摂南経済研究
第１巻 第１・２号 (2011)，19－37ページ
Stabilizing a GMM Bootstrap for Time Series: A Simulation Study

研究論文

Stabilizing a GMM Bootstrap for Time Series:
A Simulation Study
Masayuki Hirukawa

Abstract
Inoue and Shintani (2006) demonstrate that in order for their GMM bootstrap to achieve
asymptotic refinements for symmetric two-sided confidence intervals and J-statistics ofoveridentifying restrictions, a kernel of order greater than two must be employed for HAC
estimation. A well-known problem of employing a higher-order kernel for HAC estimation is thatthe resulting covariance estimate does not necessarily become positive semi-definite in finite
samples, which often leads to unsatisfactory performance of the bootstrap. This paper proposes tostabilize the bootstrap through employing the nonparametric prewhitened HAC estimator by Xiao
and Linton (2002) and Hirukawa (2006), which has the same bias property as a fourth-order kernel
but always generates a positive semi-definite estimate in finite samples. Monte Carlo results
indicate that the HAC estimator indeed stabilizes the GMM bootstrap.

Keywords: asymptotic refinements; block bootstrap; covariance estimation; generalized method of
moments; Monte Carlo simulations; nonparametric prewhitening.
JEL classification numbers: C12; C22; C32.

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摂南経済研究 第 1 巻 第 1・2 号 (2011)

1

Introduction

Since the seminal work by Hansen (1982), the generalized method of moments (“GMM”) has been aworkhorse in the empirical analysis on macroeconomics and finance. However, a number of Monte
Carlo experiments indicate rather poor finite sample performance of GMM: examples include Tauchen(1986), Kocherlakota (1990), Ferson and Foerster (1994), the articles in the special issue of the Journalof Business & Economic Statistics (vol.14, July 1996), and Smith (1999), among others. As regards
finite sample performance of GMM-based inference, these articles find that empirical coveragefrequencies of confidence intervals are often fewer than the nominal level, and that the chi-squared
approximation to the J-statistic of overidentifying restrictions is poor.
There are two possible approaches to improve finite sample performance of the GMM-based inference.
One is to provide better approximations to finite sample distributions of test statistics of interest via
bootstrapping, and the other is to rely on alternatives to GMM, such as empirical likelihood (Kitamura,1997) and exponential tilting (Kitamura and Stutzer, 1997). This paper adopts the former approach.
More specifically, the paper aims at improving the bootstrap for two-step GMM estimation of
overidentified linear models1 proposed by Inoue and Shintani (2006; abbreviated as “IS” hereafter).
IS’s bootstrap is an extension of earlier work such as Hall and Horowitz (1996) and Andrews (2002a).
While all these articles apply the technique of the moving block bootstrap (“MBB”) for weakly
dependent data by Künsch (1989) to GMM estimation, IS differ from Hall and Horowitz (1996) andAndrews (2002a) in that IS leave the dependent structure of the moment function unspecified whenestablishing asymptotic refinements of their bootstrap. In order to deal with possibly infinite-order of
dependence in the moment function, IS use the inverse of a heteroskedasticity and autocorrelationconsistent (“HAC”) covariance estimator as the optimal weighting matrix for the efficient GMM
estimation. IS further point out that the order of magnitude in the leading bias term of the HAC estimatoris crucial to asymptotic refinements for the symmetric two-sided confidence interval and the J-statistic
provided by their bootstrap. Finally, they demonstrate that a kernel of order greater than two (i.e. ahigher-order kernel) must be employed in the HAC estimation for asymptotic refinements.
However, it is well-known that if we employ a higher-order kernel for HAC estimation, the resulting
covariance estimate is not necessarily positive semi-definite (“psd”) in finite samples. In fact, IS reportin their Monte Carlo simulations that even in the presence of positive serial dependence in the moment
function, with 4-7% chances the HAC estimate with the bandwidth chosen by their automatic procedure
fails to be psd. IS describe the problem of such non-psd HAC estimates as “a very troubling issue which

1
While exclusive attention is paid to linear models, they are of particular interest in empirical macroeconomics
and finance. Examples include consumption and labor demand (Rotemberg, 1984), the expectation hypothesis of theterm structure (Campbell and Shiller, 1991), inventory models (Fuhrer, Moore and Schuh, 1995; West and Wilcox,
1996), investment (Oliner, Rudebusch and Sichel, 1996), the monetary policy reaction function (Clarida, Galí andGertler, 2000), the permanent-income hypothesis (Runkle, 1991), and the present value model of stock prices (West,1988), to name a few.

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Stabilizing a GMM Bootstrap for Time Series: A Simulation Study

is difficult to resolve (Remark 4, p.539)”.
The problem of non-psd HAC estimates may be more crucial for the GMM bootstrap as it appears.2
While the theory refers to its large sample properties, the bootstrap is primarily intended to provide a
better approximation to the finite-sample distribution of a test statistic of interest. Indeed an arbitrarycorrection of a non-psd HAC estimate is admissible in an asymptotic sense (see endnote 4 in Andrews,
2002b, for example); however, the correction tends to deteriorate the quality of the GMM estimate,which will in turn affect the performance of the bootstrap testing adversely. As we see in Section 3, theproblem of non-psd HAC estimates is more pronounced in the presence of negative serial dependence inthe moment function.
Generating a psd estimate in finite samples is a highly desirable property of a HAC estimator, andseveral efforts have been made to derive such an estimator. Leading examples are two HAC estimators,
one by Newey and West (1987) for possibly infinite-order of dependence and the other by West (1997)
for finite-order dependence. One of the main reasons why HAC estimators with the Bartlett, Parzen, andQuadratic Spectral (“QS”) kernels are most popularly applied in empirical studies is that each of thesekernels always generates a psd estimate in finite samples.3 The nonparametric prewhitened (“NPW”)
HAC estimator by Xiao and Linton (2002) and Hirukawa (2006) has been developed in line with theseachievements. NPW is a subclass of multiplicative bias corrections for spectral estimation, andequivalent bias correction techniques have been already applied in nonparametric regression (Linton andNielsen, 1994), probability density estimation (Jones, Linton and Nielsen, 1995; Hirukawa, 2010a), andhazard estimation (Nielsen, 1998; Nielsen and Tanggaard, 2001). A remarkable feature of NPW is thatwhen a second-order spectral window is employed, the resulting HAC estimator attains the same orderof magnitude in the bias term as with fourth-order kernels while not inflating the order of the varianceterm. In this sense, NPW can be viewed as an operation that implicitly constructs a fourth-order spectral
window from a second-order one. However, unlike many other operations of constructing a higher-order
spectral window, NPW does not sacrifice positive semi-definiteness of the resulting spectral estimate forthe rate improvement. Employing a second-order spectral window guarantees the NPW-HAC estimatorto be psd in finite samples by construction.
Based on these attractive properties, the NPW-HAC estimator can be a remedy to what IS call the“troubling issue”. Then, as an important application of the HAC estimator this paper proposes to apply
its inverse as the optimal weighting matrix for the efficient GMM estimation. It is anticipated that
because the NPW-HAC estimator automatically generates a psd covariance estimate, it can deliver the
GMM estimate of better quality and stabilize the performance of IS’s bootstrap. As the initial step ofinvestigating IS’s bootstrap with the NPW-HAC estimator employed (called “the NPW-HAC-GMM
bootstrap” hereafter), this paper conducts a small Monte Carlo study to examine finite sample

2

The last paragraph on p.703 in Newey and West (1987) also describes drawbacks of non-psd HAC estimates.

3

Andrews (1991) categorizes these kernels as is “-2-class”.

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摂南経済研究 第 1 巻 第 1・2 号 (2011)

performance of the bootstrap.
The remainder of this paper is organized as follows. Section 2 describes the procedure of the
NPW-HAC-GMM bootstrap. Section 3 provides the Monte Carlo results with discussions. Finally,
Section 4 concludes the paper with possible research extensions.
This paper adopts the following notational conventions: 4x5 denotes the integer part of x; ○ signifies
convolution; vec (A) denotes the column by column vectorization function of matrix A;  is used torepresent the tensor (or Kronecker) product; * signifies conjugate transpose of a complex-valued matrix,
i.e. A* = Ā' ; and Ip denotes the p-dimensional identity matrix. Kdd is the d 2×d 2 commutation matrixddthat transforms vec (A) into vec (A' ), i.e. Kdd =Σi=1Σj =1 eie'j  eje'i , where ei is the ith elementary dvector. Lastly, the expression‚ ‘XT ~YT’ is used whenever XT /YT →1 as T →∞.

2

Combining the NPW-HAC Estimator with the GMM Bootstrap

2.1

Bootstrap Algorithm

For a stationary and ergodic time series {(x't , yt , z't )' }, consider an overidentified linear regression model
E (zt ut) : = E { zt ( yt －β'0 xt)} = 0,
p

p

(1)

m

where β0 ∈ is a parameter vector, xt ∈ , and zt ∈ ( p < m) is the vector of instruments that may
0possibly include predetermined values of yt. Suppose that given T0 observations {(x't , yt , z't )'}Tt=1,we areinterested in estimating β0 based on the moment condition (1) by the two-step GMM. For some m ×m psd
weighting matrix VT , the first-step GMM estimator is given by~

βT = arg min
β

T0

'

{Σ
1

T 0 t=1zt ( yt －β'xt)

T0

} {Σ
VT

1

T 0 t=1zt ( yt －β'xt)}.
~

Based on the cross product of instruments zt and the first-step GMM residual＾
ut := yt－β'T xt, we obtain＾T, which is described in the next section. Set the bandwidth M for the HAC
the NPW-HAC estimatorΩ
,estimator in the second-step GMM estimation equal to the block length ℓ for MBB so that M = ℓ = ∟T 2/9」
1＾－
and put T = T0－ ℓ + 1. We use T observations and employΩT as the optimal weighting matrix that is

intended for asymptotic refinements for the symmetric confidence interval and the distribution of the
J-statistic. Then, the second-step GMM estimator becomesT{Σ＾T = arg min 1
β
β

T t=1

zt ( yt －β'xt)

'
1
＾－
Ω
T

T

} {Σ

}

1
zt ( yt －β'xt) .
T t=1

The GMM bootstrap considers the distributions of the studentized statistic of a linear combination of
the parameter― 22 ―

Stabilizing a GMM Bootstrap for Time Series: A Simulation Study

(

＾T c
T c'Σ

－1/2

)

(

＾T－ β0c' β)

for an arbitrary nonzero vector c ∈p and{( Σ ） （ Σ )}
T

＾
ΣT =

T

1
x z'
T t=1 t t

－1

1
z x'
T t=1 t t

－1

＾T
Ω


,

and the J-statistic
T

JT =

T

'

{ Σ

}

} {Σ

1
＾' x ) .
z ( y －βT t=1 t t T t

1＾'T xt) ＾zt ( yt －β
ΩT－1
T t=1

The bootstrap applies the overlapping MBB by Künsch (1989). Suppose that we may write T = bℓ forsome integer b. The algorithm for bootstrap resampling takes the following five steps.
Step 1:

Let N1, N2, . . . , Nb be iid uniform random variables on {0, 1, . . . , T－ℓ } and let(x '( j－1)ℓ ＋i , y( j－1)ℓ ＋i , z('j－1)ℓ ＋i***)' = ( x'

N j ＋i , yN j ＋i , z'N j ＋i)'

for 1 @ i @ℓ and 1 @ j @ b.
Step 2:

Calculate the first-step bootstrap GMM estimator as
T

T

'

{Σ

} {Σ

1
βT = arg minz*t ( y*t －β' x*t )－ µ*T VT
T t=1
β
~*

1
z*t ( y*t －β' x*t )－ µ*T
T t=1

}

,

where
T－ℓ

ℓ

1
1
＾'T xt＋i ) .
µT = －－－－－－－－
zt i ( yt＋i －β
T－ℓ ＋1 t=0 ℓ i=1 ＋Σ Σ*

Step 3:

Compute the second-step bootstrap GMM estimator as
T

β

T t=1

T

'

{Σ

＾*T = arg min 1βz*t ( y*t －β' x*t )－ µ*T

} {Σ
＾*T－1
Ω

where

― 23 ―

1
z*t ( y*t －β' x*t )－ µ*T
T t=1

}

,

摂南経済研究 第 1 巻 第 1・2 号 (2011)
ℓ

b

ℓ 

1
( z* ＾u*u*N k ＋j － µ*T )',＾T =－ µ*T ) ( z*N k ＋j ＾ΩT k=1 i=1 j=1 N k ＋i N k ＋i

ΣΣΣ

*

~

＾
u*t = y*t－β*T' x*t.
Step 4:

Obtain the bootstrap version of the studentized statistic＾*T c )T ( c'Σ－1/2*＾T － β
＾T )
c' (β

for

{( Σ ） （ Σ )}
T

＾
Σ*T =

1
x* z*'
T t=1 t t

T

* －1

＾T
Ω

1
z* x*'
T t=1 t t

－1



and the J-statistic
T

*

JT =

Step 5:

{ Σ

'

T

} {Σ

1
＾*T' x*t )－ µ*T Ω＾*T－1
z*t ( y*t －β
T t=1

}

1
＾*T' x*t )－ µ*T .
z* ( y*－β
T t=1 t t

Repeat Steps 1-4 sufficiently many times to approximate the finite-sample distributions of thestudentized statistic and the J-statistic by the empirical distributions of their bootstrap version.
The entire procedure basically follows IS’s. In particular, Steps 1 and 2 play a key role in the MBB.
To preserve the dependent structure of the moment function, the MBB resamples ℓ consecutive observations
b times and construct a bootstrap sample by gluing them together. Each ℓ consecutive observations arechosen via the uniform weighting scheme. Because bootstrap resampling does not reproduce the overidentified
moment condition that holds for the data, recentering the moment condition by demeaning is required foreach bootstrap sample. Alternatively, applying the implied probabilities from the moment condition (e.g.
Brown and Newey, 2002; Allen, Gregory and Shimotsu, 2010) can get rid of recentering the moment
condition, but this approach is not considered in this paper.
There are two remarkable differences in the NPW-HAC-GMM bootstrap from IS’s, which are
underlined in the above description. First, the inverse of the NPW-HAC estimator is used as the optimal
weighting matrix to obtain the GMM estimate. As discussed in the next section, the NPW-HAC estimatoris by construction psd and has the same order of magnitude in the bias term as the HAC estimator with afourth-order kernel. Because of its positive semi-definiteness the NPW-HAC estimator is expected toyield the GMM estimate in good quality, which is a key element for better performance of the bootstrap.
Second, while the bandwidth M is set equal to the block length ℓ as in Götze and Künsch (1996) andIS, the divergence rate of M and ℓ is O (T 2/9), which is slower than O (T 1/3) in IS’s framework. It canbe conjectured from IS’s technical discussions and properties of the NPW-HAC estimator that setting― 24 ―

Stabilizing a GMM Bootstrap for Time Series: A Simulation StudyM = O (T δ ) for some δ ∈ (2/9, 1/4) and ℓ = O (M ) will deliver asymptotic refinements for the symmetricconfidence interval and the distribution of the J-statistic.4 A rigorous proof is left for future work.
2.2

The NPW-HAC Estimator

The NPW-HAC estimator is originally proposed by Xiao and Linton (2002) and modified by Hirukawa(2006). The idea behind this estimator is to apply a multiplicative bias correction in the frequency domain
to a HAC estimator so that the bias can be accelerated by an order of magnitude while the order of thevariance remains unchanged. As a result, the NPW-HAC estimator is guaranteed to be psd in finite samples
when a second-order kernel is employed, and at the same time it has the same order of magnitude in thebias term as with fourth-order kernels.
The NPW-HAC estimator is formally defined as follows. Let vt := z t u t so that the moment condition(1) is rewritten as E (vt) = 0. Also let fundamental frequencies be λ j = 2π j/T, j = 0,±1, . . . ,± ∟ T /2」,and the periodogram at the frequency λ j be Ivv (λ j) = ζv (λ j) ζv (λ j)*, where ζv (λ j) = (2πT )－1/2ΣTt=1 vt e－ it λ j
is the finite Fourier transform of {vt} evaluated at the frequency λj. Given a kernel K(･), the bandwidthM, and the spectral window corresponding to this kernel W (θ ) = (2π)－1∫－∞∞ K (u) e－iuθd u, define the5amplitude window (Parzen, 1963) as WM (θ ) = MΣ∞j=－∞ W{M (θ + 2π j)}. Then, the (bias-uncorrected)

HAC estimator of {vt} can be represented in two ways as a weighted autocovariance and a smoothed
periodogram
~



T－1

ST =

Σ K ( M ) Γ ～ Tπ Σ W (λ ) I (λ ) ,
l

4 2

~

l

λ j∈B(0)l=－(T－1)

j

M

vv

j

~
min{T+l,T}
where Γl = T －1Σt=max{1,1＋l }vtv't－l is the lth sample autocovariance of {vt} and B (0) = {λ j|－ π < λ j < π} isthe frequency band with width 2π centered at zero frequency.
Similarly, for B (ω ) = {λ j|ω － π < λ j < ω ＋π}, the frequency band with width 2π centered at the frequency

ω , the spectral matrix of {vt} evaluated at the frequency ω ∈ (－ π , π ) can be estimated as

1
f vv (ω ) = 2π
~

T－1



Σ K ( M )Γe
l

~

l

－ilω

～

l=－(T－1)2π
TΣ W (λ －ω ) I (λ ) .

λ j∈B(ω )

M

j

vv

j

~

Also define the “square-root” of the spectral matrix f vv (ω ) as the one obtained by the unitary decomposition
4
Strictly speaking, the divergence rate of M and ℓ needs to be slightly faster than O (T 2/9 ). However, it appears
that using M = O (T 2/9 ) is not problematic in practice, because, for example, M = 4T 2/95and M = 4T 201/9005would
yield the same bandwidth for the sample sizes considered in the Monte Carlo study. Although deriving the data-based
choice rules for M and ℓ that can establish asymptotic refinements is an interesting challenge, it is beyond the scopeof this paper.
5
As described in Parzen (1963), WM (θ) ~ MW (Mθ). In particular, the approximation is replaced by the equality

when the spectral window W (θ) is band-limited such that W (θ) = 0 for |θ | ≥ π.

― 25 ―

摂南経済研究 第 1 巻 第 1・2 号 (2011)~ 1/ 2

1/ 2

f vv (ω ) = UΛ U * ,
1/2

1 /2

whereΛ1/2 = diag (λ1 , . . . , λ m ) is the diagonal matrix containing the square roots of the eigen-values of~f vv (ω ), and U is the unitary matrix, i.e. U U * = Im.6 Finally, the NPW-HAC estimator is defined as~~ 1/ 2 ~~ 1/ 2~ 1/ 2

ΩT =S T α (0 ) S T = S T{}

Σ W (λ ) 2Tπ f~ －1 / 2~ －1 / 2v v (λ j) Iv v (λ j) f v v (λ j)jM

λ j∈B(0)~ 1/ 2ST ,~＾T in the previous section is obtained by replacing
where α (0) serves as the bias correction term. Note that Ω~~'
vt = zt＾
ut = zt ( yt －βT xt), where βT is the first-step GMM estimator.
vt with＾The name of “NPW” comes from the fact that the bias correction technique reminds us of thewell-known prewhitening procedure (see Andrews and Monahan, 1992, for example), because the~~transformed periodograms f －v 1v/ 2 (λ j) Iv v (λ j) f －v 1v/ 2 (λ j) are approximately constant. Also observe that when~

W (θ) ≥ 0,∀θ∈, the NPW-HAC estimator ΩT is psd in finite samples because of its “sandwich form”
structure.

~

Hirukawa (2006) further establishes that under the standard regularity conditions ΩT has an asymptotic
expansion~

ΩT = Ω0＋＋v＋op

(

M

－4

＋

)

M
T

.

(2)

where

1
E
T→∞ T

Ω0 = lim '{(Σ ）（Σ ） }
T

t=1

T

vt

t=1

vt

~

is the long-run variance of {vt} , and  and & constitute the leading bias and variance terms of ΩT. The
leading bias term  is approximated by2

 ～ － K2 ΨM

－4

,

1/2
1/2
where K2 = lim x →0 {1 － K (x)} / | x |2 ∈ (0, ∞), Ψ =Ω0 Ф'' (0)Ω0 , Ф'' (0) = d 2Ф (ω ) / dω 2| ω =0 ,Ф(ω ) = f －v 1v/ 2 (ω ) f v''v (ω ) f －v 1v/ 2 (ω ), and f 'v'v (ω ) = d 2 f vv (ω ) / dω 2. Moreover, the leading variance termv is approximated by

Var {vec (v)} ～MT∫ t (x) dx (I ＋K )Ω Ω ,
∞

2

－∞ k

m2

mm

0

0

where t K (x) =∫－∞∞ TW (θ ) eixθd θ and TW (θ ) = 2W (θ ) － W ○ W (θ ) is the fourth-order spectral window
obtained by twicing (Stuetzle and Mittal, 1979). Observe that the bias term is of order M －4, which is
6~~

Indeed f 1v/v2 (ω ) is well-defined. The Ivv (λ j) are Hermitian and psd by construction, so is f vv (ω ) when W (θ) ≥ 0,~∀θ∈ ℝ. Hence, f vv (ω ) has positive eigenvalues λ 1, . . . , λ m. For the reference to the related argument, see Section 3.7
in Brillinger (1975).

― 26 ―

Stabilizing a GMM Bootstrap for Time Series: A Simulation Study

usually achieved under the fourth-order spectral window, whereas the order of magnitude of the varianceterm is not inflated, i.e. usual O (M/T ) is maintained.

3

Monte Carlo Experiment

3.1

Setup

This section conducts a small Monte Carlo simulation study to investigate the reliability of the
NPW-HAC-GMM bootstrap in finite samples. Following IS, consider the two-step GMM estimation of asimple linear regressionyt =β1 +β2 xt + ut .
Without loss of generality the true parameter value is set equal toβ= (β1,β2 )' = (0, 0)'. The regressor{xt} and the error term {ut} are generated independently by AR(1) and either AR(1) or MA(1) processes.
xt =ρ xt－1 + ε1t, ρ ∈ {0.5, 0.8}.

ut =AR(1) .
{φε u+ ψε+ ε ,,φψ==±ρ±ρ :: MA(1)εt(ε1t, ε2t)' ～ N (0, I2 ) .

=

t－1

2t

2t
2t－1
iid

Observe that the AR or MA coefficient in the error term has the same magnitude (although the sign may
be opposite) as the AR coefficient in the regressor. Because predetermined regressors are frequently used
as instruments in the presence of endogeneity, this experiment adopts zt = (1, xt－1, x t－2)' as the vector of
instruments. Hence, the moment condition model is overidentified with 1 degree of freedom for the J-test.
In the simulations, 5000 random samples are generated for two sample sizes (T ), namely, 64 and 128.
Our primary interest is to evaluate finite sample null rejection frequencies of the t-ratio ofβ2 (that is
intended for the two-sided test) and the J-statistic. Because there is parameter uncertainty due to thefirst-step estimation and we apply a univariate testing procedure to each element of the moment function,
it is hard to control the size of this procedure. Therefore, this Monte Carlo experiment concentrates on
the critical values for the nominal 10% level to be conservative. Furthermore, the bootstrap critical values
are calculated based on 499 replications for each sample.
The first-step GMM estimation uses the 3 × 3 identity matrix as the initial weighting matrix. In thesecond-step GMM estimation, the following HAC estimators and bandwidths (and block lengths,
whenever applicable) are employed. First, the NPW-HAC-GMM bootstrap applies NPW-HAC estimatorswith Parzen and Bohman (Bohman, 1961) kernels.7 The bandwidth M and the block length ℓ are chosen
7

The Bohman kernel is expressed as
K (x) ＝≤1.
{ (1－|x|) cos (π x)0 + sin (π|x|) /π ifif |x||x| > 1

― 27 ―

摂南経済研究 第 1 巻 第 1・2 号 (2011)

as M = ℓ = ∟ T 2/9」. Observe that M = ℓ = 2 for each sample size.
Second, for IS’s bootstrap, HAC estimators are computed using the truncated kernel, the trapezoidal
kernel (Politis and Romano, 1995) with parameter 1/2, and the Parzen (b) kernel (Parzen, 1957) of order
3. 8 Again M is set equal to ℓ , and ℓ is chosen by IS’s automatic procedure similar to thegeneral-to-specific modelling strategy: see p.540 in IS for details. Each of the three kernels does notnecessarily generate psd estimates in finite samples. When the HAC estimate is not psd, the block lengthchosen automatically is shortened until the resulting covariance estimate becomes psd.
Third, besides the bootstrap results, the first-order asymptotic results based on widely applied HAC
estimators are also obtained. For this purpose, two most popular automatic bandwidth choice rules byAndrews (1991) and Newey and West (1994) are employed for the QS and Bartlett kernels, respectively.
For the former, AR(1) reference is used. For the latter, the bandwidth for the normalized curvature(Hirukawa, 2010b, p.713) estimator using the truncated kernel is set equal to ∟ 4(T/100)2/9」. In addition,
recently Hirukawa (2010b) has proposed yet another bandwidth choice rule called the solve-the-equationplug-in rule. Bartlett and Parzen HAC estimators using this rule are also calculated. Moreover, because
prewhitening (Andrews and Monahan, 1992) is popularly applied in empirical studies, each of these
HAC estimators is computed for the prewhitened series as well as the original (= non-prewhitened)
series. The procedure of prewhitening follows Andrews and Monahan (1992) with the eigenvalues of thefitted VAR(1) coefficient matrix adjusted to being less than 0.97 in magnitude.

──────────────────────────────────────────────────The spectral windows corresponding to the Parzen and Bohman kernels are W (θ ) = 24 (1 － cos (θ / 2))2 / (πθ 4) andW (θ ) = 2 π (1 + cos (θ )) / (π 2 －θ 2)2, respectively.
8
The trapezoidal kernel and the Parzen (b) kernel of order 3 are expressed as
K (x) ＝{1

if |x| ≤ 1/2

2(1－|x|)0

if 1/2 < |x| ≤ 1 ,if |x| > 1

and3

K (x) ＝

if |x| ≤ 1 ,{ 1－|x|0
if |x| > 1

respectively.
The spectral windows corresponding to the Parzen and Bohman kernels are W (θ ) = 24 (1 － cos (θ / 2))2 / (πθ 4) andW (θ ) = 2 π (1 + cos (θ )) / (π 2 －θ 2)2, respectively.
8
The trapezoidal kernel and the Parzen (b) kernel of order 3 are expressed as
K (x) ＝{1

if |x| ≤ 1/2

2(1－|x|)0

if 1/2 < |x| ≤ 1 ,if |x| > 1

and3

K (x) ＝

if |x| ≤ 1 ,{ 1－|x|0
if |x| > 1

respectively.

― 28 ―

Stabilizing a GMM Bootstrap for Time Series: A Simulation Study

Table 1: Finite Sample Null Rejection Frequencies against Nominal 10% Level for AR(1) Error

Panel (a): t -ratio
Kernel(1) Bootstrap:
Truncated(0.8, -0.8)
64
128

16.4
23.8
(60.5) (51.3)
Trapezoidal
27.7
34.1
(16.3) (5.6)
Parzen (b)
25.3
36.6
(40.0) (23.1)
Parzen-NPW
8.3
8.8
Bohman-NPW
12.0
13.7
(2) First-Order Asymptotic:
Non-Prewhitened:
QS-A
7.5
8.5
Bartlett-NW
19.4
9.4
Bartlett-H
13.3
12.2
Parzen-H
14.5
12.4
Prewhitened:
QS-A
13.7
12.6
Bartlett-NW
18.6
15.0
Bartlett-H
13.4
12.1
Parzen-H
13.6
11.7(0.5, -0.5)
64
128(0.5, 0.5)
64
128

(%)
(0.8, 0.8)
64
128

21.0
(18.0)
21.3(15.6)
19.7
(7.8)
7.3
8.5

25.9
(8.1)
24.2
(5.0)
21.4
(2.0)
8.0
8.9

15.1
(4.1)
15.0
(2.7)
14.8
(2.5)
13.1
13.0

13.9
(3.7)
13.4
(1.7)
13.5
(1.4)
13.2
12.6

18.2
(4.1)
17.7
(1.9)
17.9
(1.6)
26.2
24.8

9.2
(3.1)
8.4
(1.0)
8.6
(0.9)
24.3
22.5

6.4
13.8
9.2
10.6

6.9
12.0
9.1
10.3

17.4
19.6
18.0
18.2

14.7
16.5
15.8
15.3

29.6
31.5
31.2
33.9

21.8
23.1
23.0
24.1

10.3
14.7
10.7
10.1

9.9
12.8
9.9
9.6

17.0
20.0
17.0
17.3

13.7
15.8
13.7
13.8

25.0
26.1
25.2
25.1

18.3
19.1
18.6
18.2

Panel (b): J -statistic
Kernel(1) Bootstrap:
Truncated(0.8, -0.8)
64
128

10.5
11.0
(60.5) (51.3)
Trapezoidal
12.7
11.7
(16.3) (5.6)
Parzen (b)
11.4
11.2
(40.0) (23.1)
Parzen-NPW
9.0
10.2
Bohman-NPW
8.910.2(2) First-Order Asymptotic:
Non-Prewhitened:
QS-A
11.5
10.7
Bartlett-NW
5.5
9.4
Bartlett-H
8.0
9.7
Parzen-H
11.9
10.8
Prewhitened:
QS-A
12.7
11.0
Bartlett-NW
12.7
11.2
Bartlett-H
12.6
11.3
Parzen-H
12.7
11.2(0.5, -0.5)
64
128(0.5, 0.5)
64
128

(%)
(0.8, 0.8)
64
128

10.8
(18.0)
10.8(15.6)
10.6
(7.8)
9.1
9.1

11.8
(8.1)
11.9
(5.0)
11.8
(2.0)
10.3
10.6

9.6
(4.1)
9.5
(2.7)
9.3
(2.5)
9.1
9.0

10.8
(3.7)
10.6
(1.7)
10.5
(1.4)
9.9
9.7

11.6
(4.1)
11.1
(1.9)
11.0
(1.6)
10.9
10.5

12.1
(3.1)
11.7
(1.0)
11.3
(0.9)
11.7
11.8

10.5
8.4
9.9
10.7

10.9
10.9
11.0
11.2

16.0
14.1
13.8
15.2

16.9
15.9
15.2
16.7

23.7
18.6
18.3
19.9

23.4
19.5
19.3
21.5

11.4
11.5
11.5
11.4

11.2
11.3
11.0
10.9

19.3
19.3
19.6
19.4

19.6
19.5
19.9
19.7

27.3
27.5
27.4
27.2

25.6
25.8
25.7
25.8

psd corrections.
Note: Numbers in parentheses are frequencies of positive
semi-definite corrections.

― 29 ―

摂南経済研究 第 1 巻 第 1・2 号 (2011)

Table 2: Finite Sample Null Rejection Frequencies against Nominal 10% Level for MA(1) Error

Panel (a): t -ratio
Kernel(1) Bootstrap:
Truncated(0.8, -0.8)
64
128

15.4
18.2
(30.2) (26.6)
Trapezoidal
15.9
19.5
(29.8) (24.7)
Parzen (b)
15.1
17.4
(15.7) (9.1)
Parzen-NPW
2.7
2.1
Bohman-NPW
3.1
2.6(2) First-Order Asymptotic:
Non-Prewhitened:
QS-A
2.5
2.2
Bartlett-NW
14.4
13.2
Bartlett-H
7.5
6.8
Parzen-H
9.2
8.8
Prewhitened:
QS-A
3.7
2.5
Bartlett-NW
14.6
13.1
Bartlett-H
6.3
5.7
Parzen-H
4.3
2.3(0.5, -0.5)
64
128(0.5, 0.5)
64
128

(%)
(0.8, 0.8)
64
128

17.3
(10.6)
17.6
(9.3)
15.3
(4.7)
5.7
6.4

19.4
(5.8)
19.1
(3.2)
15.9
(1.7)
6.2
6.7

15.1
(4.6)
15.4
(3.0)
14.9
(2.7)
11.7
11.6

13.9
(3.9)
13.6
(1.6)
13.5
(1.4)
11.8
11.3

14.2
(4.4)
13.8
(2.5)
13.5
(2.2)
12.7
11.8

11.4
(4.0)
11.3
(1.7)
11.4
(1.5)
11.1
10.2

5.4
13.1
7.4
8.2

5.2
11.1
7.6
8.0

15.5
18.2
15.4
16.5

13.6
15.7
14.3
14.5

19.3
22.9
19.6
24.3

15.7
18.4
15.6
17.6

8.0
13.4
8.2
7.8

6.6
11.1
6.9
6.4

15.4
18.9
15.6
16.1

12.9
15.4
13.1
13.5

15.2
21.6
16.0
18.7

11.5
16.4
11.8
14.0

Panel (b): J -statistic
Kernel(1) Bootstrap:
Truncated(0.8, -0.8)
64
128

9.8
11.0
(30.2) (26.6)
Trapezoidal
10.0
11.0
(29.8) (24.7)
Parzen (b)9.7
11.0(15.7) (9.1)
Parzen-NPW
8.7
10.5
Bohman-NPW
8.910.7(2) First-Order Asymptotic:
Non-Prewhitened:
QS-A
11.0
11.4
Bartlett-NW
6.6
8.0
Bartlett-H
9.5
10.6
Parzen-H
12.1
12.1
Prewhitened:
QS-A
11.7
11.5
Bartlett-NW
10.9
11.1
Bartlett-H
11.3
11.3
Parzen-H
11.4
11.3(0.5, -0.5)
64
128(0.5, 0.5)
64
128

(%)
(0.8, 0.8)
64
128

10.4
(10.6)
10.5
(9.3)
10.5
(4.7)
9.1
9.3

11.8
(5.8)
11.9
(3.2)
11.9
(1.7)
10.4
10.3

9.7
(4.6)
9.7
(3.0)
9.7
(2.7)
8.9
8.6

10.7
(3.9)
10.9
(1.6)
10.7
(1.4)
10.1
9.9

10.4
(4.4)
10.4
(2.5)
10.3
(2.2)
9.8
9.9

11.2
(4.0)
10.9
(1.7)
10.8
(1.5)
10.4
10.5

10.5
9.2
10.4
10.9

11.0
10.9
11.3
11.4

15.0
14.2
13.7
14.6

15.2
14.4
13.7
14.7

16.8
14.6
15.0
15.6

15.2
14.2
14.2
14.8

11.6
11.6
11.6
11.6

11.2
11.6
11.1
11.0

17.1
16.8
18.0
17.9

16.9
16.6
17.4
17.2

18.4
17.0
19.7
18.7

16.2
15.2
17.2
16.0

psd corrections.
Note: Numbers in parentheses are frequencies of positive
semi-definite corrections.

― 30 ―

Stabilizing a GMM Bootstrap for Time Series: A Simulation Study

Figure 1: Shapes of the Parzen and Bohman Kernels

3.2

Results

Tables 1 and 2 report finite sample null rejection frequencies of the t-ratio and the J-statistic for the caseswith AR(1) and MA(1) errors, respectively. Numbers in parentheses for the truncated, trapezoidal andParzen (b) kernels are frequencies of psd corrections. In addition, the automatic bandwidth choiceprocedures by Andrews (1991), Newey and West (1994) and Hirukawa (2010b) are denoted as A, NW
and H, respectively.
We start looking into the results for AR(1) errors. Panel (a) in Table 1 presents null rejection frequenciesof the t-ratio. At a first glance we can recognize that there is little difference in performance between two
kernel choices for the NPW-HAC estimation. This is not surprising, because their shapes look alike (seeFigure 1) and exactly the same block length (and thus exactly the same bandwidth) is used. We now closely
look at the results for each parameter setting. In the presence of negative serial dependence in the moment
function (i.e. whenφ = －0.8, －0.5), all three versions of IS’s bootstrap suffer from considerable
frequencies of psd corrections. The non-psd covariance estimates deteriorate the quality of the GMM
estimates and lead to large null rejection frequencies. Moreover, contrary to what the large sample theory
predicts, the null rejection frequencies deviate from the nominal level as the sample size increases. Since
the NPW-HAC estimator is free of the non-psd problem, both versions of the NPW-HAC-GMM
bootstrap exhibit better size properties than IS’s. Whenφ= 0.5, the NPW-HAC-GMM bootstrap still
maintains better performance, but the margin gets thinner due to infrequent psd corrections in IS’sbootstrap. Whenφ= 0.8, IS’s bootstrap at last outperforms the NPW-HAC-GMM bootstrap in a decisive
manner. However, the unsatisfactory performance of the latter is attributed mainly to a short block length:― 31 ―

摂南経済研究 第 1 巻 第 1・2 号 (2011)

see below for additional analysis. In addition, first-order asymptotic-based inference whenφ = －0.8,－0.5 is in general satisfactory. Often null rejection frequencies are quite close to the nominal level,
regardless of whether prewhitened or non-prewhitened series is chosen. On the other hand, the results
based on first-order asymptotics get worse asφ becomes positive and larger. Even prewhitening does
not resolve the issue of size distortion substantially.
Panel (b) in Table 1 displays null rejection frequencies of the J-statistic. We can find a clear advantage
of bootstraps over first-order asymptotics. The null rejection frequencies vary very little acrossbootstraps, and they are quite close to the nominal level. While first-order asymptotics perform equallywell for negativeφ’s, they tend to reject the null too often for positiveφ’s. This tendency is pronounced
when the moment function becomes persistent.
We now examine the results for MA(1) errors given in Table 2. The general tendencies shown in Panels(a) and (b) are basically the same as those for AR(1) errors. A notable difference is that whenψ = －0.8,－0.5, t-ratios from both versions of the NPW-HAC-GMM bootstrap (as well as those based on two
versions of the QS-HAC estimator) tend to reject the null too infrequently. In other words, the bootstrap
quantiles tend to inflate in this situation, but it seems that this issue is not an outcome from the quantile
breakdown by outliers (see Camponovo, Scaillet and Trojani, 2010, for example) but again due to a short
block length: see below for details.
Table 3: Finite Sample Null Rejection Frequencies of the NPW-HAC-GMM Bootstrap for MA(1)Error with (ρ, ψ) = (0.8, －0.8) , (0.5, －0.5) When M = ℓ = 3

Kernel(0.8, -0.8)
64
128

(%)
(0.5, -0.5)
64
128

9.6
11.2

8.4
10.5

11.6
12.5

11.1
12.5

10.1
9.9

11.0
11.3

10.4
10.3

11.2
11.8

t -ratio
Bootstrap:
Parzen-NPW
Bohman-NPW

J -statistic
Bootstrap:
Parzen-NPW
Bohman-NPW― 32 ―

Stabilizing a GMM Bootstrap for Time Series: A Simulation Study

Figure 2: A Plot of Null Rejection Frequencies against Block Lengths(T0 + 1 = 128; AR(1) Error with (ρ,φ) = (0.8, 0.8) )Tables 1 and 2 reveal that the NPW-HAC-GMM bootstrap performs poorly in the following two
scenarios: (i) for the MA(1) error with (ρ,ψ) = (0.8, －0.8), (0.8, －0.5); and (ii) for the AR(1) error with
(ρ,φ) = (0.8, 0.8). In each scenario only the t-ratio is affected adversely. Invoke that the block length ischosen in a non-data-dependent manner; to be more precise, it is fixed at 2 regardless of the sample size.
Therefore, we reasonably anticipate that the issue of poor performance can be resolved once a longer
block length is chosen. As regards the scenario (i), the numbers in Table 3 are obtained by using thesame bootstrap samples but setting M = ℓ = 3. Indeed the block length makes null rejection frequenciesof the t-ratio close to the nominal level, while maintaining the highly satisfactory performance of the
J-statistic.
Next, the scenario (ii) corresponds to the case of a persistent moment function. For such a persistentprocess, a large value of the block length is expected to be a remedy. Figure 2 plots null rejection
frequencies of the t-ratio and the J-statistic based on the NPW-HAC-GMM bootstrap for the sample size
128. Again the numbers are obtained by using the same bootstrap samples and changing only the valueof M and ℓ . We can see that the performance of the t-ratio is highly sensitive to the block length,
whereas that of the J-statistic is insensitive. Judging from the performance of the t-ratio, setting M = ℓ =4 appears to be the best for sample size 128.
Overall, the NPW-HAC-GMM bootstrap performs well, and it often outperforms IS’s bootstrap andfirst-order asymptotics; in other words, the NPW-HAC estimator contributes to stabilizing IS’s bootstrap.
In particular, the good performance for MA-type dependence is encouraging, because linear rational― 33 ―

摂南経済研究 第 1 巻 第 1・2 号 (2011)

expectations models often exhibit MA-type dependence in the moment function. On the other hand,
there is a pressing need for a data-based choice rule of the block length. Because the performance of the
J-statistic is insensitive to the block length, a possible strategy of deriving the choice rule would bebased on the coverage accuracy of the symmetric two-sided confidence interval.

4

Concluding Remarks

To stabilize IS’s bootstrap, this paper has proposed to employ the inverse of the NPW-HAC estimator as
the optimal weighting matrix for the second-step GMM estimation. Monte Carlo results indicate that the
HAC estimator can indeed stabilize IS’s bootstrap and improve its finite sample performance, in particular,
in the presence of either negative AR-type dependence or MA-type dependence in the moment function.
There are several important research extensions. First, the asymptotic theory for the
NPW-HAC-GMM bootstrap, including the condition for divergence rates of the bandwidth and the block
length that can provide asymptotic refinements, should be established. While IS’s asymptotic results canbe carried through in principle, an important modification should be made. IS’s theory is built on HAC
estimators of a weighted autocovariance form. Since the NPW-HAC estimator takes a more complicatedform, relevant parts of IS’s proofs should be suitably overwritten. The Edgeworth expansion of thet-ratio in the frequency domain by Velasco and Robinson (2001) may be a nice hint for the modification.
Second, an equal priority is given to a data-dependent choice rule of the bandwidth and the block length
in the framework of the NPW-HAC-GMM bootstrap. Although there are a number of block length choicerules proposed, it is important to see whether one of these rules is readily applicable to the bootstrap or a
new rule must be tailor-made.
Third, it is highly motivated to investigate finite sample performance of the bootstrap in a more
realistic framework. In particular, since West and Wilcox (1996) report that test statistics implied by theirinventory model are often poorly sized, it is of interest to see how much the NPW-HAC-GMM bootstrap
can contribute to size corrections. Furthermore, as an empirical example, the bootstrap procedure can beapplied to the monetary policy reaction function by Clarida, Galí and Gertler (2000), as well as theinventory model. Because the GMM estimate of a structural econometric model often has a policy
implication, it is essential to construct more reliable confidence intervals for key structural parameters.
All these extensions are currently under the author’s investigation, and they will be addressed in a separate
paper.

― 34 ―

Stabilizing a GMM Bootstrap for Time Series: A Simulation Study

References
Allen, J., A. W. Gregory, and K. Shimotsu (2010): “Empirical Likelihood Block Bootstrapping,”
Journal of Econometrics, forthcoming.
Andrews, D. W. K. (1991): “Heteroskedasticity and Autocorrelation Consistent Covariance Matrix
Estimation,” Econometrica, 59, 817 - 858.
Andrews, D. W. K. (2002a): “Higher-Order Improvements of a Computationally Attractive k-Step
Bootstrap for Extremum Estimators,” Econometrica, 70, 119 - 162.
Andrews, D. W. K. (2002b): “Equivalence of the Higher Order Asymptotic Efficiency of k-Step andExtremum Statistics,” Econometric Theory, 18, 1040 - 1085.
Andrews, D. W. K., and J. C. Monahan (1992): “An Improved Heteroskedasticity and Autocorrelation
Consistent Covariance Matrix Estimator,” Econometrica, 60, 953 - 966.
Bohman, H. (1961): “Approximate Fourier Analysis of Distribution Functions,” Arkiv för Matematik, 4,
99 - 157.
Brillinger, D. R. (1975): Time Series: Data Analysis and Theory. New York: Holt, Rinehart & Winston.
Brown, B. W., and W. K. Newey (2002): “Generalized Method of Moments, Efficient Bootstrapping,and Improved Inference,” Journal of Business & Economic Statistics, 20, 507 - 517.
Campbell, J. Y., and R. J. Shiller (1991): “Yields Spread and Interest Rate Movements: A Bird’s Eye
View,” Review of Economic Studies, 58, 495 - 514.
Camponovo, L., O. Scaillet, and F. Trojani (2010): “Robust Resampling Methods for Time Series,”
Working Paper, HEC Genève.
Clarida, R., J. Galí, and M. Gertler (2000): “Monetary Policy Rule and Macroeconomic Stability:
Evidence and Some Theory,” Quarterly Journal of Economics, 115, 147 - 180.
Götze, F., and H. R. Künsch (1996): “Second-Order Correctness of the Blockwise Bootstrap for
Stationary Observations,” Annals of Statistics, 24, 1914 - 1933.
Ferson, W. E., and S. R. Foerster (1994): “Finite Sample Properties of the Generalized Method ofMoments in Tests of Conditional Asset Pricing Models,” Journal of Financial Economics, 36, 29 - 55.
Fuhrer, J. C., G. R. Moore, and S. D. Schuh (1995): “Estimating the Linear-Quadratic Inventory Model:
Maximum Likelihood versus Generalized Method of Moments,” Journal of Monetary Economics, 35,
115 - 157.
Hall, P., and J. L. Horowitz (1996): “Bootstrap Critical Values for Tests Based on
Generalized-Method-of-Moments Estimators,” Econometrica, 64, 891 - 916.
Hansen, L. P. (1982): “Large Sample Properties of Generalized Method of Moments
Estimators,” Econometrica, 50, 1029 - 1054.

― 35 ―

摂南経済研究 第 1 巻 第 1・2 号 (2011)

Hirukawa, M. (2006): “A Modified Nonparametric Prewhitened Covariance Estimator,” Journal of Time
Series Analysis, 27, 441 - 476.
Hirukawa, M. (2010a): “Nonparametric Multiplicative Bias Correction for Kernel-Type Density
Estimation on the Unit Interval,” Computational Statistics & Data Analysis, 54, 473 - 495.
Hirukawa, M. (2010b): “A Two-Stage Plug-In Bandwidth Selection and Its Implementation for
Covariance Estimation,” Econometric Theory, 26, 710 - 743.
Inoue, A., and M. Shintani (2006): “Bootstrapping GMM Estimators for Time Series,” Journal of
Econometrics, 133, 531 - 555.
Jones, M. C., O. Linton, and J. P. Nielsen (1995): “A Simple Bias Reduction Method for Density
Estimation,” Biometrika, 82, 327 - 338.
Kitamura, Y. (1997): “Empirical Likelihood Methods with Weakly Dependent Processes,” Annals of
Statistics, 25, 2084 - 2102.
Kitamura, Y., and M. Stutzer (1997): “An Information-Theoretic Alternative to Generalized Method ofMoments Estimation,” Econometrica, 65, 861 - 874.
Kocherlakota, N. R. (1990): “On Tests of Representative Consumer Asset Pricing Models,” Journal of
Monetary Economics, 26, 285 - 304.
Künsch, H. R. (1989): “The Jackknife and the Bootstrap for General Stationary Observations,” Annalsof Statistics, 17, 1217 - 1241.
Linton, O., and J. P. Nielsen (1994): “A Multiplicative Bias Reduction Method for Nonparametric
Regression,” Statistics & Probability Letters, 19, 181 - 187.
Nielsen, J. P. (1998): “Multiplicative Bias Correction in Kernel Hazard Estimation,” Scandinavian
Journal of Statistics, 25, 541 - 553.
Nielsen, J. P., and C. Tanggaard (2001): “Boundary and Bias Correction in Kernel Hazard Estimation,”
Scandinavian Journal of Statistics, 28, 675 - 698.
Newey, W. K., and K. D. West (1987): “A Simple, Positive Semi-Definite, Heteroskedasticity and
Autocorrelation Consistent Covariance Matrix,” Econometrica, 55, 703 - 708.
Newey, W. K., and K. D. West (1994): “Automatic Lag Selection in Covariance Matrix
Estimation,” Review of Economic Studies, 61, 631 - 653.
Oliner, S. D., G. D. Rudebusch, and D. Sichel (1996): “The Lucas Critique Revisited: Assessing the
Stability of Empirical Euler Equations for Investment,” Journal of Econometrics, 70, 291 - 316.
Parzen, E. (1957): “On Consistent Estimates of the Spectrum of a Stationary Time Series,” Annals of
Mathematical Statistics, 28, 329 - 348.
Parzen, E. (1963): “Notes on Fourier Analysis and Spectrum Windows,” Technical Report No.48,
Department of Statistics, Stanford University. (Published in Parzen, E. (1967): Time Series Analysis
Papers. San Francisco: Holden-Day.)― 36 ―

Stabilizing a GMM Bootstrap for Time Series: A Simulation Study

Politis, D. N., and J. P. Romano (1995): “Bias-Corrected Nonparametric Spectral Estimation,” Journalof Time Series Analysis, 16, 67 - 103.
Rotemberg, J. J. (1984): “Interpreting the Statistical Failures of Some Rational Expectations Macroeconomic
Models,” American Economic Review Papers and Proceedings, 74, 188 - 193.
Runkle, D. E. (1991): “Liquidity Constraints and the Permanent-Income Hypothesis,” Journal of
Monetary Economics, 27, 73 - 98.
Smith, D. C. (1999): “Finite Sample Properties of Tests of the Epstein-Zin Asset Pricing Model,” Journalof Econometrics, 97, 113 - 148.
Stuetzle, W., and Y. Mittal (1979): “Some Comments on the Asymptotic Behavior of Robust
Smoothers,” in T. Gasser and M. Rosenblatt (eds.), Smoothing Techniques for Curve Estimation:
Proceedings of a Workshop Held in Heidelberg, April 2 - 4, 1979. Berlin: Springer-Verlag, 191 - 195.
Tauchen, G. (1986): “Statistical Properties of Generalized Method-of-Moments Estimators of Structural
Parameters Obtained from Financial Market Data,” Journal of Business & Economic Statistics, 4, 397- 425.
Velasco, C., and P. M. Robinson (2001): “Edgeworth Expansions for Spectral Density Estimates andStudentized Sample Mean,” Econometric Theory, 17, 497 - 539.
West, K. D. (1988): “Dividend Innovations and Stock Price Volatility,” Econometrica, 56, 37 - 61.
West, K. D. (1997): “Another Heteroskedasticity- and Autocorrelation-Consistent Covariance Matrix
Estimator,” Journal of Econometrics, 76, 171 - 191.
West, K. D., and D. W. Wilcox (1996): “A Comparison of Alternative Instrumental Variables Estimatorsof a Dynamic Linear Model,” Journal of Business & Economic Statistics, 14, 281 - 293.
Xiao, Z., and O. Linton (2002): “A Nonparametric Prewhitened Covariance Estimator,” Journal of Time
Series Analysis, 23, 215 - 250.

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