CAE Working Paper # Fixed-b Asymptotic Approximation of the Sampling Behavior of Nonparametric Spectral Density Estimators

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1 CAE Working Paper #06-04 Fixed-b Asymptotic Approximation of the Sampling Behavior of Nonparametric Spectral Density Estimators by Nigar Hashimzade and Timothy Vogelsang January 2006.

2 Fixed-b Asymptotic Approximation of the Sampling Behavior of Nonparametric Spectral Density Estimators Nigar Hashimzade Department of Economics, University of Exeter and Timothy J. Vogelsang Department of Economics and Department of Statistical Science, Cornell University January 3, 2006 Abstract We propose a new asymptotic approximation for the sampling behavior of nonparametric estimates of the spectral density of a covariance stationary time series. According to the standard approach, the truncation lag grows slower than the sample size. We derive rst order limiting distributions under the alternative assumption that the truncation lag is a xed proportion of the sample size. Our results extend the approach of Neave (970) who derived a formula for the asymptotic variance of spectral density estimators under the same truncation lag assumption. We show that the limiting distribution of zero frequency spectral density estimators depends on how the data is demeaned. The implications of our zero frequency results are qualitatively similar to exact results for bias and variance computed by Ng and Perron (996). Finite sample simulations indicate that new asymptotics provides a better approximation than the standard asymptotics when the bandwidth is not small. Keywords: Zero frequency, bandwidth, kernel, truncation lag, inference. We thank participants at the 2003 Summer Meetings of the Econometric Society for helpful comments and suggestions. We thank Kin-Yip Ho for carefully proofreading an earlier version of the paper. Vogelsang gratefully acknowledges nancial support from the National Science Foundation through grant SES Corresponding Author: Tim Vogelsang, Department of Economics, Uris Hall, Cornell University, Ithaca, NY , Phone: , Fax: , tjv2@cornell.edu

3 Introduction Spectral density estimation is an important and well established area of time series analysis. Much of the original work done on this topic was published in seminal papers going back over 50 years. Priestley (98) provides an excellent review and discussion. This paper adds to the now very large literature on nonparametric spectral density by extending ideas rst proposed by Neave (970). A well-known class of nonparametric spectral density estimators takes the form of weighted sums of sample autocovariances. Asymptotic theory for these estimators has been developed under the assumption that a truncation lag parameter, M (sometimes called a bandwidth) increases at a rate slower than the sample size, T. Asymptotic normality of nonparametric spectral density estimators has been established under this assumption (see Grenander and Rosenblatt (953)). In practice, a speci c truncation lag must be chosen and the value of b = M=T is positive. Although this asymptotic theory requires that b go to zero as T increases, in practice b is greater than zero and can be nontrivially di erent from zero. Therefore, the traditional asymptotic theory can be labeled as small-b asymptotics. Neave (970) argued that a more accurate asymptotic variance formula could be developed by treating b as a xed constant as T increases, thus mimicking the fact that b is not zero in small samples. Neave (970) derived asymptotic variance formulas based on this xed-b assumption. In this paper we adopt the assumption that b is a xed constant as T increases. We generalize the results of Neave (970) and derive asymptotic distributions of nonparametric spectral density estimators. The distributions we obtain are nonstandard but can be expressed as functionals of standard Wiener processes. One interesting result we nd is that asymptotic distributions of zero-frequency spectral density estimators depend on whether the data has been demeaned or detrended. This contrasts with the standard asymptotics where demeaning or detrending has no e ect on the rst order asymptotics at frequency zero. Our zero frequency results are qualitatively similar to exact results for bias and variance computed by Ng and Perron (996). Because the xed-b asymptotic approximation captures much of the bias in zero frequency estimators when b is not close to zero, it is a particularly useful result for serial correlation robust tests that use zero frequency spectral density estimators as standard errors. Capturing the bias reduces the tendency of serial correlation robust tests to over-reject when serial correlation in the errors is strong; see Kiefer and Vogelsang (2005). The paper is organized as follows. In the next section the model is given and estimators are de ned. Section 3 reviews well known asymptotic results under the standard small-b approach. Section 4 presents the new xed-b asymptotic results. This section contains the theoretical contributions of the paper. Inference regarding the spectral density is brie y discussed in Section 5, and Section 6 presents a selection of Monte Carlo simulations that compare the accuracy of the small-b

4 and the xed-b asymptotics. All proofs are given in the appendix. 2 The Model and Estimators Consider the time series y t = d t + u t ; t = ; 2; :::; T () where u t is a mean-zero covariance stationary time series with autocovariance function j = cov(u t ; u t j ). The component d t represents the deterministic part of y t. Leading cases include d t = 0 (a meanzero time series), d t = (a time series with mean ) and d t = + t (a trending time series). Let! 2 [0; ] denote frequencies and de ne the spectral density of u t as 2 3 f(!) = X j cos(!j) 5 : 2 It is assumed that 0 < f(!) <. where j= A well known class of estimators of f(!) is the nonparametric class which takes the form 2 3 bf(!) = 4b k( j 2 M )b j cos(!j) 5 ; (2) b j = T j= T X t=j+ bu t = y t b dt ; bu t bu t j ; and b d t is an estimate of d t. Estimators of the form given by (2) were rst proposed (in a slightly more general form) in the time series literature by Grenander and Rosenblatt (953). The function k(x) is a weighting function, or kernel, that typically downweights high lag sample autocovariances. k(x) is an even function with k(0) = and k(x) declines to zero as x!. Well known kernels have been proposed by Bartlett (950), Daniell (946), Parzen (96) and Tukey (949), among many others. See Priestley (98) for a general discussion. The truncation lag, M, controls the amount of downweighting that is applied to the high order sample autocovariances. Consistency of bf(!) requires that M! and M=T! 0 as T!. 2

5 3 Asymptotic Normality It has been shown that, under suitable regularity conditions, b f(!) has an asymptotically normal distribution. Su cient regularity conditions for obtaining such a result are that d t is a linear polynomial in t, u t = P j=0 j" t j where " t is an i:i:d: process with E(" t ) = 0, E[" 2 t ] <, E[" 4 t ] < and P j=0 j jj < (see Anderson (97)). Under these conditions it follows that r T bf(!) f(!)! d N(0; V (!)); M where 2f 2 (!) R V (!) = k2 (x)dx for! = 0; f 2 (!) R k2 (x)dx for 0 <! < : Note that this asymptotic approximation can be rewritten as bf(!) f(!) N bf(!) f(!) N ; M Z T 2 k 2 (x)dx ; M T 4 Fixed-b Asymptotic Approximation Z k 2 (x)dx for! = 0; ; (3) for! 6= 0: In this section an alternative asymptotic approximation for the sampling behavior of b f(!) is developed. The approach taken here is in the spirit of Neave (970) who argued that while the assumption that b = M=T! 0 is convenient mathematically and ensures consistency of b f(!), a more accurate approximation for the sampling variance of b f(!) can be obtained under the assumption that M = where b 2 (0; ]: Under this alternative assumption for M, Neave (970) proved that lim T! ( T 2f M var( f(!)) b 2 (!) R =b =b = k2 (x) ( b jxj) dx for! = 0; f 2 (!) R =b =b k2 (x) ( b jxj) dx for 0 <! < ; and these expressions become the standard formulae when b = 0. It is important to note that this result by Neave (970) does not apply to demeaned or detrended data when! = 0. Part of Neave s proof follows a proof by Parzen (957) and there appears to be a mistake (or typo) on page 340 of Parzen (957) that does not a ect Parzen s proof but a ects Neave s proof. Speci cally, to show that demeaning or detrending have no asymptotic e ects on b f(0), Parzen (957) requires a term involving the scaled integral of the kernel, k(x), to be o(). See the third equation on page 340 of Parzen (957). Parzen argues that this term is bounded from above by a constant times T " where " > 0 is a xed constant. The mistake/typo is that the bound should be proportional to (T=M) " not T ". Obviously, under the standard small-b asymptotics, T=M! as T! in which case (T=M) " = o() and Parzen s proof goes through as argued. The problem for Neave s proof is the 3

6 claim on page 72 of Neave (970) that formulas given by his equation (2.5) follow from arguments in Parzen (957) and do not depend on the condition that M=T! 0. This claim is true for! 6= 0, whereas for! = 0 it is only true if the data is know to be mean zero (d t = 0). It does not hold for! = 0 if the data is demeaned or detrended. The appeal of Neave s approach is that it provides an approximation that re ects the fact that M=T > 0 in nite samples. The limitation of Neave s result is that it only provides a formula for the variance and it does not address bias of f(!) b or indeed the distribution of f(!). b Building on the approach of Kiefer and Vogelsang (2005) who focused only on the case of! = 0 for demeaned data, it is possible to extend Neave s result to the entire distribution of f(!). b A rst order asymptotic distribution theory for f(!) b can be developed under fairly general regularity conditions. De ne the partial sum processes tx tx St c (!) = cos(!j)u j ; St s (!) = sin(!j)u j : j= j= The key requirement for our results is that the following functional central limit theorems hold for these partial sums: [rt ] T =2 S[rT c ] (0) = T X =2 u t ) p 2f(0)W (r); (4) t= [rt ] T =2 S[rT c ] () = T X =2 ( ) t u t ) p 2f()W (r); (5) t= T =2 S c [rt ] (!) ) p f(!)w (r);! 6= 0; ; (6) T =2 S s [rt ] (!) ) p f(!)w 2 (r);! 6= 0; ; (7) where W (r); W (r); W (r) and W 2 (r) are standard independent Wiener processes, [rt ] is the integer part of rt with r 2 (0; ]. Note that it trivially follows that T =2 S s [rt ] (0) = T =2 S s [rt ] () = 0 because sin(0) = sin(t) = 0. Su cient conditions under which (4) - (7) hold is that u t is a stationary Gaussian process with lim!!0 f(a!)=f(!) = for any a > 0, see Theorem 3.2 of Chan and Terrin (995). It would also be straightforward to show that (5) - (7) follow from (4). A well known su cient condition for (4) is u t = P j=0 j" t j with E(" t ) = 0, E[" 2 t ] < and P j=0 j j jj <. These regularity conditions are similar to, but are di erent from, the regularity conditions used in the standard approach. We do not require a nite fourth moment for " t ; but we need the slightly stronger condition of one-summability of the j coe cients. The asymptotic distribution of b f(0) depends on b d t. We consider three cases for d t and b d t. Case (i): d t = 0; b d t = 0: Case (ii): d t = ; b d t = y = T P T t= y t: Case (iii): d t = +t; b d t = b+ b t where b and b are the least squares estimates from a regression of y t on (; t). The limiting distribution 4

7 of b f(0) can be expressed in terms of demeaned and detrended W (r) which we denote by c W (r) and is de ned as follows. For case (i) c W (r) = W (r). For cases (ii) and (iii) cw (r) = W (r) Z r 0 Z Z g(s)ds g(s)g(s) 0 ds g(s)dw (s); 0 0 where g(s) = for case (ii) and g(s) = [; s] 0 for case (iii). Note that c W () = 0 for cases (ii) and (iii). De ne k 0 () = lim h!0 [(k() k( h)) =h] ; i.e. k 0 () is the derivative of k(x) from the left at x =. The following theorem provides the main results of the paper. The proof is given in the appendix. Theorem Let y t be given by () where u t is a mean zero stationary process that satis es (4) - (7) and suppose that 0 < f(!) < : Let M = with b 2 (0; ]. The following hold as T! :. If k 00 (x) exists and is continuous, then for! 6= 0; : bf(!) f(!) ) 2 2X i= b 2 Z Z 0 0 k 00 r s W i (r)w i (s)drds b + 2X 2 2 b W i() i= Z 0 k 0 b r W i (r)dr + W i () 2 ; for! = : bf() f() ) b 2 Z Z 0 0 k 00 r b s W (r)w (s)drds+ 2 b W () Z 0 k 0 b r W (r)dr+w () 2 ; for! = 0: bf(0) f(0) ) b 2 Z Z 0 0 k 00 r b s cw (r) c W (s)drds + 2 b c W () Z 0 k 0 b r cw (r)dr + c W () 2 : 2. If k(x) is continuous, k(x) = 0 for jxj ; and k(x) is twice continuously di erentiable everywhere except for possibly jxj =, then for! 6= 0; : bf(!) f(!) ) 2 " 2X i= ZZ b 2 jr r k 00 sjb s b W i (r)w i (s)drds + 2 Z b k0 () + 2 2X i= Z 2 b W i() k 0 b 0 b W i (r)w i (r + b)dr r b # W i (r)dr + W i () 2 ; 5

8 for! = : bf() f() ) ZZ b 2 jr and for! = 0: bf(0) f(0) ) ZZ b 2 jr r k 00 sjb r k 00 sjb s b W (r)w (s)drds + 2 Z b k0 () s b + 2 Z b W () k 0 b r b cw (r) W c (s)drds + 2 Z b k0 () + 2 Z W b c () k 0 b r b 0 b W (r)w (r + b)dr W (r)dr + W () 2 ; 0 b cw (r) c W (r + b)dr cw (r)dr + c W () 2 : 3. If k(x) = jxj for jxj and k(x) = 0 otherwise, i.e. k(x) is the Bartlett kernel, then for! 6= 0: bf(!) f(!) ) 2 for! = : 2X i= 2 b Z 0 W i (r) 2 dr Z 2 b 0 b W i (r)w i (r + b)dr Z 2 b W i() W i (r)dr + W i () 2 ; b bf() f() ) 2 b Z 0 W (r) 2 dr Z 2 b 0 b W (r)w (r + b)dr Z 2 b W () W (r)dr + W () 2 ; b and for! = 0: bf(0) f(0) ) 2 Z cw (r) 2 dr b 0 Z 2 b 0 b cw (r) c W (r + b)dr Z 2 W b c () cw (r)dr + W c () 2 : b The theorem shows that limiting distribution of f(!) b is proportional to f(!) under the assumption that M =. Similar to the standard asymptotics, di erent limits are obtained for! = 0 compared to! 6= 0. Unlike in the standard approach, the limits in the! = 0 case depend on the estimated deterministic trend, d b t. The limiting distribution theory for! = 0 was used by Kiefer and Vogelsang (2005) for demeaned data and by Bunzel and Vogelsang (2005) for detrended data to obtain a more accurate asymptotic theory for tests that use f(0) b when constructing standard errors. When f(0) b is used to estimate an asymptotic variance, consistency of f(0) b is usually the approximation used when determining how the sampling behavior of f(0) b a ects the ultimate test. Thus, f(0) b is approximated by f(0). Because f(0) b can exhibit severe downward bias, using the random variables in Theorem to approximate f(0) b yields a more accurate asymptotic approximation since some of the bias and sampling variability in f(0) b is captured by the xed-b asymptotics. A formal result along these 6

9 lines has been established by Phillips, Sun and Jin (2005)a for the case where d t = and u t is Gaussian. A recent paper by Phillips, Sun and Jin (2005)b develops an asymptotic theory analogous to the results in Theorem for the case of exponentiated kernels. Speci cally, Phillips et al. (2005)b consider estimators of f(!) where in (2) k( j M ) is replaced with k( j T ) where is a positive real number. Increasing places less weight on higher-order sample autocovariances in much the same way as does smaller values of M. Holding xed as T increases leads to an asymptotic theory that is analogous to xed-b asymptotics. In addition to providing xed- asymptotic results, Phillips et al. (2005)b also derive a more traditional asymptotic normal approximation under the assumption that increases with T but at a slower rate. 5 Finite Sample Comparison of the Normal and Fixed-b Asymptotic Approximations One standard metric by which to judge an asymptotic approximation is how well the asymptotic distribution approximates the sampling distribution of the statistic. Suppose for a given sample of size T; a particular value of M is used to construct b f(!). The question is then which asymptotic approximation, the normal approximation based on standard small-b asymptotics or the new xed-b asymptotics, is more accurate. We performed a simple Monte Carlo simulation study to compare the accuracy of the two asymptotic approximations. The data is assumed to have unknown mean so that the data is demeaned before estimating f(!) (this is case (ii) from above). We consider the data generating process y t = + u t ; (8) u t = u t + t + t ; t iid N(0; ); where u 0 = 0 = 0: The spectral density for y t is given by f(!) = + 2 cos(!) ( 2 cos(!) + 2 ) : Data was generated according to (8) and b f(!) was computed using (2). Using R = 5; 000 replications we computed the empirical cumulative distribution function, P b rob(x) of f(!)=f((!) b using! bp rob(x) = RX bf i (!) R f(!) < x ; i= where b f (!); b f 2 (!); :::; b f R (!) are the simulated realizations from b f(!) and () is the indicator function. We obtained results for large range of values for M,!,, and a group of well known 7

10 kernels that includes the Bartlett, Parzen, Daniell and quadratic spectral (QS) kernels. We report results for AR() errors with = 0:0; 0:4; 0:4; for the frequencies! = 0:0; 0:5; and b = M=T = 0:04; 0:2; 0:5. Plots of the empirical CDFs for T = 50; 00; 200 are given in Figures The gures are organized rst by kernel: Bartlett (Figures.-.8), Parzen (Figures ), Daniell (Figures ), QS (Figures ) and then by frequency:! = 0 (Figures x.-x.9) and! = 0:5 and (Figures x.0-x.8). Each gure also provides plots of the CDFs given by the normal asymptotic approximation from (3) and by the xed-b approximation from Theorem. The CDFs of the asymptotic random variables in Theorem were accurately estimated using simulation methods. The standard Brownian motions were approximated by scaled partial sums of i:i:d: standard normal random variables using,000 increments and 50,000 replications. Figures x.-x.3 provide results for i:i:d: errors at frequency zero. When a small bandwidth is used (b = 0:04) as in Figures x., we see that the nite sample empirical CDFs are similar for all three sample sizes and are close to the xed-b asymptotic CDF. The asymptotic normal CDF, on the other hand, is obviously di erent between the 0. and 0.9 percentiles. In situations where larger bandwidths are used as in Figures x.2 (b = 0:2) and x.3 (b = 0:5), the xed-b asymptotic CDF remains very close to the empirical CDFs whereas the asymptotic normal CDF systematically becomes a less accurate approximation. When the data has serial correlation as is the case in Figures x.4-x.6 ( = 0:4) and Figures x.7-x.9 ( = 0:4), the xed-b asymptotic approximation remains accurate when b = 0:2; 0:5 whereas the asymptotic normal approximation exhibits substantial inaccuracy. When b = 0:04, neither asymptotic approximation is accurate when T = 50. But, the xed-b asymptotic approximation improves as T increases whereas the asymptotic normal approximation remains less accurate. It is interesting to note that for b = 0:2; 0:5 the CDF for the asymptotic normal approximation is strictly positive at the origin. Thus, the asymptotic normal approximation attributes negative values to f(!)=f((!) b with positive probability even though f(!)=f((!) b is positive by construction for the four kernels considered here. This problem does not arise with the xed-b approximation. Now consider the frequency! = 0:5. As Figures x.0-x.8 illustrate, the di erences between the normal and xed-b approximations are smaller than when! = 0 although the patterns are similar. When b = 0:04, the CDFs of the two asymptotic approximations are very close to each other and they are reasonably close to the empirical CDFs when T = 00; 200. When T = 50 and 6= 0, neither asymptotic approximation is adequate. When b = 0:2; 0:5, the xed-b asymptotic approximation is good for all three sample sizes and all three values of whereas the asymptotic normal approximation is much less accurate. The overall picture that emerges from the gures is that the xed-b approximation can be systematically better than the asymptotic normal approximation regardless of the kernel being used. The next section provides some calculations that can shed some light on patterns exhibited 8

11 in the simulations. 6 Asymptotic Bias and Variance Many of the patterns seen in the simulations can be intuitively explained by examining the asymptotic bias and variances implied by xed-b asymptotics in comparison to the asymptotic normal approximation. We continue to focus on the case where the data is demeaned and the for the sake of concreteness we isolate attention on the Bartlett kernel. In a recent paper, Hashimzade, Kiefer and Vogelsang (2003) formally calculated the following results under xed-b asymptotics. For! 6= 0; " # bf(!) lim bias = 0; T! f(!) " # bf(!) Z =b Z lim var = b ( bjxj) k 2 (x) dx = b ( bjxj)( jxj) 2 dx T! f(!) =b = 2b 3 2 b ; Note that this variance formula exactly matches the formula obtained by Neave (970). For! = 0, the formulas are given by " # bf(!) lim bias = b + T! f(!) 3 b2 " # bf(!) lim var = 4 T! f(!) 3 b 7 3 b b b4 5b 2 (2b )5 (b > 2 ): This variance formula di ers from 4b 3 2 b ; the formula given by Neave (970), because Neave s zero frequency variance formula is only valid when the data has mean zero and the data is not demeaned. Neave s variance formula and the xed-b variance are the same at frequency zero when the data is mean zero and is not demeaned. It is instructive to compare these bias and variance formulas with the well known standard formulas for the Bartlett kernel which are given by " # bf(!) lim bias = 0; T! f(!) " # bf(!) lim var = 2 T! f(!) 3 b; 9

12 for! 6= 0; and " # bf(!) lim bias = 0 T! f(!) " # bf(!) lim var = 4 T! f(!) 3 b; for! = 0. In both sets of formulas, the bias and variance are given by polynomials in b and they have the same leading terms. The di erence is that xed-b asymptotics contains higher order terms in b. When b is small, we would expect the higher order terms to be relatively small and this suggests the normal and xed-b approximations will be similar at least with respect to bias and variance. In light of this observation, it not surprising that the CDFs of the two asymptotic approximations are very close to each other in the gures when b = 0:04. When b is not close to zero, the higher order terms in the xed-b bias and variances will matter and we would expect the approximations to be di erent. This is exactly what we see in the gures when b = 0:2; 0:5. While the above bias and variance formulas help explain the di erences between the accuracy of the normal asymptotic approximation and the xed-b asymptotic approximation, they do not explain the relative accuracy of the xed-b asymptotic approximation across di erent value of b and T. To provide an explanation, it is useful to examine the well-known spectral bias formulas derived by Parzen (957). For the Bartlett and Daniell kernels the Parzen bias is given by and f (2) (!) 2 =(6M 2 ) respectively where f (k) (!) = X j= 3 jjj k j cos(!j) 5 : f () (!)=M Because under both the normal asymptotic approximation and the xed-b asymptotic approximation, M! as T!, the Parzen bias is o() and does not appear in the asymptotic approximations (at least to rst order). The behavior of the Parzen bias term can explain why the xed-b approximation improves as b increases when the data is not i:i:d: especially when T = 50. Given T, as b increases, M increases thus reducing the Parzen bias and improving the approximation. This is why in Figures 6, 9, 5 and 8 (b = 0:5) we see that the T = 50 CDF and the xed-b CDF are close whereas in Figures 4, 7, 3 and 6 (b = 0:04) the T = 50 CDF and the xed-b CDF are not close. The Parzen bias also explains why, for a given value of b, the xed-b approximation improves as T increases. With b xed, as T increases, so does M and the Parzen bias shrinks. 0

13 7 Appendix: Proofs De ne the following partial sums: bs t c (!) = bs t s (!) = tx bu j cos(!j); j= tx bu j sin(!j): j= Note that bs s t (0) = b S s t () = 0 (9) because sin(0) = sin(j) = 0: Consider the case where b d t = b + b t. Then, from simple algebra it follows that bu t = u t (b ) ( b )t. Because T =2 (b ) = O p () and T 3=2 ( b ) = O p (), it is easy to show that for! 6= 0;, and [rt ] X T =2 t= [rt ] X T =2 t= [rt ] X T =2 t= Therefore, it directly follows from (5) - (7) that h (b ) ( b i )t cos(!t) = o p (); h (b ) ( b i )t sin(!t) = o p (); h (b ) ( b i )t ( ) t = o p (): T =2 b S c [rt ] () ) p 2f()W (r); (0) T =2 b S c [rt ] (!) ) p f(!)w (r); for! 6= 0; ; () T =2 b S s [rt ] (!) ) p f(!)w 2 (r); for! 6= 0; : (2) Obviously, the limits (0), () and (2) continue to hold when d b t = y or when d b t = 0. The remaining case is S b t c (0) = P t j= bu j, and this partial sum has a di erent type of limit because [rt ] X T =2 t= h (b ) ( b i )t = O p ():

14 Simple algebra gives [rt ] [rt ] X X h T =2 S bc [rt ] (0) = T =2 bu [rt ] = T =2 u t (b ) ( b i )t t= t= t= [rt ] X = T =2 [rt ] u t T T =2 (b ) T 3=2 ( b ) [rt ] X T 2 t ) p 2f(0) c W (r): (3) Note that the limiting result (3) is a standard result that follows from (4) and simpli es in obvious ways when b d t = y or when b d t = 0. Proof of Theorem. De ne the following functions: i j K ij = k Simple algebra gives = k i j T ; 2 K ij = (K ij K i;j+ ) (K i+;j K i+;j+ ) : 2 bf(!) = 4b = 2 T = 2 T j= i= j= i= j= j k i bu i bu j k 3 b j cos(!j) 5 j cos [!(i j)] bu i cos(!i)k ij bu j cos(!j)+ (4) + 2 T Rewrite the rst term in (4) as the following: 2 T i= j= i= j= bu i cos(!i)k ij bu j cos(!j) = 2 T bu i sin(!i)k ij bu j sin(!j) = 2 T bu i cos(!i) i= a i b i ; i= t= K ij bu j cos(!j) j= where a i = bu i cos(!i), b i = K ij bu j cos(!j): j= 2

15 Using the identity 2 a i b i = 4(b i b i+ ) i= we obtain, for a i and b i de ned above, 0 a i b i = (K K i+;j ) bu j cos(!j) S b i c (!) A + i= i= j= i= ix j= a j 3 X T 5 + b T a j ; (5) By applying the identity (5) one more time, we obtain for the rst term in (6) j= K T j bu j cos(!j) S b T c (!) : (6) (K ij K i+;j ) bu j cos(!j) = [(K ij K i+;j ) (K i;j+ K i+;j+ )] S b j c (!) + (K it K i+;t ) S b T c (!) j= j= j= = 2 K ijs bc j (!) + (K it K i+;t ) S b T c (!) : j= Similarly, for the second term in (6) we obtain K T j bu j cos(!j) = (K T j K T;j+ ) S b j c (!) + K T T S bc T (!) : j= j= Finally, noting that K ij = K ji and K T T = k(0) =, we obtain the following expression for the rst term in (4): T 2 T X De ne i= T T X T =2 S bc i (!) T 2 2 K ij T =2 S bc j (!) j= T j= [rt ] + DT (r) = T 2 k T Notice that T 2 2 K ij = D T i j T T =2 S bc j (!) T (K jt K j+;t ) T =2 S bc T (!) + 2 T =2 S bc 2 T (!) : [rt ] k T [rt ] k T. If k 00 (x) exists and is continuous, then by de nition of the second derivative. Also, in that case lim T! T (K j+;t [rt ] k T (7) lim T! D T (r) = k 00 (r) (8) K jt ) = lim T! k [rt ]+ T 3 k [rt ] T T = k 0 (r): (9)

16 For! 6= 0 the limit of (7) follows from () (for! 6= ), (0) (for! = ), (8), (9) and the continuous mapping theorem. For! = 0, (7) simpli es using the fact that S b T c (0) = 0 and the limit follows from (3). The second term in (4) is identical to (7), with S b i c (!) replaced by S b i s (!) and for! 6= 0; the limit follow using analogous arguments. For! = 0; the second term vanishes because of (9). This completes the proof of part of the theorem. If k(x) is not everywhere twice continuously di erentiable, we cannot apply (8) and (9) directly. The leading cases are kernels that truncate. Suppose k(x) is continuous, k(x) = 0 for jxj, and k(x) is twice continuously di erentiable everywhere except for possibly jxj =. We can rewrite the sums in (7) to separate the kink points. For the rst term in (7) we have: X X bs i c (!) 2 K ijs bc j (!) = T T T T i= j= ji X X + T jj< + T X i j= + T i j= ji X jj> (20) By the assumptions listed above, k(x) is continuous and twice continuously di erentiable in the range of the rst sum and is zero in the range of the last sum. Hence, the limit of the rst term follows from the arguments used in the proof of part and the last term vanishes. For the second term we have T i X j= = T j= = T = T j= X j= bs c j (!) [(K j+;j K j+;j+ ) (K j+ +;j K j+ +;j+ )] b S c j+ (!) bs j c (!) k () k T =2 S bc j (!) k () k k + T =2 b S c j+ (!) : k () bs j+ c (!) By de nition of the derivative from the left it follows that, k () k = k 0 (): (2) lim T! Similar manipulation with the third term in (20) yields T X i j= = T X j= T =2 S bc j (!) k + By de nition of the derivative from the right, it follows that k + k ( ) = k+( 0 ): lim T! 4 k ( ) T =2 S bc j+ (!) :

17 By symmetry of k(x) around x = 0, k 0 +( ) = k 0 (). Hence, second and third terms in (20) are equal and can be combined into one. Now consider the second term in (7): T T X j= bs j c (!) (K jt K j+;t ) S b T X T c (!) = T j= + T j=t (22) The rst term in (22) vanishes because for j T, j T j+ T < and, and therefore K jt = K j+;t = 0. The second term in (22) can be expressed as T j=t = T = j=t j=t bs c j (!) (K jt K j+;t ) b S c T (!) = T =2 b S c j (!) k j+ T k j T T =2 b S c T (!) : By de nition of the rst derivative, k j+ T lim T! k (j T ) r r = k 0 = k 0 b b (23) where the last equality follows from the symmetry of k(x). Collecting all terms in (20) and (22) into (7), adding corresponding terms with S b j s (!) in place of S b j c (!), and using (2) and (23) along with (0), (), (2), (3) and (9) (as needed depending on!) establishes part 2 of the theorem. To prove part 3 of Theorem we use the de nition of Bartlett kernel: i j ji jj K ij = k = ; ji jj 0; ji jj > Then it follows that and K ij 8 0; i j >< K i;j+ = ; j + i j >: ; j + i j + 0; i j + + K i+;j 8 0; i j >< K i+;j+ = ; j i j >: ; j i j + 0; i j < 2 ; j = i K ij = : ; i = j 0 otherwise 5

18 Hence, for the rst term in (7) we obtain X X T T T i= j= bs c i (!) 2 K ij b S c j (!) = 2 = 2 i= i= 2 T =2 S bc i (!) T [ ] X i= 2 T =2 S bc 2 i (!) T [ ] X j= T =2 b S c j+[ ] (!) T =2 b S c j (!) T =2 b S c i (!) T =2 b S c i+[ ] (!) T [ ] X Similarly, in the second term of (7) we have 0; j T K T;j K T;j+ = ; T j T i= T =2 b S c i (!) T =2 b S c i+[ ] (!) : (given that j takes values from to T ). Hence, (7) can be expressed as 2 2 i= 2 T =2 S bc i (!) 2 T T j=t [ ] T [ ] X i= T =2 b S c i (!) T =2 b S c i+[ ] (!) T =2 b S c j (!) T =2 b S c T (!) + 2 T =2 b S c T 2 : An analogous expression holds with S b j s (!) in place of S b j c (!) and the limit follows using (0), (), (2), (3) and (9) (as needed depending on!). This completes the proof of part 3 of the theorem. 6

19 References Anderson, T. (97), The Statistical Analysis of Time Series, Wiley, New York. Bartlett, M. (950), Periodogram Analysis and Continuous Spetra, Biometrika 37, 6. Bunzel, H. and Vogelsang, T. J. (2005), Powerful Trend Function Tests That are Robust to Strong Serial Correlation with an Application to the Prebisch-Singer Hypothesis, Journal of Business and Economic Statistics, forthcoming. Chan, N. H. and Terrin, N. (995), Inference for Unstable Long-Memory Processes with Applications to Fractional Unit Root Autoregressions, Annals of Statistics 23, Daniell, P. J. (946), Discussion on Symposium on Autocorrelation in Time Series, Journal of the Royal Statistical Society, Suppliment 8, Grenander, U. and Rosenblatt, M. (953), Statistical Spectral Analysis Arising from Stationary Stochastic Processes, Annals of Mathematical Statistics 24, Hashimzade, N., Kiefer, N. M. and Vogelsang, T. J. (2003), Moments of HAC Robust Covariance Matrix Estimators Under Fixed-b Asymptotics, Working Paper, Department of Economics, Cornell University. Kiefer, N. M. and Vogelsang, T. J. (2005), A New Asymptotic Theory for Heteroskedasticity- Autocorrelation Robust Tests, Econometric Theory 2, Neave, H. R. (970), An Improved Formula for the Asymptotic Variance of Spectrum Estimates, Annals of Mathematical Statistics 4, Ng, S. and Perron, P. (996), The Exact Error in Estimating the Spectral Density at the Origin, Journal of Time Series Analysis 7, Parzen, E. (957), On Consistent Estimates of the Spectrum of a Stationary Time Series, Annals of Mathematical Statistics 28, Parzen, E. (96), Mathematical Considerations in the Estimation of Spectra, Technometrics 3, Phillips, P. C. B., Sun, Y. and Jin, S. (2005)a, Optimal Bandwidth Selection in Heteroskedasticity- Autocorrelation Robust Testing, Working Paper, Department of Economics, UCSD. Phillips, P. C. B., Sun, Y. and Jin, S. (2005)b, Spectral Density Estimation and Robust Hypothesis Testing using Steep Origin Kernels without Truncation, International Economic Review, forthcoming. 7

20 Priestley, M. B. (98), Spectral Analysis and Time Series, Vol., Academic Press, New York. Tukey, J. W. (949), The Sampling Theory of Power Spectrum Estimates, Symposium on Applications of Autocorrelation Analysis to Physical Problems, NAVEXOS-P-735, O ce of Naval Research, Department of the Navy, Washington, D.C., pp

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