POWER MAXIMIZATION AND SIZE CONTROL IN HETEROSKEDASTICITY AND AUTOCORRELATION ROBUST TESTS WITH EXPONENTIATED KERNELS

Transcription

1 POWER MAXIMIZAION AND SIZE CONROL IN HEEROSKEDASICIY AND AUOCORRELAION ROBUS ESS WIH EXPONENIAED KERNELS By Yixiao Sun, Peter C.B. Phillips and Sainan Jin COWLES FOUNDAION PAPER NO. 34 COWLES FOUNDAION FOR RESEARCH IN ECONOMICS YALE UNIVERSIY Box 88 New Haven, Connecticut

2 Econometric heory, 7,, doi:.7/s POWER MAXIMIZAION AND SIZE CONROL IN HEEROSKEDASICIY AND AUOCORRELAION ROBUS ESS WIH EXPONENIAED KERNELS YIXIAO SUN University of California, San Diego PEER C.B. PHILLIPS Yale University, University of Auckland, University of Southampton, Singapore Management University SAINAN JIN Singapore Management University Using the power kernels of Phillips, Sun, and Jin 6, 7, we examine the large sample asymptotic properties of the t-test for different choices of power parameter. We show that the nonstandard fixed- limit distributions of the t-statistic provide more accurate approximations to the finite sample distributions than the conventional large- limit distribution. We prove that the second-order corrected critical value based on an asymptotic expansion of the nonstandard limit distribution is also second-order correct under the large- asymptotics. As a further contribution, we propose a new practical procedure for selecting the test-optimal power parameter that addresses the central concern of hypothesis testing: he selected power parameter is test-optimal in the sense that it minimizes the type II error while controlling for the type I error. A plug-in procedure for implementing the test-optimal power parameter is suggested. Simulations indicate that the new test is as accurate in size as the nonstandard test of Kiefer and Vogelsang a, b, and yet it does not incur the power loss that often hurts the performance of the latter test. he results complement recent work by Sun, Phillips, and Jin 8 on conventional and b HAC testing.. INRODUCION Seeking to robustify inference, many practical methods in econometrics now make use of heteroskedasticity and autocorrelation consistent HAC covariance he authors gratefully acknowledge partial research support from NSF under Grant no. SES Sun, SES Phillips, and NSFC under Grant no. 75 and 76 Jin. Address correspondence to Yixiao Sun, Department of Economics, University of California, San Diego, 95 Gilman Dr., La Jolla, CA ; yisun@ucsd.edu. 3 c Cambridge University Press

3 POWER MAXIMIZAION AND SIZE CONROL IN HAR ESS 3 matrix estimates. Most commonly used HAC estimates are formulated using conventional kernel smoothing techniques see den Haan and Levin 997 for an overview, although quite different approaches like wavelets Hong and Lee, and direct regression methods Phillips, 5b have recently been explored. While appealing in terms of their asymptotic properties, consistent HAC estimates provide only asymptotic robustness in econometric testing, and finite sample performance is known to be unsatisfactory in many cases, but especially when there is strong autocorrelation in the data. HAC estimates are then biased downwards and the associated tests are liberal-biased. hese size distortions in testing are often substantial and have been discussed extensively in recent work e.g., Kiefer and Vogelsang, 5, hereafter KV 5; Sul, Phillips, and Choi, 5. o address the size distortion problem, Kiefer, Vogelsang, and Bunzel hereafter KVB, Kiefer and Vogelsang a, b hereafter KV, and Vogelsang 3 suggested setting the bandwidth equal to the sample size i.e., M = in the construction of the long-run variance LRV estimation. While the resulting LRV estimate is inconsistent, the associated test statistic is asymptotically nuisance parameter-free, and critical values can be simulated from its nonstandard asymptotic distribution. We may therefore regard these procedures as falling within a general class of heteroskedastic and autocorrelation robust HAR techniques in econometrics Phillips, 5a. KV show by simulation that the nonstandard test has better size properties than the conventional asymptotic normal or chi-squared test. However, the size improvement comes at the cost of a clear power reduction. In order to reduce the power loss while maintaining the good size properties of the KVB test, KV 5 set the bandwidth to be proportional to the sample size, i.e., M = b for some b,]. heir approach is equivalent to contracting traditional kernels k to get k b x = kx/b and using the contracted kernels k b in the LRV estimation without truncation. In other work, Phillips, Sun, and Jin 6, 7 hereafter PSJ obtain a new class of kernels by exponentiating the conventional kernels using k = k, where is a power exponent parameter. For finite b and, both contracted and exponentiated kernels are designed to reduce, but not totally eliminate, the randomness of the denominator in the t-ratio and in doing so help to improve the power of the t-test. he parameter b in the KV approach and in the PSJ approach are smoothing parameters that play an important role in balancing size distortion and potential power gain. In other words, the smoothing parameters entail some inherent trade-off between type I and type II errors. Both KV and PSJ suggest plug-in procedures that rely on conventional asymptotic theories to select b or. More specifically, the plug-in selection of b or suggested in these papers minimizes the asymptotic mean squared errors AMSEs of the underlying LRV estimator. In theory, such formulations ensure that the selected values b and as, and thus the fixed b or fixed asymptotic theory is not applicable. However, to maintain good size properties or smaller type I errors in practical testing,

4 3 YIXIAO SUN E AL. KV and PSJ propose using critical values from the fixed b or fixed asymptotic distributions, which are nonstandard, and treating the estimated b or as fixed even though they are delivered by asymptotic formulas. PSJ justify this hybrid procedure on the basis of simulations that show the resulting tests have better size properties than tests that use standard normal asymptotic critical values. However, there are two remaining problems with this procedure. First, the rationale for good test size based on such plug-in values of is not rigorously established. Second, the AMSE-optimal choice of is not necessarily optimal in the context of hypothesis testing. A primary contribution of the present paper is to provide analytic solutions to both of these problems by means of asymptotic expansions that provide higherorder information about the type I and type II errors. his approach provides a rigorous justification for the recommended procedure. We consider both firstorder and second-order power kernels from PSJ 6, 7, which have been found to work very well in simulations and empirical applications see Ray and Savin, 8; Ray, Savin, and iwari, 9. o investigate the size properties of the PSJ test, we consider the Gaussian location model, which is used also in Jansson 4 and Sun, Phillips, and Jin 8 hereafter SPJ. Asymptotic expansions developed here reveal that the PSJ statistic is closer to its limit distribution when is fixed than when increases with. More specifically, the error in rejection probability ERP of the t-test based on the nonstandard limiting distribution is of order O/ while that based on the standard normal is O/ q/q+. his result relates to similar results in Jansson and SPJ, who showed that the ERP of the KVB test is of order Olog / and O /, respectively, while the ERP of the conventional test using the Bartlett kernel is at most O/, as shown in Velasco and Robinson. hese findings therefore provide theoretical support for the simulation results reported in PSJ 6, 7, Ray and Savin 8, and Ray et al. 9. he PSJ test, which is based on the nonstandard limiting distribution, is not very convenient to use in practice as the critical values have to be simulated. o design an easy-to-implement test, we develop an expansion of the nonstandard limiting distribution about its limiting chi-squared distribution. A Cornish-Fishertype expansion then leads to second-order corrected critical values. We find that the corrected critical values provide good approximations to the actual critical values of the nonstandard distribution. he PSJ test based on the corrected critical values has the advantage of being easily implemented and does not require the use of tables of nonstandard distributions. o show that the hybrid PSJ test using a plug-in exponent and nonstandard critical value is generally less size-distorted than the conventional test, we develop a higher-order asymptotic expansion of the finite sample distribution of the t-statistic as and go to infinity simultaneously. It is shown that the corrected critical values obtained from the asymptotic expansion of the nonstandard distribution are also second-order correct under the conventional limiting theory. his

5 POWER MAXIMIZAION AND SIZE CONROL IN HAR ESS 33 finding provides a theoretical explanation for the size improvement of the hybrid PSJ test compared to the conventional test. Combining the standard t-statistic and the high-order corrected critical values, we obtain a new t -test whose type I and type II errors can be approximately measured using the above asymptotic expansions. he type I and type II errors depend on the power exponent parameter used in the HAC estimation. Following Sun 9, we propose to choose this parameter to minimize the type II error subject to the constraint that the type I error is bounded. he bound is defined to be κα, where α is the nominal type I error and κ> is a parameter that captures the user s tolerance on the discrepancy between the nominal and true type I errors. he parameter κ is allowed to be sample-size dependent. For a smaller sample size, we may have a higher tolerance, while for larger sample sizes we may have lower tolerance. he new procedure addresses the central concern of classical hypothesis testing, viz., maximizing power subject to controlling size. For convenience we refer to the resulting as the test-optimal. he test-optimal is fundamentally different from the mean squared erroroptimal MSE-optimal that applies when minimizing the AMSE of the corresponding HAC variance estimate cf. PSJ 6, 7. When the tolerance factor κ is small enough, the test-optimal is of smaller order than the MSE-optimal. he test-optimal can even be O for certain choices of the tolerance factor κ. he theory provides some theoretic justification for the use of fixed rules in econometric testing. Simulation results show that the new plug-in procedure suggested in the present paper works remarkably well in finite samples. o implement the test-optimal for two-sided tests in a location model, users may proceed in 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 available. In the absence of such a value, we recommend that the user set the default discrepancy parameter value δ =.39 in below. his choice of δ ensures that the first-order power of the test is 75%.. Specify the significance level α of the test and the associated two-sided standard normal critical value z α satisfying z α = α/. Specify a tolerance parameter κ so that κα is the intended upper bound for the type I error. We suggest setting κ =. as the default value. 3. Estimate the model by ordinary least squares OLS and construct the residuals û t. Fit an AR model to the estimated residuals and compute ˆφ ˆd =, ˆσ = ˆφ û t ˆφû t, t= δ = c ˆφ / ˆσ,

6 34 YIXIAO SUN E AL. where ˆ is the OLS estimator of the autoregression AR coefficient. Set δ =.39 as a default value if the user is unsure about the alternative hypothesis. 4. Specify the kernel function to be used in HAC standard error estimation. Among the commonly used positive definite kernels, we recommend a suitable second-order kernel q = such as the Parzen or quadratic spectral QS kernel. 5. Compute the automatic power exponent parameter: ˆ opt = cδ G q/q+ δ z α,3 ] qg ˆd [ˆλ q /q+, opt G z α, G δ z α, where ˆλ opt = G δ z α, G z α, + cδ G δ z α,3 [ gdg zα,] /q q[κ α] +/q, if ˆd < z +/q α, if ˆd >, 3 { 6., Parzen kernel g =.4, QS kernel {.539, Parzen kernel, c =, 4., QS kernel and G δ,k is the probability density function pdf of a noncentral chisquared variate with k degrees of freedom and noncentrality parameter δ. 6. Compute the HAC standard error using power parameter ˆ opt and construct the usual t-statistic t ˆopt. Reject the null hypothesis if t ˆopt ẑ α, ˆopt, where ẑ α, ˆopt = z α + { π / + ˆ opt g z α + 4 } 4 z3 α. 5 In related work, SPJ 8 considered conventional kernels and selected the bandwidth to minimize a loss function formed from a weighted average of type I and type II errors. he present paper differs from SPJ in two aspects. First, while SPJ used the contracted kernel method, an exponentiated kernel approach is used here. An earlier version of the present paper followed SPJ and employed a loss function methodology to select the power exponent parameter, finding that for both LRV estimation and hypothesis testing, the finite sample performance of the exponentiated kernel method is similar to and sometimes better than that of the contracted kernel method. So, exponentiated kernels appear to have some natural advantages. A simulation study in the present paper provides some further evidence on these advantages. Second, the procedure for selecting the smoothing parameter is different in the present paper. While SPJ selected the smoothing parameter to minimize loss based on a weighted average of type I and type II

7 POWER MAXIMIZAION AND SIZE CONROL IN HAR ESS 35 errors, we minimize the type II error after controlling for the type I error. In effect, the loss function here is implicitly defined with an endogenous weight given by the Lagrange multiplier, while the loss function in SPJ is explicitly defined and thus requires a user-chosen weight. his requirement can be regarded as a drawback of the explicit loss function approach, especially when it is hard to evaluate the relative importance of type I and type II errors. Furthermore, the implicit loss function approach used here is more in line with the standard econometrics testing literature, where size control is often considered to be a priority. he rest of the paper is organized as follows. Section overviews the class of power kernels that are used in the present paper and reviews some first-order limit theory for Wald-type tests as with the power parameter fixed and as. Section 3 derives an exact distribution theory using operational techniques. Section 4 develops an asymptotic expansion of the nonstandard limit distribution under both the null and alternative hypotheses as the power parameter. Section 5 develops comparable finite sample expansions of the statistic as forafixed and as both and. Section 6 proposes a selection rule for that is suitable for implementation in semiparametric testing. Section 7 reports some simulation evidence on the performance of the new procedure. Section 8 concludes. Proofs and additional technical results are in the Appendix.. HEEROSKEDASICIY AND AUOCORRELAION ROBUS INFERENCE FOR HE MEAN Consider the location model y t = β + u t, t =,,...,, 6 with u t autocorrelated and possibly heteroskedastic, and Eu t =. o test a hypothesis about β, we consider the OLS estimator ˆβ = Ȳ = t= y t. Recentering and normalizing gives us ˆβ β = / S, 7 where S t = t τ= u τ. We impose the following convenient high-level condition e.g., KVB; PSJ, 6, 7; Jansson, 4. Assumption. he partial sum process S [r] satisfies the functional law / S [r] ωw r, r [,], where ω is the long-run variance of u t and W r is the standard Brownian motion. hus, ˆβ β ωw = N,ω. Let û τ = y τ ˆβ be the demeaned time series, and define the corresponding partial sum process Ŝ t = t τ= ûτ. Under Assumption, we have / Ŝ [r] ωv r, r [,], where V is a standard Brownian bridge process. When u t is stationary, the long-run variance of u t is

8 36 YIXIAO SUN E AL. ω = γ + j= γj, where γj = Eu tu t j. he conventional approach to estimating ω typically involves smoothing and truncation lag covariances using kernel-based nonparametric HAC estimators. HAC estimates of ω typically have the form ˆω M = j= + { k j M ˆγj, ˆγj = j t= ût+ jû t for j t= j+ ût+ jû t for j <, 8 where ˆγj are sample covariances, k is some kernel function, and M is a bandwidth parameter. Consistency of ˆω M requires that M grows with the sample size but at a slower rate so that M = o e.g., Andrews, 99; Andrews and Monahan, 99; Hansen, 99; Newey and West, 987, 994; de Jong and Davidson,. Jansson provides a recent overview and weak conditions for consistency of such estimates. o test the null H : β = β against H : β β, the standard nonparametrically studentized t-ratio statistic is of the form t ˆωM = / ˆβ β / ˆωM, 9 which is asymptotically standard normal. ests based on t ˆωM and critical values from the standard normal are subject to size distortion, especially when there is strong autocorrelation in the time series. In a series of papers, KVB and KV propose the use of kernel-based estimators of ω in which M is set equal to the sample size or proportional to, taking the form M = b. hese so called fixed-b estimates are inconsistent and tend to random quantities instead of ω, so the limit distribution of 9 is no longer standard normal. Nonetheless, use of these estimates results in valid asymptotically similar tests. For convenience and as in the Introduction, we refer to the standard approach as the contracted kernel approach. In related work, PSJ 6, 7 propose the use of estimates of ω based on power kernels or exponentiated kernels without truncation. he power kernels were constructed by taking an arbitrary integer power of conventional kernels. In this paper we consider the power kernels k x = k BAR x, k PR x, or k QS x, where { x, x k BAR x =, x >, 6x + 6 x 3, for x / k PR x = x 3, for / x,, otherwise k QS x = 5 sin6π x/5 π x 6π x/5 cos6π x/5 are the Bartlett, Parzen, and QS kernels, respectively. hese kernels have a linear or quadratic expansion at the origin: k x = gx q + o x q, as x +,

9 POWER MAXIMIZAION AND SIZE CONROL IN HAR ESS 37 where g =, q = for the Bartlett kernel, g = 6, q = for the Parzen kernel, and g = 8π /5, q = for the QS kernel. For convenience, we call the exponentiated Bartlett kernels the first-order power kernels and the exponentiated Parzen and QS kernels the second-order power kernels. Using k in 8 and letting M = gives HAC estimates of the form ˆω j = ˆγj. j= +k he associated t-statistic is given by t ˆω = / ˆβ β / ˆω. When the power parameter is fixed as, PSJ 6, 7 show that under Assumption, ˆω ω, where = k r sdvrdvs. he associated t -statistic has the nonstandard limit distribution t ˆω W /, under the null and t ˆω δ + W /, 3 under the local alternative H : β = β + c /, where δ = c/ω. When goes to at a certain rate, PSJ 6, 7 further show that ˆω is consistent. In this case, the t -statistic has conventional normal limits: Under the null t ˆω W =d N,, and under the local alternative t ˆω δ + W. hus, the t -statistic has nonstandard limit distributions arising from the random limit of the HAC estimate ˆω when is fixed as, just as the KVB and KV tests do. However, as increases the effect of this randomness diminishes, and when the limit distributions approach those of standard regression tests with consistent HAC estimates. he mechanism we develop for making improvements in size without sacrificing much power is to use a test statistic constructed with ˆω and to employ critical values that are second-order corrected. he correction is first obtained using an accurate but simple asymptotic expansion of the nonstandard distribution about its limiting chi-squared distribution that applies as. his expansion is developed in Section 4. he correction is further justified by an asymptotic expansion of the finite sample distribution in Section PROBABILIY DENSIIES OF HE NONSANDARD LIMI DISRIBUION AND HE FINIE SAMPLE DISRIBUION his section develops some useful formulas for the probability densities of the fixed limit theory and the exact distribution of the test statistic.

10 38 YIXIAO SUN E AL. First note that in the limit theory of the t-ratio test, W is independent of, so the conditional distribution of W / given is normal, with zero mean and variance. We can write = V, where the process V has probability measure P V. he pdf of t = W / can then be written in the mixed normal form as p t z = N dpv. V>, For the finite sample distribution of t = t ˆω, we assume that u t is a Gaussian process. Since u t is in general autocorrelated, ˆβ β and ˆω are statistically dependent. o find the exact finite sample distribution of the t- statistic, we decompose ˆβ and ˆω into statistically independent components. Let u = u,...u, y = y,..., y, l =,...,, and = varu. hen the generalized least squares GLS estimator of β is β = l l l y, and ˆβ β = β β + l l l ũ, 4 where ũ = I l l l l u, which is statistically independent of β β. herefore the t-statistic can be written as t = β β ˆω û + l ũ ˆω ũ. 5 It is easy to see that û = I l l l l u = I l l l l ũ. In consequence, the conditional distribution of t given ũ is N l ũ l ˆω ũ, l ˆω ũ. 6 Letting P ũ be the probability measure of ũ, we deduce that the probability density of t is p t z = N l ũ l ˆω ũ, l ˆω ũ dpũ = E N l ũ l ˆω ũ, l ˆω ũ which is a mean and variance mixture of normal distributions., 7

11 POWER MAXIMIZAION AND SIZE CONROL IN HAR ESS 39 Using ũ N, l l l l and employing operational techniques along the lines developed in Phillips 993, we can write expression 7 in the form, where q = / q, p t z = N l q l ˆω q, l ˆω q e q ũdpũ = N l q l ˆω q, l ˆω q e q { l q= } l l l q. his provides a general expression for the finite sample distribution of the test statistic t under Gaussianity. 4. EXPANSION OF HE NONSANDARD LIMI HEORY Asymptotic expansions of the limit distributions given in and 3 can be obtained as the power exponent parameter. Moreover, these expansions may be developed for both the central and noncentral chi-squared limit distributions that apply when, corresponding to the null and alternative hypotheses. he approach adopted here is to work with the expansion of the noncentral distribution and is therefore closely related to the approach used in SPJ 8 for studying standard and b type tests. Let G λ = G ; λ be the cumulative distribution function { CDF of a noncentral } χ λ variate with noncentrality parameter λ δ+w / ; then P z = P { δ + W z } = E { G δ z }. An expansion of E { G δ z } can be developed in terms of the moments of μ, where μ = E and σ = var. In particular, we have EG δ z = G δ μ z + G δ μ z E μ z E [ G δ μ z μ 3 z 6 ] + O q= {E μ 4 }, 8 as, where the O term holds uniformly for any z [M l, M u ] R + and M l and M u may be chosen arbitrarily small and large, respectively. It is easy to see that = k r,sdwrdws, where k r,s is defined by k r,s = k r s k r tdt k τ sdτ + k t τdtdτ. Since k r,s is a positive semidefinite kernel see Sun, 4, for a proof, it can be represented as k r,s = i= λ n f n r f n s by Mercer s theorem,

12 33 YIXIAO SUN E AL. where λ n > are the eigenvalues of the kernel and f n r are the corresponding eigenfunctions, i.e., λ n f n s = k r,s f n rdr. Using this representation, we can write as = n= λ n Z n, where Z n iidn,. { herefore, } the characteristic function of μ is given by φ t = E e it μ = e itμ { n= iλ n t } /. Let κ,κ,κ 3,... be the cumulants of μ. hen κ =,κ m = m m!... m k τ j,τ j+ j= dτ...dτ m, 9 where τ = τ m+ and m. hese calculations enable us to develop an asymptotic expansion of E{G δ z } as the power parameter. A full series expansion is possible using this method, but we only require the leading term in the expansion in what follows. Ignoring the technical details, we have, up to smaller-order terms, { } δ P + W / z z k r sdrds = G δ z z G δ + z 4 G δ z k r sdrds. Depending on the value of q, the integral k r sdrds has different expansions as. For first-order power kernels, direct calculation yields k r sdrds = + O. For second-order power kernels, we use the Laplace approximation and obtain π / k r sdrds = + O. g It is clear that the rate of decay of the integral depends on the local behavior of the kernel function at the origin. Plugging these expressions into, we obtain the following theorem. { } δ / HEOREM. Let F δ z := P + W z be the nonstandard limiting distribution. hen as, F δ z = G δ z + /q c q L q G δ, z + O where { L q G δ, z = + G δ z z 4 G δ z z } I q = { G δ z z 4 G δ z z } I q =, /q,

13 POWER MAXIMIZAION AND SIZE CONROL IN HAR ESS 33 π / c q = giq = + I q =, g and the remainder O term holds uniformly for any z [M l, M u ] with < M l < M u <. When δ =, we have F z = Dz + c q /q L q D, z + O /q, where D = G is the CDF of χ distribution. For any α,, let z α, R +, zα R+, such that F z α, = α, Dzα = α. hen, using a Cornish- Fisher-type expansion, we obtain the following corollary. COROLLARY. Second-order corrected critical values based on the expansion are as follows: i For the first-order power kernel, z α, = z α + 4 5z α + zα 3 + O. 3 ii For the second-order power kernel, z α, = z α + { π / + z α + g 4 } 4 z3 α + O, 4 where z α is the nominal critical value from the standard normal distribution. Consider as an example the case where α =.5, z α =.96, and PW.96 =.95. hus, for a two-sided t ˆω test, the corrected critical values at the 5% level for the Bartlett, Parzen, and QS kernels are z BAR α, = , zα, PAR.93 =.96+, z QS 3.95 α, =.96+, 5 respectively. hese are also the critical values for the one-sided test > at the.5% level. Similarly, the corrected critical values for α =. are given by z BAR α, = , zα, PAR.375 =.645+, zα, QS.85 =

14 33 YIXIAO SUN E AL. We can evaluate the accuracy of the approximate critical values by comparing them with the exact ones obtained via simulations. In all three cases, the secondorder corrected critical values are remarkably close to the exact ones. Details are available upon request. Since the limiting distributions and 3 are valid for general regression models under certain conditions on the regressors see PSJ, 6, 7, the corrected critical value z α, may be used for hypothesis testing in a general regression framework. When δ and the corrected critical values are used, we can establish the local asymptotic power in the following corollary. COROLLARY. he local asymptotic power satisfies { } δ P + W / > z α, = G δ zα c q zα 4 K δ zα /q + O /q, where 7 K δ z = G δ z D z D z G δ δ z = z G δ z,3, 8 and G δ z,3 is the pdf of the noncentral chi-square distribution with 3 degrees of freedom. Corollary shows that the asymptotic test power increases monotonically with when is large. For a given critical value z α, the function f δ = zα 4 K δ z α obtains its maximum around δ =, implying that the power improvement from choosing a large is greatest when the local alternative is in an intermediate neighborhood of the null. he default value of δ we use in developing a testoptimal smoothing parameter choice is in this intermediate neighborhood. 5. EXPANSIONS OF HE FINIE SAMPLE DISRIBUION Following SPJ 8, we now develop an asymptotic expansion of the finite sample distribution of the t-statistic in a simple location model. We start with the following weak dependence condition. Assumption. Assume u t is a mean zero stationary Gaussian process with h= h γ h <, where γ h = Eu t u t h. In what follows, we develop an asymptotic expansion of P{ ˆβ β / ˆω z} for ˆω =ˆω and for local alternatives of the form β = β +c/. A complicating factor in the development is that ˆβ β and ˆω are in general statistically dependent due to the autocorrelation structure of u t. o overcome this difficulty,

15 POWER MAXIMIZAION AND SIZE CONROL IN HAR ESS 333 we decompose ˆβ and ˆω into statistically independent components as in Section 3. After some manipulation, we obtain { } { } F,δ z := P ˆβ β / ˆω z = E G δ z ς + O 9 uniformly over z R +, where ς := ˆω/ω converges weakly to, and ω := var ˆβ β. Since ˆω = û W û = u A W A u, where W is with j,s-th element k j s/ and A = I l l /, ς is a quadratic form in a Gaussian vector. o evaluate E { G δ z ς }, we proceed to compute the cumulants of ς μ for μ := Eς. It is easy to show that the characteristic function of ς μ is given by φ t = I it A W A / ω exp { } itμ, where = Euu and the cumulant generating function is ln φ t = logdet I it A W A ω itμ := m= κ m, it m m!, 3 where the κ m, are the cumulants of ς μ. It follows from 3 that κ, = and m κ m, = m m! m ω [ m ] race A W A for m. By proving that κ m, is close to κ m in the precise sense given in Lemma A.3 in the Appendix, we can establish the following theorem, which gives the order of magnitude of the error in the nonstandard limit distribution of the t-statistic as with fixed. HEOREM. Let Assumption hold. If k vdv </ 6z, then F,δ z = F δ z + O, 3 as with fixed. he requirement k vdv </ 6z on is a technical condition in the proof. It can be relaxed but at the cost of more tedious calculations. Now F,δ z gives the power of the test under the alternative hypothesis H : β β. heorem indicates that when is fixed the power of the test can be approximated by F δ z with an error of order O/. Under the null hypothesis H : β = β, δ =, heorem shows that for fixed the ERP for tests using critical values obtained from the nonstandard limit distribution of W / is

16 334 YIXIAO SUN E AL. O. his rate is faster than the rate for conventional tests based on consistent HAC estimates. Combined with heorem, heorem characterizes the size and power properties of the test under the sequential limit in which goes to infinity first for afixed and then goes to infinity. Under this sequential limit theory, the size distortion of the t-test based on the corrected critical values is { } P ˆβ β / ˆω z α, α = O /q + O, and the corresponding local asymptotic power is { } P ˆβ β / ˆω > z α, = G δ zα c qzα 4 K δ zα /q + O /q + O. o evaluate the order of size distortion, we have to compare the orders of magnitude of /q and /. Such a comparison jeopardizes the sequential nature of the limiting directions and calls for a higher-order approximation that allows and simultaneously. A corollary to this observation is that fixed asymptotics do not provide an internally consistent framework for selecting the optimal power parameter. he next theorem establishes the required higher-order expansion. HEOREM 3. Let Assumption hold. If / + / q as, then F,δ z = G δ z + c q /q L q G δ, z gd γ G δ z z q + o q + O + /q, 3 where d γ = ω h= + h q γh. Under the local alternative hypothesis H : β β =c/, the power of the test based on the corrected critical values is F,δ z α,. heorem 3 shows that F,δ z α, can be approximated by G δ z + c q /q L q G δ, z gd γ G δ z z q, and the approximation error is of order o q + O + /q. Under the null hypothesis, δ = and G δ = D, so F, z = Dz + c q /q L q D, z gd γ D z z q + o q + O + /q. 33 Importantly, the leading term Dz + c q /q L q D, z in this expansion is the same as the corresponding expansion of the limit distribution F z given in. A direct implication is that the corrected critical values from the nonstandard limiting distribution are also second-order correct under the conventional

17 POWER MAXIMIZAION AND SIZE CONROL IN HAR ESS 335 large asymptotics. More specifically, up to smaller-order terms, F, z α, = α gd γ D zα z α q. By adjusting the critical values, we eliminate the term c q /q L q D, z, which reflects the randomness of the standard error estimator. So if the remaining term gd γ D zα z α q is of smaller order than the eliminated term, the use of the corrected critical values given in Corollary should reduce the size distortion. When increases with, the remaining term gd γ D zα z α q approximately measures the size distortion in tests based on the corrected critical values, or equivalently those based on the nonstandard limit theory, at least to order O /q. If critical values from the standard normal are used, then the ERP is given by the O /q and O q terms. o obtain the best rate of convergence of the ERP to zero, we set = O q /q+ to balance these two terms, and the resulting rate of convergence is of order O q/q+. As we discussed before, this rate is larger than that of the nonstandard test by an order of magnitude. One might argue that this comparison is not meaningful, as the respective orders of the ERP are obtained under different asymptotic specifications of : one for growing and the other one for fixed. o sharpen the comparison, we can compare the ERP under the same asymptotic specification as in SPJ 8. When is fixed, the ERP of the standard normal test contains an O term, which makes the ERP larger by an order of magnitude than the ERP of the nonstandard test. When grows with, the ERP of the nonstandard test contains fewer terms than that of the standard normal test. his is usually regarded as an advantage in the econometrics literature. If we set = O q /q+, which is the AMSE-optimal rate for the exponent, then the ERP of the standard normal test is of order O q = O q/q+. In this case, the ERP of the nonstandard test or test using the second-order corrected critical values is also of order O q/q+. herefore, compared with the standard normal test, the hybrid procedure suggested in PSJ 6, 7 does not reduce the ERP by an order of magnitude. However, the hybrid procedure eliminates the O /q term from the ERP and thus reduces the overrejection that is commonly seen in economic applications. his explains the better finite sample performances found in the simulation studies of PSJ 6, 7, Ray and Savin 8, and Ray et al. 9. For convenience, we refer to the test based on the t -statistic t ˆω and the second-order corrected critical values as the t -test. We formalize the results on the size distortion and local power expansion of the t -test in the following corollary. COROLLARY 3. Let Assumption hold. If / +/ q as, then a the size distortion of the t -test is gd γ D zα z α q + o q + O + /q ; 34

18 336 YIXIAO SUN E AL. b under the local alternative, the power of the t -test is G δ zα c qzα 4 K δ zα /q + gd γ G δ z α z q α + o q + O + /q. 35 Just as in standard limit theory, the nonstandard limit theory does not address the bias problem due to nonparametric smoothing in LRV estimation. Comparing 35 with 7, we get an additional term, reflecting the asymptotic bias of the LRV estimator. According to Corollary 3 and ignoring the higher-order terms, the type I and type II errors of the t -test can be measured by e I = α + gd γ D zα z α q, 36 e II = G δzα + c q zα 4 K δ zα /q gd γ G δ z α z α q, respectively. SPJ 8 obtained similar expressions using the contracted kernel approach with bandwidth set equal to M = b. hey showed that approximately e b I = α + gd γ D zα z α b q, e b II = G δzα + μ zα 4 K δzα b gd γ G δ z α z α b q, 37 where μ = k xdx. Clearly, the approximations in 36 and 37 are closely related. o match the orders of magnitude in the above approximations, we can let = b q. In this case, e I = eb I and e II = eb II + c q k xdx zα 4 K δzα b. hat is, in terms of the type I error, the exponentiated kernel approach and the contracted kernel approach are asymptotically equivalent. However, they differ in the type II error. For the Bartlett, Parzen, and QS kernels, we have, using zk δ z = δ G δ z,3, e b e II = II δ zα G δ z α,3b, Bartlett kernel e b II.38 δ zα G δ z α,3b, Parzen kernel. e b II +.57 δ zα G δ z α,3b, QS kernel o calibrate the numerical difference, we take z α =.645 and δ =.39. According to the first-order asymptotics, the type II error is 5%. For this choice, δ zα G δ z α,3 = So for the Bartlett kernel, there is some advantage of using the contracted kernel approach. For the Parzen kernel, the exponentiated kernel approach is better, while for the QS kernel the contracted kernel approach is better. However, the difference is small, especially when b is close to zero. he above comparison is based on higher-order expansions. It may not completely reflect the finite sample performances, especially when the sample size is small. In the simulation study, we will compare the exponential kernel approach with the conventional contracted kernel approach.

19 POWER MAXIMIZAION AND SIZE CONROL IN HAR ESS OPIMAL EXPONEN CHOICE When estimating the long-run variance, PSJ 6, 7 show there is an optimal choice of that minimizes the AMSE of the estimator and give an optimal expansion rate of O q /q+ for in terms of the sample size. he present paper attempts to provide a new approach for optimal exponent selection that addresses the central concern of classical hypothesis testing, which can be expressed as minimizing the type II error subject to controlling the type I error. Sun 9 employs this constraint minimization approach in related work. More specifically, we propose to solve min e II, s.t. e I κα, 38 where e I and e II are approximate measures of type I and type II errors given in the previous section. Here κ is a constant greater than. Ideally, the type I error is less than or equal to the nominal type I error α. In finite samples, there is always some approximation error, and we allow for some discrepancy by introducing the tolerance factor κ. For example, when α = 5% and κ =., we aim to control the type I error such that it is not greater than 6%. Note that κ may depend on the sample size. For a larger sample size, we may require κ to take smaller values. he solution to the minimization problem depends on the sign of d γ. When d γ <, the constraint e I κα is not binding, and we have the unconstrained minimization problem opt = argmin e II, whose solution for the optimal is opt = c qzα K δ z q/q+ α qgd γ G δ z α q /q+. 39 When d γ >, the constraint e I κα may be binding, and we have to use the Kuhn-ucker theorem to search for the optimum. Let λ be the Lagrange multiplier, and define L,λ = G δ zα + c qzα 4 K δ zα /q gd γ G δ z α z α q 4 + λ α + gd γ D zα z α q κα. It is easy to show that at the optimal, the constraint e I κα is indeed binding and λ>. Hence, the optimal is κ α opt = gd γ D zα q, 4 z α and the corresponding Lagrange multiplier is D zα + c [ q K δ z α gdγ D zα ] q q [κ α] +/q λ opt = G δ z α z α /q+.

20 338 YIXIAO SUN E AL. Formulas 39 and 4 can be written collectively in the form c q zα opt = K δ z q/q+ [ α qgd γ λopt D zα G δ z α ] q /q+, where { λ opt = G δ z α D zα + c [ q K δ z α gdγ D zα ] /q q[κ α] +/q, if d γ < z /q+ α, if dγ >. 4 If we ignore the nonessential constant, then the function L,λ is a weighted sum of the type I and type II errors with the weight given by the optimal Lagrange multiplier. When d γ <, the type I error is expected to be capped by the nominal type I error. As a result, the optimal Lagrange multiplier is zero, and we assign all weight to the type II error. his weighting scheme might be justified by the argument that it is worthwhile to take advantage of the extra reduction in the type II error without inflating the type I error. When d γ >, the type I error is expected to be larger than the nominal type I error. he constraint on the type I error is binding, and the Lagrange multiplier is positive. In this case the loss function is a genuine weighted sum of type I and type II errors. As the tolerance parameter κ decreases toward, the weight attached to the type I error increases. When the nonparametric bias, as measured by d γ, is positive, the optimal grows with at the rate of q /q+, which is slower than q /q+, the MSE-optimal expansion rate. When the nonparametric bias is negative, which is typical for economic time series, the expansion rate of the optimal depends on the rate at which κ approaches. When κ at a rate faster than q, the Lagrange multiplier λ opt increases with the sample size at a rate faster than q. In this case, the optimal is bounded. Fixed rules may then be interpreted as assigning increasingly larger weight to the type I error. his gives us a practical interpretation of fixed rules in terms of the permitted tolerance of the type I error. When κ at a rate slower than q/q+, the Lagrange multiplier λ opt is bounded and the optimal expands with at a rate faster than q /q+. In particular, when κ at the rate of q/q+, the optimal expands with at the MSE-optimal rate q /q+. So when κ at a rate faster than q/q+, the optimal has a smaller order of magnitude than the MSE-optimal regardless of the direction of the nonparametric bias. When opt = O / q q /q+ λ opt, the size distortion of the t -test is of order O λ opt q/q+. It is apparent that the size distortion becomes smaller if a larger weight is implicitly assigned to the type I error. In particular, when λ opt is finite, the size distortion is of order O q/q+, which is larger than O q, the size distortion for the case λ opt q. It is obvious that the use of opt involves some trade-off between two elements in the loss function 4. Note that even when λ opt is finite, the size distortion is smaller than O q/q+,

21 POWER MAXIMIZAION AND SIZE CONROL IN HAR ESS 339 which is the size distortion for the conventional t-test using the AMSE-optimal, that is when is set to be O q /q+. he formula for opt involves the unknown parameter d γ, which could be estimated nonparametrically or by a standard plug-in procedure based on a simple model like an AR see Andrews, 99; Newey and West, 994. Both methods achieve a valid order of magnitude, and the procedure is obviously analogous to conventional data-driven methods for HAC estimation. o sum up, the test-optimal that maximizes the local asymptotic power while preserving size in large samples is fundamentally different from the MSE-optimal. he test-optimal depends on the sign of the nonparametric bias and the permitted tolerance for the type I error, while the MSE-optimal does not. When the permitted tolerance becomes sufficiently small, the test-optimal is of smaller order than the MSE-optimal. 7. SIMULAION EVIDENCE his section presents some simulation evidence on the performance of the t -test based on the plug-in implementation of the test-optimal power parameter. We consider the simple location model with Gaussian autoregressive moving average ARMA, errors: y t = β + c/ + u t, where u t = φu t + ε t + θε t,ε t iidn,. Note that the long-run variance of u t is + θ / φ. On the basis of the long-run variance, we consider the following c values: c = δ + θ/ φ for δ [,5]. We consider three sets of parameter configurations for φ and θ : AR : φ,θ =.9,,.6,,.3,,.3,,.6,,.9,, MA : φ,θ =,.9,,.6,,.3,,.3,,.6,,.9, ARMA, : φ,θ =.6,.3,.3,.6,.3,.3,,..6,.3,.3,.6, and write u t ARMA[φ,θ]. We consider three sample sizes =,, and 5. For each data generating process, we obtain an estimate ˆφ of the AR coefficient by fitting an AR model to the demeaned time series. Given the estimate ˆφ, d γ can be estimated by ˆd = ˆφ/ ˆφ when q = and ˆd = ˆφ/ ˆφ when q =. We consider the tolerance factors κ =. and.. We set the significance level to be α = % and the corresponding nominal critical value for the two-sided test is z α =.645. o compute the test-optimal, we need to choose δ for the local alternative hypothesis. In principle, we can estimate δ by OLS, but the OLS estimator is not consistent. Accordingly, we follow standard practice in the optimal testing literature and propose to select δ such that the first-order power is 75%, that is, δ solves G δ.645 = 75%. he solution is δ =.39, which lies in the intermediate

22 34 YIXIAO SUN E AL. FIGURE. Size-adjusted power for different testing procedures under AR errors with Bartlett kernel. range of alternative hypotheses presented in Figures 5. Although this choice of δ may not match the true δ under the local alternative, Monte Carlo experiments show that it delivers a test with good size and power properties. For each choice of κ, we obtain ˆ opt and use it to construct the LRV estimate and corresponding t -statistic. We reject the null hypothesis if t is larger than the corrected critical values given in 6. Using, replications, we compute the empirical type I error when δ = and c =. For comparative purposes, we

23 POWER MAXIMIZAION AND SIZE CONROL IN HAR ESS 34 FIGURE. Size-adjusted power for different testing procedures under AR errors with Parzen kernel. also compute the empirical type I error when the power parameter is the optimal one that minimizes the asymptotic MSE of the LRV estimate. he formulas for this power parameter are given in PSJ 6, 7, and plug-in versions are ˆ MSE = π g 6 ˆφ ˆφ 4 /5 8/5 43

24 34 YIXIAO SUN E AL. FIGURE 3. Size-adjusted power for different testing procedures under AR errors with QS kernel. for the second-order power kernels and /3 ˆ MSE = ˆφ / ˆφ for the first-order power kernels.

25 POWER MAXIMIZAION AND SIZE CONROL IN HAR ESS 343 FIGURE 4. Size-adjusted power for different testing procedures under MA errors with Bartlett kernel. o compare the exponentiated kernel approach with the contracted kernel approach, we include the t -test based on contracted kernels. It is easy to show that the optimal b that minimizes e b II subject to the constraint that eb I κα is given by μ b opt = zα K /q+ δ z α ] qgd γ [ λ q/q+, 45 opt D zα G δ z α

26 344 YIXIAO SUN E AL. FIGURE 5. Size-adjusted power for different testing procedures under ARMA, errors with Parzen kernel. where { λ opt = G δ z α D zα + μ [ K δ z α gdγ D zα ] /q q[κ α] +/q, if d γ < z /q+ α, if dγ >. 46 We implement b opt using the same plug-in procedure for opt. In addition, we include the nonstandard test proposed by KV b, which sets b to be, the

27 POWER MAXIMIZAION AND SIZE CONROL IN HAR ESS 345 maximum value of b or equivalently, sets to be, and uses nonstandard critical values. We refer to the four testing procedures as t opt, t mse, t b opt, and t b max, respectively. ables 3 report the empirical type I error and average values of the selected smoothing parameters for the ARMA, error with sample size =, tolerance parameter κ =., and significance level α = %. Several patterns emerge for the type I error comparison. First, when the empirical type I error is around or greater than the nominal type I error, the testing-optimal plug-in procedure t opt incurs a significantly smaller type I error than the conventional plug-in procedure t mse. In other cases, the type I errors are more or less the same for the two plug-in procedures. Second, compared with the t b max -test, the t opt -test has similar size distortion except when the error process is highly persistent and second-order ABLE. Finite sample sizes and smoothing parameters of different testing procedures under AR errors =,κ =.,α =. [φ,θ] [.9,] [.6,] [.3,] [-.3,] [-.6,] [-.9,] Bartlett opt mse b opt b max Parzen opt mse b opt b max QS opt mse b opt b max Note: Average values of the selected smoothing parameter are in parentheses. opt : t -test with exponentiated kernel, test-optimal rho, and second-order corrected critical values CV. mse : t-test with exponentiated kernel, MSE-optimal, and standard normal CV. b opt : t -test with contracted kernel, test-optimal b, and second-order corrected CV. b max : t -test with b = or =, and nonstandard critical values from KV b.

28 346 YIXIAO SUN E AL. ABLE. Finite sample sizes and smoothing parameters of different testing procedures under MA errors =,κ =.,α =. [φ,θ] [,.9] [,.6] [,.3] [,-.3] [,-.6] [,-.9] Bartlett opt mse b opt b max Parzen opt mse b opt b max QS opt mse b opt b max Note: Average values of the selected smoothing parameter are in parentheses. opt : t -test with exponentiated kernel, test-optimal rho, and second-order corrected CV. mse : t-test with exponentiated kernel, MSE-optimal, and standard normal CV. b opt : t -test with contracted kernel, test-optimal b, and second-order corrected CV. b max : t -test with b = or =, and nonstandard critical values from KV b. kernels are used. Since the bandwidth is set equal to the sample size, the t b max -test is designed to achieve the smallest possible size distortion. Given this observation, we can conclude that the t opt -test succeeds in controlling for the type I error. hird, compared with the t b opt -test, the t opt -test has smaller type I error for most of the scenarios considered. Finally, the t opt -test based on the Bartlett kernel has the smallest size distortion in an overall sense. his is true even when the error autocorrelation is very high. Due to the approximation error, the bound we impose on the approximate type I error does not fully control the empirical type I error. his is demonstrated in ables 3. he quality of approximation depends on the persistence of the time series. When the time series is highly persistent, the first-order asymptotic bias of the LRV estimator may not approximate the finite sample bias very well. As a result, the approximate type I error, which is based on the first-order asymptotic

29 POWER MAXIMIZAION AND SIZE CONROL IN HAR ESS 347 ABLE 3. Finite sample sizes and smoothing parameters of different testing procedures under ARMA, errors =,κ =.,α =. [φ,θ] [-.6,.3] [.3,-.6] [.3,.3] [,] [.6,-.3] [-.3,.6] Bartlett opt mse b opt b max Parzen opt mse b opt b max QS opt mse b opt b max Note: Average values of the selected smoothing parameter are in parentheses. opt : t -test with exponentiated kernel, test-optimal rho, and second-order corrected CV. mse : t-test with exponentiated kernel, MSE-optimal, and standard normal CV. b opt : t -test with contracted kernel, test-optimal b, and second-order corrected CV. b max : t -test with b = or =, and nonstandard critical values from KV b. bias, may not fully capture the empirical type I error. So it is important to keep in mind that the empirical type I error may still be larger than the nominal type I error even if we exert some control over the approximate type I error. ables 3 show that the optimal decreases as the time series becomes more persistent. his is true for both the test-based criterion and the MSE-based criterion. Comparing ˆ opt with ˆ mse, we find that ˆ mse is larger than ˆ opt. his is consistent with our theory. For hypothesis testing, it is desirable to employ undersmoothing to achieve bias reduction, especially when the time series is persistent. Similarly, the test-optimal b increases as the series correlation becomes stronger. he above qualitative observations remain valid for other configurations such as different sample sizes and different values of κ. All else being equal, the size distortion of the t opt -test and the t b opt -test for κ =. is slightly larger than that for κ =.. his is expected, as we have a higher tolerance for the type I error when the value of κ is larger.