No CrossRef data available.
AUTOMATIC POSITIVE SEMIDEFINITE HAC COVARIANCE MATRIX AND GMM ESTIMATION
Published online by Cambridge University Press: 08 February 2005
Abstract
This paper proposes a new class of heteroskedastic and autocorrelation consistent (HAC) covariance matrix estimators. The standard HAC estimation method reweights estimators of the autocovariances. Here we initially smooth the data observations themselves using kernel function–based weights. The resultant HAC covariance matrix estimator is the normalized outer product of the smoothed random vectors and is therefore automatically positive semidefinite. A corresponding efficient GMM criterion may also be defined as a quadratic form in the smoothed moment indicators whose normalized minimand provides a test statistic for the overidentifying moment conditions.
Information
- Type
- Research Article
- Information
- Copyright
- © 2005 Cambridge University Press
1. INTRODUCTION
Consider a random vector process that may be parameter dependent and may display serial dependence and conditional heteroskedasticity. Heteroskedastic and autocorrelation consistent (HAC) estimation of the long-run covariance matrix of such processes has received considerable attention in the econometrics literature over the last two decades. The standard estimation method employs lag kernel smoothing whereby autocovariance estimators are weighted by some suitably chosen kernel function and that also incorporates a bandwidth parameter. Seminal contributions in the statistics literature to the theoretical study of such HAC estimators include Parzen (1957) and Priestley (1962). More recently, Andrews (1991) analyzes the properties of a number of HAC estimators and prescribes suitable choices of the bandwidth parameter given a particular choice of kernel function. In addition, the quadratic spectral kernel is shown to be optimal according to an asymptotic truncated mean squared error criterion. The bibliography in Andrews (1991) also provides some previous contributions to the econometrics literature on HAC covariance matrix estimation. In particular, the method of Newey and West (1987a) based on the Bartlett kernel function is now commonly adopted in many econometrics packages. A related lag kernel–based approach is discussed in Andrews and Monahan (1992) that initially prewhitens the random vector process.
This paper suggests a novel alternative class of HAC covariance matrix estimators. Rather than weight the estimated autocovariances as in the standard lag kernel method, we initially smooth the data observations on the random vector process itself using an appropriately chosen kernel function as weights. The HAC covariance matrix estimator is then defined as the normalized outer product of the smoothed random vectors. The resultant class of HAC covariance estimators belongs to the general class of quadratic estimators described in Grenander and Rosenblatt (1984, Sect. 4.1). Standard lag kernel estimators are also quadratic estimators, but their weight matrix has Toeplitz form. As shown by Grenander and Rosenblatt (1984, Sect. 4.2), for any linear random vector process, a standard lag kernel estimator may be found that has asymptotic mean squared error no larger than that of any given asymptotically unbiased quadratic estimator with non-Toeplitz weight matrix. Interestingly, however, the weight matrices for members of the class of HAC covariance matrix estimators proposed here asymptotically take Toeplitz form. Therefore, each HAC estimator corresponds implicitly to an asymptotically equivalent lag kernel estimator. Although this paper does not formally address their limiting distributional and asymptotic mean squared error properties, one might therefore suppose that members of this class of HAC covariance matrix estimators might inherit the properties of the corresponding lag kernel estimator.
The reweighting scheme adopted here may be viewed as a form of multitaper. Recall that the long-run covariance matrix of a stationary vector process is its (second-order) spectrum at frequency zero. Brillinger (1981, Theorem 5.6.4, p. 150) details the large-sample properties of a lag kernel estimator for the spectrum where the data observations have initially been smoothed using a single taper that has bandwidth parameter equal to sample size. Brillinger (1981, p. 151) notes that lag kernel estimators with tapered data may have desirable asymptotic bias properties relative to standard lag kernel estimators; see also Brillinger (1981, Sects. 3.3 and 4.6).1
An example of the application of tapers in econometrics is the nonparametric cointegration analysis of Bierens (1997).
Another recent innovation is that of Phillips (2005), which proposes a new class of HAC covariance matrix estimators obtained as the explained sum of squares from a regression of the data observations on a sequence of basis trend functions. Estimators in this class are members of the general quadratic class where the weight matrix is the orthogonal projection matrix formed from the trend function sequence. These estimators may also be regarded as multitaper estimators where the tapers are the eigenvectors of the trend function projection matrix. Like the class proposed in this paper, Phillips's estimators also correspond implicitly to an asymptotically equivalent lag kernel estimator. Interestingly, this approach has the advantage of avoiding a choice of kernel function and bandwidth (which equals the sample size) but does, however, require a choice of the number and the sequence of basis trend functions to be included in the regression. Phillips (2005, eqn. (8)) provides an automated rule, obtained via an asymptotic mean squared error analysis, for the determination of the number of trend function terms to include.
The class of HAC covariance estimators proposed here is automatically positive semidefinite as are the estimators proposed in Phillips (2005). This property is a particular advantage if a consistent estimator is required for the asymptotic covariance matrix of the limiting normal distribution of some parameter estimator. For example, the asymptotic covariance matrix estimator may then be used in the computation of t- or F-type test statistics based on the parameter estimator. Furthermore, the inverse of the HAC covariance estimator may be employed as an estimator for the efficient metric in generalized method of moments (GMM) estimation.
The standard construction of t- and F-type statistics incorporating a HAC covariance matrix estimator has been subject to some severe criticism in the literature because of their poor finite-sample properties relative to the nominal normal or chi-square asymptotic reference distributions. In the regression context, Kiefer, Vogelsang, and Bunzel (2000) suggest an alternative approach to inference that completely avoids the use of a HAC estimator and appears to possess better properties in small samples than that based on standard t- and F-type statistics. For a more recent contribution, see Phillips, Sun, and Jin (2003). A major disadvantage of these approaches, however, from which t- and F-type statistics do not suffer, is that they seem to be restricted to just-identified models only in the nonlinear GMM context; see Kiefer et al. (2000, Sect. 4, p. 702).
An efficient GMM criterion may also be formulated as a quadratic form in the smoothed moment indicators whose normalized minimand provides a test statistic for the overidentifying moment conditions similar in structure to that of Hansen (1982). Being based solely on the smoothed moment indicators this GMM criterion is similar in structure to that for observations obtained from a random sample and thus does not require separate estimation of the efficient metric for GMM estimation, which would normally be the case; merely evaluation of the outer product of the smoothed moment indicators at an initially consistent estimator for the parameters of interest is necessary. A continuous updating estimator (Hansen, Heaton, and Yaron, 1996) may also be defined based on the revised GMM criterion.
Section 2 introduces the time series setup and briefly discusses the standard method of HAC covariance matrix estimation. The class of HAC covariance matrix estimators that is the subject of this paper is then defined. Consistency for this class of covariance matrix estimators is demonstrated. An alternative GMM criterion appropriate for serially dependent and conditionally heteroskedastic time series moment conditions is given in Section 3. Consistency, asymptotic normality, and efficiency of the GMM estimator are shown together with the limiting distribution of the normalized minimand. Section 4 concludes. Proofs of the results are given in the Appendix.
The following abbreviations are used throughout the paper: w.p.a.1: with probability approaching one;
converges in probability to;
converges in distribution to; ∥.∥ is the matrix norm defined by
where λmax(·) is the maximum eigenvalue of ·; p.d.: positive definite; p.s.d.: positive semidefinite.
2. HAC COVARIANCE MATRIX ESTIMATION
Let zt, (t = 1,…,T), denote observations on a finite-dimensional stationary and strongly mixing process {zt}t=1∞. The particular focus is the random vector g(zt,β), an m-vector of known functions of the data observation zt and the p-vector β of unknown parameters, where it is assumed that m ≥ p.
Let
.
We further assume that there exists a true value β0 of the parameter vector β at which the vector gt(β) has unconditional mean zero, that is, E [gt(β0)] = 0. In the GMM estimation context (Hansen, 1982), gt(β) would denote a vector of moment indicators, and β0 would be of some inferential interest. In many circumstances, the moment restrictions E [gt(β0)] = 0 will arise from a conditional moment restriction. For such cases, zt would also need to include lagged endogenous and current and lagged values of exogenous variables.
The following assumption describes the basic properties of the observation process {zt}t=1∞.
Assumption 2.1. The observation process {zt}t=1∞ is a stationary and α-mixing sequence such that
for some ν > 1.
Our next assumption details some restrictions on the random process {g(zt,β)}t=1∞.
Assumption 2.2. (a) E [g(zt,β0)] = 0; (b)
.
These assumptions are quite standard; see Kitamura and Stutzer (1997), albeit somewhat stronger than Assumption A, p. 823, and the hypotheses in Lemma 1 of Andrews (1991, p. 824).
2.1. Some Preliminaries
The particular concern of this section is HAC estimation of the long-run covariance matrix of the random process {g(zt,β0)}t=1∞ that is defined by
where Γ(s) = E [gt+s(β0)gt(β0)′] , Γ(−s) = Γ(s)′, is the sth autocovariance of the process {g(zt,β0)}t=1∞, (s = 0,±1,…).
A HAC covariance matrix estimator is often required when estimating the asymptotic covariance matrix of the limiting normal distribution of a root-T consistent estimator for β0. Furthermore, in a time series setting with unknown serial dependence and conditional heteroskedasticity, the inverse of a HAC estimator for Ω provides a consistent estimator for the metric required for the implementation of efficient GMM. In both of these examples, the particular requirement would be a consistent estimator for the limiting variance matrix
of the normalized sample average
, that is, the long-run covariance matrix Ω as
, where
. See (2.2) and (2.3) of Andrews (1991, pp. 819–820).
The standard method for estimation of Ω (2.1) is based on smoothing consistent sample autocovariance estimators
, where
is a preliminary consistent estimator for β0. Let k*(·) be some real-valued kernel function belonging to the class of symmetric kernels
defined by
where k*(x) = supy≥x|k*(y)|; see, for example, Andrews (1991) and Andrews and Monahan (1992).2
Jansson (2002) notes that neither the square integrability condition
in Andrews (1991, (2.6), p. 821) nor the stronger absolute integrability condition
in Andrews and Monahan (1992, (2.5), p. 955) is sufficient for the consistency results claimed in those papers. The condition ∫[0,∞) k*(x) dx < ∞ is required to rule out certain pathological cases and to ensure that particular summations used in those papers converge appropriately; see Lemma 1 of Jansson (2002).
(Andrews, 1991, (2.5), p. 820), where ST is a bandwidth parameter and T/(T − p) is a finite-sample adjustment that takes into account estimation of β0.
2.2. Positive Semidefinite HAC Covariance Matrix Estimation
In contradistinction to the standard approach, we initially reweight the vectors gt(β), (t = 1,…,T), themselves to yield their smoothed counterparts
where, as before, k(·) is some kernel and ST a bandwidth parameter, both of whose properties are defined in Assumption 2.3, which follows. The redefinition (2.4) of the random vectors {g(zt,β)}t=1∞ was suggested in Smith (2001) as a means of achieving an asymptotic first-order equivalence between generalized empirical likelihood and efficient GMM estimators in the moment condition framework with serially dependent and conditionally heteroskedastic moment indicators {g(zt,β)}t=1∞. See also Smith (1997) and Kitamura and Stutzer (1997), which employ related and special cases of the class of kernels k(·) and particular choices for the bandwidth parameter ST.
The class of HAC covariance matrix estimators for Ω (eqn. (2.1)) proposed in this paper is formed directly as the normalized outer product of the smoothed random vectors gtT(β), (t = 1,…,T) (eqn. (2.4)), also evaluated at a preliminary consistent estimator
, namely,
with the divisor
as a necessary normalisation factor. Clearly
(eqn. (2.5)) is p.s.d. and is a member of the general class of quadratic estimators (Grenander and Rosenblatt, 1984, Sect. 4.1). Restriction to consideration of p.s.d. HAC covariance estimators is particularly desirable as the estimator
(eqn. (2.5)) would often form a component of a consistent estimator for the covariance matrix of the limiting normal distribution of a parameter estimator that may then be required for the construction of t- or F-type test statistics. Moreover, the property that a HAC covariance matrix estimator is p.s.d. is important if its inverse is to be used as the metric for efficient GMM estimation; see Section 3.
The next assumption introduces standard conditions on the kernel k(·) and bandwidth parameter ST. Let
, and
Assumption 2.3. (a) ST → ∞ and ST /T2 → 0; (b)
, kmax < ∞, k(0) ≠ 0, k1 ≠ 0, and is continuous at 0 and almost everywhere; (c)
.
The bandwidth parameter therefore obeys the conditions described in Andrews (1991, Theorem 1(a), p. 827). Assumption 2.3(c) is required to ensure that certain normalized sums defined in terms of the kernel k(·) converge appropriately to their integral representation counterparts; see Jansson (2002) and note 2 in the Notes section.
To gain some intuition about the suitability of
(eqn. (2.5)) as an estimator for Ω (eqn. (2.1)), consider the infeasible HAC covariance estimator
. In the generalized empirical likelihood context, Smith (2001) discusses the asymptotic equivalence of
with the estimator
, where the infeasible sample covariances
, and the implicit kernel kT*(·) is given by
. The estimator ΩT(β0) also belongs to the general quadratic class but has Toeplitz weight matrix. Therefore, we might expect that the estimator
would inherit the desirable asymptotic mean squared error properties of standard lag kernel estimators (Grenander and Rosenblatt, 1984, Sect. 4.2). The implicit kernel kT*(·) approximates the kernel k*(·) defined by
. Smith (2001) establishes that if Assumptions 2.3(b) and (c) hold then the induced k*(·) belongs to the p.s.d. class
defined in Andrews (1991, p. 822), that is,
where the class
is given in (2.2) and K*(λ) = (2π)−1∫k*(x)exp(−ιxλ) dx is the spectral window generator of the kernel k*(·).
Given a choice of k*(·), the corresponding kernel k(·) may be obtained from the relation K*(λ) = 2π|K(λ)|2, where K(·) is the spectral window generator of k(·). Smith (2001) provides examples of kernels k(·) that satisfy Assumption 2.3 and the consequent implicit kernels k*(·).
Initially we consider the infeasible covariance matrix estimator
and state a preliminary result.
3A referee pointed out that the original proof of Lemma 2.1 abstracted from that of Lemma A.3 in Smith (2001) was incomplete. The revision of Smith (2001), currently in preparation, will address this deficiency.
LEMMA 2.1. (Consistency of
). If Assumptions 2.1–2.3 hold, then
.
Lemma 2.1 parallels Proposition 1(a) of Andrews (1991, p. 825).
Let
denote some convex neighborhood of β0. The next assumption states the root-T consistency of the preliminary estimator
for β0 and bounds the expectation of the derivative matrix of the vector g(zt,β) on
.
Assumption 2.4. (a)
; (b)
.
Assumption 2.4(a) mimics Assumption B(i) of Andrews (1991), and Assumption 2.4(b) is Assumption B(iii) of Andrews (1991, p. 825) rewritten for our context.
The consistency of the feasible HAC covariance matrix estimator
then follows. Cf. Theorem 1(a) of Andrews (1991, p. 827).
THEOREM 2.1. (Consistency of
for Ω). If Assumptions 2.1–2.4 are satisfied, then
.
3. EFFICIENT GMM ESTIMATION
The next assumption is standard and states regularity conditions for the consistency of GMM estimators.
Assumption 3.1. (a)
is the unique solution to E [g(zt,β)] = 0; (b)
is compact; (c) g(zt,β) is continuous at each
with probability one; (d) Ω is p.d.
Let
. Then, if Assumptions 2.1–2.3 and 3.1 hold, from Smith (2001, Lemma A.1),
uniformly
. Therefore, given a preliminary consistent estimator
, a HAC covariance estimator
may be defined by either (2.5) as in Section 2 or as the centered version
as
. Furthermore, from Assumption 3.1(d), by Theorem 2.1, w.p.a.1
is p.d.
Consider a GMM criterion based on the smoothed random vectors gtT(β), (t = 1,…,T), defined in (2.4), with
as efficient metric; that is,
. The GMM estimator
is then defined as
The consistency of
for β0 follows.
THEOREM 3.1. (Consistency of
). If Assumptions 2.1–2.4 and 3.1 are satisfied, then
.
Asymptotic normality requires additional regularity conditions. Let G ≡ E [∂g(zt,β0)/∂β′].
Assumption 3.2. (a)
; (b) rk(G) = p.
Let Σ ≡ (G′Ω−1G)−1.
THEOREM 3.2. (Asymptotic normality of
.) If Assumptions 2.1–2.4, 3.1, and 3.2 are satisfied, then
The GMM estimator
((3.1)) shares the standard properties of efficient GMM estimators in the class of GMM estimators that minimize the quadratic form GMM criterion
, W p.d. Moreover, it is asymptotically first-order equivalent to the efficient GMM estimator that minimizes
with
, where
is defined in (2.3). The optimized criterion function statistic is likewise first-order asymptotically equivalent to the usual Hansen (1982) test statistic for overidentifying moment restrictions.
To estimate Σ consistently, consistent estimators of G and Ω are required. The former matrix may be estimated consistently by
, where
, which is an immediate by-product from the first-order conditions defining
, and the latter matrix by
(eqn. (2.5)) considered in Section 2 or as stated previously.
Although not pursued in this paper, a continuous updating GMM estimator for β0 that will share the asymptotic properties of
given in Theorems 3.1 and 3.2 may be defined using the criterion
, where
is a generalized inverse for
. The optimized criterion
. An asymptotically equivalent statistic may be computed as ((k2)/ST2(k1)2) times the (uncentered) explained sum of squares or (k2T)/(ST k1)2 times the (uncentered)
from a least squares regression of 1 on
. The factors kj may be replaced by
. See Smith (2001) for further consideration of the continuous updating estimator
for β0 that is a special case of the generalized empirical likelihood class of estimators considered there.
Test statistics for overidentifying moment restrictions and parametric restrictions on β0 may be constructed in a similar fashion to those proposed in Newey (1985) and Newey and West (1987b), respectively. See also Smith (2001) for related test statistics.
4. SUMMARY AND CONCLUSIONS
A new class of HAC covariance matrix estimators is proposed. The point of departure for these estimators is that rather than smoothing estimated sample autocovariances the random vectors themselves are smoothed using kernel function weights. Consistency of the class is shown. A corresponding GMM criterion based on the smoothed random vectors is also defined. The resultant GMM estimator is first-order asymptotically equivalent to the efficient GMM estimator (Hansen, 1982) and thus shares the same large-sample properties. The normalized GMM criterion function is also asymptotically equivalent to the standard GMM criterion function statistic for overidentifying moment restrictions.
An analysis of the higher order properties of this class of HAC covariance matrix estimators similar to that in Andrews (1991) for the standard class would seem apposite to detail the optimal choice of kernel and bandwidth and to describe automatic bandwidth estimators. It also remains to examine the finite-sample properties of the various estimators and statistics suggested in this paper.
APPENDIX: PROOFS OF RESULTS
Proof of Theorem 2.1. Given Lemma 2.1, as in the proof of Theorem 1(a) in Andrews (1991, p. 852), it is only necessary to prove that the difference
. Without loss of generality let {g(zt,β0)}t=1∞ be a scalar process.
Using a mean value expansion of
about β0
where β is on the line segment joining
and β0. Similarly to Andrews (1991, equation (A.10), p. 852), by Cauchy–Schwartz, w.p.a.1,
using Assumptions 2.2 and 2.4(b). Therefore, from equations (A.1) and (A.2),
Now
Let kT(a) = k((s − 1)/ST), (s − 1)/ST ≤ a < s/ST, if s ≤ 0, k(s/ST), (s − 1)/ST < a ≤ s/ST, if s > 0. Using the change of variables t = [ST b] and s = [ST a] , where [·] denotes the integer part of ·,
uniformly s. Hence, from equation (A.4), by Assumption 2.3(c),
Similarly, using the change of variable s = [ST a] , by the dominated convergence theorem, using Assumption 2.3(c),
Therefore, substituting equations (A.5) and (A.6) into equation (A.3), by Assumption 2.4(a),
The result then follows from equation (A.7) using Assumption 2.3(a). █
Proof of Theorem 3.1. As Ω is p.d. from Assumption 3.1(d),
is p.d. and invertible w.p.a.1. Let g(β) ≡ E [g(zt,β)] . Under Assumption 2.1, {g(zt,β0)}t=1∞ is a stationary and α-mixing sequence (White, 1984, Theorem 3.49, p. 47) and, thus, ergodic (White, 1984, Proposition 3.44, p. 46). By a uniform weak law of numbers (Smith, 2001, Lemma A.1), if Assumptions 2.1–2.3 and 3.1 hold,
and g(β) is continuous by the strictly stationary and ergodic version of Lemma 2.4 in Newey and McFadden (1994, p. 2129). Let Q(β) = g(β)′Ω−1g(β). Then, by Assumption 3.1(a), Q(β) is uniquely minimized at β0 and is continuous in
. Therefore, as
w.p.a.1 where
is the smallest eigenvalue of
,
uniformly
. The result follows by Theorem 2.1 in Newey and McFadden (1994, p. 2121). █
Proof of Theorem 3.2. As
by Theorem 3.1,
w.p.a.1. Therefore the first-order conditions
w.p.a.1, where
. By the mean value theorem,
where β lies on the line segment joining
and may differ from row to row. An application of the uniform weak law of large numbers (Smith, 2001, Lemma A.1) to
shows that
. Therefore,
. Now, by a central limit theorem (Smith, 2001, Lemma A.2),
, and the first conclusion follows.
As
. Therefore,
, where P ≡ Ω−1 − Ω−1GΣG′Ω−1. The second conclusion follows from Lemma A.2 of Smith (2001), PΩP = P, and rk(P) = m − p. █
References
REFERENCES
Andrews, D.W.K.
(1991)
Heteroskedasticity and autocorrelation consistent covariance matrix
estimation.
Econometrica
59,
817–858.Google Scholar
Andrews, D.W.K. &
J.C. Monahan
(1992)
An improved heteroskedasticity and autocorrelation consistent
covariance matrix estimator.
Econometrica
60,
953–966.Google Scholar
Bierens, H.J.
(1997)
Nonparametric cointegration analysis.
Journal of Econometrics
77,
379–404.Google Scholar
Brillinger, D.R.
(1981)
Time Series: Data Analysis and Theory.
Holden-Day.
Grenander, U. &
M. Rosenblatt
(1984)
Statistical Analysis of Stationary Time Series.
Chelsea.
Hansen, L.P.
(1982)
Large sample properties of generalized method of moments estimators.
Econometrica
50,
1029–1054.Google Scholar
Hansen, L.P.,
J. Heaton, &
A. Yaron
(1996)
Finite-sample properties of some alternative GMM estimators.
Journal of Business & Economic Statistics
14,
262–280.Google Scholar
Jansson, M.
(2002)
Consistent covariance matrix estimation for linear processes.
Econometric Theory
18,
1449–1459.Google Scholar
Kiefer, N.M.,
T.J. Vogelsang, &
H. Bunzel
(2000)
Simple robust testing of regression hypotheses.
Econometrica
68,
695–714.Google Scholar
Kitamura, Y. &
M. Stutzer
(1997)
An information-theoretic alternative to generalized method of
moments estimation.
Econometrica
65,
861–874.Google Scholar
Newey, W.K.
(1985)
Generalized method of moments specification testing.
Journal of Econometrics
29,
229–256.Google Scholar
Newey, W.K. &
D. McFadden
(1994)
Large sample estimation and hypothesis testing. In
R.F. Engle and
D. McFadden (eds.),
Handbook of Econometrics,
vol. 4,
2111–2245.
North-Holland.
Newey, W.K. &
K.D. West
(1987a)
A simple, positive semi-definite heteroskedasticity and
autocorrelation consistent covariance matrix.
Econometrica
55,
703–708.Google Scholar
Newey, W.K. &
K.D. West
(1987b)
Hypothesis testing with efficient method of moments estimation.
International Economic Review
28,
777–787.Google Scholar
Parzen, E.
(1957)
On consistent estimates of the spectrum of a stationary time series.
Annals of Mathematical Statistics
28,
329–348.Google Scholar
Phillips, P.C.B.
(2005)
HAC estimation by automated regression.
Econometric Theory
(this issue).Google Scholar
Phillips, P.C.B.,
Y. Sun, &
S. Jin
(2003)
Consistent HAC Estimation and Robust Regression Testing Using Sharp
Origin Kernels with No Truncation.
Cowles Foundation Discussion paper 1407,
Yale University.
Priestley, M.B.
(1962)
Basic considerations in the estimation of spectra.
Technometrics
4,
551–564.Google Scholar
Smith, R.J.
(1997)
Alternative semi-parametric likelihood approaches to generalized
method of moments estimation.
Economic Journal
107,
503–519.Google Scholar
Smith, R.J.
(2001)
GEL Criteria for Moment Condition Models.
Working paper,
University of Bristol.
Thomson, D.J.
(1982)
Spectrum estimation and harmonic analysis.
Proceedings of the IEEE
70,
1055–1096.Google Scholar
Walden, A.T.
(2000)
A unified view of multitaper multivariate spectral estimation.
Biometrika
87,
767–788.Google Scholar
White, H.
(1984)
Asymptotic Theory for Econometricians.
Academic Press.
You have
Access
- 24
- Cited by