Economics Letters 111 (2011) 170–172

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Economics Lettersj o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e c o l e t

How useful is yet another data-driven bandwidth in long-run variance estimation?:
A simulation study on cointegrating regressions
Masayuki Hirukawa ⁎Faculty of Economics, Setsunan University, 17-8 Ikeda Nakamachi, Neyagawa, Osaka, 572-8508, Japana r t i c l ei n f oArticle history:
Received 4 November 2009Received in revised form 12 January 2011
Accepted 2 February 2011
Available online 12 February 2011a b s t r a c t
This paper investigates how bandwidth choice rules in long-run variance estimation affect ﬁnite-sample
performance of efﬁcient estimators for cointegrating regression models. Monte Carlo results indicate that
Hirukawa's (2010) bandwidth choice rule contributes bias reduction in the estimators.
© 2011 Elsevier B.V. All rights reserved.

JEL classiﬁcation:
C13
C14
C22
Keywords:
BandwidthCointegration
KernelLong-run variance
Simulation

1. Introduction
In time series econometrics, estimating the long-run variance(“LRV”) matrix of a random vector process is essential for empiricalresearch on estimation (e.g. generalized method of moments) andtesting (e.g. standard error calculation and unit-root testing)problems. This paper focuses on a standard, kernel smoothing
approach to LRV estimation, and investigates ﬁnite-sample performance of the bandwidth choice rule newly proposed by Hirukawa(2010) via Monte Carlo simulations, in comparison with two mostpopular ones by Andrews (1991) and Newey and West (1994). The
simulation study is conducted in the context of cointegrating
regressions. While there is rich literature in simulation studies oncointegrating regressions, this paper contributes the literature in thefollowing two respects. First, while Andrews' (1991) and Newey andWest's (1994) bandwidth choice rules (“A rule” and “NW rule”) arefrequently used in simulation studies, very little is known about ﬁnitesample performance of Hirukawa's (2010) solve-the-equation plug-in
bandwidth choice rule (“SP rule”). To the best of our knowledge, thereare only two simulation results on the SP rule, both of which areavailable in Hirukawa (2010). Neither is on cointegrating regressions.
Second, the SP rule is expected to contribute the bias correction⁎ Tel.: + 81 72 839 8095; fax: + 81 72 839 8138.
E-mail address: hirukawa@econ.setsunan.ac.jp.
0165-1765/$ – see front matter © 2011 Elsevier B.V. All rights reserved.
doi:10.1016/j.econlet.2011.02.006

method by Kurozumi and Hayakawa (2009) for the fully modiﬁed
least squares (“FMLS”; Phillips and Hansen, 1990) and canonicalcointegration regression (“CCR”; Park, 1992) estimators of cointegrating vectors. When the I(1) regressors are endogenous and/or theregression errors are serially correlated in cointegrating regressions,the ordinary least squares (“OLS”) estimators of cointegrating vectors
suffer so-called the “second-order bias”. Although FMLS and CCR areproposed as devices for correcting the second-order bias nonparametrically by means of LRV estimators, these methods work poorly in thepresence of strong serial dependence in regression errors. Then,
Kurozumi and Hayakawa (2009) develop a further bias reduction
method for FMLS and CCR when regression errors obey an AR(1)model with the AR coefﬁcient moderately close to unity. However,their simulation results indicate that average lengths of bandwidths
from the A and NW rules are too long and too short for the purpose ofbias reduction. Apparently, there is a need for a bandwidth choice rulethat tends to yield intermediate lengths. The SP rule is expected to bea remedy, as suggested in the next section.
2. Three Bandwidth Formulae in LRV EstimationTo illustrate the difference in three bandwidth choice rules, consider aproblem of estimating the LRV of a zero-mean scalar process ht, where∞
the LRV is deﬁned as ω= ∑ ∞j = − ∞ γ(j) =∑ j = − ∞ E(htht − j). Given T
observations {ht}Tt = 1, a kernel k(⋅) and a bandwidth M, the kernel
estimator of ω is given by a weighted sum of sample autocovariances

M. Hirukawa / Economics Letters 111 (2011) 170–172
minfT + j;T g−1ω̂ = ∑T−1∑t = maxf1;1 + jg ht ht−j .
j = −ðT−1Þ kðj = M Þ γ̂ð jÞ, where γ̂ð jÞ =T

Each of the three rules is an estimator of the minimizer of the asymptotic
mean squared error (“AMSE”) of ω̂. The minimizer (= AMSE-optimalbandwidth) is given byM⁎ =82 91 = ð2q + 1Þ>=< qk2q RðqÞ >>:∫∞−∞;
k2 ðxÞdx>T

1 = ð2q + 1Þ;

where q(N 0) is the characteristic exponent of k(⋅) that satisﬁes
kq = limx→0 f1−kðxÞg = jxjq ∈ð0; ∞Þ, sðnÞ =∑∞j = −∞ j j jn γð jÞ, and R(q) =s(q)/s(0) is the only unknown quantity called the normalized curvaturein Hirukawa (2010). The three rules differ in how to estimate thequantity. Andrews (1991) estimates it parametrically by ﬁtting an AR(1)model to ht as a reference. In contrast, to avoid the issue of misspecifying
the process, Newey and West (1994) estimate the normalized curvature
nonparametrically using the truncated kernel. Because the use of thetruncated kernel prevents them from providing an optimal bandwidth
for the normalized curvature estimator, they implement the bandwidthfor the truncated estimator in an ad hoc manner.
The SP rule in Hirukawa (2010) is established as an analog to thebandwidth choice rule for probability density estimation by Sheather andJones (1991). Similar to Newey and West (1994), the normalized
curvature is estimated nonparametrically using the same kernel k(⋅)
and a different bandwidth b. A remarkable difference from the approach inNewey and West (1994) is that Hirukawa (2010) has derived the AMSEoptimal bandwidth b⁎ for the kernel normalized curvature estimatorqT−1R̂ ðqÞ ðbÞ = ∑T−1
j = −ðT−1Þ kðj = bÞj j j γ̂ð jÞ = ∑j = −ðT−1Þ kð j = bÞ γ̂ð jÞ. In spiritof Sheather and Jones (1991), the SP rule can be implemented bynumerically solving the following ﬁxed point problem for M⁎:
2 n o2 31 = ð2q + 1ÞðqÞ 
qk2q R̂b⁎ M ⁎67ðÞ
T 1 = 2q + 1 ;M⁎ = 45∞∫ k2 ðxÞdx b⁎ M ⁎ =8<−∞∞

α2 ðqÞ∫−∞91 = ð4q + 1Þ
=k2 ðxÞdx

:ð2q + 1Þ∫∞ jxj2q k2 ðxÞdx;M⁎ð2q + 1Þ = ð4q + 1Þ ;−∞where the unknown quantity α(q) = s(q)/s(0) − s(2q)/s(q) is estimatedby ﬁtting an AR(1) model to ht. By construction, the SP rule
incorporates both parametric and nonparametric approaches. While
it relies on a parametric ﬁtting to the unknown process ht for
implementation, it is still based on a kernel estimator of thenormalized curvature, which limits the inﬂuence of the parametricreference. As a consequence, it is anticipated that when the A and NW
rules tend to yield long and short bandwidth lengths (which is typicalfor moderately to highly persistent ht), the SP rule is likely to pick up
intermediate lengths.
3. Monte Carlo Simulations
This paper adopts the experimental design of Kurozumi andHayakawa (2009). Let the data fðyt ; xt ÞgTt = 1 ∈R2 be generated by
yt = θ0 + θ1 xt + u1t : = zt′ θ + u1t ; Δxt = u2t ;ð1Þwhere the error term ut = ðu1t ; u2t Þ′ obeys
u1t = ρu1t−1 + 1t ; u2t = 2t ;

1t
2t


iid N∼2  01;
0σ21σ211

:True parameter values of θ = ðθ0 ; θ1 Þ′ , ρ, and σ21 are θ0 = θ1 = 1,ρ ∈ {0.7, 0.8, 0.85, 0.9, 0.95}, and σ21 ∈ {0.4, 0.8}. For each combinationof parameter values, 10, 000 data sets of sample sizes T = 100 or
T = 300 are simulated.

171

Because of the endogeneity and the serially correlated regression
error in Eq. (1), θ̂ OLS (=OLS estimator of θ) is consistent but
inefﬁcient due to the second-order bias. FMLS and CCR are developed
as efﬁcient estimation methods that eliminate the second-order bias.
Let Ω and Λ be the two- and one-sided LRVs of ut . Speciﬁcally, 
Λ11 Λ12Λ1
Ω11 Ω12;Λ=Σ+Γ==;Ω = Σ + Γ + Γ′ =
Ω21 Ω22Λ21 Λ22Λ2
where Σ = E ut u′t and Γ = ∑∞j = 1 E ut u′t + j . Also let Ω̂ and Λ̂ be
the kernel estimators of Ω and Λ with ut replaced by û t = û1t ; Δxt ′ ,where û1t is the OLS residual of Eq. (1). The FMLS estimator of θ is givenbyT∑ zt zt′θ̂ FMLS =!−1!

T∑

t=1t =1þþzt yt −T Ĵ;

 

þ
−1−1′ ′.
where yþt = yt − Ω̂ 12 Ω̂ 22 Δxt and Ĵ = 0; Λ̂ 21 − Λ̂ 22 Ω̂ 22 Ω̂ 21
CCR employs the transformed data
 −1

 −1


−1x⁎
Λ̂ 2 ′ût ; yt⁎ = yt − Σ̂ Λ̂ 2 θ̂ OLS + 0; Ω̂ 12 Ω̂ 22 ′ ′ût ;t = xt − Σ̂

where Σ̂ is a consistent estimator of Σ. Deﬁning zt⁎ = ð1; xt⁎Þ′ yields theCCR estimator of θ asθ̂ CCR =!−1T∑ zt⁎ zt⁎′

t =1T!

∑ zt⁎ yt⁎ :t =1

However, FMLS and CCR work poorly when the regression error u1t
exhibits strong serial dependence. Kurozumi and Hayakawa (2009)
consider further bias reduction in FMLS and CCR for the N local-to-unity
system such that ρ is modeled as ρ=1−c/N, where N satisﬁes N→∞
but N=o(T). In addition, if M/N→dM for a bandwidth of the LRVestimators M and if another bandwidth Mc satisﬁes Mc =o(N), c can beconsistently estimated by ĉ = ðN = 2Þ ω̂Δ11 = σ̂ 11 , where σ̂ 11 =2T −1 ∑Tt = 1 û1t and minVT þ j;T tj1∑kΔ û1t Δ û1t−j :McT t¼maxV1;1 þ jt
j¼−ðT−1Þ
T−1ω̂Δ11 = ∑The bias-corrected FMLS and CCR (“FMLS-BC”
and “CCR-BC”)
estimators ofby replacing Ω̂ 21 ; Λ̂ 21 in θ̂ FMLS and θ can beobtainedθ̂ CCR with Ω̃ 21 ; Λ̃ 21 = Ω̂ 21 = κ̂; Λ̂ 21 = κ̂ , where κ̂ = ĉ dM ∫∞0 kðr Þ ×exp − ĉ dM r dr. To implement FMLS-BC and CCR-BC, Kurozumi andHayakawa (2009) suggest that extra tuning parameters are set equal todM = 1, N = M, and Mc = M2/3.
In this simulation study, θ1 is estimated by OLS, FMLS, CCR, FMLSBC, and CCR-BC for each data set. Finite-sample performance of eachestimator is evaluated by its bias and mean squared error (“MSE”).
The Parzen kernel is employed for all LRV estimators, and threebandwidths are computed via the A, NW, and SP rules. The lag lengthfor the truncated estimator of the normalized curvature in the NW
rule is set equal to ⌊4ðT =100Þ4 = 25 ⌋, where ⌊ ⋅ ⌋ denotes the integer part.
Bandwidth values are trimmed at the sample size whenevernecessary. For FMLS-BC and CCR-BC, the SP rule sometimes picks up
aIn this case, ĉ = 0 by N = M = 0, and thus zero bandwidth.
Ω̃ 21 ; Λ̃ 21 are not well-deﬁned. Since the zero bandwidth is anindication of weak serial dependence, no bias correction is made for
FMLS or CCR whenever it occurs.
4. Conclusion
Table 1 reports biases and MSEs of the ﬁve estimators of θ1.
Because the results for σ21 = 0.8 are qualitatively similar, only those172

M. Hirukawa / Economics Letters 111 (2011) 170–172

Table 1Biases and MSEs of estimators of θ1 (σ21 = 0.4).
T = 100

Bias
OLS
FMLS

CCR

FMLS-BC

CCR-BC

MSE
OLS
FMLS

CCR

FMLS-BC

CCR-BC

Bandwidth
A

NW

SP

T = 300

ρ

0.70

0.80

0.85

0.90

A
NW
SPA
NW
SPA
NW
SPA
NW
SP

0.06230.03180.03050.03120.03220.03080.03150.02160.01280.01200.02390.01640.01590.0866
0.05270.05100.0512
0.05380.0516
0.05200.04640.03700.03730.04880.04020.04070.10770.07220.07100.07040.07390.07180.07150.06770.05900.05930.07030.06190.06250.14250.10620.10700.10430.10870.10790.10590.10330.0973
0.09620.10650.09990.0995

ANW
SP
A
NW
SP
A
NW
SP
A
NW
SP

0.01370.01230.01150.01140.01220.01140.01140.01290.01210.01200.01260.01160.01150.02610.02600.02450.02460.02580.02440.02440.02680.02550.0256
0.02640.02490.02500.0401
0.04230.04040.04080.04190.04010.04050.04330.0421
0.04250.0426
0.04130.0416

0.06970.07830.07640.07770.07710.07590.07690.07970.08000.08070.07810.07860.0791

18.33
4.42
0
9.94
2.33
0
9.24
3.62
18

25.62
7.14
011.231.80
012.19
4.97
3

31.58
9.96
011.881.46014.36
6.10
0

40.88
15.49
58
12.47
1.16
017.458.07
0Mean
Std. Dev.
#{Trimmed}
Mean
Std. Dev.
#{Trimmed}
Mean
Std. Dev.
#{Zero}

0.95

0.70

0.80

0.85

0.90

0.95

0.21140.17800.18390.17750.18140.18470.17980.17660.17730.17280.18040.17940.17620.02270.00680.00700.00710.00700.00720.00720.00270.00070.00080.00330.00150.00150.03300.01280.01370.01290.01320.01400.01330.01030.00840.00840.01100.00920.00930.04280.01920.02130.01920.02000.02170.01990.01750.01660.01570.01850.01730.01670.06090.03290.03750.03250.03420.03790.03350.03180.03330.03000.03330.03400.03140.10590.07270.08300.07150.07510.08330.07340.07230.07970.07020.07470.08020.07220.15130.18280.18220.18820.17860.18110.18520.18500.19190.19450.18020.18910.19010.00190.00140.00130.00130.00140.00130.00130.00140.00130.00130.00140.00130.00130.00400.00320.00300.00300.00320.00300.00300.00320.00300.00300.00320.00300.00300.00670.00570.00540.00540.00570.00540.00540.00570.00530.00540.00570.00530.00540.01330.01250.01180.01190.01240.01170.01180.01260.01170.01190.01250.01170.01180.03920.04190.03930.04090.04140.03920.04060.04200.0396
0.04120.04150.03950.0407

24.21
3.32
0
13.48
1.48
0
13.99
3.24
034.86
5.56
0
15.07
0.84
0
19.17
4.29
044.288.00
015.77
0.63
0
23.35
5.37
0

60.74
13.20
0
16.40
0.46
0
29.99
7.46
0

98.10
28.48
0
16.98
0.33
0
43.19
13.60
056.53
26.21
738
12.96
0.92
0
21.86
10.79
0

Note: “Mean”, “Std. Dev.”, “#{Trimmed}”, and “#{Zero}” for bandwidths are averages, standard deviations, numbers of trimmed bandwidths at the sample size, and numbers of zero
bandwidths, respectively.

for σ21 = 0.4 are presented. Descriptive statistics on bandwidths arealso provided for convenience. Frequencies of trimmed and zero
bandwidths are relatively small, and thus their inﬂuences over theresults appear to be negligible.
The results are summarized as follows:

FMLS-BC and CCR-BC, as ρ gets closer to unity, the SP rule is more
likely to give the smallest bias among three bandwidth choice rules.
• Due to the structure of the error term, the experimental design isfavorable to the A rule. Nonetheless, estimators based on the A rule
are often most biased.

• FMLS and CCR are less biased than OLS, and FMLS-BC and CCR-BC areless biased than FMLS and CCR. However, the advantage in bias ofFMLS-BC and CCR-BC over FMLS and CCR appears to be diminishingas ρ increases. In addition, FMLS and FMLS-BC tend to be less biased
than CCR and CCR-BC.
• Results on MSEs are mixed. While four efﬁcient estimators do notalways reduce MSEs from OLS for σ21 = 0.4, these estimators yieldsmaller MSEs than OLS for σ21 = 0.8 (unreported).
• For each of FMLS and CCR, when ρ is close to unity, the SP rule tends
to yield the smallest bias among three bandwidth choice rules.
• In the presence of moderate to high persistence in regression errors,
average lengths of bandwidths tend to be in the order of the NW, SP,
and A rules from shortest to longest, as anticipated. For each ofAndrews, D.W.K., 1991. Heteroskedasticity and autocorrelation consistent covariancematrix estimation. Econometrica 59, 817–858.
Hirukawa, M., 2010. A two-stage plug-in bandwidth selection and its implementation
for covariance estimation. Econometric Theory 26, 710–743.
Kurozumi, E., Hayakawa, K., 2009. Asymptotic properties of the efﬁcient estimators forcointegrating regression models with serially dependent errors. Journal ofEconometrics 149, 118–135.
Newey, W.K., West, K.D., 1994. Automatic lag selection in covariance matrix estimation.
The Review of Economic Studies 61, 631–653.
Park, J.Y., 1992. Canonical cointegrating regressions. Econometrica 60, 119–143.
Phillips, P.C.B., Hansen, B.E., 1990. Statistical inference in instrumental variables
regression with I(1) processes. The Review of Economic Studies 57, 99–125.
Sheather, S.J., Jones, M.C., 1991. A reliable data-based bandwidth selection method forkernel density estimation. Journal of the Royal Statistical Society, Series B 53,683–690.

References