## A Semiparametric Approach to the

## Estimation of the Stochastic Frontier Model

## with Time-Variant Technical Efficiency

Wen- Shuenn Deng

∗

### Department of Statistics

### Tamkang University

Ta i-Hsin Huang

### Department of Money and Banking

### National Chengchi University

**Keywords: Semiparametric stochastic frontier model, Kernel estimators, Time-variant**
technical efficiency, Composed error

**JEL classification: C14, C15, C23**

∗ _{Correspondence: Wen-Shuenn Deng, Department of Statistics, Tamkang University, Taipei County}
251, Taiwan. Tel: (02) 2621-5656 ext. 3393; Fax: (02) 2620-9732; E-mail: 121350@mail.tku.edu.tw.
The authors wish to express their sincere thanks to the editor and two referees for their constructive and
valuable suggestions and comments that have greatly improved this presentation.

362H: 2008/07/22 09:20 AM page:168

### ABSTRACT

This article extends the semiparametric stochastic frontier model developed by Fan et

al. (1996) in two ways. First, it proposes a semiparametric estimation procedure suitable for the case when panel data are available. Second, the strong assumption of time-invariant technical eﬃciency is relaxed such that Battese and Coelli’s (1992) model can be applied empirically. Our procedure is particularly useful in the examination of technical eﬃciency with respect to production, cost, and proﬁt frontiers. Monte Carlo experiments and an empirical application of the proposed procedure employing panel data on 86 Taiwanese electronic ﬁrms over the period 1995-2001 are exempliﬁed.

1. INTRODUCTION

Studies on the technical efficiency (TE) of a producer have caught much attention ever since the 1970s. In the area of production efficiency, there are two main approaches to the specification and estimation of production (cost and profit) frontiers. Data envel-opment analysis (DEA), initiated by Farrell (1957), employs a mathematical program-ming technique without the need to specify an explicit functional form for either the production, cost, or the revenue frontier. Such a function-free approach avoids possi-ble specification errors that are frequently criticized due to the use of the econometric approach to the frontier analysis pioneered by Aigner et al. (1977), Meeusen and van den Broeck (1977), and Jondrow et al. (1982). However, the econometric approach is thought to be advantageous over the DEA approach thanks to its ability in dealing with statistical noise and measurement error.

To remove the restrictive assumptions on the functional forms imposed by the econometric approach, researchers in the past few decades have made efforts to re-lax these constraints. Yatchew (1998) provided an excellent review on the issue of nonparametric regression techniques in economics by offering a collection of valid techniques for analyzing nonparametric and semiparametric regression models. H¨ardle (1990) also concentrated on the statistical aspects of nonparametric regression smooth-ing from an applied point of view. Fan et al. (1996) first extended the linear stochastic frontier model proposed by Aigner et al. (1977) to a semiparametric frontier model ap-plicable to cross-sectional data. They judiciously constructed pseudo-likelihood vari-ance estimators of the two-error components of the models using a kernel estimation of the conditional mean function.

Schmidt and Sickles (1984) pointed out three difficulties with cross-sectional stochastic production frontier models. Of them, the TE of producers fails to be con-sistently estimated, i.e., the variance of the conditional mean or the conditional mode of the one-sided error, given the composed errors for each individual producer, does not converge to zero as the size of the cross-section grows. These drawbacks are avoidable if panel data are available. Having access to panel data will result in con-sistent estimators of TE for each producer as the number of time periods goes to in-finity. The assumption of the time-invariant TE, implied by a standard fixed-effects or random-effects model, appears implausible especially when the firm operates within a

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*Academia Economic Papers 36:2 (2008)*

competitive atmosphere. This form of rigidity hinders firms from optimally selecting input levels in each period. Such a firm is hardly viable. Hence, the specification of a time-varying TE tends to be preferable. The longer the panel is, the more prefer-able such specification becomes, particularly if the market structure is close to perfect competition.

This paper proposes a partially linear stochastic frontier model under the frame-work of panel data, which is especially useful in the estimation of production, cost, and profit frontiers. Furthermore, TE is allowed to change both across firms and over time for each firm. The time-varying TE model proposed by Battese and Coelli (1992) is specifically adopted, while the Kumbhakar (1990) model containing two additional unknown parameters embedded in the inefficiency term is also applicable. The adopted model is parsimonious in terms of its accommodating only one additional parameter, while the models proposed by, e.g., Cornwell et al. (1990), Kumbhakar (1990), and Lee and Schmidt (1993) take more complicated forms and introduce multiple extra parameters. Cuesta (2000) made an excellent summary of the production models with temporal variation in TE. The computational burden will not be heavy in carrying out empirical studies and the Monte Carlo simulations are tractable.

The procedure proposed herein is based on nonparametric regression techniques in an attempt to avoid specifying the functional form characterizing the firm’s technol-ogy. Hence, the potential difficulty of misspecification is avoided. In our procedure, classical assumptions imposed on both the standard parametric and nonparametric re-gression models fail to hold, since the conditional expectation of the composite error is not equal to zero. Nevertheless, nonparametric function under consideration can be consistently estimated using the modified procedure developed by Fan et al. (1996). Despite the strength of their methodology, further modifications must be made to al-low for the panel data setting with an extra parameter attached to the inefficiency term, which describes the evolution of TE.

The rest of the paper is organized as follows. In Section 2, a semiparametric stochastic frontier model is formulated in the context of panel data, and next a pseudo-likelihood function is derived, which is used to construct pseudo-pseudo-likelihood estimators of the variances characterizing the error components of the model. According to Fan et al. (1996), these estimators are√N− consistent. The nonparametric frontier itself can be consistently estimated at the usual convergence rate of kernel regression estimators. Section 3 conducts the Monte Carlo simulations to investigate the finite sample per-formances of the pseudo-likelihood estimators proposed in the previous section. The

resultant estimators perform relatively well and enjoy consistent property as number of firm and/or panel size grows, in terms of variance and mean square error. An em-pirical study of the proposed method is carried out in Section 4, using a sample of 86 manufacturing firms, while Section 5 concludes the paper.

2. M ETHODOL OGY

We begin by assuming that we have observations on n firms, indexed by i= 1, · · · , n, through T time periods, indexed by t = 1, · · · , T . A semiparametric stochastic pro-duction frontier with time-variant TE is formulated as:

y_{it}= z_{it}β+ f(x_{it}) + ε_{it}, (1)

with

ε_{it} = v_{it}− u_{it}= v_{it}− u_{i}e−γ(t−T ), (2)

where z_{it} is a K × 1 vector of random regressors excluding the intercept, and β is
the corresponding unknown vector of parameters. Without loss of generality, f(x_{it})
is assumed to be an unknown but smooth function of a scalar x_{it}, a random variable
having distribution with support. Note, x_{it} can readily be extended to be a random
vector with support. With the caveat of the “curse of dimensionality,” the number of
elements in x_{it} should not be large. The main advantage of a partially linear model
like equation (1) is that it avoids the potential problem of the curse of dimensionality
by decreasing the dimension of x_{it} and increasing the dimension of z_{it} at the same
time. The properties of the estimators will be addressed later in the text. Moreover,
vector z_{it}is legitimate to include categorical variables, whereas the smooth function f
in a pure nonparametric model is not. See Stock (1989) for details.1 Finally, random
disturbance ε_{it}= v_{it}− u_{it}is the composite error in which v_{it}represents the two-sided
statistical noise, and u_{it}is the technical inefficiency quantified by the product of a

one-1

As an additional benefit, the introduction of the nonparametric componentf(·) into the regression model enables one to use the visualized curve or surface estimate off(·) to specify a suitable parametric functional form off, when a correctly specified pure parametric model is preferred.

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*Academia Economic Papers 36:2 (2008)*

sided error u_{i} and exp[−γ(t − T )] that depicts the evolutionary path of inefficiency
(Battese and Coelli, 1992). The last term exp[−γ(t − T )] decreases (or increases) at
an exponential rate if γ > 0 (or γ < 0), or keeps constant if γ= 0.

Equation (1) deviates from the standard nonparametric regression model in that
the conditional mean of ε_{it}given x_{it}is not equal to zero. This not only gives rise to a
biased estimate of the intercept for the case of a pure parametric model, but also entails
an inconsistent estimate of the nonparametric frontier, f(x_{it}), of (1) as the intercept is
implicitly absorbed in f(·). Consequently, the subsequent estimated variances of the
composed errors may also be inconsistent owing to the contamination of the
incon-sistent estimate of f(·). Equation (1) is specified in the context of panel data and its
inefficiency term is allowed to change both across firms and over time. The model of
Fan et al. (1996) can be viewed as a special case of the current model.

Following the convention, we assume that the variables v_{it} are identically and
independently distributed (i.i.d.) as N(0, σ_{v}2), and u_{i} are also i.i.d. random variables
drawn from an N(0, σ_{u}2) distribution truncated at zero from below. Let g_{t}= e−γ(t−T )
represent the pattern of temporal variation of inefficiency. The mean and variance of
ε_{it}can be expressed respectively by:

E(ε_{it}) = −g_{t}E(u_{i}) = −g_{t}
2
πσu = −µt,
σ_{ε}2= σ2_{v}+ π− 2
π g
2
tσ2u.
(3)

The probability density function of the composite disturbance for the ith firm is:

h(ε_{i}) = 2
σ_{v}T −1σ
1− Φ
−µ∗i
σ_{∗}
_{}T
t=1
φ
ε_{it}
σ_{v}
exp
1
2
λ ST
t=1εitgt
σ
2
, (4)
where σ2= σ_{v}2+ σ_{u}2S
T
t=1gt2, λ= σu/σ_{v}, µ_{∗i}= −σ2_{u}S
T
t=1εitg_{t}/σ2, σ_{∗}2 = σ2_{u}σ_{v}2/σ2,

and φ(·) and Φ(·) are the standard normal density and distribution functions, respec-tively.

In the presence of parametric term z_{it}β, Robinson’s (1988) procedure can be
applied to yield consistent estimators of β, ˆβ, at parametric rates, i.e., ˆβ − β =

O_{p}(N−1/2), where N(= n × T ) denotes the number of observations, in spite of the
existence of the nonparametric function f(·). Taking a conditional expectation of all
variables (1) on the x-variable and then subtracting the results from (1) yield:

y_{it}− E(y|x_{it}) = [z_{it}− E(z|x_{it})]β + µ_{t}+ v_{it}− u_{it}, (5)
where µ_{t} is defined as (3). Since the regression functions E(y|x_{it}) and E(z|x_{it}) are
completely unknown, one may approximate it by the ordinary nonparametric (kernel)
estimators ˆE(y|x_{it}) and ˆE(z|x_{it}) respectively. Parameter β of (5) can thus be estimated
by the ordinary least squares (OLS) procedure in which y_{it}− ˆE(y|x_{it}) and z_{it}− ˆE(z|x_{it})
serve as the response and independent variables respectively. Robinson (1988) showed
that if ˆE(y|x_{it}) and ˆE(z|x_{it}) converge sufficiently quickly, then their substitution in the
OLS procedure does not affect the asymptotic distribution of the resultant estimator

ˆ

β.2 _{The estimator of θ}_{(x}

it) = f(xit) − µt is thus defined by ˆθ(x_{it}) = ˆE(y|x_{it}) −
ˆ

E(z|x_{it}) ˆβ.

Based on (4), the log-likelihood function can be written as:

ln L(y|β, λ, γ, σ2) = ln
n
i=1
h(ε_{i}) = Const. − n(T − 1)
2 ln σ
2
v− n_{2} ln σ2
+ Sn
i=1ln[1 − Φ(Ai)] +
1
2
n
S
i=1A
2
i +
n
S
i=1
T
S
t=1ln φ
ε_{it}
σ_{v}
= Const. − n(T − 1)
2 ln
σ2
1+ λ2 T
S
t=1g
2
t
−
n
2 ln σ
2
+ Sn
i=1ln[1 − Φ(Ai)] +
1
2
n
S
i=1A
2
i −
1+ λ2 ST
t=1g
2
t
_{n}
S
i=1
T
S
t=1ε
2
it
2σ2 ,
(6)

where A_{i} = −µ_{∗i}/σ_{∗}. If f(·) is known, then the first-order conditions (FOC) for σ2
and λ conditional on ε_{it}= y_{it}− z_{it}β− f(x_{it}) are as follows:

2

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*Academia Economic Papers 36:2 (2008)*

∂ ln L
∂σ2 =
−nT
2σ2 +
λ
2σ3
n
S
i=1
_{φ(A}
i)
1− Φ(A_{i})− Ai
_{T}
S
t=1εitgt
+
1+ λ2 ST
t=1g
2
t
S
n
i=1S
T
t=1ε2it
2σ4 = 0, (7)
∂ ln L
∂λ =
λn(T − 1) ST
t=1g
2
t
1+ λ2 T
S
t=1g
2
t
− Sn
i=1
φ(Ai)
1− Φ(A_{i})
T
S
t=1εitgt
σ
+ Sn
i=1
A_{i} ST
t=1εitgt
σ −
λ ST
t=1g
2
t
n
S
i=1
T
S
t=1ε
2
it
σ2 = 0. (8)
Using (8), we have:
n
S
i=1
_{φ(A}
i)
1− Φ(A_{i}) − Ai
ST
t=1εitgt
σ =
λn(T − 1) ST
t=1g
2
t
1+ λ2 T
S
t=1g
2
t
−
λ ST
t=1g
2
t
n
S
i=1
T
S
t=1ε
2
it
σ2 , (9)

Plugging this back into (7) yields:

− nT
2σ2 +
λ2
2σ2
n(T −1) ST
t=1g
2
t
1+λ2 T
S
t=1g
2
t
−
T
S
t=1g
2
t
n
S
i=1
T
S
t=1ε
2
it
σ2
+
1+λ2 ST
t=1g
2
t
_{n}
S
i=1
T
S
t=1ε
2
it
2σ4 = 0,
(10)
Thus,
n
S
i=1
T
S
t=1ε
2
it
σ4 =
nT
1+ λ2 ST
t=1g
2
t
− λ2_{n(T − 1)} T
S
t=1g
2
t
σ2
1+ λ2 T
S
t=1g
2
t
,

which can be solved for:
σ2=
1+ λ2 ST
t=1g
2
t
_{n}
S
i=1
T
S
t=1ε
2
it
n
T+ λ2 T
S
t=1g
2
t
. (11)

Note from (5) that:

ε_{it} = v_{it}− u_{it}= y_{it}− E(y|x_{it}) − [z_{it}− E(z|x_{it})]β − u_{t}. (12)
Using formulae σ2 _{= σ}2

v + σu2S

T

t=1gt2 and λ = σu/σv, it can be derived that σ2u =

λ2σ2/(1 + λ2S T

t=1g2t). Plugging σu2into (3), we obtain:

µ_{t}=
√
2g_{t}λσ
π
1+ λ2 T
S
t=1g
2
t
1
2
, (13)

which contains σ. Therefore, one must replace ε_{it}in (11) by (12) and (13) to form a
quadratic equation of σ and solve for σ again. Through a tedious manipulation, this
leads to:
ˆ
σ= −b +
√
b2_{− 4ac}
2a , (14)
where a= 1 − 2λ2S
tgt2/πTT , TT = T + λ2S
tg2t, b= 23/2λ
(1 + λ2
S
tgt2)/πS
i
S

teitg_{t}/nTT , e_{it} = y_{it} − ˆE(y|x_{it}) − [z_{it} − ˆE(z|x_{it})] ˆβ and c = −(1 + λ2S

tgt2)

S

iS

te2it/nTT . In (14) we add a “ ˆ ” on σ since the nonparametric and OLS estimators
of ˆE(.|x_{it}) and ˆβ are used to replace their respective true counterparts.

When setting γ = 0, g_{t} reduces to unity. Only under this condition can the
termS

iS

teitgt in b approach to zero in a large sample, making the solution of ˆσ a simpler form. Since γ = 0 by assumption in this article,S

iS

teitgtmay not vanish asymptotically. This implies that term b is not bounded in probability like the one in Fan et al. (1996). Therefore, the solution of (14) takes a much more complicated form than that of Fan et al. (1996).

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*Academia Economic Papers 36:2 (2008)*

The conditional expectations E(y|x_{it}) and E(z|x_{it}) can be estimated by
apply-ing the existapply-ing nonparametric regression techniques. The current paper adopts the
most studied nonparametric kernel estimators. Let ˆE(y|x_{it}) and ˆE(z|x_{it}) be the kernel
estimators of E(y|x_{it}) and E(z|x_{it}) respectively. ˆE(y|x_{it}) can be expressed by:

ˆ
E(y|x_{it}) =
n
S
j=1
T
S
t=1yjtK
x_{it}− x_{jt}
h
n
S
j=1
T
S
t=1K
x_{it}− x_{jt}
h
, (15)

where K is the kernel function and h is the smoothing parameter or bandwidth, which
is used to control the smoothness of the kernel estimates. ˆE(z|x_{it}) can be similarly
obtained by substituting z_{it} for y_{it} in (15). It is important to note that [ ˆE(y|x_{it}) −
E(y|x_{it})] and [ ˆE(z|x_{it}) − E(z|x_{it})] have an order larger than O_{p}(N−1/2), i.e., the
pointwise convergence rate of the kernel regression estimator is slower than the rate
of N−1/2. On the other hand, Fan et al. (1996) argued that ˆσ2 is a√N− consistent
estimator of σ2under mild conditions.

We summarize the new semiparametric stochastic frontier estimation procedures as follows.

Step 1: For each i = 1, · · · , n and t = 1, · · · , T , produce the kernel estimates of the
conditional expectations E(y|x_{it}) and E(z|x_{it}) employing formula (15).
Step 2: Use the results in Step 1 to concentrate out σ2as in (14).

Step 3: Maximize the concentrated log-likelihood function ln L(λ, γ) over the two
un-known parameters λ and γ, where ln L(λ, γ) is the same as in (6) with the
ex-ception that ε_{it}must be replaced by ˆε_{it}= y_{it}− ˆE(y|x_{it}) − [z_{it}− ˆE(z|x_{it})] ˆβ +
µ(λ, γ) and σ must be replaced by ˆσ given by (14).

Note here that the above strategy enables one to estimate the value of E(ε_{it}) =
−µt, and one can thus have a modified version of partially linear regression model (1):
y_{it}− z_{it}βˆ+ ˆµ_{t}= f(x_{it}) + ε_{it}, (16)
where ε_{it} = ε_{it} + ˆµ_{t}(ˆλ, ˆγ) has a mean value that converges to zero due to the
con-sistency property of ˆλ and ˆγ. Hence the nonparametric component f(x_{it}) and its first
derivative f(x_{it}) can be estimated by the usual nonparametric smoothing technique.
For example, if x_{it}is a scalar, then choosing a to minimize:

S(a) = Sn
j=1
T
S
t=1(yjt− zit
ˆ
β+ ˆµ_{t}− a)2K
x_{it}− x_{jt}
h
, (17)

will yield the local constant type kernel estimator of f(x_{it}). The resultant estimator
is similar to that of Fan et al. (1996) and is defined as ˆf(x_{it}) = ˆθ(x_{it}) + ˆµ_{t}(ˆλ, ˆγ),
where ˆθ(x_{it}) = ˆE[y|x_{it}] − ˆE[z|x_{it}] ˆβ, i = 1, · · · , n, and t = 1, · · · , T . It is
well-known that ˆλ and ˆγ converge to λ and γ, respectively, at a parametric rate N−1/2,
while estimator ˆf(x_{it}) converges to f(x_{it}) for each (i, t) at a slower rate than N−1/2.
For slope estimation, the Taylor formula (e.g. for 2 dimensional x_{it}) f(x1,jt, x2,jt) ≈

f(x1, x2) + f1(x1, x2)(x1,jt − x1) + f2(x1, x2)(x2,jt− x2) = a + b1(x1,jt− x1) +

b2(x2,jt− x2) suggests that one can choose a, b1and b2to minimize

S(a, b) = Sn j=1 T S t=1[yjt− zjtβ+ ˆµt− a − b1(x1− x1,jt) − b2(x2− x2,jt)] 2 × K x1− x1,jt h K x2− x2,jt h ,

and then use the solutions ˆb1(x1, x2) and ˆb2(x1, x2) to estimate f1(x1, x2) and f2(x1, x2) respectively. See, for example, the monograph by Fan and Gijbels (1996) for a detailed introduction of the above nonparametric estimation of function and its derivatives.

3. M ONTE C ARLO SIMULATIONS

In this section, a variety of simulation studies are carried out to gain insight into the finite sample performance of the proposed estimators. We specifically consider the following semiparametric regression model:

y_{it}= z1,itβ1+ √z2,it+ ln(1 + xit) + vit− uie−γ(t−T ), (18)
where z1,it, z2,it, xit are randomly drawn from a normal distribution N(2, 0.252),
a uniform distribution U(2, 5), and a chi-square distribution χ2

(1), respectively. The
values of v_{it} and u_{i} are drawn respectively from N(0, σ2_{v}) and |N(0, σ2_{u})|, a
half-normal distribution. The smooth function f(z2,it, xit) = √z2,it+ ln(1 + xit) serves
as the nonparametric component of model (18). In the process of simulation, we set

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*Academia Economic Papers 36:2 (2008)*

γ = 0.025 and β1 = 3, which together with σu, σv, σ2 = σ_{v}2+ σ2_{u}S
T

t=1g2t and

λ= σ_{u}/σ_{v} are the parameters of interest. The kernel function used through out this
paper is the Epacnechnikov kernel function K(u) = 0.75(1 − u2)I_{[−1,1]}(u) and the
bandwidths is taken as h1= sd(x)(nT )−1/(p+4)and h2= sd(z2)(nT )−1/(p+4), where

p= 2 is the number of regressors in the smooth function f. To calculate all the kernel estimates for the case of 2 regressors in f that are needed throughout the process of pseudo-likelihood estimations, henceforth, we use formula (15) with the product kernel to calculate the kernel estimate ˆE[·|(z2, x)]. For example, the kernel estimate of E(y|z2,it, xit) is expressed by

ˆ
E(y|z2,it, xit) =
n
S
i=1
T
S
t=1K
x_{it}− x_{jt}
h1
K
z2,it− z2,jt
h2
y_{it}
n
S
i=1
T
S
t=1K
x_{it}− x_{jt}
h1
K
z2,it− z2,jt
h2
.

The formula for E(z1|z2,it, xit) can be similarly obtained by substituting z1,itfor yitin the above formula.

The purposes of the Monte Carlo simulations are fourfold. First, as one of the
primary objectives in the forgoing semiparametric pseudo-likelihood estimation
strat-egy, one can sidestep the possible specification error that may be caused by assuming
an ad hoc parametric functional form for, e.g., a production technology and/or a cost
function. We will be able to examine if pseudo-likelihood estimators ˆγ, ˆλ, and ˆσ2
are robust to functional form specification. To be more specific, two arbitrary
func-tional forms are chosen, estimated, and compared with their parametric counterparts.
Second, we plan to gain further insights on the effect of sample size by changing the
number of firms(n) and the sample period (T ). As mentioned in the previous section,
the parametric estimators converge at a rate of N−1/2, despite the employment of a
nonparametric kernel regression estimator. It is important to see whether the
perfor-mance of ˆγ, ˆλ, and ˆσ2 _{are considerably influenced by the slow convergence rates of}

ˆ

E(y|x_{it}) and ˆE(z|x_{it}) for finite samples with different combinations of (n, T ). This
is empirically meaningful under the framework of the panel data setting, where a
non-linear trend is introduced to capture the evolution of the inefficiency term. Both can
facilitate modern research in the area of, e.g., efficiency and productivity analysis from
the micro-economic and macro-economic points of view, where panel data are growing
more popular and are perhaps the main contribution of the current paper. Thus, we

at-tempt to detect the minimum values of the(n, T ) combination at which our estimators
can be accurate enough as the parametric counterparts as possible, in terms of mean
squared error (MSE). The third objective is to understand the likely effect of various
values of λ and γ on the performance of estimators ˆλ, ˆσ2_{, and ˆ}_{γ. Sets of prescribed }
val-ues on λ and γ are investigated, where for ease of comparison we adopt the same sets
of values as listed in Table 2 of Aigner et al. (1977). The same sets of values were also
utilized by Fan et al. (1996). The last objective is concerned with the effects of
exoge-nous variables on the technical efficiency, pioneered by Huang and Liu (1994), Battese
and Coelli (1995), and Schmidt and Wang (2002). Besides the time trend, we further
include an additional time-varying variable w_{t}into the composed error to explain the
variations in technical efficiency by letting ε_{it}= v_{it}− u_{i}exp[−γ(t − T ) + δw_{t}]. We
will also examine the effects of various sample sizes on the accuracy of estimators by
changing the number of firms(n) and the sample period (T ).

During simulation, 1,000 samples are generated for each case. The numerical
searches for optimal values of λ and γ (starting from 0.5 and 0.0001, respectively,
throughout this section) are all implemented by the “Maximum Likelihood”
applica-tion routine in the Gauss software, version 6.3 In the first simulation exercise we
choose n= 50 and T = 6. Following Fan et al. (1996) and Aigner et al. (1977), we
fix(σ2, λ) to be (1.88, 1.66), (1.63, 1.24), and (1.35, 0.83), respectively. Olson et al.
(1980) offered precise and detailed reasons for anchoring the two parameters. We now
report the simulated bias and MSE for each estimator based on the 1,000 replications.
Table 1 shows the results of (18). Note that the estimates ˆσ2, ˆλ, ˆγ, ˆσ2_{u}, ˆσ2_{v}, and ˆβ1
are in general compatible with those of Aigner et al. (1977) and Fan et al. (1996). It
is seen that the MSEs of ˆσ2_{, ˆ}_{λ, and ˆ}_{σ}2

u increase as the true parameters(σ2, λ) grow,
while the reverse trend is found for ˆσ_{v}, but its magnitudes of MSE are quite small.
The foregoing is consistent with Aigner et al. (1977) and Fan et al. (1996). The
introduction of the time-variant effect inflates σ2 = σ2_{v} + σ_{u}2S

T

t=1gt2 by the factor S

T

t=1gt2 which contains an extra unknown parameter γ to be estimated. These may contribute to the larger values of MSE of ˆσ2 in comparison with the preceding two papers. As far as the additional trending parameter γ is concerned, both the MSEs and biases are nearly negligible, implying that this parameter can be accurately estimated even by a small sample with n= 50 and T = 6.

3

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*Academia Economic Papers 36:2 (2008)*

**Table 1 Monte Carlo Simulation Results for Model (18):**
y_{it}= z1,itβ1+ √z2,it+ ln(1 + xit) + vit− uie−γ(t−T )
_{n}_{= 50}

T = 6

_{(σ}2_{, λ}_{)=(1.88, 1.66) (σ}2_{, λ}_{)=(1.63, 1.24) (σ}2_{, λ}_{)=(1.35, 0.83)}

MSE Bias MSE Bias MSE Bias

σ2 _{0.8758} _{0.6561} _{0.2930} _{0.3583} _{0.1563} _{0.0056}
λ 0.1223 −0.2375 0.0791 −0.1458 0.0647 −0.0781
γ 0.0020 0.0007 0.0038 0.0014 0.0133 0.0041
σ2_{u} 0.0254 −0.1185 0.0218 −0.1010 0.0214 −0.0786
σ_{v}2 0.0006 −0.0125 0.0012 −0.0276 0.0026 −0.0443
β1 0.0237 0.0002 0.0235 −0.0006 0.0256 −0.0017

**Table 2 Monte Carlo Simulation Results for Model (19):**
y_{it}= z1,itβ1+ √z2,itβ2+ ln(1 + xit) + vit− uie−γ(t−T )
_{n}_{= 50}

T = 6

_{(σ}2_{, λ}_{)=(1.88, 1.66) (σ}2_{, λ}_{)=(1.63, 1.24) (σ}2_{, λ}_{)=(1.35, 0.83)}

MSE Bias MSE Bias MSE Bias

σ2 _{2.1402} _{1.2230} _{0.9003} _{0.7158} _{0.3010} _{0.2913}
λ 0.0883 −0.1258 0.0657 −0.0792 0.0579 −0.0428
γ 0.0016 0.0014 0.0031 0.0023 0.0102 0.0045
σ2
u 0.0154 −0.0547 0.0148 −0.0471 0.0175 −0.0363
σ_{v}2 0.0004 −0.0004 0.0005 −0.0068 0.0008 −0.0136
β1 0.0215 0.0014 0.0212 0.0007 0.0229 0.0002
β2 0.0237 0.0020 0.0235 0.0063 0.0254 0.0041

Fan et al. (1996) claimed that their pseudo-likelihood estimators are robust to
functional form specification, which heuristically prompts the next simulation. Here,
we keep model (18) intact except to move z2,it from the nonparametric part to the
parametric part with the coefficient being set to unity, that is, by setting f(x_{it}) =
ln(1 + x_{it}),

y_{it}= z_{1,it}β1+ √z2,itβ2+ ln(1 + xit) + vit− uie−γ(t−T ). (19)
This specification allows us to estimate an extra parameter β2(= 1), while the

dimen-sion of the nonparametric component decreases to one. We see that all the MSEs of the
parameter estimators in Table 2 fall short of, but do not substantially differ from those
in Table 1 except for σ2. This may be attributed to the use of a simpler nonparametric
smooth function. One is led to conclude that these estimators are also robust to the
functional form specification like the ones proposed by Fan et al. (1996). Estimator ˆσ2
performs relatively less well in this case due to its higher measures of MSE and bias.
However, this potential deficiency is of no consequence, as its components of σ_{u}2, σ_{v}2,
and γ are estimated fairly accurately.

To sum up, our proposed estimators are quite satisfactory even for relatively small-sized panel data, on the basis of Tables 1 and 2. The following two Monte Carlo experiments evaluate the effects of sample size and the variance ratios λ, for a given value of γ, on the accuracies of our proposed estimators.

To probe the sample and panel size effects on the properties of the proposed esti-mators, we consider n= 50, 100, 200 and T = 6, 10, 20. For each of the nine (n, T ) combinations, we draw 1,000 random samples based on the following data generation process:

y_{it}= z1,itβ1+ z2,it· 2 + √z2,it· xit+ ln(1 + xit) + vit− uie−γ(t−T ). (20)
where β1, β2 and γ are respectively fixed at 3, 2, and 0.025 and the smooth function

f(z2,it, xit) = z2,it·2+√z2,it· xit+ln(1+xit). Following Olson et al. (1980) and Fan
et al. (1996), we fix λ= 1 and σ_{ε}2(T ) = 1. Variable x_{it}is randomly drawn from the
chi-square related distribution of 2+ χ2_{(1)}, and other variables are respectively drawn
from the same population as those in (18). Table 3 presents the sample size effects. It is
seen that the simulated bias, variance, and MSE measures of our estimators generally
decrease fast as the sample and/or panel size grows. This confirms the assertions stated
in Section 2 that the semiparametric pseudo-likelihood estimators are able to converge
at the same pace N−1/2 as the parametric estimators. The finite sample performance
of the semiparametric-based estimators is encouraging since the outcomes are well
compatible with those of Olson et al. (1980) and Fan et al. (1996). Naturally, our
esti-mation strategies can also result in consistent estimators like the maximum likelihood
semiparametric method proposed by Fan et al. (1996).

For each of 1,000 dataset samples, we calculate the average prediction errorS n i=1 S

T

362H: 2008/07/22 09:20 AM page:182

*Academia Economic Papers 36:2 (2008)*

**T**
**able**
**3**
**E**
**ff**
**ects**
**of**
**Sample**
**Siz**
**e**
**for**
**M**
**odel**
**(20**
**):**
yit
=
z1,i
t
β1
+
z2,it
·2
+
√ z
2,it
·xit
+
ln
(1
+
xit
)+
vit
−
ui
e
−
γ(
t−
T
)
(n,
T
)
σ
2 u
σ
2 v
γ
=
0
.025
MAPE
B
ias
V
ar
M
SE
B
ias
V
ar
M
SE
Bi
as
V
ar
MSE
ˆ f(z
2
,x
)
fz2
(z2
,x
)
fx
(z2
,x
)
(50
,6
)
−
0.1347
0.1878
0
.0534
−
0.0511
0.0389
0.0041
0.0014
0.0965
0.
0
093
1.
804
2
0
.3
117
0.9623
(50
,10
)
−
0.1187
0.1459
0
.0354
−
0.0379
0.0289
0.0023
0.0005
0.0315
0.
0
010
1.
109
6
0
.2
440
0.7672
(50
,20
)
−
0.0780
0.1223
0
.0210
−
0.0136
0.0215
0.0007
0.0003
0.0088
7
.7
×
10
−
5
0.
651
5
0
.2
137
0.6521
(10
0
,6
)
−
0.0978
0.1331
0
.0273
−
0.0400
0.0263
0.0023
0.0007
0.0530
0.
0
028
0.
891
4
0
.2
293
0.7085
(10
0
,10
)
−
0.0820
0.1087
0
.0185
−
0.0284
0.0209
0.0012
0.0006
0.0204
0.
0
004
0.
573
5
0
.1
819
0.5723
(10
0
,20
)
−
0.0452
0.0835
0
.0090
−
0.0066
0.0153
0.0003
0.0002
0.0058
3
.4
×
10
−
5
0.
369
0
0
.1
577
0.4880
(20
0
,6
)
−
0.0741
0.0935
0
.0142
−
0.0296
0.0192
0.0012
0.0014
0.0336
0.
0
011
0.
475
7
0
.1
719
0.5327
(20
0
,10
)
−
0.0577
0.0743
0
.0088
−
0.0197
0.0146
0.0006
−
0.0002
0.0132
0.
0
002
0.
305
0
0
.1
399
0.4363
(20
0
,20
)
−
0.0266
0.0062
0
.0046
−
0.0016
0.0106
0.0001
0.0002
0.0040
1
.7
×
10
−
5
0.
192
4
0
.1
294
0.3977
Note:
MAPE
of
ˆ f(z
2
,x
)
is
the
mea
n
v
alue
of
1,
00
0
datase
ts
’
av
erage
prediction
errors
of
ˆ f(z
2
,x
)
de
fi
ned
as
S
n i=
1
S
T t=
1
(
ˆ f(z
2
,it
,x
it
)
−
f
(z2,it
,xit
))
2/nT
.
M
A
P
E’
s
o
f
ˆ f(zz2
2
,x
)
and
ˆ f(zx
2
,x
)
ar
e
defined
simil
ar
ly
.

**Table 4 The Case with an Exogenous Variable and the Sample Size Effect**

(n, T ) σ2u σv2 γ = 0.025 d= 1.25

Bias MSE Bias MSE Bias MSE Bias MSE

(50, 6) −0.9789 0.0900 0.0184 0.0037 0.0057 0.0301 0.1228 3.0934
(50, 15) −0.0170 0.0225 0.0501 0.0039 0.0017 0.0002 0.0625 0.1025
(100, 6) −0.0499 0.0333 0.0316 0.0036 2.6 × 10−5 _{0.0043} _{0.0289 0.4292}
(100, 15) 0.0078 0.0125 0.0526 0.0038 0.0014 0.0001 0.0571 0.0493
(200, 6) −0.0615 0.0189 0.0384 0.0036 0.0004 0.0019 −0.0186 0.1653
(200, 15) 0.0396 0.0096 0.0560 0.0039 0.0013 5.2 × 10−5 _{0.0562 0.0241}
ˆ

f_{Z}2(z2, x) and ˆfx(z2, x) are defined and calculated similarly. The mean values of

1,000 average prediction errors for each of the three estimators are reported in the last three column of Table 3. The mean average prediction errors (MAPE) for 9 different (n, T ) combinations demonstrate that three MAPE’s decrease as the number of firms n and/or time period T increase. Hence the proposed estimators have the property of consistency (though at the rate slower than the parametric rate N−1/2).

Next, the composed error of (20) is slightly revised to allow for an additional
exogenous variable by setting v_{it}− u_{i}exp[−γ(t − T ) + δw_{t}], where δ = 1.25 and
w_{t} is a normal random variable distributed as N(0, 0.252). The results summarized
in Table 4 are quite similar to those in Table 3. As the sample size and/or panel size
grows, the simulated MSEs generally decrease accordingly. The coefficient estimates
of the additional exogenous variable, ˆδ, exhibits similar accuracy as those of existing
parameters. Our one-step procedure, which outperforms the two-step alternative as
demonstrated by Schmidt and Wang (2002), has been proved to be applicable.

We finally examine the effect of the variance ratios on the finite sample properties of our proposed estimators. We hold onto model (20) except for letting γ = −0.025 and fixing n= 150 and T = 10. All the data-generating processes of the variables in (20) are in line with the previous three tables. Following Olson et al. (1980) and Fan et al. (1996), we select: λ= 10−1, 10−3/5, 10−2/4, 10−1/4, 1, 101/4, 102/4, 103/4, and 101. We also count the number of Type I failure (the counts of opposite signs, i.e.

ˆ

λ < 0, ˆγ > 0) and Type II failure(λ = ∞) for our proposed estimators.

Recall that the number of replications is 1,000 for all cases studied. No Type II failure is encountered by the Monte Carlo simulations. Table 5 shows that a few Type I failures of λ occur, but are negligible, when the true value of λ is less than or equal

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*Academia Economic Papers 36:2 (2008)*

**Table 5 Effects of the Variance Ratio**

λ (number of Type I failures)

σ2

u σ2v _{(number of Type I failures)}γ = −0.025

Bias Var MSE Bias Var MSE Bias Var MSE

0.100 (43) 0.0234 0.0316 0.0322 −0.0309 0.0004 0.0013 −0.0701 1.9397 1.9446 (344) 0.178 (22) −0.0080 0.0339 0.0339 −0.0301 0.0004 0.0013 −0.0134 0.4827 0.4829 (395) 0.316 (1) −0.0184 0.0299 0.0302 −0.0286 0.0004 0.0012 −0.0010 0.1360 0.1360 (418) 0.562 (0) −0.0411 0.0125 0.0142 −0.0245 0.0004 0.0010 −0.0010 0.0023 0.0023 (271) 1.000 (0) −0.0708 0.0095 0.0199 −0.0133 0.0002 0.0003 −0.0007 0.0001 0.0001 (102) 1.778 (0) −0.1017 0.0095 0.0199 0.0120 0.0002 0.0003 −0.0004 0.0001 0.0001 (13) 3.162 (0) −0.1208 0.0106 0.0252 0.0526 0.0001 0.0029 −0.0003 4.4 × 10−5 4.5 × 10−5 (0) 5.623 (0) −0.1292 0.0112 0.0279 0.1049 0.0001 0.0111 −0.0002 2.3 × 10−5 2.3 × 10−5 (0) 10.000 (0) −0.1322 0.0115 0.0290 0.1594 0.0002 0.0256 −0.0002 1.6 × 10−5 1.6 × 10−5 (0)

**Table 6 Sample Statistics**

Variable Mean Standard Deviation

Output* 4,290,681.0578 1.2113× 107

Labor(x1) 1,484.739 2,415.883

Land, buildings, and plants(x2)* 2,266,484.675 6,626,951.264

Capital(x3)* 2,818,975.108 1.1152× 107

Note: Thousands of New Taiwan Dollars with base year 1996.

to 0.316, while no Type I failure occurs when λ exceeds 0.316. On the contrary, Type I failure of γ is more troublesome when λ falls short of 0.316, but decreases rapidly. The failure vanishes when λ exceeds 1.778. This may be attributed to the fact that, differing from Fan et al. (1996), we maximize a concentrated log-likelihood function with respect to two parameters, λ and γ, which complicate the optimization process and easily leads to unreasonable estimates for the two parameters.

It deserves specific mention that the percentages of Type I failure of λ are still much less than those of Olson et al. (1980) and possibly Fan et al. (1996). In addi-tion, all the MSEs in Table 5 are considerably less than those of Table 6 in Fan et al. (1996).4 Viewed from this angle, we claim that our estimation strategy is likely to be more preferable for cases with larger variance ratios. On the basis of our unreported

4

Two reasons may be responsible, i.e., the current paper applies a larger sample with 1,500(150×10) observations and the information on the panel structure is explicitly taken into account by the likelihood function.

simulation results, our proposed method can provide an estimator of λ with accuracy similar to that of Fan et al. (1996), when the inefficiency term is restricted to be time-invariant, i.e. γ is set to be equal to 0. The foregoing evidence enables us to conclude that our procedure performs at least as well as the parametric MLE or COLS method used by Olson et al. (1980) and the semiparametric method used by Fan et al. (1996). However, our approach allows for analyzing the growing popularity of panel data with a temporal variation of inefficiency, which permits researchers to closely characterize a firm’s true production process. More reliable estimation results and inferences based on the estimation results can then be obtained.

4. A N EM PIRICAL EXE MPLIFIC ATION

To illustrate our semiparametric estimation approach, a sample of 86 firms in the elec-tronics and information industry was collected from the Taiwan Economic Journal’s financial database over the period 1995 to 2001. All firms are listed on the Taiwan Stock Exchange. This is a balanced panel dataset with a total of 602 observations. It is noticeable that unbalanced panel data are applicable, as well. The electronics industry in Taiwan can be characterized as highly competitive and capital intensive with a huge amount of R&D expenditure.

A firm’s value-added product is defined as the output. Three inputs are identified from the databank, i.e., annual number of full-time equivalent employees(x1), land, buildings, and plants(x2), and physical capital (x3). Both x2 and x3 are taken from the book values of the respective assets net of depreciation. The wholesale price index of the manufacturing industry is used to deflate nominal values, with the base year as 1996. Table 6 summarizes the sample statistics for the output and inputs. Note that in what follows, the output and all input variables are all transformed into logarithms.

For the parametric model, a standard translog production function with trends is specified, as it is a flexible functional form widely utilized by empirical researchers in the area of efficiency and productivity analysis. Table 7 presents the estimation results. At least at the 10% level, 8 of 18 parameter estimates differ significantly from zero. Technical efficiency of the sample firms is found to deteriorate over time, as the estimated value of γ is negative and reaches statistical significance at the 5% level. Recall that the sample includes both the pre- and post-crisis periods of the Asian financial crisis which began July 2, 1997 wherein Taiwan was also affected. Three

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*Academia Economic Papers 36:2 (2008)*

**Table 7 Parameter Estimates of the Semiparametric and Parametric Models**
Variable Semiparametric Model Parametric Model

Parameter Estimate Standard Error Parameter Estimate Standard Error

Intercept N/A N/A 9.8316*** 1.7706

ln x1 1.4261*** 0.4279 1.0700* 0.5650
ln x2 N/A N/A 0.2413 0.3649
ln x3 N/A N/A −0.8288** 0.3430
T 0.3849*** 0.1337 0.3303*** 0.1048
0.5× ln x1× ln x1 0.0363 0.0957 0.2137 0.1372
0.5× ln x2× ln x2 N/A N/A 0.0079 0.0596
0.5× ln x3× ln x3 N/A N/A 0.0311 0.0473
0.5× t2 0.0195 0.0131 0.0284*** 0.0101
ln x1× ln x2 −0.0876* 0.04649 −0.1208 0.0740
ln x1× ln x3 0.0082 0.0362 −0.0185 0.0702
ln x2× ln x3 N/A N/A 0.0535 0.0417
t× ln x1 0.0157 0.0204 0.0143 0.0173
t× ln x2 −4.55276 ×10−5 0.0184 −0.0105 0.0133
t× ln x3 −0.0377*** 0.0135 −0.2150** 0.0107
γ −0.0042 0.0296 −0.0523** 0.0243
σ2_{u} N/A N/A 0.7842*** 0.1741
σ2
v N/A N/A 0.1477 0.2689
λ 1.3706*** 0.1550 N/A N/A
Log-Likelihood N/A −372.8800

Note: *: significant at the 10% level of significance; **: significant at the 5% level of signifi-cance; ***: significant at the 1% level of significance.

years later Taiwan faced another economic contraction, caused mainly by political instability. The adverse business environment confronted by the sample firms may be responsible for the negative value of γ. The average technical efficiency measure in 1995 is 0.6085, while six year later it decreases to 0.5195 in 2001. The overall mean TE score for the entire 602 observations is 0.5645. A representative firm can merely produce about half as much as an efficient firm using the same amount of input quantities.

As far as the semiparametric model is concerned, the nonparametric part of the production function is arbitrarily specified as a function of inputs x2 and x3. There are five parameter estimates that are significantly estimated out of eleven parameters, where γ is also found to be negative, but they are quite small in absolute value and insignificant. This makes the variation of the TE measures over time negligible. More

specifically, the average TE measure in 1995 is 0.8875, while the same measure in 2001 is 0.8852. The overall mean TE score for the entire sample is 0.8863, which is much higher than that of the parametric model. Huang and Wang (2004) constructed a Fourier flexible cost function, a form of a semiparametric function, to evaluate the economic efficiency of Taiwan’s banking industry. They found that the TE scores from the Fourier function are greater than their translog functions. This finding may be justified in that the semiparametric functional form is able to fit the data closer than the translog form. The introduction of flexibility in the semiparametric functional form can considerably reduce the possible confounding of specification error with inefficiency.

The function values and partial derivatives with respective to x2 and x3, respec-tively, of the nonparametric component f can be estimated using suitable nonparamet-ric smoothers. To characterize function f we first suggest a mental visualization of its surface over the(x2, x3) plane. Figure 1 shows the estimated surface of f(x2, x3). Note that the surface is the result of re-smoothing the estimated ˆf(x2, x3)’s values based on the double smoothing technique proposed by Chu and Deng (2003).

On the basis of the figure we are able to observe the relationship between the two inputs and output y. For example, it is seen that the output is almost uniquely non-decreasing when input labor and/or capital increases, implying that the marginal products of land and capital are both non-negative over the bulk of the(x2, x3) plane. This makes economic sense. The parameter estimates of the parametric model are also used to calculate marginal products of the three inputs for each sample point. The average values of the three marginal products are 0.7017, 0.1914, and−0.6949 for inputs labor, land, and capital, respectively. Both parametric and nonparametric models appear to attain similar implications in economics on the signs of the three marginal products, except for capital.

We next calculate the marginal products of land and capital. For purposes of comparison, we evaluate the slope estimates of f(x2, x3) for both nonparametric and parametric models at some selected pairs of(x2, x3) values. Table 8 lists the marginal product of land and capital evaluated both at their 1st, 2nd and 3rd quartiles, respec-tively. It is seen that the marginal products estimates of the two inputs are all positive for the nonparametric model as suggested by Figure 1, while the estimated marginal products of capital are negative even at its first and second quartiles for the paramet-ric counterpart. Thus, the former model appears to be more appealing than the latter model by making better economic sense.

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*Academia Economic Papers 36:2 (2008)*

^f (ln(Land), ln(Capi tal)) 4 8 12 16 20

### 9

### 13

### ln (Land)

### 17

### ln (Capital)

### 17

### 13

**Figure 1 Estimated Surface of f**(x2, x3)

**Table 8 Estimates of the Marginal Products (MP) of Land and Capital**

Nonparametric Model Q1 Q2 Q3 MP of Land 0.6059 1.0537 0.8075 MP of Capital 0.0426 0.1271 0.2652 Parametric Model Q1 Q2 Q3 MP of Land 0.2162 0.1874 0.1711 MP of Capital −0.7694 −0.7101 0.6287

Note: Marginal products of land and capital are evaluated both at their 1st, 2nd and 3rd quar-tiles(Q1, Q2, Q3), respectively.

5. C ONCLUDING REMA RKS

The current paper is devoted to a generalization of the semiparametric stochastic fron-tier model developed by Fan et al. (1996) and applicable to cross-sectional data, within a panel data setting, which contains more information than does a single cross-section.

Access to panel data broadens the usefulness of the model, allows for time-varying inefficiency, and results in estimates of technical efficiency that have more desirable statistical properties. The model proposed by Battese and Coelli (1992) is particularly illustrated, while the specifications of Kumbhakar (1990) and Lee and Schmidt (1993) may also be applicable. It is important to note that the professed assumption whereby technical efficiency changes over time is preferable, as it is in accordance with real-ity, where the operating environment faced by firms is usually competitive. Firms can hardly survive if technical efficiency remains constant over many years.

To justify the proposed semiparametric pseudo-likelihood estimators, a set of Monte Carlo experiments is carried out. Based on the Monte Carlo outcomes, the proposed estimators perform quite well in small and moderate samples in terms of the measures of MSE and bias, and the estimators converge to their respective true values at the same rate as the parametric models. As a semiparametric frontier model relaxes parametric restrictions, our estimators are robust to possible misspecifications of the production frontier. Through minor modifications, our model can readily be extended to the examination of technical efficiency with respect to cost and profit frontiers.

An empirical application of the proposed procedure exploiting panel data on 86 electronics firms over the period 1995-2001 is exemplified and compared with the re-sults from the parametric counterpart. The pure parametric translog production func-tion tends to substantially underestimate the TE score, due potentially to specificafunc-tion error and lack of flexibility. The extension of our model is straightforward with re-gard to the case of the one-sided error with a more complicate distribution such as a truncated normal or a gamma distribution by referring to Fan et al. (1996).

REFERENCES

Aigner, D. J., C. A. K. Lovell, and P. Schmidt (1977), “Formulation and Estimation of
*Stochas-tic Frontier Production Function Models,” Journal of Econometrics, 6, 21–37.*

Battese, G. E. and T. J. Coelli (1992), “Frontier Production Functions, Technical Efficiency and
*Panel Data:With Application to Paddy Farmers in India,” Journal of Productivity Analysis,*
3, 153–169.

Battese, G. E. and T. J. Coelli (1995), “A Model for Technical Inefficiency Effects in a
*Stochas-tic Frontier Production Function for Panel Data,” Empirical Economics, 20, 325–332.*
Chu, C. K. and W. S. Deng (2003), “An Interpolation Method for Adapting to Sparse Design in

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*Multivariate Nonparametric Regression,” Journal of Statistical Planning and Inference,*
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