Stochastic Nonparametric Envelopment of Panel Data

38 Pages·2007·0.28 MB·English

Stochastic Nonparametric Envelopment of Panel Data:

Frontier Estimation with Fixed and Random Effects Approaches

Timo Kuosmanen
Economic Research Unit, MTT Agrifood Research Finland
Luutnantintie 13, 00410 Helsinki, Finland
E-mail. [email protected].

Abstract
Stochastic nonparametric envelopment of data (StoNED) combines the virtues of data envelopment
analysis (DEA) and stochastic frontier analysis (SFA) into a unified framework of frontier estimation.
StoNED melds the nonparametric piece-wise linear DEA-type frontier with stochastic SFA-type
inefficiency and noise terms. We show that the StoNED model can be estimated in the panel data
setting in a fully nonparametric fashion. Both fixed and random effects approaches are adapted to the
StoNED framework. To disentangle changes in technology and efficiency, a dynamic semiparametric
variant of the StoNED model is developed. An application to the wholesale and retail industry illustrates
the approach.

Key Words: data envelopment analysis (DEA), nonparametric least squares, nonparametric regression,
productive efficiency analysis, stochastic frontier analysis (SFA)

JEL Classification: C14, C51, D24
1. Introduction
The literature of productive efficiency analysis has been divided between two main approaches: the
nonparametric data envelopment analysis (DEA: Farrell 1957, Charnes et al. 1978) and the parametric
stochastic frontier analysis (SFA: Aigner et al 1977; Meeusen and van den Broeck 1977). The main
appeal of DEA lies in its nonparametric treatment of the frontier, which does not assume a particular
functional form but relies on the standard axioms of production theory: monotonicity, convexity, and
homogeneity. The weakness of DEA is that it attributes all deviations from the frontier to inefficiency,
and completely ignores any stochastic noise in the data. The key advantage of SFA is its stochastic
treatment of residuals, decomposed into a non-negative inefficiency term and an idiosyncratic error term
that accounts for measurement errors and other random noise. However, SFA builds on the parametric
regression techniques, which require a rigid ex ante specification of the functional form. Since the
economic theory does not justify a particular functional form, the flexible functional forms, such as the
translog or generalized McFadden, are frequently used in the SFA literature. The problem with the
flexible functional forms is that the estimated frontiers often violate the monotonicity, concavity/convexity
and homogeneity axioms. On the other hand, imposing these axioms will sacrifice the flexibility (e.g.,
Diewert and Wales, 1987; Sauer, 2006). In summary, it is generally accepted that the virtues of DEA lie
in its nonparametric treatment of the frontier, consistent with the axioms of production theory, while the
virtues of SFA lie in its stochastic, probabilistic treatment of inefficiency and noise (e.g., Bauer, 1990;
Seiford and Thrall, 1990).
To bridge the gap between SFA and DEA, a large and growing number of stochastic semi- or
nonparametric frontier models have been developed (e.g., Simar, 1992; Park and Simar, 1994; Fan et
al., 1996; Kneip and Simar, 1996; Park et al., 1998, 2003, 2006; Post et al., 2002; Griffin and Steel,
2004; Henderson and Simar, 2005; Kuosmanen et al., 2007; and Kumbhakar et al., 2007). While these
studies come a long way in combining some of the virtues of DEA and SFA, the conceptual link
between the parametric and non-parametric branches is still missing: none of these techniques can be
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viewed as a stochastic extension of DEA in the same way as SFA extends the classic deterministic
econometric frontier models by Aigner and Chu (1968), Timmer (1971), Richmond (1974), and others.
Furthermore, while the assumptions required by the previous semi- and nonparametric SFA models are
relatively weak, there is no guarantee that these models satisfy the axioms of production theory.
Therefore, there is an evident need for semi- and nonparametric stochastic frontier approaches that
satisfy the standard axioms and thus combine the virtues of DEA and SFA in a unified framework of
frontier estimation.
In the cross-sectional setting, Banker and Maindiratta (1992) were the first to propose an
amalgam of DEA and SFA that combines a DEA-style nonparametric, convex, piecewise linear frontier
with a SFA-style parametric composite error term consisting of noise and inefficiency components.
However, their constrained maximum likelihood estimation procedure is extremely difficult to implement;
no operational computational procedure or empirical applications have been reported. Recently,
Kuosmanen (2006) and Kuosmanen and Kortelainen (2007) introduced the stochastic nonparametric
envelopment of data (StoNED) model, which is an additive variant of Banker and Maindiratta’s model.
Kuosmanen and Kortelainen showed how the StoNED model can be estimated in practice by applying a
two-stage procedure. In the first stage, the average practice frontier is estimated by means of
nonparametric least squares subject to shape constraints (monotonicity, concavity, and/or
homogeneity). In the second-stage, the standard deviations of the inefficiency and noise terms are
estimated by the method of moments or pseudolikelihood techniques, and the conditional expected
values of the inefficiency terms are computed. While the frontier is estimated nonparametrically, the
procedure as a whole can be more precisely described as semiparametric.
Panel data offers generally better possibilities for disentangling inefficiency from stochastic
noise than a cross-section: when each firm is observed many times, the effects of random noise can be
averaged out. In the parametric SFA literature, the two main approaches for dealing with panel data are
the fixed effects approach (Schmidt and Sickles 1984) and the random effects approach (Lee and Tyler
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1978). In both these approaches, the distributional assumptions about the inefficiency and error terms
can be relaxed. In the nonparametric literature, the opportunities provided by the panel data have been
largely ignored, with a notable exception of Ruggiero (2004).
The purpose of this paper is to adapt the cross-sectional StoNED model to the panel data
setting. The panel data enables us to average out the noise without any parametric assumptions about
the distributions of the inefficiency and noise terms: we present the first fully nonparametric variant of
the StoNED model. However, we also note that certain parametric assumptions about technical
progress and the inefficiency term may prove useful for disentangling the frontier shifts from efficiency
improvements over time.
Practical estimation of the StoNED model requires adapting the shape constrained
nonparametric least squares estimator to the panel data setting. To our knowledge, this is the first paper
to apply the fixed and random effects approaches to the shape constrained nonparametric least squares
regression. We also present one of the first empirical applications of the shape constrained
nonparametric least squares in the general multiple regression setting.
The remainder of the paper is organized as follows. Section 2 introduces the StoNED model in
the cross-sectional and panel data settings. Section 3 discusses the fixed effects approach to estimating
the StoNED model in the panel data setting. Section 4 describes the random effects approach. Section
5 illustrates the approach by means of a simulated example. Section 6 extends the StoNED model to
account for intertemporary changes in the technology and efficiency, and derives a semiparametric
estimator. Section 7 illustrates the semiparametric estimation by means of an application to industry-
level panel data of wholesale and retail sectors in 14 OECD countries over the period 1975-2003.
Section 8 draws the concluding remarks. For compactness, the proofs of the mathematical theorems
are presented in Appendix A. Appendix B provides a GAMS code used for computing the CNLS
problem of the application.

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2. StoNED model
2.1 Cross sectional model
To gain intuition, we start from the cross-sectional StoNED model introduced by Kuosmanen (2006) and
Kuosmanen and Kortelainen (2007). The M-dimensional input vector is denoted by x and the scalar
output by y. The production technology is represented by the production function y = f(x), where function
f belongs to the class of continuous, monotonic increasing, and concave functions, denoted by F . In
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contrast to the SFA literature, no specific functional form for f is assumed a priori; the production
function is specified along the lines of the DEA literature.
The observed output y of firm i may differ from f(x ) due to inefficiency and noise. We follow
i i
the SFA literature and introduce a composite error term ε =v −u , which consists of the inefficiency
i i i
term u >0 and the idiosyncratic error term v , formally,
i i
y =f(x )+ε =f(x )−u +v , i =1,...,n. (1)
i i i i i i
Terms u and v (i =1,...,n) are assumed to be statistically independent of each other as well as of
i i
inputs x . Kuosmanen and Kortelainen (2007) follow the standard SFA practice and assume
i
u ∼ N(0,σ2) and v ∼ N(0,σ2). Banker and Maindiratta’s (1992) model differs from (1) in that the
i u i v
i.i.d i.i.d
composite errors are multiplicative (i.e., yi =f(xi)evi−ui) and the distribution of the inefficiency term ui is
truncated normal.
Model (1) is referred to as the cross-sectional stochastic nonparametric envelopment of data
(StoNED) model. It can be thought of as a generalization of the classic SFA and DEA. Specifically, if f is
restricted to some specific functional form (instead of the class F ), model (1) boils down to the SFA
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model by Aigner et al. (1977). On the other hand, if we impose the restriction σ2 =0 and relax the
v
distributional assumption concerning the inefficiency term, we obtain the DEA model by Banker et al.
(1984). In this sense, both SFA and DEA can be seen as special cases of the more general StoNED
framework. For estimation of model (1), a reader is referred to Kuosmanen and Kortelainen (2007).
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2.2. Panel data model
We next adapt the cross-sectional StoNED model to the panel data setting where we assume a
balanced data of n firms in T time periods. The panel variant of the StoNED model can be formally
defined as
y =f(x )−u +v , i =1,...,n; t =1,...,T , (2)
it it i it
where u ≥0 is the inefficiency term of firm i and v is the idiosyncratic error of firm i in period t.
i it
Production function f is assumed to be monotonic increasing and concave as above; we assume no
particular functional form for f. We assume that the idiosyncratic errors v are uncorrelated random
it
variables with E(v )=0 ∀i,t and Var(v )=σ2 <∞ ∀i,t (i.e., the Gauss-Markov assumptions).
it it v
Importantly, we impose no parametric assumptions about the distributions of u and v : the panel
i it
variant (2) is a fully nonparametric model.
It is worth to note that, similar to the standard panel data treatments in the SFA literature, we
here assume that the production function f and the inefficiency terms u do not change over time. This
i
allows us to estimate model (2) by means of the standard fixed effects and random effects techniques,
to be considered next. A dynamic, semiparametric variant of the StoNED model that allows the
production function and inefficiency terms change over time is introduced in Section 6 below.

3. Fixed effects estimation
In the fixed effects approach the inefficiency terms u are taken as unknown firm-specific constants.
i
Since the idiosyncratic errors v are the only source of random variation, we can estimate the StoNED
it
model (2) by nonparametric least squares subject to monotonicity and concavity constraints (Hildreth,
1954; Hanson and Pledger, 1976; Groeneboom et al., 2001) [here more shortly convex nonparametric
least squares (CNLS)]. CNLS is particularly well suited for the estimation of the StoNED model because
it draws its power from the monotonicity and concavity conditions (which are the maintained
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assumptions of both StoNED and DEA models) without any further assumptions about the functional
form or its smoothness. This approach avoids the bias-variance tradeoff associated with other
nonparametric regression techniques (such as kernel or spline techniques) (e.g., Yatchew 2003). The
essential statistical properties of the CNLS estimators are nowadays well understood. The maximum
likelihood property of the CNLS estimator was noted already by Hildreth (1954). Hanson and Pledger
(1976) proved consistency of estimator (7) in the single regression case. Nemirovskii et al. (1985),
Mammen (1991) and Mammen and Thomas-Agnen (1999) have established convergence rates and
Groeneboom et al. (2001) derived the asymptotic distribution at a fixed point. In the case of m inputs,
the NLS estimator (7) achieves the standard nonparametric rate of convergence O (n-1/(2+m)).
P
All known treatments of CNLS focus on the cross-sectional estimation. To estimate model (2),
we need to adapt the CNLS estimator to the panel data setting. Introducing fixed effects u, the panel
i
variant of the nonparametric least squares problem can be formally stated as
T n
min∑∑(y −(f(x )−u ))2
it it i
f,u
t=1 i=1
s.t. (3)
f ∈F
2
In words, the CNLS problem selects production function f ∈F and inefficiency terms u to minimize the
2 i
L -norm of the residuals. Problem (3) does not restrict beforehand to any particular functional form of f,
2
but searches the best-fit function from the family F , which includes an infinite number of functions. This
2
makes problem (3) generally hard to solve. In statistics, efficient algorithms for solving CNLS problems
in the single regressor (i.e., single input) cross-sectional case have been developed (e.g., Fraser and
Massam, 1989; Meyer, 1999). These algorithms require that the data is sorted in ascending order
according to the regressor x. However, such a sorting trick is not possible in the general multiple
regression (i.e., multi-input) setting where x is a vector rather than scalar.
To solve the CNLS problem (3) in the general multi-input panel data setting, we utilize the
insights from Afriat’s Theorem, following Banker and Maindiratta (1992), Matzkin (1994), and
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Kuosmanen (2006). We convert the infinite dimensional optimization problem (3) into the following finite
quadratic programming (QP) problem:
T n
min∑∑v2
it
α,β,u,v
t=1 i=1
y =α +β′x −u +v ∀i =1,...,n;t =1,...,T (4)
it it it it i it
α +β′x ≤α +β′ x ∀h,i∈{1,...,n};s,t∈{1,...,T}
it it it hs hs it
β ≥0 ∀i =1,...,n;t =1,...,T
it
The first constraint of this problem can be interpreted as the regression equation, the second constraint
enforces concavity analogous to the Afriat inequalities, and the third constraint ensures monotonicity.
The analogy of model (4) with the conventional parametric regression models is useful for econometric
model building. Note that (4) differs from the classic OLS problem in that the coefficients α,β are here
i i
firm-specific. In this respect, model (4) is structurally similar to the varying coefficient (VC) regression
models (also referred to as random parameters models) (e.g., Fan and Zhang, 1999; Greene, 2005;
Tsionas, 2002), which typically assume a conditional linear structure. However, while the random
parameters models estimate n different production functions of the same a priori specified functional
form, the CNLS regression (4) estimates n tangent hyper-planes to one unspecified production function.
The slope coefficients β represent the marginal products of inputs (i.e., the sub-gradients ∇f(x )).
it it
Problems (3) and (4) are equivalent in the following sense:

Proposition 1: Let s2 be the minimum sum of squares of problem (3) and let s2 be the minimum sum
F A
of squares of problem (4). Then for any real-valued data, s2 =s2.
F A

The absolute inefficiency levels are unidentifiable from (3) and (4). Following Gabrielsen (1975)
and Greene (1980) (see also Schmidt and Sickles 1984), we may use the best observed practice in the
sample as a benchmark: given the CNLS estimates uˆ from (4), we compute the relative inefficiency
i
estimates as
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ˆ
uˆ ≡uˆ − min (uˆ ). (5)
i i h∈{1,...,n} h
Note that uˆ can be negative. If the sampling procedure is such that fully efficient firms (i.e., firm i such
i
ˆ
that u =0) are observed with a strictly positive probability, then uˆ is a consistent estimator of u .
i i i
ˆ
Given the estimated coefficients αˆ ,β from model (4), we may estimate the production
it it
function f by the following piece-wise linear function
fˆ(x)≡ min (αˆ +βˆ′x). (6)
i∈{1,...,n},t∈{1,...,T} it it
This piece-wise linear estimator is legitimized by the following result.

Proposition 2: Denote the set of functions that minimize problem (3) by F∗ :F∗ ⊂F . For any real-
2 2 2
valued data, fˆ∈F∗.
2

ˆ ˆ
The representor f and its coefficients (αˆ ,β ) have a compelling economic interpretation.
it it
ˆ ˆ
Vector β represents the marginal products of firm i in period t. Moreover, function f provides a first
it
order Taylor series approximation to any f ∈F∗ in the neighborhood of the observed points. This
2
ˆ
justifies the use of the representor f for forecasting the output values in unobserved points within the
observed range of input values.
ˆ
Coefficients β can also be used for nonparametric estimation of the marginal properties and
it
elasticities. We can calculate the marginal rate of substitution between inputs k and m in point x as
it
∂fˆ(x ) ∂x βˆ
it k = kit , (7)
∂fˆ(x ) ∂x βˆ
it m mit
and further, the elasticity of substitution as
βˆ x
e (x )= kit ⋅ mit . (8)
k,m it βˆ x
mit kit
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ˆ
These substitution rates and elasticities are simple to compute given the estimated β coefficients.
it
The piece-wise linear structure of the estimator (6) closely resembles that of the DEA frontier.
Although problem (4) includes n different firm-specific coefficients α,β , the number of different
i i
ˆ
hyperplane segments in f(x) is typically much lower than n as in DEA. One could easily impose further
assumptions about returns to scale as in DEA: if function f exhibits constant returns to scale, we may
ˆ
simply set α =0 in problem (4). This will guarantee that function f passes through the origin. Another
it
ˆ ˆ
similarity is that the estimator f(x) and its coefficients αˆ ,β are not necessarily unique. To test for
i i
ˆ
uniqueness, one could construct upper and lower bounds for function f(x) along the lines of Varian
(1984). Finally, one can draw statistical inference about the inefficiency estimates or the coefficients
ˆ
αˆ ,β by applying the bootstrap approach by Efron (1979, 1982) (see also Simar, 1992; and
i i
Kuosmanen and Kortelainen, 2007)
According to Greene (1999, 2005), a problem of the fixed effects approach is that it attributes
any time-invariant heterogeneity across firms to the inefficiency term. In our view, heterogeneity is not a
problem as such if it arises endogenously from the decisions of the firm management (e.g. how much to
invest in capital, or where to locate the firm). However, if firms are heterogeneous in attributes that are
exogenously given and beyond their control, then labelling impacts of heterogeneity as inefficiency (or
lack of it) is both unfair and misleading. Note that including these attributes as inputs (or as other
explanatory variables) in the model does not help because the lack of variation in these attributes
makes their coefficients unidentifiable from the fixed effects. The random effects approach can provide
a remedy for this problem.

4. Random effects estimation
While in the fixed effects model the inefficiency terms were taken as firm-specific constants, the random
effects approach views inefficiency terms as random variables with E(u )=µ>0 ∀i,t and
i
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