆω / ω c/ ω / l ũ/ ω E z ˆω / ω c/ ω / E[ϕz ˆω / ω c/ ω l ũ/ ω ] + O P{ β β/ ω + c/ ω z ˆω / ω }+O

14 86 Y. SUN, P. C. B. PHILLIPS, AND S. JIN uniformly over z R where and ϕ are the cdf and proaility density function of the standard normal distriution, respectively. he second to last equality follows ecause ˆω is quadratic in ũ and thus E[ϕz ˆω / ω c/ ω l ũ]. In a similar fashion we find that P{ ˆβ β / ˆω z} P{ β β/ ω + c/ ω z ˆω / ω }+O uniformly over z R herefore, 3 F δ z : P { ˆβ β / ˆω z } E{G δ z ˆω / ω }E{G δz ς }+O uniformly over z R +,whereς : ˆω /ω converges weakly to Ξ. Setting µ Eς,wehave 3 F δ z G δ µ z + G δ µ z Eς µ z G δ µ z Eς µ 3 z 6 + OEς µ 4 + O where the O term holds uniformly over z R +. By developing asymptotic expansions of µ and Eς µ m for m 3 4 we can estalish a higherorder expansion of the finite sample distriution for the case where and at the same time. his expansion validates, for finite samples, the use of the second-order corrected critical values given in the previous section that were derived there on the asis of an expansion of the nonstandard limit distriution. HEOREM 4: Let Assumptions and 3 hold. If as and then 33 F δ z G δ z +[c G δ µ z z 4 c G δ z z ] g q d q G δ z z q + o{ + q }+O where d q ω h h q γh, ω l Ω l, and the o and O terms hold uniformly over z R + Under the null hypothesis, δ andg δ D so 34 F z Dz +[c D z z 4 c D z z ] g q d q D z z q + o{ + q }+O

15 BANDWIDH SELECION IN ROBUS ESING 87 he leading two terms up to O in this expansion are the same as those in the corresponding expansion of the limit distriution F z given in aove. hus, use of the second-order corrected critical values given in 3, which take account of terms up to O should lead to size improvements when q q+ he third term in the expansion 34 iso q when is fixed. When decreases with this term provides an asymptotic measure of the size distortion in tests ased on the use of the first two terms of 34 or, equivalently, those ased on the nonstandard limit theory, at least to O. hus, the third term of 34 approximately measures how satisfactory the second-order corrected critical values given in 3 are for any given values of and. Under the local alternative hypothesis, the power of the test ased on the second-order corrected critical values is F δ z α heorem 4 shows that F δ z α can e approximated y G δ z α +[c G δ z α z4 α c G δ z α z4 α ] g qd q G δ z z q with an approximation error of order o q + + O Note that the aove approximation depends on δ only through δ so it is valid under oth H : β β c/ and H : β β c/ hese results on size distortion and local power are formalized in the following corollary. COROLLARY 5: Let Assumptions and 3 hold. If as and, then: a he size distortion of the two-sided t-test ased on the second-order corrected critical values is 35 F z α α g q d q D z α z α q +o q ++O Under the local alternative H : β β c/, the power of the twosided t-test ased on the second-order corrected critical values is F δ z α G δ z α c z 4 α K δz α 36 + g q d qg δ z α z α q + o q + + O 5. ACCURACY OF HE NONSANDARD APPROXIMAION he previous section showed that when goes to zero at a certain rate, the ERP of the standard normal or chi-squared test is at least O q/q+ for typical economic time series. his section estalishes a related result for the nonstandard test when is fixed and then compares the ERP of these two tests under the same asymptotic specification, that is, either is fixed or As in the previous section, we focus on the Gaussian location model.

16 88 Y. SUN, P. C. B. PHILLIPS, AND S. JIN 37 In view of 3, the error of the nonstandard approximation is given y F z F z : P { ˆβ β/ ˆω z } P { WΞ / z } EDz ς EDz Ξ + O By comparing the cumulants of ς Eς with those of Ξ EΞ we can show that EDz ς EDz Ξ O which, comined with the aove equation, gives heorem 6 elow. he requirement </6 kx dx on that appears in the theorem is a technical condition in the proof that facilitates the use of a power series expansion. he requirement can e relaxed, ut at the cost of more extensive and tedious calculations. HEOREM 6: Let Assumptions and 3 hold. If </6 kx dx, then 38 F z F z + O uniformly over z R + when with fixed Under the null hypothesis H : β β,wehaveδ In this case, heorem 6 indicates that the ERP for tests with fixed and using critical values otained from the nonstandard limit distriution of WΞ / is O he theorem is an extension to the result of Jansson 4, who considered only the Bartlett-type kernel with and proved that the ERP is O log It is an open question in Jansson 4 whether the log factor can e omitted. heorem 6 provides a positive answer to this question. In the previous section, we showed that when such that 39 F z Dz + O q/q+ for typical economic time series. Comparing 38 with39, one may conclude that the error of the nonstandard approximation is smaller than that of the standard normal approximation y an order of magnitude. However, the two O terms are otained under different asymptotic specifications. he O term in 38 holds for fixed, while the O term in 39 holds for diminishing Since the O term in 38 does not hold uniformly over ] the two O terms cannot e directly compared, although they are oviously suggestive of the relative quality of the two approximations when is small, as it typically will e in practical applications. Indeed, F z and Dz are just different approximations to the same quantity F z o compare the two approximations more formally, we need to evaluate F z F z and F z Dz under the same asymptotic specification, that is, either is fixed or

17 BANDWIDH SELECION IN ROBUS ESING 89 First, when is fixed, we have 4 F z Dz F z F z + F z Dz O his is ecause F z F z O/ asshowninheorem6 and F z Dz O.Comparing4with38, we conclude that when is fixed, the error of the nonstandard approximation is smaller than that of the standard approximation y an order of magnitude. Second, when O q/q+ we have 4 F z F z [F z Dz ] [F z Dz ] g q d q D z z q + o{ + q } + O wherewehaveusedheorems and 4. herefore, when d q > which is typical for economic time series, the error of the nonstandard approximation is smaller than that of the standard normal approximation, although they are of the same order of magnitude for this choice of. We can conclude from the aove analysis that the nonstandard distriution provides a more accurate approximation to the finite sample distriution regardless of the asymptotic specification employed. here are two reasons for the etter performance: the nonstandard distriution mimics the randomness of the denominator of the t statistic and it accounts for the ias of the LRV estimator that results from the unoservaility of the regressor errors. As a result, the critical values from the nonstandard limiting distriution provide a higher-order correction on the critical values from the standard normal distriution. However, just as in the standard limiting theory, the nonstandard limiting theory does not deal with another source of ias, that is, the usual ias that arises in spectral density estimation even when a time series is known to e mean zero and oserved. his second source of ias manifests itself in the error of approximation given in OPIMAL BANDWIDH CHOICE It is well known that the optimal choice of that minimizes the asymptotic mean squared error in LRV estimation has the form O q/q+.however, there is no reason to expect that such a choice is the most appropriate in statistical testing using nonparametrically studentized statistics. Developing an optimal choice of for semiparametric testing is not straightforward and involves some conceptual as well as technical challenges. In what follows we provide one possile approach to constructing an optimizing criterion that is ased on alancing the type I and type II errors.

18 9 Y. SUN, P. C. B. PHILLIPS, AND S. JIN In view of the asymptotic expansion 35, we know that the type I error for a two-sided test with nominal size α can e expressed as 4 F z α α + g q d q D z α z α q + o q + + O Similarly, from 36, the type II error has the form 43 G δ z + c α z 4 K α δz g α qd q G δ z α z α q + o q + + O A loss function for the test can e constructed ased on the following three factors: i the magnitude of the type I error, as measured y the second term of 4; ii the magnitude of the type II error, as measured y the O and O q terms in 43; and iii the relative importance of the type I and type II errors. For most economic time series we can expect that d q >, and then oth g q d q D z α z > andg α qd q G δ z α z α > Hence, the type I error increases as decreases. On the other hand, the q term in 43 indicates that there is a corresponding decrease in the type II error as decreases. Indeed, for δ> the decrease in the type II error will generally exceed the increase in the type I error ecause G δ z α >D z for δ 7 5 and z α α , or 58. he situation is further complicated y the fact that there is an additional O term in the type II error. As we saw earlier, K δ z α >, so that the second term of 43 leads to a reduction in the type II error as decreases. hus, the type II error generally decreases with for two reasons one from the nonstandard limit theory and the other from the typical downward ias in estimating the long-run variance. he case of d q < usually arises where there is negative serial correlation in the errors and so tends to e less typical for economic time series. In such acase,4 shows that the type I error increases with while the type II error may increase or decrease with depending on which of the two terms in 43 dominates. hese considerations suggest that a loss function may e constructed y taking a suitale weighted average of the type I and type II errors given in 4 and 43. Setting 44 e I α + g qd q D z z q e II G δz α + c z 4 α K δz α g qd q G δ z α z α q we define the loss function to e 45 L; δ z α w δ + w δ ei + + w δ eii

19 BANDWIDH SELECION IN ROBUS ESING 9 where w δ is a function that determines the relative weight on the type I and II errors, and this function is allowed to depend on the sample size and δ Oviously, the loss L; δ z α is here specified for a particular value of δ and this function could e adjusted in a simple way so that the type II error is averaged over a range of values of δ with respect to some prior distriution over alternatives. We focus elow on the case of a fixed local alternative, in which case we can suppress the dependence of w δ on δ and write w w δ. o sum up, the loss function we consider is of the form 46 L; δ z α [ g q d q {w D z α G δ z α }z α q + c z 4 α K δz α ] + w + C where C [w α + G δ z ]/[ + w α ], which does not depend on In the rest of this section, we consider the case w D z α G δ z α > which holds if the relative weight w is large enough. It turns out that the optimal choice of depends on whether d q > or d q < We consider these two cases in turn. When d q > the loss function L; δ z α is minimized for the following choice of : 47 { qgq d q [w D z α opt G δ z] } /q+ α c zα K δzα q/q+ herefore, the optimal shrinkage rate for is O q/q+ when w is a fixed constant. If w as, we then have 48 { qgq d q D z α opt } /q+ /q+ w zα c K δ zα q Fixed- rules may then e interpreted as assigning relative weight w O q in the loss function so that the emphasis in tests ased on such rules is a small type I error, at least when we expect the type I error to e larger than the nominal size of the test. his gives us an interpretation of fixed- rules in terms of the loss perceived y the econometrician using such a rule. Within the more general framework given y 47, may e fixed or shrink with up to an O q/q+ rate corresponding to the relative importance that is placed in the loss function on the type I and type II errors. Oserve that when Ow / q /q+, size distortion is Ow q/q+ rather than O as it is when is fixed. hus, the use of opt for a finite w involves some compromise y allowing the error order in the rejection proaility to e somewhat larger so as to achieve higher power. Such compromise is an inevitale consequence of alancing the two elements in the loss function 46. Note that even in this case, the order of ERP is smaller than

20 9 Y. SUN, P. C. B. PHILLIPS, AND S. JIN O q/q+ which is the order of the ERP for the conventional procedure in which standard normal critical values are used and is set to e O q/q+ For Parzen and QS kernels the optimal rate of is /3 whereas for the Bartlett kernel the optimal rate is / herefore, in large samples, a smaller should e used with the Parzen and QS kernels than with the Bartlett kernel. In the former cases, the ERP is at most of order /3 and in the latter case the ERP is at most of order / he O /3 rate of the ERP for the quadratic kernels represents an improvement on the Bartlett kernel. Note that for the optimal, the rate of the ERP is also the rate for which the loss function L; δ z α approaches C from aove. herefore, the loss L; δ z α is expected to e smaller for quadratic kernels than for the Bartlett kernel in large samples. Finite sample performance may not necessarily follow this ordering, however, and will depend on the sample size and the shape of the spectral density of {u t } at the origin. he formula for opt involves the unknown parameter d q which could e estimated nonparametrically e.g., Newey and West 994 or y a standard plug-inprocedureasedonasimple model like AR e.g., Andrews 99. Both methods achieve a valid order of magnitude and the procedure is oviously analogous to conventional data-driven methods for HAC estimation. When w D z α G δ z>andd α q < L; δ z α is an increasing function of. o minimize loss in this case, we can choose to e as small as possile. Since the loss function is constructed under the assumption that and the choice of is required to e compatile with these two rate conditions. hese considerations lead to a choice of of the form J / for some J that goes to infinity ut at a slowly varying rate relative to In practice, we may set J log so that log / o sum up, for typical economic time series, the value of that minimizes the weighted type I and type II errors has a shrinkage rate of O q/q+ his rate may e compared with the optimal rate of O q/q+ that applies when minimizing the mean squared error of estimation of the corresponding HAC estimate, ˆω itself Andrews 99. hus, the AMSE optimal values of for HAC estimation are smaller as than those that are most suited for statistical testing. In effect, optimal HAC estimation tolerates more ias so as to reduce variance in estimation. In contrast, optimal selection in HAC testing undersmooths the long-run variance estimate to reduce ias and allows for greater variance in long-run variance estimation through higher order adjustments to the nominal asymptotic critical values or y direct use of the nonstandard limit distriution. his conclusion highlights the importance of ias reduction in statistical testing. In Monte Carlo experiments not reported here, we have compared the finite sample performance of the new plug-in procedure with that of the conventional plug-in procedure given in Andrews 99. he findings from these simulations indicate that the new plug-in procedure works well in terms of incurring a smaller loss than the conventional plug-in procedure. Detailed results of this experiment are reported in Sun, Phillips, and Jin 6.

21 BANDWIDH SELECION IN ROBUS ESING CONCLUDING DISCUSSION Automatic andwidth choice is a long-standing prolem in time series models when the autocorrelation is of unknown form. Existing automatic methods are all ased on minimizing the asymptotic mean squared error of the standard error estimator, a criterion that is not directed at statistical testing. In hypothesis testing, the focus of attention is the type I and type II errors that arise in testing, and it is these errors that give rise to loss. Consequently, it is desirale to make the errors of incorrectly rejecting a true null hypothesis and failing to reject a false null hypothesis as small as possile. While these two types of errors may not e simultaneously reduced, it is possile to design andwidth choice to control the loss from these errors. his paper develops for the first time a theory of optimal andwidth choice that achieves this end y minimizing a weighted average of the type I and type II errors. he results in this paper suggest some important areas of future research. he present work has focused on two-sided tests for the Gaussian location model, ut the ideas and methods explored here can e used as a foundation for tackling andwidth choice prolems in nonparametrically studentized tests and confidence interval construction in general regression settings with linear instrumental variale and Generalized Method of Moments GMM estimation. Results for these general settings will e reported in later work. he additional complexity of the formulae in the general case complicates the search for an optimal truncation parameter, ut the main conclusion of the present findings are the same, namely, that when the goal is testing or confidence interval construction, it is advantageous to reduce ias in HAC estimation y undersmoothing. Dept. of Economics, University of California, San Diego, La Jolla, CA 993, U.S.A.; yisun@ucsd.edu, Cowles Foundation, Yale University, Box 88, New Haven, C 65, U.S.A.; University of Auckland, Auckland, New Zealand; and University of York, York, U.K.; peter.phillips@yale.edu, and Guanghua School of Management, Peking University, Beijing, 87, China; sainan.jin@gsm.pku.edu.cn. Manuscript received Decemer, 4; final revision received June, 7. REFERENCES ANDREWS, D. W. K. 99: Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation, Econometrica, 59, [75,77,79,9] GONÇALVES, S., AND. J. VOGELSANG 5: Block Bootstrap Puzzles in HAC Roust esting: he Sophistication of the Naive Bootstrap, Working Paper, University of Montreal and Cornell University. [76] GRENANDER, U., AND M.ROSENBLA 957: Statistical Analysis of Stationary ime Series. New York: Wiley. [75,85]

22 94 Y. SUN, P. C. B. PHILLIPS, AND S. JIN JANSSON, M. 4: On the Error of Rejection Proaility in Simple Autocorrelation Roust ests, Econometrica, 7, [76,79,84,88] KIEFER, N. M., AND. J. VOGELSANG a: Heteroskedasticity Autocorrelation Roust esting Using Bandwidth Equal to Sample Size, Econometric heory, 8, [76,78] : Heteroskedasticity Autocorrelation Roust Standard Errors Using the Bartlett Kernel Without runcation, Econometrica, 7, [76,78] 5: A New Asymptotic heory for Heteroskedasticity Autocorrelation Roust ests, Econometric heory,, [78,8] KIEFER, N. M.,. J.VOGELSANG, AND H. BUNZEL : Simple Roust esting of Regression Hypotheses, Econometrica, 68, [76] NEWEY, W. K., AND K. D. WES 987: A Simple, Positive Semidefinite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix, Econometrica, 55, [75,79] 994: Automatic Lag Selection in Covariance Estimation, Review of Economic Studies, 6, [75,79,9] PHILLIPS, P. C. B. 98: Finite Sample heory and the Distriutions of Alternative Estimators of the Marginal Propensity to Consume, Review of Economic Studies, 47, [85] PHILLIPS, P.C.B.,Y.SUN, AND S. JIN 5a: Improved HAR Inference Using Power Kernels Without runcation, Mimeographed, Yale University. [8] 5: Balancing Size and Power in Nonparametric Studentized esting with Quadratic Power Kernels Without runcation, Working Paper, Department of Economics, UCSD. [8] 6: Spectral Density Estimation and Roust Hypothesis esting Using Steep Origin Kernels Without runcation, International Economic Review,, [76,8] 7: Long-Run Variance Estimation and Roust Regression esting Using Sharp Origin Kernels with No runcation, Journal of Statistical Planning and Inference, 37, [76,8] SUN, Y., P. C. B. PHILLIPS, AND S. JIN 6: Optimal Bandwidth Selection in Heteroskedasticity Autocorrelation Roust esting, Working Paper, Department of Economics, UCSD. [75,78,9] 7: Supplement to Optimal Bandwidth Selection in Heteroskedasticity Autocorrelation Roust esting, Econometrica Supplementary Material, 76, econometricsociety.org/ecta/supmat/558proofs.pdf. [78,8] VELASCO, C., AND P. M. ROBINSON : Edgeworth Expansions for Spectral Density Estimates and Studentized Sample Mean, Econometric heory, 7, [85]

23 Econometrica Supplementary Material SUPPLEMEN O OPIMAL BANDWIDH SELECION IN HEEROSKEDASICIY AUOCORRELAION ROBUS ESING Econometrica, Vol. 76, No., January 8, BY YIXIAO SUN,PEER C. B. PHILLIPS, AND SAINAN JIN HIS APPENDIX provides technical results and proofs for the paper. A.. echnical Lemmas and Supplements LEMMA : Define c 4 kv dv Let Assumption hold. hen the cumulants κ m of Ξ µ satisfy A. κ m m m!c m for m and its moments α m EΞ µ m satisfy A. α m m m!c m for m PROOF OF LEMMA : o find the cumulants of Ξ µ,wewrite A.3 Ξ where k r s is defined y k r s k r s + k r s dw r dw s k r tdt k t τdt dτ k τ sdτ It can e shown that the cumulant generating function of Ξ µ is A.4 ln φt m it m κ m m! where κ andform, A.5 κ m m m! j m k τ j τ j+ dτ dτ m

24 Y. SUN, P. C. B. PHILLIPS, AND S. JIN with τ τ m+. Now m k τ j τ j+ dτ dτ m j A.6 k τ τ k τ τ 3 k τ m τ m k τ m τ dτ dτ m A.7 sup τ k τ τ k τ τ 3 k τ m τ m dτ dτ m k τ τ dτ sup τ k τ τ 3 k τ 3 τ 4 k τ m τ m dτ dτ m k τ τ dτ A.8 sup τ 3 sup s k τ τ 3 dτ sup τ m k r s dr m [k τ m τ m ] dτ m But A.9 sup s k r s dr 4sup s 4sup s [ ] 4 c k r s dr s s k v dv k v dv As a result, A. m k τ j τ j+ dτ dτ m c m j

25 BANDWIDH SELECION IN ROBUS ESING 3 so A. κ m m m!c m Note that the moments {α j } and cumulants {κ j } satisfy the relationship A. α m π m! j! m j! m jl! m l m!m! m l! κ j j π where the sum is taken over the elements A.3 π [j j }{{} j j j }{{} l j l ] }{{} m times m times m l times for some integer l, sequence{j i } l i such that j >j > >j l and m l i m ij i. Comining the preceding formula with A. gives A.4 α m < m m!c m π m m!c m j m j m jl m l m!m! m l! where the last line follows ecause A.5 j m j m jl m l π m!m! m l! π m!m! m l! < m Q.E.D. LEMMA : Let Assumptions and 3 hold. When for a fixed, we have A.6 µ µ + O ; { } m! m A.7 κ m κ m + O c m uniformly over m ; { } m! m A.8 α m Eς µ m α m + O c m uniformly over m.

26 4 Y. SUN, P. C. B. PHILLIPS, AND S. JIN PROOF OF LEMMA : We first calculate µ ω raceω A W A. Let W A W A. hen the i jth element of W is A.9 So k i j i j k q p k q j i p k + p q p q k A. But A. raceω A W A raceω W { γr r k r r } h r r r r γh k r + h h r h + r h r r + h k r h r + h k h r q r p h r h r +h p r γh k r + h r h q r k + r k r p r p q r p k p q r q k q r p q k p q h k + k + Ch

27 BANDWIDH SELECION IN ROBUS ESING 5 r r s r s h k + k + Ch r k r { } h + k k + Ch where Ch is a function of h that satisfies Ch h. Similarly, A. r h k r r + h r r k r { } h + k k + Ch herefore, raceω A W A is equal to A.3 h + γh + O h + r γh + q h + γh r k r + r h + r + q g q r k r h + { } h γh k k { } kh/ k h q γh + O h/ q r k r h q γh + o + O h Using A.4 h + γh ω + O

28 6 Y. SUN, P. C. B. PHILLIPS, AND S. JIN and A.5 r r k r k r r dr + O we now have A.6 µ k r r dr q g q + O ω h q γh + o h By definition, µ EΞ k r rdr and thus µ µ + O as desired. We next approximate race[ω A W A m ] for m>. he approach is similar to the case m ut notationally more complicated. Let r m+ r, r m+ r,andh m+ h.hen A.7 where A.8 race[ω A W A m ] m γr j r j k rj r j+ I r r r m+ j r r r 4 r m h r h h r m j I + II + r 4 h r 4 h r h r m m h m r m j γh j k rj r j+ + h j+ h r m j h + h r h γh j k rj r j+ h m r m γh j k rj r j+ + h j+ h m h m h m r m + + h m r m h m h m r m h m

29 BANDWIDH SELECION IN ROBUS ESING 7 and A.9 II O { h h r + h r h m } hj+ γh j j h m h m r m + h m r m h m Here we have used k rj r j+ + h j+ A.3 o show this, note that A.3 p k rj r j+ O p rj+ h j+ k h j+ p h j+ k p hj+ p rj+ p rj+ hj+ k + O and A.3 so that A.33 k rj r j+ h j+ rj r j+ k O hj+ rj k r j+ + h j+ k rj r j+ rj r j+ h j+ + k rj r j+ hj+ k + O k rj r j+ + O hj+ he first term I can e written as A.34 I h r h m r m h r h + h m r m h m+ h h r h m h m r m

30 8 Y. SUN, P. C. B. PHILLIPS, AND S. JIN m j π { rj γh j k r } j+ h r m h m r m j { rj γh j k r } j+ where h j r j is one of the three choices hj h j r j h r,and π,,or h j r j h j r j h j + is the summation over all possile cominations of h m r m. he 3 m summands in A.34 can e divided into two groups: the first group consists of the summands all of whose r indices run from to and the second group consists of the rest. It is ovious that the first group can e written as h m γh j j r { rj k r j+ } he dominating terms in terms of the order of magnitude in the second group are of the forms h r h p r p h p+ m h m r m j { rj γh j k r } j+ or h r h p h p r p m h m r m j { rj γh j k r } j+ hese are the summands with only one r index not running from to.both of the aove terms are ounded y h r [ sup r 4 r h p h m r m j m γh j h p k rj r j+ j p r k r ] m 4 m γh j γh p h p h j h p using the same approach as in A.6. Approximating the sum y an integral and noting that the second group contains m terms, all of which are of the same order of magnitude as the aove typical dominating terms, we

31 BANDWIDH SELECION IN ROBUS ESING 9 conclude that the second group is O[m m c m ] uniformly over m. Asa consequence, A.35 I h m γh j j r { rj k r j+ } + O{m m c m } uniformly over m. he second term II is easily shown to e om m c m uniformly over m. herefore, A.36 race[ω A W A m ] m { rj γh k r } j+ + O{m m c m } h r and A.37 κ m m m! m ω m race[ω A W A m ] { m m! m rj k r [ ]} j+ m + O c m r m m! { m j { } m! m κ m + O c m [ ] } m k τ j τ j+ dτ j dτ j+ + O c m uniformly over m. Finally, we consider α m. Note that α Eς µ and A.38 α m π m! κ j! m j! m jk! m j k m!m! m k! j π where the summation π is defined in A.. Comining the preceding formula with A.7 gives { m α m α m + O } m! A.39 c m m π!m! m k! { } m! m α m + O c m

32 Y. SUN, P. C. B. PHILLIPS, AND S. JIN uniformly over m, where the last line follows ecause π < m. m!m! m k! Q.E.D. LEMMA 3: Let Assumptions and 3 hold. If and such that, then A.4 A.4 µ κ for m 3 and 4, A.4 k r r dr q g q + O ; κ m O m + O ω h q γh + o h k r s dr ds + o + O PROOF OF LEMMA 3: WehaveprovedA.4 in the proof of Lemma ecause equation A.6 holds for oth fixed and decreasing. It remains to consider κ m for m, 3, and 4. We first consider κ ω 4 race[ω A W A ]. As a first step, we have ; A.43 race[ω A W A ] { r k r 3 r4 k r } γr r γr 3 r 4 r r r 3 r 4 r r 4 r h r r 4 h r 4 γh γh h h r + { r k r 4 + h : I + I + I 3 + I 4 { k r h r h r 4 + h h h r 4 r4 k r + h r4 k r } + h + h r 4 h } γh γh

33 BANDWIDH SELECION IN ROBUS ESING where and I h h h h r r 4 I I 3 I 4 h h r h r 4 h γh γh h { r k r 4 + h r4 k r } + h γh γh h h r h r r 4 γh γh { k r h r h h r 4 h γh γh r 4 + h r4 k r } + h { r k r 4 + h r4 k r } + h { k r r 4 + h r4 k r } + h We now consider each term in turn. Using equation A.33, we have r k r 4 + h k r r 4 h A.44 + O and A.45 r4 k r + h k r4 r + O h It follows from A.44andA.45 that { r I k r 4 + h r4 k r } + h A.46 γh γh h h r r 4 [ + O h + h + h h ] γh γh h h { h h r r 4 + O r k r 4 + h r4 k r } + h γh γh

34 Y. SUN, P. C. B. PHILLIPS, AND S. JIN { h h r r 4 { + O r k r 4 r4 k r } γh γh + O k r r 4 h + h h h r r 4 } { h h r r 4 γh γh Following the same procedure, we can show that A.47 A.48 I I 3 { h r h r 4 { h r r r 4 r k r 4 r4 k r } γh γh + O r k r 4 r4 k r } γh γh + O r k r 4 r4 k r } γh γh + O and A.49 I 4 { h r h r 4 + O r k r 4 r4 k r } γh γh As a consequence, A.5 race[ω A W A ] r r4 { r k r } 4 γh + O h and A.5 κ ω 4 race[ω A W A ] k r s dr ds + o + O he proof for κ m for m 3 and 4 is essentially the same except that we use Lemma to otain the first term O m. he details are omitted. Q.E.D.