Econometrics and Statistics 20 (2021) 109–130

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Econometrics and Statistics
journal homepage: www.elsevier.com/locate/ecosta

Bias correction for local linear regression estimation using
asymmetric kernels via the skewing method
Benedikt Funke a, Masayuki Hirukawa b,∗
ab

Department of Mathematics, Technical University of Dortmund, Vogelpothsweg 87, Dortmund 44227, Germany
Faculty of Economics, Ryukoku University, 67 Tsukamoto-cho, Fukakusa, Fushimi-ku, Kyoto 612-8577, Japana r t i c l e

i n f oArticle history:
Received 27 March 2019
Revised 27 January 2020
Accepted 28 January 2020
Available online 17 February 2020
JEL classiﬁcation:
C13
C14
Keywords:
Beta kernel
Bias correction
Boundary bias
Curve estimation
Gamma kernel
Local linear regression estimationa b s t r a c t
The skewing method, which has been originally proposed as a bias correction device for
local linear regression estimation using standard symmetric kernels, is extended to thecases of asymmetric kernels. The method is deﬁned as a convex combination of three local
linear estimators. It is demonstrated that the skewed estimator using asymmetric kernels
with properly chosen weights can accelerate the bias convergence from O(b) to O(b2 ) asb → 0 under suﬃcient smoothness of the unknown regression curve while not inﬂating thevariance in an order of magnitude, where b is the smoothing parameter and the regressor
is assumed to have at least one boundary. As a consequence, the estimator has optimal
pointwise convergence of n−4/9 when best implemented, where n is the sample size. It isnoteworthy that these properties are the same as those for a local cubic regression estimator. Finite-sample properties of the skewed estimator are assessed in comparison withlocal linear and local cubic estimators. An application of the skewed estimation to real datais also considered.
© 2020 EcoSta Econometrics and Statistics. Published by Elsevier B.V. All rights reserved.

1. Introduction
The aim of this paper is to reduce the order of magnitude in bias of local linear (“LL”) regression estimators (Fan and Gijbels, 1992; Fan, 1993) smoothed by asymmetric kernels while not inﬂating the one in variance. The key trick for accelerating
the bias convergence is the skewing method, which has been proposed originally for standard nonnegative symmetric kernels by Choi and Hall (1998). The method is deﬁned as a convex combination of three LL estimators, namely, one standard(i.e., centered) and two shifted (i.e., left- and right-shifted) LL estimators.
Before proceeding, we should explain why we concentrate on a nonstandard smoothing technique using asymmetrickernels. Throughout it is assumed that the support of the regressor has at least one boundary (e.g., [0, 1] or R+ ). Indeed,
there are many economic and ﬁnancial variables that have a natural boundary at the origin; examples include wages, incomes, short-term interest rates, and insurance claims (or ﬁnancial losses), to name a few. Often the variables also haveright-skewed distributions. Asymmetric kernels are expected to work well for such distributions because of their freedom ofboundary bias and adaptive smoothing property via changing shapes automatically across design points. Among all asymmetric kernels proposed so far, our main focuses are on the beta and gamma kernels proposed by Chen (1999, 20 0 0b),∗

Corresponding author.
E-mail addresses: benedikt.funke@mathematik.tu-dortmund.de (B. Funke), hirukawa@econ.ryukoku.ac.jp (M. Hirukawa).

https://doi.org/10.1016/j.ecosta.2020.01.004
2452-3062/© 2020 EcoSta Econometrics and Statistics. Published by Elsevier B.V. All rights reserved.

110

B. Funke and M. Hirukawa / Econometrics and Statistics 20 (2021) 109–130

respectively.1 Our preference is based on empirical relevance of the kernels in economics and ﬁnance; see Table 1.1 of
Hirukawa (2018) for a variety of applications of the beta and gamma kernels to estimation and testing problems.
Up until recently, the bias reduction in asymmetric kernel smoothing has been actively studied within the framework ofdensity estimation. In fact, many authors have examined different kinds of bias correction methods that can reduce the order of magnitude in bias from O(b) to O(b2 ) as b → 0 while the one in variance is left unchanged, where b is the smoothing
parameter. Interested readers may consult, for instance, Hirukawa (2010), Hirukawa and Sakudo (2014, 2015), and Funke andKawka (2015). Hirukawa (2018, Chapter 3) also provides a survey. On the other hand, little is investigated on nonparametric
regression estimation smoothed by asymmetric kernels. A few exceptions are Chen (20 0 0a, 20 02) as theoretical contributions and Gospodinov and Hirukawa (2012) as an application.2 Speciﬁcally, Chen (20 0 0a) examines the Gasser-Müller type
regression estimator (Gasser and Müller, 1979) smoothed by the beta kernel. Subsequently, Chen (2002) explores statisticalproperties of LL estimators smoothed by the beta and gamma kernels. Moreover, Gospodinov and Hirukawa (2012) apply the
local constant (“LC”) regression estimation using the gamma kernel to nonparametric estimation of time-homogeneous drift
and diffusion functions in continuous-time models that can be used to describe the underlying dynamics of spot interest
rates.
However, no attempts have been made to improve the bias convergence in asymmetric kernel regression estimation, tothe best of our knowledge. In this respect, our work may be viewed as a natural extension of Chen (2002) in the spirit of theaforementioned articles on bias reduction in density estimation. Following Choi and Hall (1998), we investigate two versionsof the skewed estimator smoothed by asymmetric kernels. The ﬁrst version is a convex combination of one standard andtwo shifted LL estimators. The second version, constructed as a convex combination of two shifted LL estimators only, canbe viewed as a limiting form of the ﬁrst version. It is demonstrated that under suﬃcient differentiability of the underlying
regression curve, each version can accelerate the bias convergence from O(b) to O(b2 ) while the order of magnitude invariance is preserved. As a consequence, when best implemented, the estimators have optimal pointwise convergence rateof n−4/9 where n is the sample size. It is an improvement from the n−2/5 rate in the usual case with no additional smoothing
as in Chen (2002).
There are some advantages in incorporating asymmetric kernels into the skewing method. First, unlike Choi and Hall’s(1998) original skewed estimator using symmetric kernels, it is possible to make the skewed estimator using asymmetrickernels free of edge effects. Details of the method are available in Remark 2. Second, asymmetric kernels possess the robustness property to sparse design problems. The kernels spread out as the design point moves away from a boundary, andas a result, they tend to collect more data points to smooth in the areas with fewer observations. The property continues tohold when asymmetric kernels are combined with the skewed estimation. This is documented in Theorem 2 and conﬁrmed
numerically in Section 3.
There are a few other approaches that can reduce the bias from O(b) to O(b2 ) in nonparametric regression estimation.
Examples include (i) local polynomial (e.g., local cubic) regression smoothing by Ruppert and Wand (1994) and (ii) the
multiplicative bias correction method for nonparametric regression estimation by Linton and Nielsen (1994) and Jones et al.
(1995, Section 4.2). Some readers may wonder why we do not instead investigate these. For (i), the reason why we donot turn to the local cubic estimation is that because higher-order polynomial estimators are subject to potential problems
caused by sparse design, we must prepare for a guard against near-singularity in large matrices in practice, as argued in
Choi and Hall (1998, Remark 2.3). For (ii), while the skewing method is a version of additive bias correction techniques,
the multiplicative bias correction method is based on the identity h( · ) ≡ g( · ){h( · )/g( · )} := g( · )r( · ), where h( · ) is either thedensity or the conditional mean (i.e., the regression function), g( · ) is its initial estimate, and r( · ) serves as a correction
factor. The method works well in density estimation as long as the underlying density is strictly positive at a given design
point or on a certain interval, because in this case the initial density estimate g( · ) is also likely to be strictly positive. On
the other hand, the underlying regressand can take any real value. Therefore, its bias-corrected estimator is not well deﬁned
when the regression curve touches or passes through the horizontal axis.3
The remainder of this paper is organized as follows. Section 2 delivers convergence properties of the skewed estimators
using asymmetric kernels, including the bias acceleration of the estimators. Section 3 conducts Monte Carlo simulations to
examine ﬁnite-sample properties of the skewed estimators. Section 4 applies the skewed estimation to a US industry-level
data and estimates a linear-homogenous production function nonparametrically. Section 5 summarizes the main results ofthe paper. All proofs are provided in the Appendix.
∞This paper adopts the following notational conventions: for a > 0, (a ) = 0 t a−1 exp (−t )dt is the gamma function; for 1 p−1q−1p, q > 0, B( p, q ) = 0 y(1 − y ) dy denotes the beta function; and 1{ · } signiﬁes an indicator function. The expressiond‘X = Y ’ reads “A random variable X obeys the distribution Y”. The expression ‘Xn ∼ Yn ’ is used whenever Xn /Yn → 1 as n → ∞.
Lastly, in order to describe different asymptotic properties of an asymmetric kernel estimator across positions of the de1
Extensions to the cases of other asymmetric kernels are straightforward and thus not pursued any further in this paper; see Section 1.2.2 of Hirukawa(2018) for a non-exhaustive list of asymmetric kernels.
2
It is known that asymmetric kernel density estimators are not integrated to unity in general. This comes from the fact that roles of the data and design
points in asymmetric kernels are not exchangeable. The lack of normalization in asymmetric kernels may cause a problem in some applications of density
estimation, whereas it is not an issue when they are employed for regression estimation.
3
Two ad hoc solutions to this problem are available in Glad (1998), actually. These are: (i) clipping the correction factor below 1/10 and above 10; and(ii) shifting all response data a distance a. Recently Scholz et al. (2015) have applied them to ﬁnancial data.

B. Funke and M. Hirukawa / Econometrics and Statistics 20 (2021) 109–130

111

sign point x ∈ R+ (x ∈ [0, 1]), we denote by “interior x” and “boundary x” a design point x that satisﬁes x/b → ∞ (x/b and(1 − x )/b → ∞) and x/b → κ (x/b or (1 − x )/b → κ ) for some κ ∈ (0, ∞) as n → ∞, respectively.
2. Estimators and Their Large-Sample Properties
2.1. The Estimator
Suppose that we are interested in estimating an unknown conditional meanm(x ) := E ( Y |X = x )nonparametrically using i.i.d. bivariate random variables {(Yi , Xi )}ni=1 . Throughout the corresponding nonparametric regression modelY = m (X ) + 
2is examined, where  is the regression error that satisﬁes E ( |X ) = 0. Also denote the conditional variance of  as σ (· ) :=2
E  X =· .
Often the economic and ﬁnancial data have a boundary. Our particular focus is on the cases in which the support of
the regressor X has at least one boundary. More speciﬁcally, supp (X ) is assumed to be either [0, 1] or R+ . Accordingly, we
employ asymmetric kernels to estimate m(x) nonparametrically.
Our aim is to improve the bias property of the LL regression estimator of m(x) smoothed by an asymmetric kernel while
the order of magnitude in its variance remains unchanged. For this purpose, we ﬁnd (α , β ) that jointly minimize the local
least squares problemn{Yi − α − β (Xi − x )}2 K j (x,b) (Xi ),i=1

where Kj(x,b) is some asymmetric kernel indexed by j that depends on the design point x and the smoothing parameter b.



Let


αˆ j (x ), βˆ j (x ) be the minimizer for a given x. It is well known thatαˆ j (x ) =βˆ j (x ) =s j2 (x )t j0 (x ) − s j1 (x )t j1 (x )ands j0 (x )s j2 (x ) − s2j1 (x )s j0 (x )t j1 (x ) − s j1 (x )t j0 (x )s j0 (x )s j2 (x ) − s2j1 (x )

,

where

1
(Xi − x ) p K j (x,b) (Xi ) and
n

(1)

1
Yi (Xi − x ) p K j (x,b) (Xi )
n

(2)

n

s jp (x ) =i=1
nt jp (x ) =i=1

for p = 0, 1, 2 . . .
ˆ j (x ) = αˆ j (x ). In what follows, a nonstandard
The standard approach is to estimate m(x) by the (classical) LL estimator m
LL estimatorˆ j ( u|x ) := αˆ j (x ) + βˆ j (x )(u − x )m(3)ˆ j ( u|x ) can be interpreted as a shifted LL estimator in which smoothing is made at
is instead considered. The estimator ma slightly off-centered point, i.e., smoothing is made at x, which is located slightly to the left or right of the center u. Also
ˆ j ( x|x ) = m
ˆ j (x ) = αˆ j (x ),observe that putting u = x in (3) (i.e., the case in which the center coincides with x) reduces to mwhich is the classical LL estimator.
The skewed estimator is deﬁned as a convex combination of three such LL estimators. Given weight parameters (called“weights” hereinafter) λ1 , λ2 > 0 and shift parameters (called “shifts” hereinafter) 1 , 2 > 0, the general form of the skewed
estimator of m(x) can be expressed as

˜ j,λ1 ,λ2 , 1 , 2 (x ) :=
m









λ1 mˆ j x|x − 1 b1/2 + mˆ j ( x|x ) + λ2 mˆ j x|x + 2 b1/2,
λ1 + 1 + λ2




(4)

ˆ j x|x − 1 b1/2 and m
ˆ j x|x + 2 b1/2 are called the left- and right-shifted LL estimators, respectively, hereinafter.
where m112

B. Funke and M. Hirukawa / Econometrics and Statistics 20 (2021) 109–130

Unlike standard symmetric kernels, exploring statistical properties of nonparametric estimators smoothed by asymmetrickernels relies on kernel-speciﬁc and thus diversiﬁed approaches. The skewed estimator is not an exception. Then, to save
space, we specialize in two asymmetric kernels, namely, the beta kernel ( j = B)

KB(x,b) (u ) =ux/b (1 − u )(1−x )/b1{u ∈ [0, 1]}
B{x/b + 1, (1 − x )/b + 1}

for x ∈ [0, 1] by Chen (1999) and the gamma kernel ( j = G)

KG(x,b) (u ) =ux/b exp (−u/b)
1 {u ≥ 0 }bx/b+1 (x/b + 1 )for x ∈ R+ by Chen (20 0 0b), due to their popularity in empirical studies in economics and ﬁnance.
Furthermore, when x is located in the vicinity of a boundary, a certain choice of shifts ( 1 , 2 ) may preclude us fromdeﬁning either the left- or right-shifted LL estimator. As a result, the skewed estimator is not well-deﬁned, either. Therefore,
the subsequent asymptotic results for boundary x should be interpreted as those on condition that the skewed estimator iswell-deﬁned; see Remark 2 for more details.
2.2. Convergence Properties of the Gamma Skewed Estimator
2.2.1. Regularity Conditions
We start from exploring convergence properties of the gamma skewed estimator; those of the beta skewed estimator will
be delivered shortly as an extension. To develop their convergence properties, we impose the following common regularity
conditions across the two estimators:
Assumption 1. The i.i.d. random sample {(Yi , Xi )}ni=1 is drawn from a bivariate distribution having support on R × supp (X ),where supp (X ) is either [0, 1] or R+ .
Assumption 2. Let f( · ) be the marginal pdf of X. Then, there is a neighborhood N around the design point x ∈ supp (X )such that σ 2 ( · ), the second-order derivative of f( · ) and the fourth-order derivative of m( · ) are Hölder-continuous of orderς ∈ (0, 1] on N . In addition, f(x) > 0 and σ 2 (x) < ∞.

Assumption 3. The smoothing parameter b(= bn > 0 ) satisﬁes b + nb3

−1→ 0 as n → ∞.

Assumption 1 refers to random sampling, which makes it easier to derive the dominant terms in bias and variance ofthe skewed estimators.4 Fourth-order smoothness of the regression curve in Assumption 2 is crucial for bias acceleration of
the skewed estimation. While only the continuity of f( · ) is required for LC or LL estimation, the bias expansion developed
below involves f(2) ( · ) and thus its local Hölder-continuity in Assumption 2 is a key requirement. Local Hölder-continuity off(2) ( · ) in the neighborhood of x, for example, implies that there is a constant C ∈ (0, ∞) such that

 (2 )
 f (u ) − f (2) (v ) ≤ C |u − v|ς , ∀u, v ∈ N .


−1

The condition “ nb3→ 0” in Assumption 3 is imposed typically in the literature on bias reduction in asymmetric kernel density estimation (e.g., Hirukawa, 2010; Hirukawa and Sakudo, 2014, 2015). It is required to control the remainder term−1in the bias approximation of the skewed estimators, whereas a milder one “ nb2→ 0” suﬃces for LC or LL estimation;
see equation (2.3) of Chen (2002) for moredetails.
It will be shown shortly
that
the MSE-optimal smoothing parameter foreach skewed estimator becomes b∗ = O n−2/9 for interior x and b† = O n−1/5 for boundary x (only if it is well-deﬁned);these convergence rates are indeed within the required range.
2.2.2. Bias Approximation
Below we document the bias approximation of the gamma skewed estimator, conditional on X = x. In addition, as a byproduct of the analysis, a second-order bias expansion (i.e., the bias approximation up to the O(b2 ) term) of the gamma LL
ˆ G ( x|x ) = m
ˆ G (x ) has been derived. The result is summarized in a separate corollary.
estimator m
Theorem 1. Suppose that for a given design point x > 0, weights (λ1 , λ2 ) and shifts ( 1 ,λ1 = λ2 = λ and 1 = 2 = = (λ, x ) =2 ) in (4) are chosen as

2λ + 1x.
2λ(5)Denote the gamma skewed estimator with the above weights and shifts plugged into (4) by

˜ G,λ (x ) =
m





λmˆ G x|x − b1/2 + mˆ G ( x|x ) + λmˆ G x|x + b1/2.
2λ + 1

4
We do not consider the case in which residuals are correlated. However, it is conjectured that asymptotic results in this paper will continue to holdfor weakly dependent and strictly stationary data by imposing some mixing condition (e.g., α -mixing with a suitable size), although an extension in thisdirection is beyond the scope of this paper.

B. Funke and M. Hirukawa / Econometrics and Statistics 20 (2021) 109–130

113

 2

˜ G,λ (x ) can be approximated by Bias m
˜ G,λ (x ) = BG,λ (x; m, f )b2 + o b ,If Assumptions 1-3 hold, then, as n → ∞, the bias of m
where



BG,λ (x; m, f ) = −1 + 2x













f (1 ) ( x )x2 f ( 2 ) ( x )5f (1 ) ( x )x2
1+m (2 ) ( x ) + x+xm (3 ) ( x ) +1+
m (4 ) ( x ) .
f (x )
2 f (x )3f (x )44λˆ G (x ) can be expanded up to the second order as
Corollary 1. Under Assumptions 1-3, as n → ∞, the bias of m 

x2 f ( 2 ) ( x )x
f (1 ) ( x )ˆ G (x ) = m(2 ) (x )b + − 1 + x
Bias m−+ x2
2
f (x )
2 f (x )f (1 ) ( x )f (x )2m (2 ) ( x )

 
x
x2
+ m(3 ) (x ) + m(4 ) (x ) b2 + o b2 .
3
8
The theorem states that under the symmetric weights and shifts given in (5), the gamma skewed estimator can attain
˜ G,λ (x ) is well-deﬁned). Suchthe bias convergence of O(b2 ) regardless of the position of the design point x (as long as mweights and shifts are determined by setting coeﬃcients on O(b) and O(b3/2 ) bias terms of (4) equal to zero; see the Proofof Theorem 1 in Appendix for more details. Observe that the bias acceleration in the skewed estimator can be attained by aﬁxed λ. In Section 2.3 below, we let λ → ∞ and explore convergence properties of the skewed estimator in limit.
ˆ G (x ) are O(b) and O(b2 ) for
Chen (2002) shows that orders of magnitude in bias of the (standard) gamma LL estimator m
˜ G,λ (x ) indeed improves from that of the LL estimator.
interior and boundary x, respectively. Therefore, the bias property of m
While the same order of magnitude in bias can be attained by local cubic smoothing, the skewing method is expected to be
more robust against sparse design due to its LL-based structure.
Remark 1. There are a few noteworthy differences in the choices of shifts and weights between skewed estimators using
symmetric and asymmetric kernels. The ﬁrst difference is whether the shift 1 = 2 = that can accelerate the bias convergence of the skewed estimator depends on the design point x. As shown in Theorems 1 and 3, shifts of the gamma and beta
skewed estimators vary with x. This is a sharp contrast to the result when a standard nonnegative symmetric kernel K(u) is
employed. Choi and Hall (1998) demonstrate that as in our case, symmetric weights and shifts can lead to bias accelerationfrom O(h2 ) to O(h4 ) in their skewed estimation using symmetric
However, for the weight kernels, where h is the bandwidth.

λ1 = λ2 = λ, they derive the shift as 1 = 2 = = (λ ) = (2λ + 1 )κ2 /(2λ ), where κ2 = u2 K (u )du; to put it in anotherway, the shift remains unchanged regardless of the position of x in Choi and Hall’s (1998) original skewed estimator.
The second difference lies in sensitivity to the choice of λ. Our Monte Carlo study in Section 3 compares ﬁnitesample performances of asymmetric kernel-based skewed estimators with those of the skewed estimator using the symmetric, Epanechnikov kernel. The results indicate that skewed estimators smoothed by asymmetric kernels are insensitive
to the choice of λ as long as it is large enough. This is not true for the Epanechnikov skewed estimator. Performances of
the Epanechnikov estimator appear to be largely affected by the choice of λ. Choi and Hall (1998, Remark 2.4) provide
λ∗ = 0.0352 as the minimizer of the scaling factor V(λ) in the leading variance term of the Epanechnikov estimator, andthey use this value exclusively in their simulation study. Indeed, once the weight deviates from λ∗ , performances of thisestimator tend to deteriorate.5 From this viewpoint, we also employ the optimal λ∗ for the Epanechnikov skewed estimatorin our simulation study.
Third, the points where smoothing is made in left- and right-shifted LL estimators appear to take different forms. Theseare x ± h in Choi and Hall’s (1998) original skewed estimator and x ± b1/2 in skewed estimators using asymmetric kernels.
Such ‘seemingly’ different expressions turn out to be equivalent if Chen’s (1999, 20 0 0b) original notation b is replaced by
h2 .6 The notational change also enables us to understand that the bias acceleration from O(b) to O(b2 ) via the asymmetrickernel-based skewed estimation is equivalent to the one from O(h2 ) to O(h4 ) due to Choi and Hall’s (1998) original skewed
estimation.
Remark 2. A referee asks us how to choose (λ, ) for the gamma skewed estimator so that Theorem 1 holds when xlies in the vicinityof the origin. To consider this issue,we ﬁrst focus on the case in which the left-shifted LL estimatorˆ G x|x − b1/2 can be still well-deﬁned. Given that = (2λ + 1 )x/(2λ ), we must have x − b1/2 ≥ 0 in this case. It followsmthat a possible choice of the weight λ would be any number satisfying λ ≥ b/{2(x − b)}, provided that x > b; in other words,
for x ∈ (b, ∞), the bias acceleration occurs unless a very small λ is chosen. Therefore, x ∈ [0, b] is the truly problematic case.
All we can do in this case would be to deﬁne the (standard) gamma LL estimator as the gamma skewed estimator; in fact,
˜ G,λ (0 ) (i.e., the gamma skewed estimator at the origin) collapses to m
ˆ G (0 ) (i.e., the gamma LL estimator at the origin).
m
Our Monte Carlo study in Section 3 adopts this practice. Clearly, this practice can make the gamma skewed estimator freeof edge effects. It is also noteworthy that the practice does not contradict the order of magnitude in bias of the skewed
ˆ G (x ) for boundary x itself has a O(b2 ) bias. The Monte Carlo study deals
estimator documented in Theorem 1, because m
5
A referee ﬁnds this aspect in his/her own simulation study, and our preliminary simulation results conﬁrm it. We thank him/her for running simulations
and sharing results.
6
Jones and Henderson (2007) adopt this type of notational change in developing asymmetric kernels with support on [0, 1].

114

B. Funke and M. Hirukawa / Econometrics and Statistics 20 (2021) 109–130

with yet another skewed estimator smoothed by the beta kernel with support on [0, 1], and the practice of switching to thebeta LL estimator in the vicinity of 0 or 1 is again applied to this estimator.
2.2.3. Variance Approximation
˜ G,λ (x ), conditional on X = x, is presented.
Next, the variance approximation of the gamma skewed estimator m
˜ G,λ (x ) can be approximated by
Theorem 2. Under Assumptions 1-3, as n → ∞, the variance of m⎧2
⎨ nb11/2 4√5πσ√(xxf) x + o n−1 b−1/2( )˜ G,λ (x )} =V ar {m⎩  −1 −1 O nf or interior x.

bf or boundary x

The analytical expression of the leading variance term for boundary
 x is ofless importance, and thus we only document
˜ G,λ (x ) are O n−1 b−1/2 and O n−1 b−1 for interior and boundary x,
its order of magnitude. Orders of magnitude in V ar m

ˆ G (x ) . A unique feature of the variance property is that the variancerespectively, and these are the same as those for V ar m
coeﬃcient for interior x is inversely proportional to x1/2 and thus decreases as x increases, as in other gamma density
and regression estimators. This reﬂects that the gamma kernel tends to spread out as x increases for a given value of thesmoothing parameter b; in other words, the kernel can collect more data points (or increase its effective sample size) tosmooth in the areas with fewer observations. This property is particularly advantageous to estimating the regression curves
whose regressors have distributions that have a long tail with sparse data, such as those of the economic and ﬁnancial
variables mentioned in Section 1.
˜ G,λ (x ) for interior x is independent of λ (although higherInterestingly, the coeﬃcient of the dominant term in V ar morder terms may depend on it), and it follows from

ˆ G (x ) =V ar m1σ 2 (x )+ o n−1 b−1/2
√ √
1
/
2
nb 2 π x f (x )˜ G,λ (x ) /V ar m
ˆ G (x ) → 5/2 for interior x. At a ﬁrst glance, this result may give us the impression that while thethat V ar m
gamma skewed estimator does not inﬂate the order of magnitude in variance from the gamma LL estimator,
 the former
inﬂates the variance coeﬃcient; indeed, the increase can be attributed to the fact that variations of ∓ b1/2 βˆG x ± b1/2 inthe left- and right-shifted LL estimators (3) introduce additional variability, as the proof in Appendix indicates.
However, the above eﬃciency comparison is based on the implicit assumption that the same smoothing parame˜ G,λ (x ) for interior x
ter b is chosen across two estimators. Rather, observe that the mean squared error (“MSE”) of m
˜ G,λ (x ) = O b4 + n−1 b−1/2 . It follows that the optimal smoothing parameter b∗G,λ that minimizes the dominantis MSE m









two terms in the MSE is b∗G,λ = O n−2/9 , which yields the variance convergence of O n−8/9 . On the other hand, because 2

ˆ G (x ) = O b + n
MSE m−1 −1/2b −4/5 which leads to the variance convergence of O n

for interior x, the optimal smoothing parameter for the LL estimator is b∗G,LL = O n−2/5 ,˜ G,λ (x ) /V ar m
ˆ G (x ) =. As a result, when best implemented, V ar m

˜ G,λ (x ) can be asymptotically more eﬃcient than m
ˆ G (x ) for interior x.
O n−4/45 → 0, or m

It also follows that the skewed estimator has optimal pointwise convergence of n−4/9 for interior x. This convergence ratecoincides with Stone’s (1980) optimal nonparametric one under fourth-order smoothness of the unknown regression curve.
2.3. A Limit Case
So far we have seen that under a suitably chosen weighting scheme with a ﬁxed weight λ, the skewing method canˆ G (x ) in an order of magnitude. Some readers may then wonder whether the
reduce the bias of the gamma LL estimator mvalue of λ can be optimized under some criterion. However, the answer is negative. The asymptotic variance for interior x
does not depend on λ, nor the integrated variance based on the trimming argument as in Chen (1999, 20 0 0b). As a result,when the mean integrated squared error (“MISE”) is considered as a global performance measure, the squared bias alone
varies with λ. Moreover, the leading bias term suggests the choice of a preferably large value of λ because of the reciprocal
occurrence of λ in BG,λ (x; m, f ).
Instead, as in Choi and Hall (1998, Remarks 2.1 and 2.4), we let λ diverge to inﬁnity and denote the limit gamma leftˆ G,− (x ) and m
ˆ G,+ (x ), respectively. These are formally expressed as
and right-shifted LL estimators by m

ˆ G,± (x ) := m
ˆ G x|x ±
m

 

xb ,



√
ˆ G x|x ± b1/2 . The corresponding limit gamma skewed estimator
where limλ→∞ (λ, x ) = x is substituted in place of in mtakes the form of an average of the two shifted LL estimators
˜ G,∞ (x ) :=m1
ˆ G,− (x ) + m
ˆ G,+ (x ) .
m
2This estimator can be interpreted as a limit form of the convex combination of three LL estimators.

B. Funke and M. Hirukawa / Econometrics and Statistics 20 (2021) 109–130

115

ˆ G,± (x ) and m
˜ G,∞ (x ), again conditional on
The next proposition provides approximations to the bias and variance of m
X = x.
ˆ G,± (x ) can be approximated by
Proposition 1. (i) Under Assumptions 1-3, as n → ∞, the bias and variance of m√
ˆ G,± (x ) = ∓ x
Bias m







 
3
f (1 ) ( x )x+xm(2 ) (x ) + m(3 ) (x ) b3/2 + BG,∞ (x; m, f )b2 + o b2 ,2f (x )
3

where

BG,∞ (x; m, f ) = lim BG,λ (x; m, f )
λ→∞



=−
and









f (1 ) ( x )x2 f ( 2 ) ( x )5f (1 ) ( x )x2
1 + 2x+m (2 ) ( x ) + x+xm (3 ) ( x ) + m (4 ) ( x ) ,f (x )
2 f (x )3f (x )
4⎧
2⎨ nb11/2 4√3πσ√(xxf) x + o n−1 b−1/2( )ˆ G,± (x ) =V ar m⎩  −1 −1 O nbf or interior x.
f or boundary x

˜ G,∞ (x ) can be approximated by(ii) Under Assumptions 1-3, as n → ∞, the bias and variance of m 

˜ G,∞ (x )} = BG,∞ (x; m, f )b2 + o b2 , andBias{m⎧
2⎨ nb11/2 4√5πσ√(xxf) x + o n−1 b−1/2( )˜ G,∞ (x )} =V ar {m⎩  −1 −1 O nbf or interior x.
f or boundary x

ˆ G,± (x ) alone can reduce the bias from O(b) for the standard LL estimator to O(b3/2 ) for interior x.
Observe that m
Coeﬃcients on their leading O(b3/2 ) bias terms have opposite signs with the same magnitude, and thus they are offset
˜ G,∞ (x ) can accelerate the bias convergence up until O(b2 ). It can be also
by taking an average of the two. It follows that m
ˆ G,± (x ) /V ar m
ˆ G (x ) → 3/2 for interior x and a given b.
deduced that V ar m
2.4. An Extension: The Beta Skewed Estimator
Our analysis is now extended to the skewed estimators smoothed by the beta kernel. After convergence properties (i.e.,˜ B,λ (x ) and the second-order expansion of the bias of the beta LL
the bias-variance trade-off) of the beta skewed estimator mˆ B ( x|x ) = m
ˆ B (x ) are presented, those of the limit beta left- and right-shifted LL estimators mˆ B,± (x ) and the limit
estimator mˆ B,∞ (x ) are documented. These results are again conditional on X = x. It can be immediately foundbeta skewed estimator mthat all the previous discussions on the asymptotic results for the gamma case continue to hold for the beta case.
Theorem 3. Suppose that for a given design point x ∈ (0, 1) , weights (λ1 , λ2 ) and shifts ( 1 ,λ1 = λ2 = λ and

1 =

2 =

= (λ, x ) =2 ) in (4) are chosen as

2λ + 1x(1 − x ).
2λDenote the beta skewed estimator with the above weights and shifts plugged into (4) byλmˆ B x|x − b1/2 + mˆ B ( x|x ) + λmˆ B x|x + b1/2
˜ B,λ (x ) =m.
2λ + 1 

˜ B,λ (x ) can be approximated by Bias m
˜ B,λ (x ) = BB,λ (x; m, f )b2 + o b2 ,If Assumptions 1-3 hold, then, as n → ∞, the bias of m
where



BB,λ (x; m, f ) = −f (1 ) ( x )1
f (2 ) (x )1 − 6x ( 1 − x ) + 2x ( 1 − x ) ( 1 − 2x )+ x2 (1 − x )2m (2 ) ( x )f (x )2f (x )

+ x (1 − x )˜ B,λ (x ) can be approximated byMoreover, the variance of m⎧2
⎨ nb11/2 √ √5σ (x)+ o n−1 b−1/2
4 π x(1−x ) f (x )˜ B,λ (x )} =V ar {m⎩  −1 −1 
O n

b





5
f (1 ) ( x )11m (3 ) ( x ) +1+
x2 (1 − x )2 m(4 ) (x ) .
( 1 − 2x ) + x ( 1 − x )3f (x )
4
4λf or interior x.
f or boundary x

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B. Funke and M. Hirukawa / Econometrics and Statistics 20 (2021) 109–130

ˆ B (x ) can be expanded up to the second order as
Corollary 2. Under Assumptions 1-3, as n → ∞, the bias of m

 

5
1
f (1 ) ( x )ˆ B (x ) = x(1 − x )m(2 ) (x )b + − 1 − x(1 − x ) + x(1 − x )(1 − 2x )Bias m2
2
f (x )1f (2 ) (x )− x2 (1 − x )2
+ x2 (1 − x )22
f (x )f (1 ) ( x )f (x )2m (2 ) ( x )



 
1
1
x(1 − x )(1 − 2x )m(3 ) (x ) + x2 (1 − x )2 m(4 ) (x ) b2 + o b2 .
3
8

+



ˆ B,± (x ) := m
ˆ B x|x ±Proposition 2. (i) Under Assumptions 1-3, as n → ∞, the bias and variance of mx(1 − x )b

can be ap-

proximated by3
f (1 ) (x )
ˆ B,± (x ) = ∓ x(1 − x )Bias mm (2 ) ( x )( 1 − 2x ) + x ( 1 − x )2f (x ) 1+ x(1 − x )m(3 ) (x ) b3/2 + BB,∞ (x; m, f )b2 + o b2 ,


3

where

BB,∞ (x; m, f ) = lim BB,λ (x; m, f )
λ→∞




+ x (1 − x )and

1 − 6x ( 1 − x ) + 2x ( 1 − x ) ( 1 − 2x )

= −





5
f (1 ) ( x )1m(3 ) (x ) + x2 (1 − x )2 m(4 ) (x )( 1 − 2x ) + x ( 1 − x )3f (x )
4⎧
2⎨ nb11/2 √ √3σ (x)+ o n−1 b−1/2
4 π x(1−x ) f (x )
ˆ B,± (x ) =V ar m⎩  −1 −1 O nf (1 ) (x )1
f (2 ) ( x )+ x2 (1 − x )2m (2 ) ( x )f (x )2f (x )bf or interior x.
f or boundary x

˜ B,∞ (x ) := m
ˆ B,− (x ) + m
ˆ B,+ (x ) /2 can be approximated by(ii) Under Assumptions 1-3, as n → ∞, the bias and variance of m ˜ B,∞ (x )} = BB,∞ (x; m, f )b2 + o b2 , andBias{m 1

˜ B,∞ (x )}=V ar {m5σ ( x )√2

nb1/2 4√πx(1−x ) f (x )

 −1 −1 O n

+ o n−1 b−1/2bf or interior x.

f or boundary x

Remark 3. Seifert and Gasser (1996, Theorem 1) argue that ﬁnite-sample variances of local polynomial regression estimators
using kernels with compact support may be unbounded. This problem typically occurs when smoothing is made in sparse
regions. As a remedy, Seifert and Gasser (1996) propose to increase the bandwidth locally in such regions. However, Chen
(2002, Lemma 1) demonstrates that the pth-order local polynomial regression estimator using the beta kernel is immune
to this problem, as long as at least p + 1 different data points are not on the boundary of [0, 1]. It follows that in ﬁnite
samples the beta LL estimator has ﬁnite variance with probability 1, and the same is true for the beta skewed estimation.
3. Finite-Sample Performance
In this section, ﬁnite-sample properties of the skewed estimators are assessed in comparison with other competing nonparametric regression estimators. Our Monte Carlo study starts from assessing the gamma skewed estimation and thenproceeds to the beta skewed estimation. For each case, the data are generated from the regression model Y = m (X ) +  ,dwhere the regressor X obeys some marginal distribution that will be speciﬁed shortly, X   , and  = N 0, 0.052 .
3.1. On the Gamma Skewed Estimator
3.1.1. Monte Carlo DesigndOur ﬁrst Monte Carlo study focuses on a regressor with support on R+ . Let X = G(2, 1 ). Also let the regression curvem(x) take one of the following functional forms:

Model A : m(x ) = exp (−x ) + exp −4(x − 1 )2 .

Model B : m(x ) = −2

x
sin
1+x

4π
.
√
1+x

B. Funke and M. Hirukawa / Econometrics and Statistics 20 (2021) 109–130

117

The shape of each curve can be found in Fig. 2 below. The sample size is n ∈ {10 0, 20 0, 40 0}, and 10 0 0 replications are
drawn for each combination of the model and sample size. It is worth remarking that our Monte Carlo design with Model A
exactly mimics the one in Chen (2002). Hence, our results for this model are directly comparable with the results therein.
The simulation study compares ﬁnite-sample performance of the following seven regression estimators for m(x): (i) the LL
ˆ E (x ); (ii) the skewed estimator with λ∗ = 0.0352 smoothed byestimator smoothed by the symmetric, Epanechnikov kernel m
˜ E,λ∗ (x ); (iii) the LL estimator smoothed by the gamma kernel m
the Epanechnikov kernel mˆ G (x ); (iv) the local cubic estimatorˆ G,C (x ) = e SG (x )−1 TG (x ), where e = 1 0 0 0 ,smoothed by the gamma kernel m⎡

sG0 ( x )⎢sG1 (x )SG ( x ) = ⎣sG2 ( x )sG3 ( x )sG1 ( x )sG2 ( x )sG3 ( x )sG4 ( x )sG2 ( x )sG3 ( x )sG4 ( x )sG5 ( x )⎤⎡⎤sG3 ( x )tG0 ( x )sG4 (x )⎥⎢tG1 (x )⎥, and TG (x ) = ⎣;sG5 (x )⎦tG2 (x )⎦sG6 ( x )tG3 ( x )˜ G,0.1 (x ); (vi) the skewed estimator with λ = 1 (=(v) the skewed estimator with λ = 0.1 smoothed by the gamma kernel m
˜ G,1 (x ); and (vii) the limit skewed estimator smoothedequally-weighted skewed estimator) smoothed by the gamma kernel m
˜ G,∞ (x ). The estimators (i)-(iv) can be viewed as competitors to the gamma skewed estimators (v)by the gamma kernel m(vii). The Epanechnikov skewed estimator (ii) is exactly the same as employed in the simulation study of Choi and Hall
(1998). As mentioned in Remark 1, this estimator is sensitive to the choice of λ, and λ∗ has some optimality. On the other
˜ G,λ to the choice
hand, we consider three different gamma skewed estimators (v)-(vii) in order to check the sensitivity of mof λ. Furthermore, as recommended in Remark 2, if a gamma skewed estimator is not well-deﬁned for a very small x, thenit is replaced by the gamma LL estimator evaluated at the same design point.
Performance of an estimator m̄(x ) is assessed via both global and local measures. The global measures include the root
integrated squared error (“RISE”)

RISE {m̄(x )} =

#

∞

0

{m̄(x ) − m(x )}2 dx,

and the integrated absolute deviation (“IAD”)

IAD{m̄(x )} =# ∞0|m̄(x ) − m(x )|dx,

where each integral is approximated by the trapezoidal rule on an equally-spaced grid of 551 points over the interval [0.0, 5.5]. In addition, as the local measures, the squared bias and variance evaluated at twelve design points x ∈{0.0, 0.5, . . . , 5.0, 5.5} over 10 0 0 Monte Carlo samples are computed.
Furthermore, how to choose the tuning parameter is an important practical issue in kernel smoothing. We obtain the
optimal global tuning parameter of an estimator by an oracle method (Case #1) and then by the cross-validation (“CV”)
method (Case #2). Details of each method are discussed in the corresponding section below.
3.1.2. Case #1: Oracle Tuning Parameters
By the oracle tuning parameter for an estimator m̄(x ) we mean the optimal global tuning parameter (i.e., the bandwidth hfor the Epanechnikov kernel or the smoothing parameter b for the gamma kernel) that minimizes RISE {m̄(x )}. The minimizer
is found on an equally-spaced grid of 40 points over the interval [0.01, 0.40].
Table 1 presents averages (“Ave”) and standard deviations (“SD”) of global performance measures and oracle tuning parameter values over 10 0 0 Monte Carlo samples. Assessing an estimator with the oracle tuning parameter value plugged incan be interpreted as evaluating its potential.
It can be immediately found that global measures of each estimator shrink with the sample size, which indicates consistency. A closer look reveals that two Epanechnikov estimators perform inferior to other estimators for n = 100. Althoughˆ G,C (x ), m
˜ G,1 (x ) and m
˜ G,∞ (x ) even for
their performances improve with the sample size, they do not surpass those of mn = 400. It is also worth remarking that average lengths of their tuning parameters are much larger than those of gamma
estimators. Choosing a global bandwidth for a symmetric kernel implies that amount of smoothing is ﬁxed everywhere. It
seems that to take care of sparseness of data points on the right tail, both Epanechnikov estimators pick out ﬁxed, long
bandwidths for extrapolation to the right.
ˆ G,C (x ), m
˜ G,0.1 (x ),Now we turn to gamma estimators. As expected, each of four bias-corrected gamma estimators (i.e., m
˜ G,1 (x ), and m
˜ G,∞ (x )) more or less outperforms the gamma LL estimator m
ˆ G (x ). Comparing global measures of three gammam
˜ G,1 (x ) and m
˜ G,∞ (x ) look alike and are in general superior to those ofskewed estimators reveals that performances of m
˜ G,0.1 (x ). This can be attributed to the fact that a small weight λ inﬂates the leading bias coeﬃcient of m
˜ G,λ (x ), whereasm
˜ G,1 (x ) and m
˜ G,∞ (x ) perform better than m
ˆ G,C (x ) for Modelits leading variance coeﬃcient is insensitive to λ. Furthermore, mA, and the results are reversed for Model B. However, margins are small for each case, and thus it may be safe to say thatthese estimators perform equally well. Also observe that average tuning parameter values of gamma estimators are muchsmaller than those of Epanechnikov estimators.
Following Chen (2002), we visualize local measures of the estimators in Fig. 1. The results from Models A and B are
qualitatively similar, and thus only the former is presented. Notice that the vertical axis in each panel is the natural logarithm of either squared bias or variance. Moreover, to avoid too many lines in a panel, we include only the limit skewed

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B. Funke and M. Hirukawa / Econometrics and Statistics 20 (2021) 109–130
Table 1Monte Carlo Results (Gamma Case; Oracle).
Model

n

Estimator

Ave

SD

Ave

SD

Ave

SD

A

100

ˆ E (x )m
˜ E,λ∗ (x )m
ˆ G (x )mˆ G,C (x )m
˜ G,0.1 (x )m
˜ G,1 (x )m
˜ G,∞ (x )m

0.10420.10070.08120.06890.07070.06270.0627(0.0711)(0.0688)(0.0216)(0.0277)(0.0235)(0.0208)(0.0205)

0.14830.14500.13830.11970.12080.10910.1093(0.0489)(0.0537)(0.0338)(0.0451)(0.0367)(0.0322)(0.0312)0.27970.27250.02250.07360.02300.02690.0279(0.0797)(0.0763)(0.0073)(0.0385)(0.0064)(0.0088)(0.0090)200ˆ E (x )m
˜ E,λ∗ (x )m
ˆ G (x )mˆ G,C (x )m
˜ G,0.1 (x )m
˜ G,1 (x )m
˜ G,∞ (x )m

0.06750.06400.06540.04680.0541
0.04440.0443(0.0319)(0.0327)(0.0113)(0.0138)(0.0132)(0.0119)(0.0117)0.11030.1034
0.10920.08100.09250.07780.0777(0.0247)(0.0282)(0.0174)(0.0211)(0.0201)(0.0174)(0.0171)0.24650.24010.0206
0.05400.02220.02280.0234(0.0650)(0.0586)(0.0025)(0.0208)(0.0042)(0.0050)(0.0055)400

ˆ E (x )m
˜ E,λ∗ (x )m
ˆ G (x )mˆ G,C (x )m
˜ G,0.1 (x )m
˜ G,1 (x )m
˜ G,∞ (x )m

0.04730.04290.05830.03480.04800.03410.0337(0.0122)(0.0130)(0.0068)(0.0099)(0.0107)(0.0080)(0.0078)

0.08300.0735
0.0936
0.05900.07950.0591
0.0583(0.0146)(0.0160)(0.0115)(0.0143)(0.0167)(0.0123)(0.0119)0.21820.22020.02020.04350.02280.02160.0217(0.0449)(0.0398)(0.0013)(0.0154)(0.0045)(0.0037)(0.0038)100

ˆ E (x )m
˜ E,λ∗ (x )m
ˆ G (x )mˆ G,C (x )m
˜ G,0.1 (x )m
˜ G,1 (x )m
˜ G,∞ (x )m

0.76460.73560.12360.0706
0.09140.08910.0905(0.4999)(0.4812)(0.0676)(0.0364)(0.0751)(0.0742)(0.0739)0.73070.69840.19920.11200.12320.11890.1213(0.5103)(0.4966)(0.0596)(0.0468)(0.0567)(0.0516)(0.0512)0.38410.38400.02080.07870.02400.02530.0250(0.0360)(0.0352)(0.0036)(0.0404)(0.0062)(0.0073)(0.0070)200

ˆ E (x )m
˜ E,λ∗ (x )m
ˆ G (x )mˆ G,C (x )m
˜ G,0.1 (x )m
˜ G,1 (x )m
˜ G,∞ (x )m0.22210.21290.0871
0.04630.04950.04870.0497(0.2709)(0.2588)(0.0205)(0.0193)(0.0253)(0.0249)(0.0248)0.24200.21840.15870.07310.07840.07560.0773(0.1959)(0.1915)(0.0263)(0.0251)(0.0229)(0.0206)(0.0204)

0.31430.31390.02020.05840.02290.02300.0228(0.0658)(0.0652)(0.0013)(0.0282)(0.0046)(0.0048)(0.0047)400

ˆ E (x )m
˜ E,λ∗ (x )m
ˆ G (x )mˆ G,C (x )m
˜ G,0.1 (x )m
˜ G,1 (x )m
˜ G,∞ (x )m

0.07550.06860.07750.0336
0.03630.03440.0350(0.0715)(0.0711)(0.0084)(0.0148)(0.0106)(0.0102)(0.0101)

0.1146
0.09490.14700.05130.0611
0.05490.0560(0.0502)(0.0494)(0.0180)(0.0152)(0.0156)(0.0122)(0.0120)0.21580.21850.0201
0.04860.02330.02170.0216(0.0559)(0.0534)(0.0011)(0.0173)(0.0047)(0.0039)(0.0038)

B

RISE

IAD

Tuning Parameter

estimator among all gamma skewed estimators.7 Notice that although the limit gamma skewed estimator for a very small
x is replaced with the gamma LL estimator, their local measures differ because of difference in their smoothing parameter
˜ G,∞ (0 ) is replaced by m
ˆ G (0 ). Nonetheless, their squared bias and variance differ (although the gaps
values. For example, m
are very small and thus almost invisible in Fig. 1).
Fig. 1 indicates that the gamma LL estimator tends to have large squared bias. Both the gamma local cubic and limit
gamma skewed estimators improve the LL estimator in terms of bias in the sense that squared bias of the former in generallies below that of the latter. We can also see that variance of each Epanechnikov estimator tends to inﬂate as the design
point x moves away from the origin. In contrast, three gamma estimators have similar variance properties, and their variancereduction on the right tail is often substantial. These ﬁndings coincide with those in Chen (2002).
To illustrate how well each estimator can capture shapes of true regression curves, we plot pointwise averages of curve
estimates over 10 0 0 Monte Carlo samples for n = 200 in Fig. 2. The set of estimators used here is the same as in Fig. 1. It
can be found that the gamma local cubic and limit gamma skewed estimators can capture shapes of entire regression curves(including peaks and troughs near the origin and right tails) equally well. On the other hand, two Epanechnikov estimators

7
Based on our preliminary analysis, local performances of gamma skewed estimators with λ = 1 and λ = 0.1 are similar and inferior to those of thelimit skewed estimator, respectively.

B. Funke and M. Hirukawa / Econometrics and Statistics 20 (2021) 109–130

119

Fig. 1. Squared Bias and Variance of Curve Estimates (Gamma Case [Model A]; Oracle).
Note: Dash-dot, thin solid, dot, dash, and solid lines correspond to the Epanechnikov LL, Epanechnikov skewed, gamma LL, gamma local cubic, and limit
gamma skewed estimates, respectively.

are not so impressive as the gamma estimators. Despite a long bandwidth and the sample size of 200, they yield wiggly(and often severely biased) curve estimates on the right tail. As a result of oversmoothing via the long bandwidth, they also
fail to precisely capture the peak or trough in the vicinity of the origin.
In sum, unless a small weight is chosen, the gamma skewed estimator has nice ﬁnite-sample properties. Its ﬁnite-sample
performance in general dominates those of the Epanechnikov LL and skewed estimators and is comparable with that of the
gamma local cubic estimator. Based on these ﬁndings, we focus exclusively on the gamma skewed with λ = 1 and limit
gamma skewed estimators in the analysis below and Section 4. The gamma LL estimator is also employed as a benchmark.
3.1.3. Case #2: Cross-Validated Tuning Parameters
We proceed to investigate curve estimates with data-based tuning parameter values plugged in. The leave-one-out CV is
exclusively considered as the data-driven implementation method. CV is popularly employed for LL regression smoothing,
and Choi and Hall (1998, Remark 2.6) themselves anticipate that it may serve as a straightforward implementation method
for their skewed estimation. While a plug-in method based on the leading two terms in the MISE could be another option,
it does not look attractive because in practice it requires to estimate many derivatives of m and f (as well as themselves), as
indicated in Theorems 1 and 2.
Details of the CV employed in this study are as follows. Let m̄−i,b (· ) be a regression estimate using the smoothing parameter b and the sample with the ith observation eliminated. Then, the CV criterion function becomesCV (b) =n

{Yi − m̄−i,b (Xi )}2 .

i=1

The data-driven optimal global smoothing parameter is deﬁned as the minimizer of CV(b). As in the oracle case, the minimizer is found on an equally-spaced grid of 40 points over the interval [0.01, 0.40].

120

B. Funke and M. Hirukawa / Econometrics and Statistics 20 (2021) 109–130

Fig. 2. Pointwise Averages of Curve Estimates (Gamma Case; Oracle; n = 200).
Note: Thin dash lines represent true regression curves. Dash-dot, thin solid, dot, dash, and solid lines correspond to the Epanechnikov LL, Epanechnikov
skewed, gamma LL, gamma local cubic, and limit gamma skewed estimates, respectively.

Table 2 and Fig. 3 report the global and local performance measures of the gamma LL, gamma skewed with λ = 1 andlimit gamma skewed estimators. The performance measures are exactly the same as in Table 1 and Fig. 1. Again the resultsof local measures from Models A and B are qualitatively similar, and thus only the former is presented in Fig. 3. Notice thatin this case the random number generator is initialized using the same seed number as in Case #1. Therefore, Monte Carlosamples for two cases are identical and thus we may compare the results in Tables 1 and 2 or Figs. 1 and 3 directly.
It can be found in Table 2 that the CV-based smoothing parameter values are fairly close enough to the corresponding
oracle ones. Because kernel estimators are sensitive to choices of tuning parameter values, both global and local performancemeasures become worse. However, thanks to small difference between oracle and CV-based smoothing parameter values,
deterioration in the measures by switching from the former to the latter is minimal at best. As indicated in the previous
section, global measures of two skewed estimators look alike again and dominate those of the gamma LL estimator. In termsof local measures, there is not much difference in variance among the three estimators. It is not surprising that variances of
two skewed estimators are close to each other. This simply reﬂects that dominant terms of their variances are identical andindependent of λ. Also observe that the skewed estimators tend to have better bias properties as the sample size increases.
All in all, Monte Carlo results indicate that gamma skewed estimators work well in practice and tend to outperform the
gamma LL estimator for the sample sizes that are typically used in empirical applications. To implement the estimators, we
may rely on CV.
3.2. On the Beta Skewed Estimator
The second Monte Carlo study deals with a regressor with support on [0, 1]. The regressor X is generated as absolutevalues of N(0, 0.32 ) random variables truncated on [−1, 1]. The true regression curve is m(x ) = (x − 1/2 )2 . The sample size
is n ∈ {10 0, 20 0, 40 0}, and the number of replications is 10 0 0. This design is again the same as in the one in Chen (2002),
and thus our results are directly comparable with the results therein.

B. Funke and M. Hirukawa / Econometrics and Statistics 20 (2021) 109–130

121

Table 2Monte Carlo Results (Gamma Case; CV).
Model

n

Estimator

Ave

SD

Ave

SD

Ave

SD

A

100

ˆ G (x )m
˜ G,1 (x )m
˜ G,∞ (x )m

0.09290.07380.0735(0.0356)(0.0386)(0.0384)

0.15080.12090.1203(0.0449)(0.0482)(0.0469)

0.02880.03200.0321(0.0195)(0.0199)(0.0202)200

ˆ G (x )m
˜ G,1 (x )m
˜ G,∞ (x )m

0.08070.05500.0534(0.0214)(0.0209)(0.0204)

0.12880.09400.0912(0.0308)(0.0317)(0.0298)

0.03070.03420.0338(0.0143)(0.0154)(0.0153)400

ˆ G (x )m
˜ G,1 (x )m
˜ G,∞ (x )m

0.08210.05050.0478(0.0187)(0.0184)(0.0177)

0.12410.08430.0800(0.0268)(0.0286)(0.0269)

0.03470.03800.0375(0.0130)(0.0138)(0.0138)100

ˆ G (x )m
˜ G,1 (x )m
˜ G,∞ (x )m0.14240.10540.1063(0.0798)(0.0899)(0.0900)

0.23390.13490.1366(0.0927)(0.0728)(0.0723)0.02710.03110.0308(0.0136)(0.0162)(0.0164)200

ˆ G (x )m
˜ G,1 (x )m
˜ G,∞ (x )m

0.11400.06380.0639(0.0382)(0.0407)(0.0401)

0.21290.0941
0.0943(0.0659)(0.0430)(0.0416)0.02980.03390.0333(0.0109)(0.0131)(0.0129)400

ˆ G (x )m
˜ G,1 (x )m
˜ G,∞ (x )m

0.11770.05450.0541(0.0318)(0.0296)(0.0289)

0.22850.08260.0815(0.0637)(0.0407)(0.0387)

0.03430.03800.0374(0.0110)(0.0126)(0.0125)

B

RISE

IAD

Tuning Parameter

Fig. 3. Squared Bias and Variance of Curve Estimates (Gamma Case [Model A]; CV).
Note: Dot, dash, and solid lines correspond to the gamma LL, gamma skewed with λ = 1, and limit gamma skewed estimates, respectively.

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B. Funke and M. Hirukawa / Econometrics and Statistics 20 (2021) 109–130
Table 3Monte Carlo Results (Beta Case; Oracle).
n

Estimator

RISE

IAD

Tuning Parameter

Ave

SD

Ave

SD

Ave

SD

100

ˆ E (x )m
˜ E,λ∗ (x )m
ˆ B (x )mˆ B,C (x )m
˜ B,0.1 (x )m
˜ B,1 (x )m
˜ B,∞ (x )m

0.03850.03920.02700.03540.0266
0.02630.0264(0.0247)(0.0246)(0.0219)(0.0249)(0.0221)(0.0226)(0.0228)0.02580.02540.01850.02270.01800.01760.0176(0.0135)(0.0139)(0.0126)(0.0150)(0.0128)(0.0132)(0.0133)

0.31380.30170.08810.25410.08640.08540.0844(0.1159)(0.1122)(0.0571)(0.1493)(0.0564)(0.0545)(0.0538)200

ˆ E (x )m
˜ E,λ∗ (x )m
ˆ B (x )mˆ B,C (x )m
˜ B,0.1 (x )m
˜ B,1 (x )m
˜ B,∞ (x )m

0.02370.02480.01980.02290.01930.01900.0191(0.0154)(0.0167)(0.0154)(0.0178)(0.0157)(0.0162)(0.0163)0.01720.01650.01380.01430.01310.01260.0127(0.0087)(0.0095)(0.0086)(0.0096)(0.0087)(0.0090)(0.0091)0.31940.2976
0.07620.27960.07540.07350.0725(0.0894)(0.0810)(0.0440)(0.1363)(0.0426)(0.0417)(0.0407)400

ˆ E (x )m
˜ E,λ∗ (x )m
ˆ B (x )mˆ B,C (x )m
˜ B,0.1 (x )m
˜ B,1 (x )m
˜ B,∞ (x )m

0.01690.01780.01510.01480.01450.01430.0143(0.0097)(0.0115)(0.0101)(0.0112)(0.0103)(0.0108)(0.0109)

0.01290.01180.01080.00940.00990.00950.0095(0.0059)(0.0066)(0.0055)(0.0057)(0.0056)(0.0058)(0.0059)

0.28790.2572
0.06390.2886
0.0642
0.06190.0612(0.0874)(0.0710)(0.0350)(0.1315)(0.0337)(0.0329)(0.0319)

Table 4Monte Carlo Results (Beta Case; CV).
nEstimator

RISE

IAD

Tuning Parameter

Ave

SD

Ave

SD

Ave

SD

100

ˆ B (x )m
˜ B,1 (x )m
˜ B,∞ (x )m

0.05400.04660.0472(0.0976)(0.0601)(0.0726)

0.03110.02780.0280(0.0410)(0.0272)(0.0328)

0.05390.07260.0728(0.0331)(0.0381)(0.0372)200

ˆ B (x )m
˜ B,1 (x )m
˜ B,∞ (x )m

0.03820.03040.0306(0.0493)(0.0304)(0.0306)

0.02140.01790.0180(0.0195)(0.0134)(0.0135)

0.03940.06130.0603(0.0217)(0.0275)(0.0271)400

ˆ B (x )m
˜ B,1 (x )m
˜ B,∞ (x )m

0.03100.02190.0221(0.0511)(0.0178)(0.0188)0.01620.01290.0129(0.0142)(0.0079)(0.0082)0.02940.05180.0508(0.0151)(0.0211)(0.0207)ˆ E (x ) (the Epanechnikov LL estimator), m
˜ E,λ∗ (x ) (the Epanechnikov skewed estimator with
Estimators for m(x) include m
λ∗ = 0.0352), mˆ B (x ) (the beta LL estimator), mˆ B,C (x ) (the beta local cubic estimator), m˜ B,0.1 (x ) (the beta skewed estimator
˜ B,1 (x ) (the beta skewed estimator with λ = 1), and m
˜ B,∞ (x ) (the limit beta skewed estimator). If a beta
with λ = 0.1), m
skewed estimator is not well-deﬁned for x in the vicinity of 0 or 1, then it is replaced by the beta LL estimator evaluated at
the same design point. To implement all these estimators, we choose their tuning parameters via the oracle and CV methods
as before.
The global performance measures that assess an estimator include the RISE and IAD, where each integral is approximatedby the trapezoidal rule on an equally-spaced grid of 101 points over the interval [0, 1]. As the local performance measures,the squared bias and variance evaluated at eleven design points x ∈ {0.0, 0.1, . . . , 0.9, 1.0} over 10 0 0 Monte Carlo samples
are computed.
Results of global performance measures are summarized in Tables 3 (oracle tuning parameters) and 4 (CV tuning parameters). To save space, we only report Fig. 4, which presents local measures of the estimators using oracle tuning parameters,
as in Chen (2002).
The results in Tables 3 and 4 conﬁrm our ﬁndings in Tables 1 and 2. Each estimator numerically indicates consistency
in that their global performance measures shrink with the sample size. The three beta skewed estimators outperform the
˜ B,1 (x ) and m
˜ B,∞ (x ) look alike and are superior to those of m
˜ B,0.1 (x ). The only difference
others. Again, performances of mis that the beta cubic estimator does not perform well for n = 10 0, 20 0. Its performances are rather close to those of two
Epanechnikov estimators. We now check the local performance measures in Fig. 4. As in Fig. 1, the vertical axis in eachpanel of Fig. 4 is the natural logarithm of either squared bias or variance. The ﬁgure shows that the beta cubic estimator
has the smallest bias over almost the entire unit interval, whereas the limit beta skewed estimator attains the smallest
variance around [0.4, 1]. The beta cubic estimator tends to have by far the largest variance around [0.1, 0.4], which appears

B. Funke and M. Hirukawa / Econometrics and Statistics 20 (2021) 109–130

123

Fig. 4. Squared Bias and Variance of Curve Estimates (Beta Case; Oracle).
Note: Dash-dot, thin solid, dot, dash, and solid lines correspond to the Epanechnikov LL, Epanechnikov skewed, beta LL, beta local cubic, and limit beta
skewed estimates, respectively.

to result in its mediocre global performance measures for n = 10 0, 20 0.8 Furthermore, Table 4 indicates that to implement
the beta skewed estimation, CV works well in practice.

4. An Empirical Illustration
4.1. The ModelIn this section the skewed estimators are applied to a real data set. Inspired by Tripathi and Kim (2003), we concentrateon the problem of estimating a production function with homogeneity of degree one (or linear homogeneity) imposed.
Suppose that there is the technological relationship in which two factors, namely, capital and labor inputs (K, L), can produce
output Y. This relationship can be expressed as a production function and formulated in a regression format asg(K, L ) = E ( Y |K, L ) ⇒ Y = g(K, L ) + ε
for an unknown function g and the regression error ε . With no further restrictions on g, we would have to run a multiple
nonparametric regression of Y on (K, L ) = (k, l ) to estimate g(k, l). However, as is often the case with estimating production
functions, if it is known a priori (or it may be reasonably assumed) that g is homogeneous of degree one in (K, L) (i.e., thetechnology exhibits constant returns to scale), then we can rewrite the estimation problem as that of a simple regression.

8
It seems that the mediocre performance of the beta cubic estimator is attributed to sparseness of the data on the right boundary. The lesson learnt hereis that a guard against sparseness such as the one used in Choi and Hall (1998, Section 3.2) should be considered to stabilize local polynomial estimation,
regardless of the choice of a kernel.

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B. Funke and M. Hirukawa / Econometrics and Statistics 20 (2021) 109–130
Table 5
Summary Statistics (in millions).
Variable

Ave

SD

SK

Min

Med

Max

W
Z

0.13190.11810.17290.07704.15672.82970.00940.02810.0793
0.09601.58390.6552

Note: Ave = average; SD = standard deviation; SK = skewness; Min = minimum; Med = median; and Max = maximum.

Speciﬁcally, the linear homogeneity implies

K

g(K, L ) = L · gL, 1 := L · mK Lso that the original regression can be reformulated as

 

Y
K
=m
L
L

+

ε
L

⇒ Z = m(W ) + 

by letting (Z, W,  ) := (Y/L, K/L, ε /L). The entire estimation procedure then takes two steps. The ﬁrst step is to run a simple
nonparametric regression of Z on W = w = k/l to estimate m(w ). Let m̄(w ) be a nonparametric estimate of m(w ). Then,in the second step, it follows from the relation w = k/l that g(k, l) can be estimated as ḡ(k, l ) = l · m̄(w ). Tripathi and Kim(2003) investigate LL estimation of m(w ) using standard symmetric kernels and develop statistical properties of the estimator.
4.2. The Data
The National Bureau of Economic Research (“NBER”) makes an industry-level panel data set public on its web page(http://www.nber.org/nberces/). The data set, the NBER-CES Manufacturing Industry Database, is provided in two versions,depending on the classiﬁcation of manufacturing industries; see Bartelsman and Gray (1996) and Becker et al. (2016) for
more details of the database. We choose the version with industries classiﬁed by the six-digit level of the North American
Industry Classiﬁcation System (“NAICS”) in 1997.
The 1997 NAICS version contains annual industry-level data of 473 industries from 1958 to 2011. The sample in 20 0 0
is used throughout. Because the database includes a few candidates for each of Y, K and L, we follow Kumbhakar et al.
(2012), for instance, and deﬁne VADD (= total value added in $1m), CAP (= total real capital stock in $1m) and EMP (= total
employment in 10 0 0s) as Y, K and L, respectively. Subsequently two variables (W, Z ) = (K/L, Y/L ) can be constructed.
Before proceeding, we ﬁnd a few observations with extremely large values of Z. Because they are highly likely to inﬂuence
our estimation results, it is desirable to eliminate them as outliers. For this purpose, we employ the generalized boxplot
by Bruffaerts et al. (2014).9 With its aid, two observations are detected as outliers and eliminated from the sample. The
eliminated industries are “Flavoring Syrup and Concentrate Manufacturing” (NAICS Code: 311930) with (W, Z ) = (0.24, 1.18 )and “Cigarette Manufacturing” (NAICS Code: 312221) with (W, Z ) = (0.43, 2.19 ). As a result, the sample size is 471.
Table 5 presents summary statistics of (W, Z), where their units of measurement are converted in millions. The regressor
W is nonnegative by construction and has a positive skewness; indeed, the scatter plot of data points in Fig. 5 indicatesconcentration of observations of W in the vicinity of the origin and a long right-tail of its marginal distribution. Asymmetrickernel smoothing is likely to work well with such data, and thus we are motivated to estimate the unknown curve m (w )by our gamma skewed estimators.
4.3. Estimation Results
In what follows, the gamma skewed with λ = 1 and limit gamma skewed estimators, as well as the gamma LL estimator,
are employed for the curve estimation. Their smoothing parameter values are chosen via CV. The values for the gamma LL,
gamma skewed with λ = 1, and limit gamma skewed estimators are 0.0733, 0.0818 and 0.0843, respectively.
Furthermore, researchers often specify the production function g in a simple, parametric form. The Cobb-Douglas production function (Cobb and Douglas, 1928) is among the most popular ones. When it is homogenous of degree one, it canbe expressed in a multiplicative error regression model as

Y = AK α L1−α u,where A > 0 and α ∈ (0, 1) are parameters and u > 0 is the regression error. A simple manipulation yields

ln
9

Y 
L

= ln A + α ln

K L+ ln u ⇒ ln Z = β0 + β1 ln W + v

The generalized boxplot is designed for skewed and/or heavy-tailed distributions and thus ﬁts into our context. It is available on Stata as the command

gboxplot.

B. Funke and M. Hirukawa / Econometrics and Statistics 20 (2021) 109–130

125

Fig. 5. Estimates of a Linear Homogeneous Production Function.

by letting (β0 , β1 , v ) := (ln A, α , ln u ), and parameters (β 0 , β 1 ) can be estimated by running the ordinary least squares(“OLS”) from the regression of ln Z on (1, ln W). The OLS estimates of (β 0 , β 1 ) are βˆ0 , βˆ1 = (−1.1537, 0.4590 ). We then$Z .10 More speciﬁcally, the ﬁnal parametric estiemploy the swearing estimate by Duan (1983) to back-transform Zˆ from ln ˆˆ CD (w ) = γˆ exp βˆ0 · wβ1 , where γˆ is the average ofmate of m(w ) is m%

exp ln Zi − βˆ0 − βˆ1 ln Wi

&n

i=1

.

A parametric and three nonparametric curve estimates of m(w ), as well as a scatter plot of data points, are drawn
in Fig. 5. It can be immediately found that shapes of three nonparametric estimates considerably differ from the one ofthe parametric one, which is globally concave by construction. In contrast, each of the nonparametric estimates appears
to be roughly monotone with an inﬂection point from concavity to convexity around w = 0.8. Moreover, as deduced from
the Monte Carlo results in Section 3, two skewed estimates are nearly identical. Although it is impossible to judge which
estimate is closest to the true curve, the substantial difference between nonparametric and parametric estimates indicatesthat the parametric, Cobb-Douglas speciﬁcation may be incorrect.
5. Conclusion
This paper has demonstrated that under a suitable choice of weights and shifts, the skewing method by Choi and Hall
(1998) can accelerate bias convergences of LL regression estimators using asymmetric kernels from O(b) to O(b2 ) as b → 0
while not inﬂating their variances in an order of magnitude. A remarkable difference can be found in the choice of the shiftthat can lead to a faster bias convergence when symmetric and asymmetric kernels are employed. While for a given weight
λ, the shift is constant regardless of the position of the design point for symmetric kernels, it varies with the design point
10
We have also applied the back-transformation method proposed in Chapter 6 of Wooldridge (2013), which is valid under the assumption of homoscedastic normal errors. Because the result is qualitatively similar, it is unreported in Fig. 5.

126

B. Funke and M. Hirukawa / Econometrics and Statistics 20 (2021) 109–130

for asymmetric kernels. Two versions of the skewed estimator (i.e., the case with a ﬁxed λ and the limit case with λ → ∞)are studied, and their large- and ﬁnite-sample properties are explored. In particular, Monte Carlo results indicate that the
gamma (beta) skewed with λ = 1 and limit gamma (beta) skewed estimators have equally nice ﬁnite-sample properties.
In fact, their ﬁnite-sample performances are superior to those of the Epanechnikov LL, Epanechnikov skewed and gamma(beta) LL estimators, and they are comparable with those of the gamma (beta) local cubic estimator. CV is found to be
useful for implementing the skewed estimation. Finally, the skewed estimator is applied to the US manufacturing industrylevel data, and a linear-homogeneous production function is estimated nonparametrically. We ﬁnd that the nonparametriccurve estimate substantially differs from the one implied by the popular, Cobb-Douglas speciﬁcation.
Declaration of Competing Interest
All authors have participated in (a) conception and design, or analysis and interpretation of the data; (b) drafting thearticle or revising it critically for important intellectual content; and (c) approval of the ﬁnal version. This manuscript has
not been submitted to, nor is under review at, another journal or other publishing venue. The authors have no aﬃliation
with any organization with a direct or indirect ﬁnancial interest in the subject matter discussed in the manuscript.
Acknowledgements
The authors would like to thank the co-editor Ana Colubi, an associate editor, two anonymous referees, Artem Prokhorov,and the participants of CMStatistics 2018, DAGStat 2019 and EcoSta 2019 for their constructive comments and suggestions.
The ﬁrst author gratefully acknowledges ﬁnancial support from the German Science Foundation (DFG) through the Collaborative Research Center SFB 823. The second author gratefully acknowledges ﬁnancial support from Japan Society for the
Promotion of Science (grant numbers 15K03405 and 19K01595).
A Appendix
A.1. A Useful Lemma
Before proceeding, we summarize useful results on the moments of certain gamma and beta random variables as alemma, which is frequently applied for the bias and variance approximations.
A.1.1. Lemma A.1rd(i) Let θ jx = G( jx/b + 1, b/ j ) for x ∈ R+ and j = 1, 2. Then, the moments ν j,r = E θ jx − x for r = 1, 2, . . . , are:bjb
ν j,2 = 2 ( jx + 2b);j

ν j,1 = ;ν j,3 =
ν j,4 =b2
(5 jx + 6b);j3


b2  2 2
3 j x + 26 jxb + 24b2 ; and4j ν j,r = O b3 f or r ≥ 5.

d(ii) Let η jx = Beta{ jx/b + 1, j (1 − x )/b + 1} for x ∈ [0, 1] and j = 1, 2. Then, as b → 0, the moments υ j,r = E η jx − x
1, 2, . . . , can be approximated by:υ j,1 =
υ j,2 =
υ j,3 =
υ j,4 =

 1 − 2x jb−

% 2 1 − 2x &() 2

%x 1 − x &
(
)

b+

j

 
2 − 11x + 11x2
b2 + O b3 ;2j

% 5x 1 − x 1 − 2x &
(
)(
) 2


j2



 

b + O b3 ; 
3x2 (1 − x )2 2
b + O b3 ; andj2

 υ j,r = O b3 f or r ≥ 5.

 

b + O b3 ;j2

r

for r =

B. Funke and M. Hirukawa / Econometrics and Statistics 20 (2021) 109–130

127

A.1.2. Proof of Lemma A.1
These results immediately follow from evaluating the moments of the gamma and beta random variables (for (i) and (ii))
and expanding them around b = 0 (for (ii) only). 
A.2. Proof of Theorem 1
The proof closely follows the Proof of Theorem 2.1 in Choi (1998). For brevity, the subscript G and the design point x are
˜ λ, = m
˜ G,λ, (x ),suppressed wherever no confusions may occur. In addition, we introduce short-handed notations such as mˆ∗ = mˆ G ( x|x∗ ), μ p = E sGp (x ) , μ∗p = E sGp (x∗ ) , m( p) = m( p) (x ), m( p)∗ = m( p) (x∗ ), f ( p) = f ( p) (x ), f ( p)∗ = f ( p) (x∗ ), and somon, where sGp (x) is deﬁned in (1). The proof takes the following steps:
1. Approximate the bias of each of three LL estimators, conditional on X = x, up to the O(b2 ) term. ˜ λ, .
2. Find the weights (λ1 , λ2 ) and shifts ( 1 , 2 ) that can eliminate the O(b) and O(b3/2 ) terms in Bias m˜ λ, .
3. Substitute the results in Step 2 into the O(b2 ) term in Bias mˆ ∗ asStep 1. By a similar argument to the proof of Theorem 1 in Chen (2002), we can expand E mˆ ∗ = m∗ + (x − x∗ )m(1 )∗ +E mQ ∗ + R∗
+ O n−1 b−1 ,∗∗2
μ μ2 − μ1∗0where R∗ = R(x∗ ) is the remainder term in a smaller order of magnitude by Taylor’s theorem, and Q ∗ = Q (x∗ ) is in the form
of

Q∗ =


m(2 )∗  ∗2μ2 − μ∗1 μ∗3 + (x − x∗ )(μ∗0 μ∗3 − μ∗1 μ∗2 )2m(3 )∗+{(μ∗2 μ∗3 − μ∗1 μ∗4 ) + (x − x∗ )(μ∗0 μ∗4 − μ∗1 μ∗3 )}6m(4 )∗+{(μ∗2 μ∗4 − μ∗1 μ∗5 ) + (x − x∗ )(μ∗0 μ∗5 − μ∗1 μ∗4 )}.
24

Moreover, μ∗p admits the expansionμ∗p = f ∗ ν1∗,p + f (1)∗ ν1∗,p+1 +
f (2 )∗ ∗ν+ O ν1∗,p+3 ,2 1,p+2dwhere ν1∗,r = E (θx∗ − x∗ ) for θx∗ = G(x∗ /b + 1, b). Then, it follows from Lemma A.1(i) that approximations to μ∗p for p =0, 1, . . . , 5 are:
r



 
x∗
μ∗0 = f ∗ + f (1)∗ + f (2)∗ b + O b2 ;2 ∗ ∗ (1)∗   (1)∗ 5 ∗ (2)∗  2 ∗
μ1 = f + x f
b+ 2f
+ x f
b + O b3 ;
2


 
3
μ∗2 = x∗ f ∗ b + 2 f ∗ + 5x∗ f (1)∗ + x∗2 f (2)∗ b2 + O b3 ;
2


 
μ∗3 = 5x∗ f ∗ + 3x∗2 f (1)∗ b2 + O b3 ; μ∗4 = 3x∗2 f ∗ b2 + O b3 ; and 
μ∗5 = O b3 .

Putting x∗ = x + b1/2 , taking Taylor expansions of m(p)∗ and μ∗p around x∗ = x (i.e.,





 

b1/2 = 0), doing some tedious but

straightforward calculations, and recognizing that O n−1 b−1 = o b2 due to Assumption 3, we ﬁnally have



ˆ
E m

∗




1
=m+
x − 2 m (2 ) b −
2



 

x2 f ( 2 )f (1 )
− 1+x
−
+ x2
f
2 f

+


+

x
f (1 )− 2 2+x3fm (3 ) +2
3
f (1 )+xm(2 ) + m(3 ) b3/2
2
f
3

f (1 )
f

'

2

+ 2

2
4
x2
− x−
8
4
8



f (1 )+xf'f (2 )−ff (1 )f2((m (2 ) m(4 ) b2 + o b2 .



ˆ G x|x − 1 b1/2 , m
ˆ G ( x|x ) and m
ˆ G x|x + 2 b1/2 immediately follow from letting = − 1 , 0, 2 on
The bias expansions of m
ˆ G ( x|x ) = m
ˆ G (x )the right-hand side, respectively. In particular, a second-order bias expansion of the gamma LL estimator m

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B. Funke and M. Hirukawa / Econometrics and Statistics 20 (2021) 109–130

takes the form of 

x2 f ( 2 )xf (1 )
ˆ G ( x ) = m (2 ) b + − 1 + x
Bias m
−
+ x2
2
f
2 f

2

f (1 )fxx
m (2 ) + m (3 ) +
3

2

8

)
m (4 ) 

b2 + o b2 .

Step 2. By the results in Step 1,



˜ λ,
E m



=m+



1
(λ1 + 1 + λ2 )x − λ1 21 + λ2 22 m(2) b2(λ1 + 1 + λ2 )





1
3
f (1 )1
++x
m (2 ) +λ2 32 − λ1 31 m(3) b3/2
(λ2 2 − λ1 1 )λ1 + 1 + λ2
2
f
3
 

x2 f ( 2 )f (1 )+ − 1+x
−
+ x2
f
2 f
1
−
λ1 + 1 + λ2




λ

λ

f (1 )+ 2+x
m (3 )
f

λ

2
1 1 +

+

2
2 2

''

f (1 )

21 1

2

f (1 )ff

m (2 ) +

'
+x

2
x
+ 1
4
8

x ( 3 ) x2 ( 4 )
m + m
3
8

f (2 )
−
f

+λ

2
2 2

f (1 )
f

2

2
x
+ 2
4
8

((
m (2 )

m (4 ) 

b2 + o b2 .

Therefore, both O(b) and O(b3/2 ) terms vanish if and only if(λ1 + 1 + λ2 )x − λ1 21 + λ2 22 = 0.
λ2 2 − λ1 1 = 0
λ2 32 − λ1 31 = 0

Solving this system for (λ1 , λ2 ,λ1 = λ2 = λ and

1 =

1,

2 =

2 ) yields= (λ, x ) =2λ + 1
x.
2λ



˜ λ,
Step 3. Substituting (λ, ) into the coeﬃcient on b2 in Bias m



x2 f ( 2 )f (1 )
1+x
−
+ x2
f
2 f

−
+



1 + 2xﬁnally givesxx2
m (2 ) + m (3 ) + m (4 ) − x
38f (1 )
1
2λ + 1m (3 ) + x+
m (4 )
f
4
16λ

2+x

=−

f (1 )
f


2





f (1 )+xf'f (2 )−ff (1 )f2(m (2 )

f (1 )
x2 f ( 2 )5
f (1 )x2
1+
m (2 ) + x+xm (3 ) +1+m (4 ) .
f
2 f
3
f
4
4λ

Denoting the right-hand side by BG,λ (x; m, f ) completes the proof. 
A.3. Proof of Corollary 1
This has been already established in Step 1 of the proof of Theorem 1. 
A.4. Proof of Theorem 2
The proof closely follows the Proof of Theorem 2.2 in Choi (1998). In this proof, we mean by the symbols “Var”
and “Cov” the conditional variance and covariance, conditional on X = x, respectively. In addition to the short-handed˜λ = m˜ G,λ (x ), v0 = V ar m
ˆ G ( x|x ) , c∗† =notations in the proof of Theorem 1, we use short-handed notations such as m

ˆ G ( x|x∗ ), m
ˆ G x|x†
Cov m







,


 



αˆ ∗ , βˆ ∗ = αˆ G (x∗ ), βˆG (x∗ ) , s∗p = sGp (x∗ ), t p∗ = tGp (x∗ ), σ 2 = σ 2 (x ), and so on, where x∗ , x† =x + b1/2 , x − b1/2 and sGp (x) and tGp (x) are deﬁned in (1) and (2), respectively. Then,˜ λ) =V ar (mλ2 v∗ + v0 + λ2 v† + 2λc0∗ + 2λc0† + 2λ2 c∗†.
(2λ + 1 )2

We start from working on the case for interior x and concentrate on approximating v∗ . Observe that

 


 
v∗ = V ar αˆ ∗ + 2 bV ar βˆ ∗ − 2 b1/2Cov αˆ ∗ , βˆ ∗ := A1 + A2 − 2A3 (say),B. Funke and M. Hirukawa / Econometrics and Statistics 20 (2021) 109–130

where

129

 ∗ ∗
−2 ∗2∗∗ ∗∗ ∗μ0 μ2 − μ∗2μ2 V ar (t0∗ ) + μ∗21
1 V ar (t1 ) − 2μ1 μ2Cov (t0 , t1 ) + R1 ,

−2 2
∗
∗2
∗
∗ ∗
∗ ∗
A2 = μ∗0 μ∗2 − μ∗2b μ∗210 V ar (t1 ) + μ1 V ar (t0 ) − 2μ0 μ1Cov (t0 , t1 ) + R2 ,
 ∗ ∗



−2 1/2
∗ ∗
∗ ∗
∗
∗ ∗
∗
A3 = μ0 μ2 − μ∗2bμ∗0 μ∗2 + μ∗2
1
1 Cov (t0 , t1 ) − μ1 μ2V ar (t0 ) − μ0 μ1V ar (t1 ) + R3 ,A1 =and it can be shown that the remainder terms R1 , R2 and R3 are in smaller orders of magnitude. By the proof of
Theorem 1 and Assumption 3,μ∗0 ∼ f ∗ , μ∗1 ∼ f ∗ + x∗ f (1)∗ b and μ∗2 ∼ x∗ f ∗ b

so that∗ ∗2μ∗0 μ∗2 − μ∗21 ∼ x f b.

In addition,

Cov t p∗ , tq∗ =1
E σ 2 (X )(X − x∗ ) p+q KG2(x∗ ,b) (X ) {1 + o(1 )},
n

where

E σ 2 (X )(X − x ) p+q KG2(x,b) (X ) = Bb (x )E (θ2x − x )p+q

σ 2 ( θ2 x ) f ( θ2 x ) ,andBb (x ) :=b−1/2
(2x/b + 1 )∼ √ √b22x/b+1  2 (x/b + 1 )2 π x

by Chen (20 0 0b). It also follows from Lemma A.1(i) that⎧ 2
⎨σ % f (1 ) &p+q
E (θ2x − x ) σ 2 (θ2x ) f (θ2x ) ∼ 12 σ 2 f + x σ 2 f
b
⎩x 2
σ fb
2

if p + q = 0
if p + q = 1 .
if p + q = 2

After combining these results, taking a Taylor expansion around x∗ = x (i.e., b1/2 = 0) and doing some tedious calculations,
we can ﬁnd that

A1 =

1
σ2
√ √ {1 + o(1 )},1/
2
nb 2 π x f

A2 =

1
σ2
√ √
1
/
2
nb 2 π x f



2

2x

{1 + o(1 )},

and A3 = o n−1 b−1/2 . Therefore,

v∗ =

1
σ2
√ √
nb1/2 2 π x f

1+

2

2x

{1 + o(1 )}.

Expanding other terms similarly, we have the following results for interior x:

v† =

1
σ2
√
√
nb1/2 2 π x f

v0 =

1
σ2
√ √ {1 + o(1 )};1/
2
nb 2 π x f

c 0∗ =

1
σ2
√ √
1
/
2
nb 2 π x f1+

c 0† =

1
σ2
√ √
1
/
2
nb 2 π x f

1+

c∗† =

1
σ2
√ √
1
/
2
nb 2 π x f

1+

Using these and

=



1+

2

2x

2

2x
2

2x

{1 + o(1 )};{1 + o(1 )};{1 + o(1 )}; and

4
3 2
+ 2
2x
x

{1 + o(1 )}.

(2λ + 1 )x/(2λ ) ﬁnally establishes that for interior x,

130

B. Funke and M. Hirukawa / Econometrics and Statistics 20 (2021) 109–1301σ21˜ λ) =V ar (m√ √ {1 + o(1 )}nb1/2 2 π x f(2λ + 1 )2
+ 2λ 1 +=2

2x+ 2λ 1 +2

2x
λ2 1 ++ 2λ2 1 +2

2x

+ 1 + λ2 1 +

4
3 2
+ 2
2x
x



2

2x

1
5σ 2
√ √ {1 + o(1 )}.
nb1/2 4 π x f

On the other hand, for boundary x, each of Bb (x), Bb (x∗ ) and Bb (x†) are shown to be O b−1 , where the coeﬃcient on

 −1 

each O bterm depends on (κ , λ). Modifying the expansions of μp and E (θ2x − x ) −1 −1 κ b{1 + o(1 )} leads to V ar (m˜ λ ) = O n bp+q

σ 2 (θ2x ) f (θ2x ) in light of x =. 

A.5. Proof of Proposition 1
The results can be obtained by taking limits of the corresponding parts as λ → ∞ in the proofs of Theorems 1 and 2. 
A.6. Proof of Theorem 3
Replacing Part (i) of Lemma A.1 in the proofs of Theorems 1 and 2 with Part (ii) and doing some tedious but straightforward calculations can establish the theorem. 
A.7. Proof of Corollary 2This can be established in the middle of the proof of Theorem 3. 
A.8. Proof of Proposition 2
The proof strategy is exactly the same as in the proof of Proposition 1. References
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