doi:10.1111/j.1467-9892.2006.00477.x

A MODIFIED NONPARAMETRIC PREWHITENEDCOVARIANCE ESTIMATOR
By Masayuki Hirukawa
Concordia University and CIREQFirst version received June 2005
Abstract. This paper proposes a fully modiﬁed version of the spectral matrix estimator(and the long-run variance estimator as a special case) proposed originally by Xiao and
Linton [Journal of Time Series Analysis (2002) Vol. 23, pp. 215–250], and derives its
asymptotic results. A striking feature of the modiﬁed spectral matrix estimator is toachieve the convergence rate of O(T 8/9) in the mean squared error (MSE), which isusually achieved under the fourth-order spectral window. However, this estimator does
not sacriﬁce the positive deﬁniteness of the resulting estimate for the rate improvement; it
is Hermitian and positive deﬁnite in ﬁnite samples by construction. The faster convergence
rate is established by a multiplicative bias correction of the crude spectral estimator under
the second-order spectral window. The approximations to some sensible deﬁnitions of the
MSE of the estimator and the bandwidths that minimize the asymptotic MSEs are also
derived. Monte Carlo results indicate that for a wide variety of processes the modiﬁed
spectral matrix estimator reduces the bias without inﬂating the variance and thus improves
the MSE, compared with the crude, bias-uncorrected estimator.
Keywords. Spectral density estimation; covariance estimation; kernel smoothing;multiplicative bias correction; prewhitening.
JEL classiﬁcation numbers. C13; C22; C32.

1. INTRODUCTION
The aims of this paper are to propose a fully modiﬁed version of the spectral
matrix estimator (and the long-run variance estimator as a special case) proposed
originally by Xiao and Linton (2002; XL hereafter) and to derive its asymptotic
results. XL’s estimator is designed for general multivariate time-series models
whose autocorrelation structure is not necessarily parameterized. It is intended toachieve an improvement in bias by an order of magnitude without increasing theorder of magnitude of the variance. Unlike the case with the fourth-order spectral
window, the rate improvement is achieved without sacriﬁcing positive deﬁnitenessof the resulting estimate; it is established by multiplying a bias correction term on
the crude nonparametric spectral estimate in the frequency domain. The
multiplicative bias correction (MBC) technique applied in XL reminds us of thewell-known prewhitening procedure, and thus the resulting estimator is called
the nonparametric prewhitened (NPW) spectral (or covariance) matrix estimator
hereafter. The same MBC technique has already been applied in nonparametric0143-9782/06/03 441–476
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M. HIRUKAWA

regression (Linton and Nielsen, 1994), probability density estimation (Jones et al.,1995), and hazard estimation (Nielsen, 1998; Nielsen and Tanggaard, 2001); it
may therefore be viewed as their analogue in spectral density estimation.
There are two other classes of nonparametric density and regression estimatorsthat reduce bias by an order of magnitude while maintaining positive deﬁniteness.
The ﬁrst class, non-negative adaptation, was proposed originally by Terrell andScott (1980), and was generalized and reinterpreted as an MBC technique by
Koshkin (1988) and Jones and Foster (1993), respectively. The second class is a
subclass of the MBC estimators, but either the crude estimate or the bias
correction term belongs to a parametric class (e.g. see Hjort and Glad, 1995 andHjort and Jones, 1996 for density estimation, and Glad, 1998 for regression).
Nonetheless, NPW in XL is preferable in the context of spectral matrix
estimation. The idea behind the ﬁrst class is the additive bias correction to thelogarithm of the density. Hence, we face diﬃculty in applying this technique tospectral matrix estimation at nonzero frequency, because oﬀ-diagonal elements ofthe matrix are complex-valued. The second class is subject to misspeciﬁcation by
its parametric nature. In contrast, NPW is easily extended to the matrix context,
and is expected to be robust to misspeciﬁcation.
However, several questions still remain in XL. First, although XL argue that theresulting spectral density estimator is guaranteed to be nonnegative deﬁnite (lines
26–27, p. 217), the expressions (4) and (5) in XL do not necessarily lead to a nonnegative deﬁnite estimate. Second, whereas the approximation to the mean squared
error (MSE) of each of other MBC estimators in the same class (Theorem in Linton
and Nielsen, 1994; Theorem 1 in Jones et al., 1995; Theorem 2 in Nielsen, 1998; andTheorem 3 in Nielsen and Tanggaard, 2001) involves the roughness of the kernel
obtained by twicing (Stuetzle and Mittal, 1979) in the variance term, the asymptotic
variance in Theorem 1 in XL does not have an exact twiced form of the spectral
window. Third, since the spectral matrix estimator is complex-valued in general, theMSE in XL (p. 224) does not necessarily yield a non-negative squared bias orvariance term. In other words, some extra restriction should be imposed on theweighting scheme for the MSE so that the resulting squared bias and variance terms
become non-negative for every frequency.
It is demonstrated that the spectral matrix estimator, given as a smoothed
periodogram in XL, is inconsistent with the corresponding spectral matrix estimator
as weighted autocovariances in the time domain. To ensure the positive deﬁnitenessof the resulting estimate in ﬁnite samples, a modiﬁed deﬁnition of the NPW spectral
matrix estimator is proposed. A key feature of this estimator is the sandwich form
of the crude spectral matrix estimate and the bias correction term; by construction,
the estimator yields a Hermitian and positive-deﬁnite estimate in ﬁnite samples.
Some alternative weighting schemes for the MSE of the estimator are also proposed
to guarantee non-negativity of the squared bias and variance terms for every
frequency. For each deﬁnition of the MSE, the approximation to the MSE and the
bandwidth that minimizes the asymptotic mean squared error (AMSE) are derived.
It is demonstrated that the variance term in the MSE indeed involves the roughness
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of the twiced spectral window. When the second-order spectral window and the
AMSE-optimal bandwidth are employed, the NPW spectral matrix estimator has
the convergence rate of O(T 8/9) in MSE, not O(T 4/5), where T is the sample size.
Although the spectral matrix estimator under the fourth-order spectral window canalso achieve the same convergence rate, the positive deﬁniteness in the resulting
estimate is not guaranteed. Moreover, it is shown that when the underlying spectral
density has suﬃcient smoothness, NPW can be iterated further to obtain the rate ofconvergence arbitrarily close to the parametric one, while the positive deﬁniteness of
the resulting estimate is maintained.
Monte Carlo results indicate that for a wide variety of processes the NPW
estimator reduces the bias without inﬂating the variance and thus improves theMSE, compared with the crude, bias-uncorrected estimator. It also appears to
capture the shape of the spectral density more accurately by removing spurious
peaks or troughs that the bias-uncorrected estimator might generate.
The remainder of this paper is organized as follows: Section 2 modiﬁes thedeﬁnition of the NPW spectral matrix estimator; in Section 3 the asymptotic
results based on the modiﬁed estimator are explained; Section 4 displays Monte
Carlo results; Section 5 concludes; proofs are given in the Appendix.

2. MODIFIED NPW SPECTRAL MATRIX ESTIMATOR
Let fxt gTt¼1 2 Rd be a zero-mean stationary vector time series that has a positivedeﬁnite spectral matrix. To estimate the long-run variance of fxtg deﬁned by
2
!
!T 3
T
T
X
X
1
X ¼ lim E4
xt
xt 5;
T !1 Tt¼1
t¼1
we usually construct a weighted sum of sample autocovariances by applying akernel method
0
1
 
 
minfT
þl;T g
T 1
T 1
X
X
X
l
l
1
¼
^
@
ð1Þ
X
k
k
xt xTtl A;
CðlÞ
¼
M
M
T
t¼maxf1;1þlgl¼ðT 1Þ
l¼ðT 1Þ
where k(Æ) is a kernel (or a lag window), and M 2 Rþ is the bandwidth.
There is yet another expression of the covariance estimator that is Let fundamental frequencies be kj ¼ 2pj/T, j ¼
asymptotically equivalent to X.
0,±1, . . . , ±[T/2], and the periodogram at the frequency kj be    Ixx kj ¼ fx kj fx kj ; where fx kj ¼ ð2pT Þ1=2 RTt¼1 xt eitkj is the ﬁnite Fourier transform of fxtg
evaluated at the frequency kj. Moreover, forÓ 2006 The Author
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M. HIRUKAWA

K ð hÞ ¼

1
2p

Z 1

k ðuÞeiuh du;1the spectral window corresponding to the kernel k(Æ), deﬁne the amplitude window(Parzen, 1963) as
1
X

K M ð hÞ ¼ M

KðM ðh þ 2pjÞÞ:j¼1

Then, the long-run variance estimator based on a smoothed periodogram can be
expressed as2 X   ^ ¼ 4p
KM kj Ixx kj ;X
T k 2Bð0Þ

ð2Þ

jwhere B(0) ¼ fkj|p < kj < pg is the frequency band with width 2p centred atzero frequency. As described in Parzen (1963),
K M ð hÞ ¼ M

1XKðM ðh þ 2pjÞÞ ’ MK ð MhÞ:

ð3Þ

j¼1

In particular, the approximation is replaced by the equality when the
spectral window K(h) is band-limited such that K(h) ¼ 0 for jhj  p. In
this case, let m ¼ 2[T/(2M)] þ 1 be the number of fundamental frequenciesin B(0) ¼ kjjp/M < kj < p/M. Then, eqn (2) is rewritten as2X    4p2 X   ^ ¼ 4p M
K Mkj Ixx kj ’
K Mkj Ixx kj :X
T k 2Bð0Þ
m k 2Bð0Þ
j

ð4Þ

j

This estimator is a modiﬁed version of the expression (2) in XL, and isasymptotically equivalent to eqn (1), as mentioned below; the original expression
does not establish this property.
Since X ¼ 2pfxx(0), eqn (2) gives the estimator of fxx(0) as   2p X
f^xx ð0Þ ¼
KM kj Ixx kj :T k 2Bð0Þj

In general, using the frequency band centred at the frequency x, B(x) ¼
fkjjx  p < kj < x þ pg, we can estimate the spectral matrix of fxtg evaluatedat the frequency x 2 (p, p) as
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MODIFIED NPW COVARIANCE ESTIMATOR

445


  
2p X
KM kj  x Ixx kj :f^xx ðxÞ ¼
T k 2BðxÞ

ð5Þ

j

The asymptotic equivalence of the smoothed periodogram estimator (5) and thecorresponding weighted covariance estimator
 
T 1
1 X
l ^

k
CðlÞeilx ;fxx ðxÞ ¼
2p l¼ðT 1Þ M
and hence the asymptotic equivalence of eqn (2) [or (4)] and (1), is well known; see
the expressions (2.4) and (2.5) in Robinson (1991), for example.
In applying the MBC technique in Linton and Nielsen (1994), Jones et al.
(1995), Nielsen (1998), and Nielsen and Tanggaard (2001) to eqn (5), consider that
the Ixx(kj) are Hermitian and positive deﬁnite by construction,1 so is f^xx ðxÞ if
K(h)  0, 8h 2 R. Then, f^xx ðxÞ has positive eigenvalues k1, . . . , kd, and thusf^xx1=2 ðxÞ is well deﬁned by the unitary decompositionf^xx1=2 ðxÞ ¼ U K1=2 U  ;
1=2

ð6Þ

1=2

where K1=2 ¼ diagðk1 ; . . . ; kd Þ is the diagonal matrix containing the squareroots of the eigenvalues, and U is the unitary matrix, i.e. UU ¼ Id. Using eqn (6),this paper proposes the modiﬁed NPW spectral matrix estimator of fxx(x) as
f~xx ðxÞ ¼ f^xx1=2 ðxÞ~aðxÞf^xx1=2 ðxÞ
8
9
< X
=








2p ^ 1=2
KM kj  x
kj Ixx kj f^xx1=2 kj f^xx1=2 ðxÞ;
¼ f^xx1=2 ðxÞfxx:k 2BðxÞ
;
T

ð7Þ

j

where ~
aðxÞ serves as the bias correction term. Similarly, the modiﬁed NPW
covariance matrix estimator becomes
8
9
< X
=








2p ^ 1=2^ 1=2 : ð8Þ
~ ¼X^ 1=2 ~
^ 1=2
^ 1=2 ¼ X
X
KM kj
kj Ixx kj f^xx1=2 kj Xfxx
að 0Þ X
:k 2Bð0Þ;Tj

The term NPW is derived from the fact that the MBC technique in eqns (7)and (8) reminds us of theprocedure, because the transformed prewhitening
 
periodograms f^xx1=2 kj Ixx kj f^xx1=2 kj are approximately constant.
The NPW estimators (7) and (8) are modiﬁed versions of the expressions (4)and (5) in XL. A key feature is that ~aðxÞ and f~xx ðxÞ constitute a sandwich form.
Therefore, for a non-negative spectral window K(h), eqns (7) and (8) are
Hermitian and positive deﬁnite in ﬁnite samples: ~aðxÞ is Hermitian and positivedeﬁnite by recognizing that        
f^xx1=2 kj Ixx kj f^xx1=2 kj ¼ f^xx1=2 kj Ixx kj f^xx1=2 kj ;
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M. HIRUKAWA

so isaðxÞf^xx1=2 ðxÞ ¼ f^xx1=2 ðxÞ~aðxÞf^xx1=2 ðxÞ :f~xx ðxÞ ¼ f^xx1=2 ðxÞ~Note that neither the expression (4) nor (5) in XL becomes Hermitian or positive^ is.
deﬁnite in ﬁnite samples, even if f^xx ðxÞ or X

3. ASYMPTOTIC RESULTS
3.1. The main result
In this section, an approximation to the MSE of the NPW spectral matrix
estimator f~xx ðxÞ is derived, and it is demonstrated that the approximation iscompatible with corresponding approximations for analogous MBC estimators.
These results require three deﬁnitions of smoothness which were ﬁrst introduced
by Parzen (1957) and are frequently applied in spectral density estimation.
Definition 1. For a kernel k(Æ) and a positive number r, the rth generalized
derivative of kernel k(Æ) at the origin is deﬁned as
kr ¼ lim

x!0

1  k ð xÞ:j xjr

Definition 2. A kernel k(Æ) is said to have the characteristic exponent q if it has
the following properties.
8
if r < q<¼ 0
if r ¼ q:
kr 2 ð0; 1Þ
:¼1
if r > q
Definition 3.

The rth generalized derivative of spectral density
1
1 X
f ðxÞ ¼CðjÞeijx
2p j¼1is deﬁned asf ðrÞ ðxÞ ¼

1
1 X
j jjr CðjÞeijx :2p j¼1

It is known that if r is an even integer, the generalized derivative of the kernel atthe origin and the generalized derivative of the spectral density obey
1 dr kðxÞ
kr ¼ 
;ð9Þ
r! dxr x¼0
and
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MODIFIED NPW COVARIANCE ESTIMATOR

f ðrÞ ðxÞ ¼ ð1Þr=2

dr f ðxÞ
:
dxr

447
ð10Þ

We consider second-order kernels exclusively to ensure the positive deﬁniteness of
the resulting estimator; thus q ¼ 2 throughout.2 To establish the asymptotic
properties of f~xx ðxÞ, we make the following assumptions.
Assumption 1.

The time series fxtg is fourth-order stationary with1
X

jlj4 kCðlÞk < 1

l¼1

and
1
1
1
X
X
X

jji1 i2 i3 i4 ð j; k; lÞj < 1;j¼1 k¼1 l¼1where ji1 i2 i3 i4 (Æ,Æ,Æ) is the fourth-order cumulant of (xi1t, xi2tþj, xi3tþjþk, xi4tþjþkþl) andxit is the ith element of xt.
R 1Assumption 2. The spectral window K(h) is even and non-negative withKðhÞdh ¼ 1. In addition, the kernel k(x) satisﬁes jkðxÞj  kðxÞ such thatR110 k ð xÞdx < 1, where k ð xÞ is even and monotonically decreasing on [0,1).
Assumption 3.

The bandwidth M satisﬁes 1/M þ M4/T ! 0 as T ! 1.

These assumptions are standard in the literature of spectral density estimation;indeed, they follow the assumptions made for Theorem 5.1 in Robinson (1991).
4
The condition R1
l¼1 jlj kCðlÞk < 1 in Assumption 1 implies four uniformly
bounded derivatives of the spectral matrix for every frequency; without thiscondition, the bias correction of the NPW estimator is not established. The
non-negativity of K(h) in Assumption 2 is required forR 1 the positive deﬁniteness off~xx ðxÞ. Because of the non-negativity, the condition 1 jKðhÞjdh < 1 in A2(0) of
Robinson (1991) is guaranteed. There are many examples of kernels that satisfy
Assumption 2: the Parzen, Quadratic Spectral, Bohman, and Gaussian kernels, toname a few.
Under Assumptions 1, 2 and 3, f~xx ðxÞ has an asymptotic expansion
f~xx ðxÞ ¼ fxx ðxÞ þ B~ðxÞ þ V~ðxÞ þ R þ op M

4

 1=2 !
T
þ
:
M

ð11Þ

The detailed expressions of B~ðxÞ, V~ðxÞ and R and the derivation of eqn (11) are
given in the Appendix. Since the kj(¼Ixx(kj)  fxx(kj)) are asymptoticallyindependent and have zero asymptotic mean, B~ðxÞ and V~ðxÞ constitute theÓ 2006 The Author
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448

M. HIRUKAWA

leading bias and variance terms of f~xx ðxÞ, respectively. On the other hand, R is theleading remainder part.
To deﬁne the MSE of f~xx ðxÞ, we must take extra care of the choice of theweighting scheme. Since the spectral matrix estimator is complex-valued ingeneral, the MSE in XL (p. 224), which follows the deﬁnition in Andrews(1991), does not necessarily yield a non-negative squared bias or variance term
for an arbitrarily chosen d 2  d 2 real-valued positive semideﬁnite weighting
matrix W.3 It is clear that for each ﬁxed x, (i) ABias 2 R or (ABias)2 2 Rþ,and (ii) AVar 2 Rþ are the requirements for a sensible AMSE-optimal
bandwidth choice, where ABias and AVar are the leading bias andvariance terms in the MSE. Then, the following two alternative deﬁnitionsof the MSE are considered.
Definition 4.
n 


T

o
MSE1 f~xx ðxÞ; fxx ðxÞ ¼ E vec f~xx ðxÞ  fxx ðxÞ W ðxÞvec f~xx ðxÞ  fxx ðxÞ ;where W(x) ¼ diagfw1(x), . . . , wd2(x)g and the weight assigned to (i, j) element off~xx ðxÞ is such that0
if i ¼ jwð j1Þdþi ðxÞ¼0
if i 6¼ j
for x 6¼ 0, and w(j1)dþi(0)  0, 8i, j ¼ 1, . . . , d for x ¼ 0.

Definition 5.


 
 2
MSE2 f~xx ðxÞ; fxx ðxÞ ¼ E vT f~xx ðxÞ  fxx ðxÞ v ;where v 2 Rd is some weighting vector.
The restriction on the weighting matrix W(x) for nonzero frequency in
Deﬁnition 4 is required to establish (ABias)2, AVar 2 Rþ. If the restriction isrelaxed so that the same non-negative weights are assigned to (i, j) and (j, i)elements of f~xx ðxÞ (simply because they are complex conjugates), thenthe resulting (ABias)2 is shown to be real-valued but not necessarily
non-negative, as discussed in the Appendix. In the case of zero frequency,
the weighting scheme nests the one under a natural choice of weighting matrix
Id2. On the other hand, Deﬁnition 5 is applied to the long-run varianceestimation in Newey and West (1994) and Hirukawa (2005), and has some
ﬂexibility.
Theorems 1 and 2 ﬁnd the asymptotic approximation to each deﬁnition of MSE
and the bandwidth that minimizes the corresponding AMSE.
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MODIFIED NPW COVARIANCE ESTIMATOR

Theorem 1.

Under Assumptions 1, 2 and 3, Deﬁnition 4 is approximated by
MSE1 f~xx ðxÞ; fxx ðxÞ’Z
k24 vecðWðxÞÞT W ðxÞvecðWðxÞÞ M 1 2þt ð xÞdx
M8
T 1 kno trW ðxÞfxx ðxÞ  fxx ðxÞT þ1ðx ¼ 0ÞtrW ð0ÞKdd fxx ð0Þ  fxx ð0Þ ;

ð12Þ

where
WðxÞ ¼ fxx1=2 ðxÞU00 ðxÞfxx1=2 ðxÞ;UðxÞ ¼ fxx1=2 ðxÞfxx00 ðxÞfxx1=2 ðxÞ;andtk ð xÞ ¼

Z 1

TK ðhÞeixh dh

1

is the kernel corresponding to TK(h) ¼ 2K(h)  K s K(h), the fourth-order spectral
window obtained by twicing (Stuetzle and Mittal, 1979). In this deﬁnition(ABias)2 2 Rþ and AVar 2 Rþ are guaranteed. The bandwidth that minimizeseqn (12) is
1=9

M1opt ðxÞ ¼ c1 ðxÞT
91=9
8
=
<
8k24 vecðWðxÞÞT W ðxÞvecðWðxÞÞ
¼ R
T 1=9 :
: 1 t2 ð xÞdx trW ðxÞf ðxÞ  f ðxÞT þ1ðx ¼ 0ÞtrW ð0ÞK f ð0Þ  f ð0Þ ;
xx
xx
dd xx
xx
1 k

At the optimum,~
MSEopt1 fxx ðxÞ;fxx ðxÞnT4’ T 8=9 c81 ðxÞk2 vecðWðxÞÞ W ðxÞvecðWðxÞÞ
Z 1
þ c1 ð x Þ
tk2 ð xÞdx trW ðxÞfxx ðxÞfxx ðxÞT þ1ðx¼ 0ÞtrW ð0ÞKdd fxx ð0Þfxx ð0Þ:

1

Theorem 2.

Under Assumptions 1, 2 and 3, Deﬁnition 5 is approximated by
MSE2 f~xx ðxÞ; fxx ðxÞ

2
Z
 T
2 1 2k 4 vT WðxÞv
M
ð
1
þ
1
ð
x
¼
0
Þ
Þ
v
þ
f
ð
x
Þv
tk ð xÞdx
’ 2
xx
T
M8
1

ð13Þ

for the same W(x) and tk(x) as in Theorem 1. In this deﬁnition ABias 2 R and
AVar 2 Rþ are guaranteed. The bandwidth that minimizes eqn (13) isÓ 2006 The Author
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450

M. HIRUKAWA(
1=9
M2opt ðxÞ ¼ c2 ðxÞT ¼

2
8k24 vT WðxÞv

)1=9

R1
ð1 þ 1ðx ¼ 0ÞÞðvT fxx ðxÞvÞ2 1 tk2 ð xÞdxT 1=9 :At the optimum,
MSEoptf~xx ðxÞ; fxx ðxÞ
2

Z

2
 T
2 1 28=9
84 T
c2 ðxÞk2 v WðxÞv þc2 ðxÞð1 þ 1ðx ¼ 0ÞÞ v fxx ðxÞvtk ð xÞdx :’T
1Each theorem demonstrates that although the second-order spectral window isemployed, the squared bias term is of order M8, which is usually achieved under
the fourth-order spectral window. The variance term involves the roughness of thetwiced spectral window. These are compatible with the analogous results in
Linton and Nielsen (1994), Jones et al. (1995), Nielsen (1998), and Nielsen andTanggaard (2001). As mentioned in XL, the divergence rate of the optimal
bandwidth is O(T1/9) for each deﬁnition of the MSE. As a result, the convergence
rate of the MSE at the optimum is O(T 8/9), which is faster than O(T 4/5), therate usually achieved under the second-order kernel.4
In particular, two deﬁnitions of the MSE coincide in the scalar case, and the
AMSE is given by
MSE f~xx ðxÞ; fxx ðxÞ
 00
00 2

Z 1
k24
fxx
M
’ 8 fxx ðxÞðx Þþ ð1 þ 1ðx ¼ 0ÞÞ
tk2 ð xÞdxfxx2 ðxÞ:
T
M
fxx
1
This AMSE takes the same form as of the expression (2.1) in Jones et al.
(1995). It is also natural to compare the theoretical performance of f~xx ðxÞ withthat of f€xx ðxÞ, the spectral matrix estimator under the fourth-order spectral
window TK(Æ) (or the corresponding kernel tk(Æ)). The AMSEs of two
estimators have an identical variance term [i.e. AVar in eqn (12) or (13),depending on the deﬁnition], whereas the squared bias term in the AMSE off€xx ðxÞ isT
k24 vec fxx0000 ðxÞ W ðxÞvec fxx0000 ðxÞ
for Definition 4; orM82
k24 vT fxx0000 ðxÞvfor Definition 5,
M8

ð14Þ

as shown in the Appendix. Hence, the two estimators diﬀer only in thedependence of the bias term on the spectral density and its derivatives. This result
is also consistent with the one in Jones et al. (1995).
Recognizing that X ¼ 2pfxx(0) gives Corollary 1 on the asymptotic results of~
the long-run variance estimator X.
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MODIFIED NPW COVARIANCE ESTIMATOR

Corollary 1.

~ is approximated by
Under Assumptions 1, 2 and 3, the MSE of X
~ X
MSE1 X;’451

k24 vecðWÞT W vecðWÞ M
þ
M8
T

Z 1
1

tk2 ð xÞdxftr W ðId 2 þ Kdd ÞðX  XÞg

ð15Þ

for Deﬁnition 4, orZ k24 vT Wv 2
M  T 2 1 2
~
v
þ
2
Xv
tk ð xÞdx
MSE2 X; X ’
T
M8
1


ð16Þ

for Deﬁnition 5, where W ¼ X1/2U00 (0)X1/2 and W ¼ W(0). The bandwidth thatminimizes eqn (15) or (16) is M1opt ð0Þ or M2opt ð0Þ.
Unlike the case of nonzero frequency, there is no problem of complex-valued
elements in the squared bias or variance term for the case of zero frequency.
Hence, in reality, the AMSE (15) can be also interpreted as the one foran arbitrarily chosen d 2  d 2 real-valued positive semideﬁnite weighting matrix
W.

3.2. Iteration
It is of theoretical interest to see what happens if NPW is iteratively applied tospectral matrix estimation. To answer this question, deﬁne n-times iterated NPW
spectral matrix estimator as
1=2
1=2f~xx;n ðxÞ ¼ f~xx;n1 ðxÞ~an ðxÞf~xx;n1 ðxÞ
8
9
< X
=








2p ~ 1=2
1=2
1=2
1=2
¼ f~xx;n1 ðxÞK M kj  x
fxx;n1 kj Ixx kj f~xx;n1 kj f~xx;n1 ðxÞ;:k 2BðxÞ
;Tjwhere f~xx;0 ðxÞ ¼ f^xx ðxÞ. Clearly, f~xx;n ðxÞ is Hermitian and positive deﬁnite. To
derive the asymptotic expansion of f~xx;n ðxÞ, Assumptions 1 and 3 are modiﬁed asfollows:
Assumption 10 .
1
X
l¼1

The time series fxtg is fourth-order stationary with

jlj2nþ2 kCðlÞk < 1

and

1
1
1
X
X
X

jji1 i2 i3 i4 ð j; k; lÞj < 1;j¼1 k¼1 l¼1where n 2 N is the number of NPW iterations, and ji1 i2 i3 i4 (Æ,Æ,Æ) is deﬁned inAssumption 1.
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452

M. HIRUKAWA

Assumption 30 . For the same n deﬁned in Assumption 10 , the bandwidth M
satisﬁes 1/M þ M2nþ2/T ! 0 as T ! 1.
Theorem 3 ﬁnds the asymptotic expansion of the NPW spectral matrix
estimator after NPW is iterated further.
Theorem 3.
expansion

Under Assumptions 10 , 2 and 30 , f~xx;n ðxÞ has an asymptotic

f~xx;n ðxÞ ¼ fxx ðxÞ þ B~n ðxÞ þ V~n ðxÞ þ op Mð2þ2nÞ

 1=2 !
T
þ
:
M

ð17Þ

The leading bias term B~n ðxÞ is approximated by
M 2þ2n B~n ðxÞ ’ ð1Þn k2nþ1 Wn ðxÞ;where
Wn ðxÞ ¼ fxx1=2 ðxÞU00n ðxÞfxx1=2 ðxÞ;Un ðxÞ ¼ fxx1=2 ðxÞWn1 ðxÞfxx1=2 ðxÞ;andW0 ðxÞ ¼ fxx00 ðxÞ:The leading variance term V~n ðxÞ is approximated by
( R1
 

for x 6¼ 0
2p 1 Kn2 ðhÞdhfxx ðxÞ  fxx ðxÞTT~var vec V n ðxÞ ’R1 2
M
2p 1 Kn ðhÞdhðId 2 þ Kdd Þfxx ð0Þ  fxx ð0Þ
for x ¼ 0
(R1
kn2 ð xÞdxfxx ðxÞ  fxx ðxÞTfor x 6¼ 0
’ R1
;
1
2
2
k
ð
x
Þdx
ð
I
þ
K
Þf
ð
0
Þ

f
ð
0
Þ
for
x¼0
dd xx
xx
d
1 n
where
Kn ðhÞ ¼ K0 ðhÞ þ Kn1 ðhÞ  K0  Kn1 ðhÞ;Z 1
kn ð xÞ ¼
Kn ðhÞeixh dh:

K0 ðhÞ ¼ K ðhÞ;

and

1

Theorem 3 demonstrates that the NPW iteration reduces the order ofmagnitude of the bias without inﬂating that of the variance. In the scalar case,the approximation to the leading bias term becomes
d2n f 00
M 2þ2n B~n ðxÞ ’ ð1Þn k2nþ1 fxx ðxÞ 2n xx ðxÞ :dx
fxx
In particular, for n ¼ 2,
M 6 B~2 ðxÞ ’ k23 fxx ðxÞ

 00
0000
fxx
ðx Þ
;
fxx

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MODIFIED NPW COVARIANCE ESTIMATOR

which takes the same form as of Theorem 4 in Nielsen and Tanggaard (2001).
Moreover, the spectral window Kn(h) is exactly the same as that Hössjer and
Ruppert (1995) arrived at while using their transformation bias-reducing
technique n times. The same form as K2(h) also appears in Theorem 4 in Nielsen
and Tanggaard (2001).
Propositions 1 and 2 are natural outcomes of Theorem 3. Since they are thegeneralizations of Theorems 1 and 2, their proofs are omitted. Observe that after
NPW is iterated further, the resulting spectral matrix estimator achieves theconvergence rate arbitrarily close to the parametric one.
Proposition 1.

Under Assumptions 10 , 2 and 30 , Deﬁnition 4 is approximated by
MSE1 f~xx;n ðxÞ; fxx ðxÞ’Z
k22nþ2 vecðWn ðxÞÞT W ðxÞvecðWn ðxÞÞ M 1 2þk ð xÞdx
T 1 nM 4nþ4no tr W ðxÞfxx ðxÞ  fxx ðxÞT þ1ðx ¼ 0Þtr W ð0ÞKdd fxx ð0Þ  fxx ð0Þ :ð18Þ

In this deﬁnition (ABias)2 2 Rþ and AVar 2 Rþ are guaranteed. The bandwidththat minimizes eqn (18) is
1

opt
M1;n
ðxÞ ¼ c1;n ðxÞT 4nþ5
8
9 1<
=4nþ5 1
ð4n þ 4Þk22nþ2 vecðWn ðxÞÞT W ðxÞvecðWn ðxÞÞ
¼ R1
T 4nþ5 ::k 2 ð xÞdx tr W ðxÞf ðxÞ  f ðxÞT þ1ðx ¼ 0Þtr W ð0ÞK f ð0Þ  f ð0Þ ;
1 n

xx

xx

dd xx

xx

At the optimum,
MSEoptf~xx;n ðxÞ;fxx ðxÞ1n4nþ4ð4nþ4ÞðxÞk22nþ2 vecðWn ðxÞÞT W ðxÞvecðWn ðxÞÞ’T 4nþ5 c1;n
Z 1
þc1;n ðxÞkn2 ð xÞdx tr W ðxÞfxx ðxÞfxx ðxÞT þ1ðx¼0Þtr W ð0ÞKdd fxx ð0Þfxx ð0Þ:

1

Proposition 2.

Under Assumptions 10 , 2 and 30 , Deﬁnition 5 is approximated by
MSE2 f~xx;n ðxÞ; fxx ðxÞ

2
Z
 T
2 1 2k22nþ2 vT Wn ðxÞv
M
ð
1
þ
1
ð
x
¼
0
Þ
Þ
v
’
þ
f
ð
x
Þv
kn ð xÞdx:xxT
M 4nþ41

ð19Þ

In this deﬁnition ABias 2 R and AVar 2 Rþ are guaranteed. The bandwidth thatminimizes eqn (19) isÓ 2006 The Author
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454

M. HIRUKAWA
1

4nþ5
opt
M2;n
ðxÞ ¼ c2;n ðxÞT
¼

(

1
)4nþ5

2
ð4n þ 4Þk22nþ2 vT Wn ðxÞv1
T 4nþ5 :R1
ð1 þ 1ðx ¼ 0ÞÞðvT fxx ðxÞvÞ2 1 kn2 ð xÞdxAt the optimum,f~xx;n ðxÞ; fxx ðxÞ
MSEopt2Z22 1 24nþ4
ð4nþ4Þ
’ T 4nþ5 c2;nðxÞk22nþ2 vT Wn ðxÞv þc2;n ðxÞð1 þ 1ðx ¼ 0ÞÞ vT fxx ðxÞvkn ð xÞdx :1

4. MONTE CARLO RESULTS
4.1. Description of data-generating processes and estimators
In this section, small Monte Carlo experiments investigate the accuracy of theestimate of the long-run variance X ¼ 2pfxx(0) by the NPW spectral matrix
estimator, in comparison with alternative spectral estimators. As the datagenerating processes (DGPs), the following univariate zero-mean ARMA(1,1),
MA(2) and AR(2) models are chosen.
ARMA(1,1):
xt ¼ qxt1 þ t þ wt1 ;

i:i:d:

t  N ð0; 1Þ;ðq; wÞ ¼ ð0:8; 0Þ; ð0:5; 0Þ; ð0:5; 0Þ; ð0:8; 0Þ; ð0; 0:8Þ; ð0; 0:5Þ; ð0; 0:5Þ;ð0; 0:8Þ; ð0:5; 0:8Þ; ð0:5; 0:5Þ; ð0:5; 0:8Þ; ð0:2; 0:2Þ; ð0:2; 0:2Þ;ð0:5; 0:8Þ; ð0:5; 0:5Þ; ð0:5; 0:8Þ:MA(2):
xt ¼ t þ w1 t1 þ w2 t2 ;i:i:d:

t  N ð0; 1Þ;ðw1 ; w2 Þ ¼ ð0:67; 0:33Þ; ð1:0; 0:2Þ; ð1:9; 0:95Þ; ð1:3; 0:5Þ;ð1:0; 0:9Þ; ð0; 0:9Þ; ð0; 0:9Þ:
AR(2):
i:i:d:

xt ¼ q1 xt1 þ q2 xt2 þ t ; t  N ð0; 1Þ;
q1
0:4; 0:8; 1:2; 1:6 for x ¼ p=6
q1 ¼:; q2 ¼
0:4; 0:8; 1:2
for x ¼ p=4q1  4 cos x

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MODIFIED NPW COVARIANCE ESTIMATOR

455

The parameter settings of the ARMA(1,1) model basically follow XL. This model
includes four pure AR(1) and four pure MA(1) models as special cases. However,the spectral density of an ARMA(1,1) model over the frequency band [0, p] ismonotone. Then, MA(2) and AR(2) models are considered to evaluate theperformances of the NPW estimator when the spectral density has a peak or a troughat nonzero frequency. The parameter settings of the MA(2) and AR(2) models
depend on Hirukawa (2005) and Phillips et al. (2003), respectively. The spectral
densities for the MA(2) models over the frequency band [0, p] have the following
shapes: close to monotone decreasing (with a slight trough in high frequency part)for (w1, w2) ¼ (0.67, 0.33); monotone increasing for (w1, w2) ¼ (1.0, 0.2); close tomonotone increasing (with a slight trough in low frequency part) for (w1, w2) ¼(1.9, 0.95), (1.3, 0.5); with a shallow trough near p/3 for (w1, w2) ¼ (1.0, 0.9);
with an obvious trough near p/2 for (w1, w2) ¼ (0, 0.9); and with an obvious peaknear p/2 for (w1, w2) ¼ (0, 0.9). On the other hand, the spectral density of eachAR(2) model has a peak at the frequency x ¼ p/6 or p/4. In all experiments, thesample size is 256, and the number of replications is 5,000.
As in XL, the long-run variance estimates are calculated by four spectral
estimators, namely: (i) the NPW estimator (NPW); (ii) the conventional, biasuncorrected estimator, deﬁned as eqn (5) (BUC); (iii) the AR(1)-prewhitened
estimator in Andrews and Monahan (1992) (AM); and (iv) the trapezoidal lagwindow estimator in Politis and Romano (1995) (PR). A reasonable bandwidth
choice for each estimator is the key for meaningful comparison. From theviewpoint of separating the issue of estimator quality from the bandwidth
selection problem, the oracle AMSE-optimal bandwidths are used for NPW and
BUC. In particular, the AMSE-optimal bandwidth of BUC is given by( 002 )1=5
4k22
fxx
opt
R1
MBUC ðxÞ ¼
ðx Þ
T 1=5
ð1 þ 1ðx ¼ 0ÞÞ 1 k 2 ð xÞdx fxx
for univariate time series fxtg, and under this bandwidth BUC achieves theconvergence rate of O(T 4/5) in MSE.
AM is thought of as the best competitor to NPW in the literature of
econometrics. The estimation procedure of AM is as follows. First, an AR(1)model xt ¼ /xt1 þ gt is ﬁtted to univariate time series fxtg, regardless of its true
DGP. Second, eqn (5) is estimated for the AR(1)-prewhitened process^ t1 , where /^ is the least squares estimate of /. Finally, AM is given^
gt ¼ xt  /x
by recolouring the AR(1)-prewhitened spectral density estimate f^^g^g ðxÞ so thatf^^g^g ðxÞf^AM ðxÞ ¼ 
2 :^ ix 
1  /e(For more detailed implementation, see Andrews and Monahan, 1992.) AM
usually has the convergence rate of O(T 4/5) in MSE, but it achieves theparametric rate of O(T 1) in MSE when the true DGP is an AR(1) model.
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456

M. HIRUKAWAFor NPW, BUC and AM, the Gaussian kernel k(x) ¼ exp(x2/2)pﬃﬃﬃﬃﬃ
isﬃ
employed. Since its corresponding spectral window K ðhÞ ¼ expðh2 =2Þ= 2p
has unbounded support, the amplitude window is approximated by
KM(h) ’ MK(Mh) as in eqn (3). For convenience, some characteristic numbersof the Gaussian kernel are given below.
1
k2 ¼ ;2

Z 1

pﬃﬃﬃ
k ð xÞdx ¼ p;

1

2

Z 1
1

tk2 ð xÞdx ¼

!
pﬃﬃﬃ
pﬃﬃﬃ
4 2
1
p 4  pﬃﬃﬃ þ pﬃﬃﬃ :3
2

On the other hand, PR is well known as a bias-corrected spectral density
estimator in the literature of statistical time series. It is expressed as a linear
combination of two spectral estimators under the Bartlett kernelf^PR ðxÞ ¼ ðh þ 1Þf^1 ðxÞ  hf^2 ðxÞ;where h ¼ M2/(M1  M2) and M1 and M2(<M1) are the bandwidths for f^1 ðxÞand f^2 ðxÞ, respectively. Since f^PR ðxÞ is not necessarily non-negative, Politis andRomano (1995) deﬁne PR as the clipped estimator
n
o
þ
f^PR
ðxÞ ¼ max f^PR ðxÞ; 0 :PR has the convergence rate of O(logT/T) and O(T 1) in MSE for ARMA
models and m-dependent processes, respectively. However, unlike three otherestimators concerned, the issue of optimal bandwidth choice for PR is yet to be
fully solved. Then, the bandwidth choice follows XL. Based on the suggestion in
Politis and Romano (1995), set h ¼ 1 so that M1 ¼ 2M2. Let M2 take ﬁve values
such that M2 ¼ 4, 8, 12, 16, 20, and denote the corresponding PR estimators as
PR1, PR2, PR3, PR4 and PR5, respectively.

4.2. Simulation results
Tables I, II and III display the Monte Carlo results for ARMA(1,1), MA(2) andAR(2) models, respectively. The root mean squared errors (RMSEs) of spectral
estimators are exhibited in the ﬁrst row of a given DGP. The numbers in thesecond row (in parenthesis) are the biases of the estimators. For convenience, thetrue value of the long-run variance X is also provided.
The Monte Carlo results indicate superior ﬁnite sample performances of NPW
for a wide variety of DGPs. Moreover, the MBC technique in NPW does not leadto inﬂations in variance, as opposed to alternative bias reduction techniques forspectral estimation. For many DGPs in Table I, the RMSE of NPW is not only
smaller than that of BUC but also the smallest among all. In general, the bias of
NPW is smaller in magnitude than that of BUC. Table II demonstrates that theseÓ 2006 The Author
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457

MODIFIED NPW COVARIANCE ESTIMATOR
TABLE I
RMSEs and Biases of Long-Run Variance Estimates for ARMA(1,1) Models
q

w

X

NPW

BUC

AM

PR1

PR2

PR3

PR4

PR5

0.8

0.0 25.0000 10.6541
9.82139.85359.47559.3699 11.1310 12.6877 13.9998(5.0036) (5.9529) (0.5427) (7.9805) (4.4592) (4.3080) (5.0591) (5.9867)
0.5 0.0 4.0000 1.0470
1.0022
0.94291.05781.5154
1.85782.11862.3129(0.3574) (0.5312) (0.0249) (0.2768) (0.3856) (0.5637) (0.7442) (0.9152)
0.5 0.0 0.4444 0.0745
0.07920.05430.12700.17550.20930.23550.2569(0.0269) (0.0359) (0.0020) (0.0091) (0.0347) (0.0519) (0.0694) (0.0877)
0.8 0.0 0.3086 0.0661
0.05910.03840.11990.12330.14740.16750.1838(0.0451) (0.0290) (0.0022) (0.0730) (0.0010) (0.0279) (0.0447) (0.0577)
0.0

0.8 3.2400

0.49810.5664(0.2725) (0.3016)
0.0 0.5 2.2500 0.3389
0.3838(0.1663) (0.1956)
0.0 0.5 0.2500 0.0626
0.0667(0.0056) (0.0224)
0.0 0.8 0.0400 0.0196
0.0208(0.0042) (0.0087)
0.5

0.8 12.9600

1.51290.90981.28791.53561.73781.9041(1.2093) (0.1552) (0.3022) (0.4442) (0.5839) (0.7132)
0.8794
0.63740.88701.06471.20061.3061(0.6755) (0.1071) (0.2200) (0.3219) (0.4204) (0.5093)
0.20860.08110.10020.11910.13490.1482(0.1985) (0.0088) (0.0193) (0.0308) (0.0417) (0.0519)
0.29540.03700.02600.02430.02480.0256(0.2903) (0.0076) (0.0026) (0.0008) (0.0012) (0.0030)

3.57593.4543 11.4281
3.5816
5.17196.22077.00007.6323(1.1764) (1.7770) (9.2036) (0.9033) (1.1806) (1.7245) (2.2854) (2.8324)
0.5 0.5 9.0000 2.4473
2.37096.97672.45373.44784.17604.7376
5.1720(0.8460) (1.2583) (5.5619) (0.6569) (0.8650) (1.2472) (1.6302) (2.0139)
0.5 0.8 0.1600 0.0731
0.07480.5426
0.09300.08180.08680.09390.1008(0.0104) (0.0283) (0.5325) (0.0375) (0.0038) (0.0035) (0.0105) (0.0176)
0.2 0.2 2.2500 0.4278
0.43400.59450.62740.87941.0444
1.17981.2917(0.1351) (0.2231) (0.3090) (0.1119) (0.2055) (0.3028) (0.4039) (0.5018)
0.2 0.2 0.4444 0.0843
0.08790.09510.13110.17560.20760.23190.2527(0.0467) (0.0335) (0.0686) (0.0179) (0.0380) (0.0569) (0.0761) (0.0948)
0.5 0.8 1.4400 0.1788
0.21850.40470.40880.57400.68850.77930.8535(0.0090) (0.1705) (0.2891) (0.0775) (0.1375) (0.1943) (0.2553) (0.3140)
0.5 0.5 0.1111 0.0281
0.03280.09400.05590.04930.05490.0611
0.0662(0.0052) (0.0137) (0.0894) (0.0207) (0.0049) (0.0104) (0.0155) (0.0206)
0.5 0.8 0.0178 0.0137
0.01470.13070.06460.02760.02100.01840.0172(0.0057) (0.0078) (0.1288) (0.0420) (0.0108) (0.0063) (0.0048) (0.0040)

ﬁndings are robust to a wide variety of shapes of spectral densities: NPW indeed
achieves the smallest RMSE for all but one DGP. The RMSE and bias of BUC
are smaller for most of DGPs in Table III, whereas NPW achieves the smallest
RMSE for two extreme DGPs with (x, q1) ¼ (p/6, 1.6),(p/4, 1.2).
On the other hand, AM is subject to misspeciﬁcation by ﬁtting the AR(1) ﬁlter.
As far as the ﬁlter correctly speciﬁes DGPs (i.e. for the ﬁrst four DGPs in Table I),
AM performs best. In contrast, it performs quite poorly for all other DGPs: an
MA or extra AR term makes the AM estimate imprecise, and biased upwardly inmost cases. PR estimators (PR1 and PR2, in particular) appear to have a better
bias property for a wide variety of DGPs, whereas their RMSEs for a given DGP
are in general greater than that of NPW or BUC.5
For illustrative purposes, Figure 1 displays the plots of spectral density
estimates by NPW, BUC, AM and PR1 with true densities over the frequency
band (0, p]. The spectral densities for four selected DGPs have a variety of shapes:(a) downward sloping; (b) upward sloping; (c) with a trough at nonzeroÓ 2006 The Author
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458

M. HIRUKAWATABLE II
RMSEs and Biases of Long-Run Variance Estimates for MA(2) Models
w2

w1

NPW

X

BUC

AM

PR1

PR2

PR3

PR4

PR5

0.67 0.33 4.0000

0.73640.82642.42921.1252
1.59731.93012.18192.4011(0.3368) (0.4460) (1.9870) (0.2094) (0.3835) (0.5660) (0.7418) (0.9026)
1.0 0.2 0.0400 0.0176
0.01960.18650.03700.02610.02430.02470.0257(0.0053) (0.0092) (0.1832) (0.0080) (0.0026) (0.0013) (0.0010) (0.0023)
1.9 0.95 0.0025 0.0217
0.02150.19420.07530.04300.03390.02840.0258(0.0116) (0.0117) (0.1911) (0.0445) (0.0265) (0.0213) (0.0184) (0.0169)
1.3 0.5 0.0400 0.0168
0.01950.10310.04540.02980.02600.02540.0261(0.0009) (0.0125) (0.1008) (0.0118) (0.0054) (0.0021) (0.0002) (0.0016)
1.0 0.9 0.8100 0.2075
0.21480.38510.22030.31490.38070.43050.4715(0.1403) (0.1180) (0.3786) (0.0372) (0.0745) (0.1107) (0.1439) (0.1768)
0.0 0.9 3.6100 0.7454
0.8292
1.73650.98581.41071.70481.9302
2.1089(0.4355) (0.4635) (1.6945) (0.1855) (0.3570) (0.5476) (0.7191) (0.8629)
0.0 0.9 0.0100 0.0190
0.02031.80740.07620.04470.03460.03000.0268(0.0070) (0.0096) (1.7875) (0.0421) (0.0243) (0.0186) (0.0164) (0.0142)

TABLE III
RMSEs and Biases of Long-Run Variance Estimates for AR(2) Models
x

q1

X

NPW

BUC

AM

PR1

PR2

PR3

PR4

PR5

p/6 0.4 1.8737

0.37750.27440.7601
0.52840.7386
0.89451.00551.0898(0.0096) (0.1032) (0.5990) (0.0806) (0.1581) (0.2420) (0.3163) (0.3878)
0.8 3.9954
0.9546
0.76373.95151.1364
1.55971.88872.14822.3610(0.0961) (0.0170) (3.5477) (0.1223) (0.3632) (0.5574) (0.7322) (0.8925)
1.2 9.1821
3.07622.5160 20.1334
2.87373.85174.47235.03725.4579(1.9532) (0.6436) (18.5009) (0.4995) (0.9547) (1.1338) (1.5696) (1.9573)
1.6 14.9857
5.3853
5.9783 28.8981 10.6293
8.84387.40618.34499.2074(1.7173) (3.0992) (24.9134) (8.2406) (5.5838) (0.2788) (2.3109) (3.6329)
p/4 0.4 1.7100
0.3506
0.28810.82030.49410.67250.80890.91721.0005(0.0371) (0.0278) (0.6945) (0.0874) (0.1613) (0.2346) (0.3029) (0.3711)
0.8 2.8304
0.83800.67633.24030.8722
1.11851.32871.4884
1.6183(0.5373) (0.2092) (3.0054) (0.2454) (0.2457) (0.3794) (0.5032) (0.6211)
1.2 3.4690
1.01081.09674.43102.60451.4202
1.74871.87772.0734(0.2424) (0.5319) (4.0859) (2.3419) (0.0685) (0.5596) (0.4373) (0.6689)

frequency; and (d) with a peak at nonzero frequency. Although each panelpresents the plots based on a typical realization for a given DGP with the sample
size 256, it provides a feeling about the general tendency of spectral density
estimates concerned. First, we see how NPW works. When BUC underestimates(overestimates) the density, NPW corrects the estimate in an upward (adownward) direction. The bias correction in a downward direction might besomewhat excessive, whereas the one in an upward direction not.
Second, NPW appears to capture the shape of a spectral density moreaccurately. Although BUC mistakenly estimates a peak in the low frequency partof Panel (c), NPW draws the U-shape successfully. On the other hand, PR1 have atendency to generate a spurious peak or trough. Because of the AR(1) ﬁlter, thedensity estimate by AM is in general misleading. In particular, AM tends toÓ 2006 The Author
Journal compilation Ó 2006 Blackwell Publishing Ltd
JOURNAL OF TIME SERIES ANALYSIS Vol. 27, No. 3

459

MODIFIED NPW COVARIANCE ESTIMATOR(b) ARMA(1, 1) with ( , y ) = (0.0, –0.8)
0.8
0.7

0.7

0.8

(a) ARMA(1,1) with ( , y ) = (0.0, 0.8)

0.2

Spectrum
0.3 0.4 0.5

0.6

Truth
NPW
BUC
AM
PR1

0.1
0.8

1.2
1.6
2.0
Frequency (radians)

2.4

2.8

3.2

(c) MA(2) with ( y1, y 2 ) = (0.0, 0.9)

0.7
0.0

0.1

0.2

Spectrum
0.3 0.4 0.5

0.6

Truth
NPW
BUC
AM
PR1

0.0

0.4

0.8

1.2
1.6
2.0
Frequency (radians)

2.4

2.8

3.2

0.0

0.4

Spectrum
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.0

0.8

0.0

0.1

0.2

Spectrum
0.3 0.4 0.5

0.6

Truth
NPW
BUC
AM
PR1

0.0

0.4

0.8

1.2
1.6
2.0
Frequency (radians)

2.4

2.8

3.2

(d) MA(2) with ( y1, y 2 ) = (0.0, –0.9)
Truth
NPW
BUC
AM
PR1

0.0

0.4

0.8

1.2
1.6
2.0
Frequency (radians)

2.4

2.8

3.2

Figure 1. Spectral density estimates for selected DGPs.

estimate the spectral density of such a DGP as in Panels (c) or (d) as of the one forthe white noise process!

5. CONCLUSION
This paper has modiﬁed the NPW spectral matrix estimator (and the long-run
variance estimator as a special case) proposed by XL, and derived its asymptotic
results. Because of its sandwich form of the crude spectral matrix estimate andthe bias correction term, the modiﬁed NPW estimator is shown to be Hermitianand positive deﬁnite in ﬁnite samples. Some sensible deﬁnitions of the MSE of theestimator are also proposed. For each deﬁnition, the approximation to the MSE
and the bandwidth that minimizes the asymptotic MSE are derived. It isdemonstrated that the variance term in the MSE indeed involves the roughness ofthe twiced spectral window. The estimator establishes the convergence rate ofO(T 8/9) in MSE when best implemented under the second-order spectral
window. NPW can be iterated further to obtain better rates of convergence,
provided that the underlying spectral density has suﬃcient smoothness. MonteÓ 2006 The Author
Journal compilation Ó 2006 Blackwell Publishing Ltd
JOURNAL OF TIME SERIES ANALYSIS Vol. 27, No. 3

460

M. HIRUKAWA

Carlo results indicate that for a wide variety of DGPs the NPW estimator reduces
the bias without inﬂating the variance and thus improves the MSE, comparedwith the bias-uncorrected estimator.

APPENDIX
A.1. MATHEMATICAL NOTATIONS

Notation

Description
Integer part of x 2 R
Real and imaginary parts of x 2 C
Indicator function that takes 1 if S is true
Convolution
Tensor (or Kronecker) productT
Conjugate transpose of a complex-valued matrix, i.e. A ¼ A
Euclidean norm of the matrix A, i.e. jjAjj ¼ [tr(A A)]1/2
Column by column vectorization function of the matrix A
p-dimensional identity matrix.
d 2  d 2 commutation matrix that transforms vec(A) into vec(AT),i.e. Kdd ¼ Rdi¼1 Rdj¼1 ei eTj  ej eTi , where ei is the ith elementary d-vector
second-order derivative of a d  d spectral matrix f(x) ¼ ffij(x)gdd,i.e. f00 (x) ¼ fd2fij(x)/dx2gddFourth-orderderivative of a d  d spectral matrix f(x) ¼ ffij(x)gdd,0000
i.e. f (x)¼fd4fij(x)/dx4gdd
XT ¼ YT þ op(YT)[x]
Re(x), Im(x)
1(S)
s




jjAjj
vec(A)Ip
Kdd
f00 (x)0000

f (x)XT ’ YT

A.2.

DETAILED EXPRESSIONS OF B~ðxÞ, V~ðxÞ AND R IN EQN (11)

Let
 
 2p
and
wxkj ¼ KM kj  x
T


 
 
kj ¼ Ixx kj  fxx kj

for notational simplicity. In addition, observe thatkj ¼ x þ2pj; j ¼ 0; 1; . . . ; N ¼
T

N
2



holds if kj 2 B(x). Then, B~ðxÞ and V~ðxÞ, the leading bias and variance terms of f~xx ðxÞ, aregiven by^ x fxx1=2 ðxÞ  fxx1=2 ðxÞ
B~ðxÞ ¼ B

X ^ kj fxx1=2 ðxÞ
wxkj fxx1=2 kj B

jj N j^ x  fxx1=2 ðxÞþ fxx1=2 ðxÞBX 
^ kj fxx1=2 kj fxx1=2 ðxÞ;wxkj BjjN jÓ 2006 The Author
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JOURNAL OF TIME SERIES ANALYSIS Vol. 27, No. 3

461

MODIFIED NPW COVARIANCE ESTIMATOR

V~ðxÞ ¼ fxx1=2 ðxÞ

X

 
 
wxkj fxx1=2 kj kj fxx1=2 kj fxx1=2 ðxÞ þ fxx1=2 ðxÞV^x þ V^x fxx1=2 ðxÞjj N j fxx1=2 ðxÞX  1  1   1=2   1 1=2
V^kj fxx1=2 kj
wxkj fxx1=2 kj
fxx kj fxx kj
fxx ðxÞ

jjN j fxx1=2 ðxÞX  1   1=2   1
  1 1=2
V^kj fxx1=2 kj
wxkj fxx1=2 kj
fxx kj fxx kj
fxx ðxÞ;jjN jwhere^ x ¼ f 1=2 ðxÞ  f 1=2 ðxÞ;B
xx
xx

V^x ¼ f^xx1=2 ðxÞ  fxx1=2 ðxÞ;and fxx1=2 ðxÞ ¼

X 
wxkj fxx1=2 kj :jj N jTo save space, among all 206 terms in the remainder part R, only 12 leading terms aredescribed below:
X
 
 
^x
wxkj fxx1=2 kj kj fxx1=2 kj fxx1=2 ðxÞBjj N j fxx1=2 ðxÞX
jjN jþ fxx1=2 ðxÞX

jjN j fxx1=2 ðxÞ

X

 
 
 
^ kj fxx1=2 kj kj fxx1=2 kj fxx1=2 ðxÞ
wxkj fxx1=2 kj B
 
 
^x
wxkj fxx1=2 kj kj fxx1=2 kj B
 
 
 
^ kj f 1=2 kj f 1=2 ðxÞ
wxkj fxx1=2 kj kj fxx1=2 kj Bxx
xx

jjN j^x^ 2x  BþBX ^ kj fxx1=2 kj fxx1=2 ðxÞ
wxkj Bjj N jþ fxx1=2 ðxÞXX  2 1=2   1=2
 ^ k fxx^ kj B^x
wxkj fxx1=2 kj B
kj fxx ðxÞ  fxx1=2 ðxÞ
wxkj fxx1=2 kj BjjjN jþ fxx1=2 ðxÞX

jjN jþ fxx1=2 ðxÞX

jjN jwxkj fxx1=2

X
 
 
 
^ kj f 1=2 kj B
^ kj f 1=2 ðxÞ  B^ kj f 1=2 ðxÞ^x
kj B
wxkj fxx1=2 kj B
xx
xx
xx
jjN j

X
 
 
 
^ kj fxx1=2 kj fxx1=2 ðxÞ  fxx1=2 ðxÞ^x:^ kj fxx1=2 kj B
^ kj fxx1=2 kj B
wxkj B
wxkj BjjN jjjN jA.3.

DERIVATION OF EQN (11)The derivation of eqn (11) requires Lemmata 1 and 2.
Lemma 1.

Let
^ x ¼ fxx1=2 ðxÞ  fxx1=2 ðxÞ;B
V^x ¼ f^xx1=2 ðxÞ  fxx1=2 ðxÞ;X fxx1=2 ðxÞ ¼
wxkj fxx1=2 kj

andjjN jThen,
Assumptions
1, 2 and 3,
 under 2 
^ 
uniformly in x.
(a) Bx ¼ O M(b) V^x  ¼ op ððT =M Þ1=2 Þ uniformly in x.
Ó 2006 The Author
Journal compilation Ó 2006 Blackwell Publishing Ltd
JOURNAL OF TIME SERIES ANALYSIS Vol. 27, No. 3

462

M. HIRUKAWA

The standard result on the bias term of the spectral density estimator

Proof of Lemma 1.
implies thatfxx ðxÞ  fxx ðxÞ ¼ O M 2

uniformly in x, wherefxx ðxÞ ¼

X 
wxkj fxx kj :jjN jTheorem 7.7.4 in Brillinger (1975, p. 265) also implies that there is a d > 0 such that 1=2d !

 X

T
f^xx ðxÞ  fxx ðxÞ ¼ 

wxkj kj  ¼ op
M
jjN j
uniformly in x. Using an approximation such asfxx1=2 ðxÞ  fxx1=2 ðxÞ ’ fxx1=2 ðxÞ þ fxx1=2 ðxÞ1 

fxx ðxÞ  fxx ðxÞuestablishes the lemma.
Lemma 2.

The bias correction term ~aðxÞ in eqn (7) can be rewritten as~aðxÞ ¼ a0 ðxÞ þ aB ðxÞ þ aV ðxÞ þ R;

where
a0 ðxÞ ¼ Id þ

X

 
 
wxkj fxx1=2 kj kj fxx1=2 kj ;jjN jaB ðxÞ ¼ 

X

 
 
  X
 
^ kj fxx1=2 kj kj fxx1=2 kj ^ kjwxkj fxx1=2 kj B
wxkj fxx1=2 kj B

jj N j

þX

jjN jþ

X

jjN jX

jj N jwxkj fxx1=2

þjjN jþ

X

jjN j

X

jjN j

X

jjN jþ

X

jjN j

þ

X

 
 
 
 
^ kj f 1=2 kj B
^ kj f 1=2 kj kj f 1=2 kj
kj B
xx
xx
xx

 
 
^ kj fxx1=2 kj B
^ kjwxkj fxx1=2 kj B
 
 
  X
 
^ kj fxx1=2 kj ^ kj fxx1=2 kj
wxkj fxx1=2 kj kj fxx1=2 kj B
wxkj BjjN jX

ð20Þ

jjN jwxkj fxx1=2

 
 
 
 
^ kj f 1=2 kj kj f 1=2 kj B
^ kj f 1=2 kj
kj B
xx
xx
xx

  2 1=2  ^ k fxx
wxkj fxx1=2 kj B
kj
j
 
 
 
 
 
^ kj fxx1=2 kj B
^ kj fxx1=2 kj kj fxx1=2 kj B
^ kj fxx1=2 kj
wxkj fxx1=2 kj B 
  2 1=2  ^ kj f 1=2 kj B
^ fwxkj fxx1=2 kj B
kj
xx
kj xx
 
 
 
 
^ kj f 1=2 kj B
^ kj f 1=2 kj
wxkj fxx1=2 kj kj fxx1=2 kj B
xx
xx
 
 
^ kj fxx1=2 kj^ kj fxx1=2 kj B
wxkj BjjN jÓ 2006 The Author
Journal compilation Ó 2006 Blackwell Publishing Ltd
JOURNAL OF TIME SERIES ANALYSIS Vol. 27, No. 3

MODIFIED NPW COVARIANCE ESTIMATOR

X
jjN j

X

jjN jþ

X

jjN j

þ

X

463

 
 
 
 
 
^ kj fxx1=2 kj kj fxx1=2 kj B
^ kj fxx1=2 kj B
^ kj fxx1=2 kj
wxkj fxx1=2 kj B
  2 1=2   ^ f^ kj f 1=2 kj
wxkj fxx1=2 kj B
kj B
kj xx
xx
 
 
 
 
 
 
^ kj f 1=2 kj B
^ kj f 1=2 kj kj f 1=2 kj B
^ kj f 1=2 kj B
^ kj f 1=2 kj
wxkj fxx1=2 kj B
xx
xx
xx
xx
xx
 
  2 1=2   ^ kj fxx1=2 kj B
^ k fxx^ kj fxx1=2 kj ;wxkj fxx1=2 kj B
kj B
jjjN jaV ðxÞ ¼ X

  1  1  1
wxkj fxx1=2 kj
kj fxx1=2 kj
V^kj fxx1=2 kj

jjN jX  1  1   1=2   1V^kj fxx1=2 kj
wxkj fxx1=2 kj
fxx kj fxx kjjj N jX  1  1  1
wxkj fxx1=2 kj
kj fxx1=2 kj
V^kj fxx1=2 kj

jj N jX  1   1=2   1
  1V^kj fxx1=2 kj
wxkj fxx1=2 kj
fxx kj fxx kjjj N jþX  1  1  1  1
wxkj fxx1=2 kj
kj fxx1=2 kj
V^kj fxx1=2 kj
V^kj fxx1=2 kj

jj N jþX  1  1   1=2   1
  1V^kj fxx1=2 kj
V^kj fxx1=2 kj
wxkj fxx1=2 kj
fxx kj fxx kj;jj N jand R ¼ op(M4þ(T/M)1/2).
Proof of Lemma 2.

Rewrite ~aðxÞ as~aðxÞ ¼ a0 ðxÞ þ faðxÞ  a0 ðxÞg þ f~aðxÞ  aðxÞg;

ð21Þ

wherea0 ðxÞ ¼

X    
wxkj fxx1=2 kj Ixx kj fxx1=2 kj ;ð22Þ  1   1=2   1
wxkj fxx1=2 kj
Ixx kj fxx kj:ð23ÞjjN jandaðxÞ ¼

X
jjN jThree terms in eqn (21) are approximated separately as follows. First, eqn (3) and thechange of variable hj ¼ M(kj  x) yield the Riemann sum
Xjj N jwxkj ’

Z 1X  2pM ¼
K M kj  x
K hj Dhj ’K ðhÞdh ¼ 1:
T
1
k 2BðxÞ
h 2ð1;1Þ
X

j

ð24Þ

j

Using eqn (24) and kj ¼ Ixx(kj)  fxx(kj) in eqn (22) gives the expression of a0(x).
Ó 2006 The Author
Journal compilation Ó 2006 Blackwell Publishing Ltd
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464

M. HIRUKAWA

Second, by a geometric series expansion,  1 n    o1 1=2  ¼ Id þ fxx1=2 kj fxx1=2 kj  fxx1=2 kj
fxx
kjfxx1=2 kj








 
¼ fxx1=2 kj  fxx1=2 kj fxx1=2 kj  fxx1=2 kj fxx1=2 kj
 
 
 
 
 
 
 
þ fxx1=2 kj fxx1=2 kj  fxx1=2 kj fxx1=2 kj fxx1=2 kj  fxx1=2 kj fxx1=2 kj

 
 3
þ O fxx1=2 kj  fxx1=2 kj  ;where the order of the remainder term is O(M6) ¼ op(M4) by Lemma 1(a). Substituting
this in eqn (23), expanding it, and using
 
 
 
 
^ kj ¼ fxx1=2 kj  fxx1=2 kj
Band kj ¼ Ixx kj  fxx kj ;we see that the leading term is aB(x) and the remainder term is op(M4).
Lastly, by the same geometric series expansion as above,   1 1=2    1 1=2   1f^xx1=2 kj ¼ Id þ fxx1=2 kj
f^xx kj  fxx1=2 kj
fxx kj  1  1 1=2   f^xx kj  fxx1=2 kj¼ fxx1=2 kj
 fxx1=2 kj

 
 2
þ O f^xx1=2 kj  fxx1=2 kj  ;  1fxx1=2 kj

where the order of the remainder term is
op ðT =M Þ1 ¼ op ðT =M Þ1=2
by Lemma 1(b). Substituting this to ~aðxÞ, expanding it, and using
 
 
 
 
and kj ¼ Ixx kj  fxx kj ;V^kj ¼ f^xx1=2 kj  fxx1=2 kj
we see that the leading term is aV(x) and the remainder term is op((T/M)1/2).

u

Derivation of eqn (11). It is easy to see that^ x þ V^x :f^xx1=2 ðxÞ ¼ fxx1=2 ðxÞ þ B

ð25Þ

Substituting eqn (25) and Lemma 2 into eqn (7) and doing some tedious but
straightforward calculations establish the expansion to f~xx ðxÞ.
u

A.4.

PROOF OF THEOREM 1

The proofs of Theorems 1 and 2 require Lemmata 3–5.
Lemma 3.

Under Assumptions 1, 2 and 3, the leading bias term of f~xx ðxÞ is approximatedby
M 4 B~ðxÞ ’ k22 WðxÞ;Ó 2006 The Author
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JOURNAL OF TIME SERIES ANALYSIS Vol. 27, No. 3

MODIFIED NPW COVARIANCE ESTIMATOR

465

where
WðxÞ ¼ fxx1=2 ðxÞU00 ðxÞfxx1=2 ðxÞ andUðxÞ ¼ fxx1=2 ðxÞfxx00 ðxÞfxx1=2 ðxÞ:Proof of Lemma 3. The Riemann sum (24) implies that ﬁrst two terms in M 4 B~ðxÞ areapproximated by
8
9
<
=
X


^ kj fxx1=2 ðxÞ^ x fxx1=2 ðxÞ  fxx1=2 ðxÞ
wxkj fxx1=2 kj B
M4 B:;
jjN jX ^ x  fxx1=2 kj M 2 B^ kj fxx1=2 ðxÞ:’ M 2 fxx1=2 ðxÞ
wxkj fxx1=2 ðxÞM 2 B
ð26Þ
jjN jUsing the standard result on the bias term of the spectral density estimator and eqn (10),
M 2 fxx ðxÞ  fxx ðxÞ ’ k2 fxxð2Þ ðxÞ ¼ k2 fxx00 ðxÞ:Then, by eqn (20), for a large T,^ x ¼ M 2 f 1=2 ðxÞ  f 1=2 ðxÞ
B
xx
xx
1


’ M 2 fxx ðxÞ  fxx ðxÞ fxx1=2 ðxÞ þ fxx1=2 ðxÞ’ k2 fxx00 ðxÞ 2fxx1=2 ðxÞ
¼

1

k2 00
f ðxÞfxx1=2 ðxÞ:
2 xx

ð27Þ

Substituting eqn (27) into (26) gives
8
9
<
=
X
 
4 ^
1=2
1=2
1=2
1=2
^ kj fxx ðxÞ
M Bx fxx ðxÞ  fxx ðxÞ
wxkj fxx
kj B:;
jj N j’M 2 k2 1=2
f ðx Þ
2 xx nX    owxkj fxx1=2 ðxÞfxx00 ðxÞfxx1=2 ðxÞ  fxx1=2 kj fxx00 kj fxx1=2 kj fxx1=2 ðxÞ:ð28Þjj N jLet     
U kj ¼ fxx1=2 kj fxx00 kj fxx1=2 kj :Since each element of the spectral matrix fxx(Æ) has four continuous derivatives,

2

 1

2
 
U kj ¼ UðxÞ þ U0 ðxÞ kj  x þ U00 ðxÞ kj  x þo kj  x :2

ð29Þ

By the symmetry of KM(Æ) about the origin,X 2p 

kj  x ¼ 0:
wxkj kj  x ¼
KM kj  x
TjjN jk 2BðxÞ
X

ð30Þ

jÓ 2006 The Author
Journal compilation Ó 2006 Blackwell Publishing Ltd
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466

M. HIRUKAWA

Substituting eqns (29) and (30) into (28), we have, by eqn (3) and the change of variablehj ¼ M(kj  x),
8
9
<
=
X


^ kj fxx1=2 ðxÞ^ x fxx1=2 ðxÞ  fxx1=2 ðxÞ
wxkj fxx1=2 kj B
M4 B:;
jjN j
’

X  
 
2 2pM
k2 1=2fxx ðxÞU00 ðxÞfxx1=2 ðxÞK M kj  x M kj  x
4Tk 2BðxÞjX
 k2
h2j K hj Dhj¼  fxx1=2 ðxÞU00 ðxÞfxx1=2 ðxÞ4
hj 2ð1;1Þ
Z 1
k2 1=2’  fxx ðxÞU00 ðxÞfxx1=2 ðxÞ
h2 K ðhÞdh
4
1
k2¼  2 fxx1=2 ðxÞU00 ðxÞfxx1=2 ðxÞ
2k22¼  WðxÞ:2
The next to last expression is established because
Z 1
K ðhÞeixh dhk ð xÞ ¼
1

and eqn (9) imply thatZ 1

h2 K ðhÞdh ¼ k 00 ð0Þ ¼ 2k2 :

ð31Þ

1

Similarly, ﬁnal two terms in M 4 B~ðxÞ are also approximated by k22 WðxÞ=2, which
completes the proof.
uLemma 4. Under Assumptions 1, 2 and 3, the leading variance term of f~xx ðxÞ isapproximated by
( R1
 

2p 1 TK2 ðhÞdhfxx ðxÞ  fxx ðxÞTfor x 6¼ 0
Tvar vec V~ðxÞ ’R1
M
2p 1 TK2 ðhÞdhðId 2 þ Kdd Þfxx ð0Þ  fxx ð0Þ
for x ¼ 0
(R1
tk2 ð xÞdxfxx ðxÞ  fxx ðxÞTfor x 6¼ 0
;
’ R1
1 2
for x ¼ 0
1 tk ð xÞdxðId 2 þ Kdd Þfxx ð0Þ  fxx ð0Þ
where TK(h) ¼ 2K(h)K s K(h) is the fourth-order spectral window obtained by twicing(Stuetzle and Mittal, 1979), andZ 1
TK ðhÞeixh dht k ð xÞ ¼
1

is the kernel corresponding to the spectral window TK(h).
Proof  of Lemma 4. Using  kj ’ x for a large T and the ﬁrst-order Taylor expansion givesfxx kj ’ fxx ðxÞ and f^xx kj ’ fxx ðxÞ. Then, by eqn (20), second and third terms in V~ðxÞ
are approximated byÓ 2006 The Author
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467

MODIFIED NPW COVARIANCE ESTIMATOR

fxx1=2 ðxÞV^x þ V^x fxx1=2 ðxÞ¼ fxx1=2 ðxÞ f^xx1=2 ðxÞ  fxx1=2 ðxÞ þ f^xx1=2 ðxÞ  fxx1=2 ðxÞ fxx1=2 ðxÞ  1f^xx ðxÞ  fxx ðxÞ’ fxx1=2 ðxÞ f^xx1=2 ðxÞ þ fxx1=2 kj
1

þ f^xx ðxÞ  fxx ðxÞf^xx1=2 ðxÞ þ fxx1=2 ðxÞ
fxx1=2 ðxÞ
X
1 X
1
wxkj kj þ
wxkj kj 2fxx1=2 ðxÞ
fxx1=2 ðxÞ’ fxx1=2 ðxÞ 2fxx1=2 ðxÞ¼

X

jj N jjjN jwxkj kj :jjN jSimilarly, ﬁnal two terms in V~ðxÞ are approximated byX  1  1   1=2   1 1=2
fxx1=2 ðxÞV^kj fxx1=2 kj
wxkj fxx1=2 kj
fxx kj fxx kj
fxx ðxÞ
jjN jþ fxx1=2 ðxÞX  1   1=2   1
  1 1=2
V^kj fxx1=2 kj
wxkj fxx1=2 kj
fxx kj fxx kj
fxx ðxÞ

jjN j’ fxx1=2 ðxÞX

wxkl fxx1=2 ðkl ÞV^kl fxx1=2 ðxÞ þ fxx1=2 ðxÞlj N j’ fxx1=2 ðxÞXX

wkl kj kj

X

wxkl

f^xx1=2 ðkl Þ þ fxx1=2 ðkl Þ

¼

X X

1

fxx1=2 ðxÞ1 X

wkl kj kj fxx1=2 ðkl Þfxx1=2 ðxÞjjN jwxkl fxx1=2 ðkl Þlj N jþ fxx1=2 ðxÞf^xx1=2 ðkl Þ þ fxx1=2 ðkl ÞjjN jX
ljN j’ fxx1=2 ðxÞ

wxkl V^kl fxx1=2 ðkl Þfxx1=2 ðxÞljN jwxkl fxx1=2 ðkl Þlj N jþ fxx1=2 ðxÞXX

wkl kj kj

2fxx1=2 ðkl Þ

1

fxx1=2 ðxÞjjN jX

wxkl

2fxx1=2 ðkl ÞljN j1 X

wkl kj kj fxx1=2 ðkl Þfxx1=2 ðxÞjjN jwxkl wkl kj fxx1=2 ðxÞfxx1=2 ðkl Þkj fxx1=2 ðkl Þfxx1=2 ðxÞ:jjN j lj N jThen, V~ðxÞ is approximated by
X
X
 
 
wxkj fxx1=2 ðxÞfxx1=2 kj kj fxx1=2 kj fxx1=2 ðxÞ þ
wxkj kj
V~ðxÞ ’
jjN j

X X

jjN jwxkl wkl kj fxx1=2 ðxÞfxx1=2 ðkl Þkj fxx1=2 ðkl Þfxx1=2 ðxÞ:jj N j lj N j

Since kj ’ x and kl ’ x for a large T and the kj are asymptotically independent with zero
asymptotic mean, we have
0
12
X
X
 

  
TT@2wxkj 
wxkl wkl kj A var vec kj :var vec V~ðxÞ ’ð32Þ
2pM
2pM jjN jljN jUsing the standard result on the variance of the ﬁnite Fourier transform and kj ’ x fora large T givesÓ 2006 The Author
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468

M. HIRUKAWA  
var vec kj ’

fxx ðxÞ  fxx ðxÞTfxx ð0Þ  fxx ð0Þ

for x 6¼ 0:
for x ¼ 0

ð33ÞAlso by eqn (3),X  2pM
T X:
wxkl wkl kj ’
K ð M ðkl  xÞÞK M kj  kl
2pM ljN j
T
kl 2Bðkj Þ
Consider the change of variable tl ¼ M(kj  kl). Then, by M(kl  x) ¼ M(kj  x)  tl,X  2pMK ð M ðkl  xÞÞK M kj  kl
T
kl 2Bðkj Þ
X
 


¼
K M kj  x  tl K ðtl ÞDtltl 2ð1;1Þ
Z 1

 


K ðtÞK M kj  x  t dt
1
 

¼ K  K M kj  x :’Hence, again by the change of variable hj ¼ M(kj  x),
0
12
X
T X@
2wxkj wxkl wkl kj A
2pM jjN jlj N jX    2 2pM
2K M kj  x  K  K M kj  x
T
kj 2BðxÞ
X    2¼
2K hj  K  K hj Dhj’’¼

hj 2ð1;1Þ
Z 1

f2K ðhÞ  K  K ðhÞg2 dh

Z1
1
1

TK2 ðhÞdh:ð34Þ

Substituting eqns (33) and (34) into (32) yields Tvar vec V~ðxÞ ’
2pM

R1 2
TK ðhÞdhfxx ðxÞ  fxx ðxÞTR1
1
2
1 TK ðhÞdhðId 2 þ Kdd Þfxx ð0Þ  fxx ð0Þ

for x 6¼ 0;
for x ¼ 0

and thus the ﬁrst expression is established. Moreover, let tk(x) be the kernel corresponding
to the spectral window TK(h). Then,Z 1
TK ðhÞeixh dh;
tk ð xÞ ¼
1

and thus the second expression is established by Parseval’s relation.
Lemma 5.

Under Assumptions 1, 2 and 3, R ¼ op(M4 þ (T/M)1/2).

Ó 2006 The Author
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MODIFIED NPW COVARIANCE ESTIMATOR

469

Proof of Lemma 5. Applying the same method as in the proof of Lemma 2 and using
Lemma 1(a) and (b), it can be shown that each of the following six paired terms is op(M4):
^x
B

X

 
 
wxkj fxx1=2 kj kj fxx1=2 kj fxx1=2 ðxÞjjN j fxx1=2 ðxÞ
fxx1=2 ðxÞ

X

 
 
 
^ kj f 1=2 kj kj f 1=2 kj f 1=2 ðxÞ;wxkj fxx1=2 kj B
xx
xx
xx

jjN j  ^xwxkj fxx1=2 kj kj fxx1=2 kj B

jjN j fxx1=2 ðxÞ^x^2  B
B
x

X

X

X

 
 
 
^ kj fxx1=2 kj fxx1=2 ðxÞ;wxkj fxx1=2 kj kj fxx1=2 kj B

jjN j ^ kj f 1=2 kj f 1=2 ðxÞ;wxkj Bxx
xx

jj N jfxx1=2 ðxÞXX  2 1=2   1=2
 ^ f^ kj B^x;
wxkj fxx1=2 kj B
kj fxx ðxÞ  fxx1=2 ðxÞ
wxkj fxx1=2 kj B
kj xx

jjN jfxx1=2 ðxÞX

jjN jfxx1=2 ðxÞX

jjN jwxkj fxx1=2

X
 
 
 
^ kj fxx1=2 kj B
^ kj fxx1=2 ðxÞ  B^ kj fxx1=2 ðxÞ;^xkj B
wxkj fxx1=2 kj B
jj N j

X
 
 
 
^ kj fxx1=2 kj fxx1=2 ðxÞ  fxx1=2 ðxÞ^x:^ kj fxx1=2 kj B
^ kj fxx1=2 kj B
wxkj B
wxkj BjjN jjj N jIt is easy to see that all other 194 terms are either op(M4) or op((T/M)1/2) (although they
are not given explicitly). Therefore, the lemma is established.
uProof of Theorem 1. Theorem 4 of Chapter 12 in Magnus and Neudecker (1999; MN
hereafter), implies that
MSE1 f~xx ðxÞ; fxx ðxÞ
 
T
 

W ðxÞE vec f~xx ðxÞ  fxx ðxÞ¼ E vec f~xx ðxÞ  fxx ðxÞ þ tr W ðxÞvar vec f~xx ðxÞ  fxx ðxÞ :ð35Þ

The proof is split into two cases, depending on the frequency.
For x 6¼ 0: By Lemmata 3 and 5, the squared bias term is approximated byd
k24 vecðWðxÞÞT W ðxÞvecðWðxÞÞ
k4 X
¼ 28
wði1Þdþi ðxÞW2ii ðxÞ:
8
M
M i¼1

ð36Þ

Observe thatUðxÞ ¼ fxx1=2 ðxÞf 00 ðxÞfxx1=2 ðxÞ ¼ fxx1=2 ðxÞf 00 ðxÞfxx1=2 ðxÞis Hermitian (and positive deﬁnite), so isWðxÞ ¼ fxx1=2 ðxÞU00 ðxÞfxx1=2 ðxÞ ¼ fxx1=2 ðxÞU00 ðxÞfxx1=2 ðxÞ :Then, Wii(x) 2 R, 8i, and thus (ABias)2 2 Rþ.
Ó 2006 The Author
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JOURNAL OF TIME SERIES ANALYSIS Vol. 27, No. 3

470

M. HIRUKAWA

The reason why w(j1)dþi(x) ¼ 0, 8i 6¼ j must be imposed is demonstrated as follows.
Suppose that the restriction on weights is relaxed so thatwð j1Þdþi ðxÞ ¼ wði1Þdþj ðxÞ  0;

8i 6¼ j:Then, eqn (36) becomes
k24 vecðWðxÞÞT W ðxÞvecðWðxÞÞ
8
( M
)
d
d X
d
4
X
X
k
¼ 28
wði1Þdþi ðxÞW2ii ðxÞ þwð j1Þdþi ðxÞ W2ij ðxÞ þ W2ji ðxÞ :M
i¼1
i¼1 j¼1
Since Wji ðxÞ ¼ Wij ðxÞ, we can merely show that





W2ij ðxÞ þ W2ji ðxÞ ¼ 2 Re2 Wij ðxÞ  Im2 Wij ðxÞ 2 R:
As a result, the sign of (ABias)2 is not necessarily non-negative!
Lemmata 4 and 5 imply that the variance term is approximated byZ
M 1 2t ð xÞdx trW ðxÞ fxx ðxÞ  fxx ðxÞT :T 1 kð37Þ

The trace part is rewritten as
tr W ðxÞfxx ðxÞ  fxx ðxÞT ¼

dXwði1Þdþi ðxÞfii2 ðxÞ  0;

i¼1

because fxx(x) is Hermitian and thus fii(x) 2 R, 8i. Hence, AVar 2 Rþ is established.
For x ¼ 0: It is not hard to show that W(0) is real-valued. Then, by Lemmata 3 and 5,
the squared bias term is approximated byd X
dk24 vecðWð0ÞÞT W ð0ÞvecðWð0ÞÞ
k24 X¼
wð j1Þdþi ð0ÞW2ij ð0Þ  0;
M8
M 8 i¼1 j¼1

or (ABias)2 2 Rþ is established.
Lemmata 4 and 5 imply that the variance term is approximated byZ
M 1 2t ð xÞdx tr W ð0ÞðId 2 þ Kdd Þfxx ð0Þ  fxx ð0Þ
T 1 kZ 1M
t2 ð xÞdxftr W ð0Þfxx ð0Þ  fxx ð0Þ þ trW ð0ÞKdd fxx ð0Þ  fxx ð0Þg:¼
T 1 kð38Þ

ð39Þ

The ﬁrst trace part istrW ð0Þfxx ð0Þ  fxx ð0Þ ¼

d X
d
X

wð j1Þdþi ð0Þfii ð0Þfjj ð0Þ  0;

i¼1 j¼1

because fxx(0) is positive deﬁnite and thus fii (0) 2 Rþ, 8i. Since fxx (0) is symmetric, thispart can be rewritten as tr W(0)fxx(0)  fxx(0)T.
By the deﬁnition of Kdd and the expression 2-(4) of Chapter 2 in MN, the second trace
part isÓ 2006 The Author
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471

MODIFIED NPW COVARIANCE ESTIMATOR

tr W ð0ÞKdd fxx ð0Þ  fxx ð0Þ ¼ tr W ð0Þ

d X
d
X

ei eTj  ej eTi ðfxx ð0Þ  fxx ð0ÞÞ

i¼1 j¼1¼ tr W ð0Þ

d X
d
X



ei eTj fxx ð0Þ  ej eTi fxx ð0Þ

i¼1 j¼1

¼

d X
d
X

wð j1Þdþi ð0Þfij ð0Þfji ð0Þ

i¼1 j¼1

¼

d X
d
X

wð j1Þdþi ð0Þfij2 ð0Þ  0;

i¼1 j¼1

because fxx(0) is real-valued and symmetric. Hence, AVar 2 Rþ is established.
Substituting eqns (36), (37), (38) and (39) into (35) yields (12). Taking the ﬁrst-ordercondition to the right-hand-side gives the AMSE-optimal bandwidth M1opt ðxÞ, and thus thetheorem is established.
u

A.5.

PROOF OF THEOREM 2

By the expressions 4-(3)(4) of Chapter 2 in MN, it can be shown that xTAx ¼(x  x)Tvec(A) for a d-dimensional (possibly complex-valued) vector x and a d  d(possibly
matrix A. Using this and Theorem 4 of Chapter 12 in MN,
 complex-valued)
MSE2 f~xx ðxÞ; fxx ðxÞ is approximated by
MSE2 f~xx ðxÞ; fxx ðxÞ
n 
T

o
¼ E vec f~xx ðxÞ  fxx ðxÞ ðv  vÞðv  vÞT vec f~xx ðxÞ  fxx ðxÞ
 
T
 

¼ E vec f~xx ðxÞ  fxx ðxÞðv  vÞðv  vÞT E vec f~xx ðxÞ  fxx ðxÞ þ trðv  vÞðv  vÞT var vec f~xx ðxÞ  fxx ðxÞ


 2
 

¼ vT E f~xx ðxÞ  fxx ðxÞ v þtrðv  vÞT var vec f~xx ðxÞ ðv  vÞ:

ð40Þ

Bias Term: By Lemmata 3 and 5, the bias term is approximated byd X
dk 2 vT WðxÞv
k22 X
¼

vi vj Wij ðxÞ
 2
M4
M 4 i¼1 j¼1
(
)
d
d X
d
X


k22 X
2
¼ 4
vi Wii ðxÞ þvi vj Wij ðxÞ þ Wji ðxÞ :M
i¼1
i¼1 j¼iþ1

ð41Þ

Since W(x) is Hermitian, Wii(x) 2 R, 8i, and
Wij ðxÞ þ Wji ðxÞ ¼ Wij ðxÞ þ Wij ðxÞ ¼ 2Re Wij ðxÞ 2 R;

8i 6¼ j:Hence, ABias 2 R is established.
Variance Term for x 6¼ 0: Lemmata 4 and 5 imply that the variance term isapproximated by
Z
M 1 2
ð42Þ
t ð xÞdx trðv  vÞT fxx ðxÞ  fxx ðxÞT ðv  vÞ:T 1 kÓ 2006 The Author
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JOURNAL OF TIME SERIES ANALYSIS Vol. 27, No. 3

472

M. HIRUKAWA

By the expression 2-(4) of Chapter 2 in MN, the trace part istrðv  vÞT fxx ðxÞ  fxx ðxÞT ðv  vÞ ¼ ðv  vÞT fxx ðxÞ  fxx ðxÞT ðv  vÞ

¼ vT fxx ðxÞv vT fxx ðxÞT v :Since fxx(x) is Hermitian, so is fxx(x)T¼fxx(x). Then,vT fxx ðxÞv ¼d
X

v2i fii ðxÞ þ

d X
d
X

i¼1

¼

d
X



vi vj fij ðxÞ þ fji ðxÞ

i¼1 j¼iþ1

v2i fii ðxÞ þ 2

d X
d
X

i¼1



vi vj Re fij ðxÞ 2 R;

i¼1 j¼iþ1because fii (x) 2 R, 8i, and fji (x) ¼ fij(x), 8i 6¼ j. In addition,d
XvT fxx ðxÞT v ¼

v2i fii ðxÞ þ

i¼1
d
X

¼

d X
d
X

vi vj fij ðxÞ þ fji ðxÞ

i¼1 j¼iþ1

v2i fii ðxÞ þ 2

d X
d
X

i¼1



vi vj Re fij ðxÞ

i¼1 j¼iþ1

T¼ v fxx ðxÞv;because fii (x) 2 R, 8i, and fij(x) þ fji(x) ¼ fij(x) þ fji(x) ¼ 2Re(fij(x)), 8i 6¼ j. Then, eqn(42) is rewritten asZZ2 1 2M 1 2
M Ttk ð xÞdx trðv  vÞT fxx ðxÞ  fxx ðxÞT ðv  vÞ ¼
tk ð xÞdx; ð43Þv fxx ðxÞv
T 1
T
1
and thus AVar 2 Rþ is established.
Variance Term for x ¼ 0: By Lemmata 4 and 5, the variance term is approximated byZ
M 1 2t ð xÞdx trðv  vÞT ðId 2 þ Kdd Þfxx ð0Þ  fxx ð0Þðv  vÞT 1 k
Z
n
o
2
M 1 2
tk ð xÞdx vT fxx ð0Þv þtrðv  vÞT Kdd ðfxx ð0Þ  fxx ð0ÞÞðv  vÞ :
ð44Þ
¼
T 1
By the deﬁnition of Kdd and the expression 2-(4) of Chapter 2 in MN, the second trace partistrðv  vÞT Kdd ðfxx ð0Þ  fxx ð0ÞÞðv  vÞ
¼ ðv  v ÞT

d X
d
X

ei eTj  ej eTi ðfxx ð0Þ  fxx ð0ÞÞðv  vÞ

i¼1 j¼1

¼

d X
d
X



vT ei eTj fxx ð0Þv  vT ej eTi fxx ð0Þv

i¼1 j¼1

¼

d X
d
X

vT ei eTj fxx ð0Þv

 T Tv ej ei fxx ð0Þv

i¼1 j¼1

Ó 2006 The Author
Journal compilation Ó 2006 Blackwell Publishing Ltd
JOURNAL OF TIME SERIES ANALYSIS Vol. 27, No. 3

MODIFIED NPW COVARIANCE ESTIMATOR

¼

d X
d
X

vi

i¼1 j¼1!
vk fjk ð0Þ

k¼1

d X
d
X

¼

d
X

!

vi vl fil ð0Þ

i¼1 l¼1

vj

d
X!
vl fil ð0Þ

l¼1
d X
d
X

473

!

vj vk fjk ð0Þ

j¼1 k¼1

2¼ vT fxx ð0Þv  0:

ð45ÞHence, AVar 2 Rþ is established.
Substituting eqns (41), (43), (44) and (45) into (40) yields (13). The AMSEoptimal bandwidth M2opt ðxÞ follows the proof of Theorem 1, and thus the theorem isestablished.
h

A.6.

DERIVATION OF EQN (14)The standard result on the bias term of the spectral density estimator and eqn (10) yield M 4 E f€xx ðxÞ  fxx ðxÞ ’ tk;4 fxxð4Þ ðxÞ ¼ tk;4 fxx0000 ðxÞ;
ð46Þwhere tk,4 is the fourth generalized derivative of tk(Æ) at the origin. On the other hand,
Lemma in Stuetzle and Mittal (1979) implies that
Z 1
2
Z 1
4
2
h TK ðhÞdh ¼ 6h K ðhÞdh :ð47Þ
Using tk ð xÞ ¼

R1

1

1

ixh
1 TK ðhÞe dh and eqn (9),Z 1
h4 TK ðhÞdh ¼ tk0000 ð0Þ ¼ 24tk;4 :

ð48Þ

1

Substituting eqns (31) and (48) into eqn (47) establishes tk;4 ¼ k22 . Therefore, eqn (46) isrewritten as M 4 E f€xx ðxÞ  fxx ðxÞ ’ k22 fxx0000 ðxÞ;and thus eqn (14) follows Theorems 1 and 2.

A.7.

u

PROOF OF THEOREM 3

This theorem can be shown by induction. Lemmata 3, 4 and 5 have already shown the caseof n ¼ 1. Next, suppose that that eqn (17) is true for some n(1). Then, by a similar
technique to the derivation of eqn (11),f~xx;nþ1 ðxÞ ¼ fxx ðxÞ þ B~nþ1 ðxÞ þ V~nþ1 ðxÞ þ Rnþ1 ;
where
1

1=2
ðxÞ þ fxx1=2 ðxÞ
fxx1=2 ðxÞ
B~nþ1 ðxÞ ¼ B~n ðxÞ f~xx;nX    1=2    1 1=2
wxkj fxx1=2 kj B~n kj f~xx;nkj þ fxx1=2 kj
fxx ðxÞ
 fxx1=2 ðxÞjjN jÓ 2006 The Author
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474

M. HIRUKAWA
1

1=2
þ fxx1=2 ðxÞ f~xx;nðxÞ þ fxx1=2 ðxÞ
B~n ðxÞX   1   1=2   1=2
1=2
wxkj f~xx;nkj þ fxx1=2 kj
kj fxx ðxÞ;B~n kj fxx
 fxx1=2 ðxÞjjN jV~nþ1 ðxÞ ¼ fxx1=2 ðxÞ

X

 
 
wxkj fxx1=2 kj kj fxx1=2 kj fxx1=2 ðxÞjjN j1

1=2
1=2
f~xx;nðxÞ þ fxx1=2 ðxÞðxÞ þ fxx1=2 ðxÞ
V~n ðxÞ þ V~n ðxÞ f~xx;n
X
1
 
1=2
wxkj f~xx;nðxÞ þ fxx1=2 ðxÞ
V~n ðxÞfxx1=2 kj fxx1=2 ðxÞ fxx1=2 ðxÞþ fxx1=2 ðxÞ1

fxx1=2 ðxÞjjN j fxx1=2 ðxÞ

X

1
 
1=2
wxkj fxx1=2 kj V~n ðxÞ f~xx;nðxÞ þ fxx1=2 ðxÞ
fxx1=2 ðxÞ;jjN jand Rnþ1 is the remainder term.
    
To approximate the bias term, let Unþ1 ðxÞ ¼ fxx1=2 kj Wn kj fxx1=2 kj . Then, by theassumption of induction and the same argument as in the proof of Lemma 3, each of ﬁrst
two
and
ﬁnal
two
terms
in
M 2þ2ðnþ1Þ B~nþ1 ðxÞisapproximatedbyð1Þnþ1 k2nþ2 fxx1=2 ðxÞU00nþ1 ðxÞfxx1=2 ðxÞ. Letting Wnþ1 ðxÞ ¼ fxx1=2 ðxÞU00nþ1 ðxÞfxx1=2 ðxÞ establishes
the approximation to the bias term for the case of n þ 1. Also note that Wn(x) is Hermitianand positive deﬁnite, so is Unþ1(x) and thus Wnþ1(x).
On the other hand, applying the same argument as in the proof of Lemma 4, for large Tsuch that kj ’ x, we have
0 0
11
X
X
 

 
T
T
wxkj kj þ V~n ðxÞ 
wxkj V~n kj AA:var vec V~nþ1 ðxÞ ’var@vec@
2pM
2pM
jj N jjj N jThe kj are asymptotically independent with zero asymptotic mean, and by the standard
result on the variance of the ﬁnite Fourier transform (33) holds. In addition, V~n ðxÞ is theweighted sum of the kj, and by the assumption of induction, each weight can beapproximated by Kn(h)dh. Hence, Tvar vec V~nþ1 ðxÞ
2pM (R1
fK0 ðhÞ þ Kn ðhÞ  K0  Kn ðhÞg2 dhfxx ðxÞ  fxx ðxÞT’ R1
1
fK0 ðhÞ þ Kn ðhÞ  K0  Kn ðhÞg2 dhðId 2 þ Kdd Þfxx ð0Þ  fxx ð0Þ
( R1
1
2
Knþ1
ðhÞdhfxx ðxÞ  fxx ðxÞTfor x 6¼ 0
¼ R1
1
2
for x ¼ 0
1 Knþ1 ðhÞdhðId 2 þ Kdd Þfxx ð0Þ  fxx ð0Þ

for x 6¼ 0
for x ¼ 0

by letting Knþ1(h) ¼ K0(h) þ Kn(h)  K0 s Kn(h). Finally, using the Parseval’s relation

establishes the approximation to the variance term for the case of n þ 1.
Lastly, by the same argument as in the proof of Lemma 5, the remainder term is shown
to be Rnþ1 ¼ op(M(2þ2(nþ1)) þ (T/M)1/2). Therefore, the proof by induction iscompleted.
hÓ 2006 The Author
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JOURNAL OF TIME SERIES ANALYSIS Vol. 27, No. 3

MODIFIED NPW COVARIANCE ESTIMATOR

475

ACKNOWLEDGEMENTS

I would like to thank Bruce Hansen, Gordon Fisher, Oliver Linton, Zhijie Xiao,
and Victoria Zende-Walsh for constructive discussions and suggestions at earlier
stages of this research. I also thank an anonymous referee, Herman Bierens,
Chuan Goh, Carlos Martins-Filho, Taisuke Otsu, Joon Park, Shinichi Sakata,
Kenneth West, and participants at the 2005 MEG, CESG, and CIREQ Time
Series Conferences for helpful comments.

NOTES

1. For the reference to the subsequent argument, see Section 3.7 in Brillinger(1975).
2. We are not motivated to apply the MBC technique to the estimator underthe ﬁrst-order kernel such as the Bartlett kernel, because the resulting estimator merely attains the convergence rate that is usually attained under thesecond-order kernel.
3. As mentioned later, the general positive semideﬁnite weighting matrix W
works only for the case of zero frequency.
4. In the absence of serial dependence in fxtg, the optimal bandwidth becomes^ k ¼ 0; 8k and thus
any ﬁxed non-negative number. To see this, observe that B
B~ðxÞ ¼ 0. In other words, the bias-uncorrected estimator f^xx ðxÞ is alreadyunbiased, and thus no bias correction is required. Hence, only the varianceterm survives in each deﬁnition of the MSE. Therefore, for each ﬁxed
bandwidth M, the convergence rate of the MSE is O(T 1).
5. A disturbing issue on PR is zero estimates. The frequency of zero estimates depends on the size of the long-run variance for a given choice ofthe bandwidth. As far as the true long-run variance is relatively large,
zero estimates rarely (but not never) occur. However, the issue becomessevere when the truth is close to zero: in an extreme case (MA(2) with
(w1, w2) ¼ (1.9, 0.95)), nearly 43% of PR1 estimates are zeros!
Corresponding author: M. Hirukawa, Department of Economics, Concordia
University, 1455 de Maisonneuve Boulevard West, Montreal, Québec H3G
1M8, Canada. Tel.: (þ1) 514-848-2424 (Ext. 3921); Fax: (þ1) 514-848-4536;
E-mail:mhirukaw@alcor.concordia.ca;
Website:http://artsandscience.
concordia.ca/economics/hirukawa.html.

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