Economics Letters 68 2000 235–244
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Evaluating the semi-nonparametric Fourier, AIM, and neural networks cost functions
a b
c ,
Adrian R. Fleissig , Terry Kastens , Dek Terrell
a
Department of Economics , California State University–Fullerton, Fullerton, CA 92834-6848, USA
b
Department of Agricultural Economics , 304F Waters Hall, Kansas State University, Manhattan, KS 66506, USA
c
Department of Economics , 2114 CEBA, Baton Rouge, LA 70806, USA
Received 26 April 1999; accepted 14 December 1999
Abstract
This study compares how well three semi-nonparametric functions, the Fourier flexible form, asymptotically ideal model, and neural networks, approximate simulated production data. Results show that higher order series
expansions better approximate the true technology for data sets that have little or no measurement error. For highly nonlinear technologies and much measurement error, lower order expansions may be appropriate.
2000 Elsevier Science S.A. All rights reserved.
Keywords : Semi-nonparametric; Fourier; AIM; Neural networks
JEL classification : C14; D12
1. Introduction
Semi-nonparametric SNP forms represent the latest development in a trend toward using functional forms that can globally asymptotically approximate more complex data generating
functions, a property Gallant 1981 defines as Sobolev-flexibility. In applications, however, these
1
SNP functions can give different results and implications for policymakers and firms. Our goal is to evaluate the approximations of the SNP functions when the data have as much as 10 and 20
measurement error. Given data with considerable measurement error, it is possible for the SNP
Corresponding author. Tel.: 11-225-388-3785; fax: 11-225-388-3807. E-mail address
: mdterreunix1.sncc.lsu.edu D. Terrell
1
For example, Fleissig et al. 1997 compare estimates of the elasticity of substitution between capital and labor for U.S. manufacturing using three SNP forms. They find that although all forms predict that capital and labor are substitutes in
production, the magnitudes of elasticity estimates vary considerably across SNP forms. 0165-1765 00 – see front matter
2000 Elsevier Science S.A. All rights reserved.
P I I : S 0 1 6 5 - 1 7 6 5 0 0 0 0 2 4 1 - X
236 A
.R. Fleissig et al. Economics Letters 68 2000 235 –244
functions to provide a poor approximation by fitting the ‘noise’ around the true data generating
2
process. Furthermore, SNP forms are also compared to two Diewert-flexible functional forms. While
3
previous studies compare one SNP function to parametric flexible forms, this paper provides the first comparison of how well three SNP functions, the Fourier flexible form, asymptotically ideal model
AIM and neural networks NN, approximate data generated from different technologies.
2. Semi-nonparametric forms
Economists must choose among various functional forms to approximate an unknown technology. In this study, we examine the ability of functional forms to approximate a unit cost function. Output is
omitted which is equivalent to approximating a constant returns to scale cost function. Once the appropriate functional form is determined for the unit cost function, additional terms can be appended
to address returns to scale.
2.1. The Fourier cost function The Fourier approximation of the true cost function developed by Gallant 1982 is:
A J
1 ]
9 9
g x, u 5 u 1 b9x 1
x9Cx 1
O
u 1 2
O
u cos j
lk x 2 v sin jlk x s
d s
d
S
f g
D
k a
j a
a j
a a
T
2
a 51 j 51
2 A
9
where u , b, u , u , . . . ,v , v , . . . are the parameters to be estimated, C 5 l
o u k k and
01 02
11 12
a 51 a a a
x 5 ln p 1 ln a is a logarithmic transformation of prices. The constant a chosen so that x is strictly
i i
i i
i 25
positive ln a 5 2 minln p 1 10 and the location shift does not affect the results. The scaling
i i
factor l is chosen a priori, l 5 6 maxx , to ensure that all x are within the interval 0,2p see
i i
Gallant, 1982 for details. The sequence of multi-indices hk j represents the partial derivatives of the
a
Fourier cost function with each set producing a particular Fourier series expansion. The cost function
n n
is linearly homogeneous in price if o
b 5 1 and o
k 5 0 and these restrictions are imposed in
i 51 i
i 51 i
a
estimation. The Fourier cost share Eqs. are:
A J
9 9
9
s 5 b 2 l
O
u lk x 1 2
O
u sin j
lk x 1 v cos jlk x k
s d
s d
S
f g
D
i a
a j
a a
j a
a a
a 51 j 51
for i 5 1, . . . ,n and system of n 21 cost shares is estimated. Estimation requires selecting the order of the Fourier expansion which depends on A and J. For
reliable asymptotics, Huber 1981 shows that the number of parameters required is about two-thirds the root of the effective sample size. With two Eqs. and 100 observations this amounts to estimating
about 35 parameters. With A55 and J 53, the Fourier has 37 parameters which are used in estimation. However, when the data are measured with error, higher order Fourier expansions may fit
2
Diewert 1974 defines a flexible form as any function that gives a local second order approximation to the data generating function.
3
For example, see Chalfant and Gallant 1985, Terrell 1995, Jensen 1997 and Fleissig and Swofford 1997.
A .R. Fleissig et al. Economics Letters 68 2000 235 –244
237
the ‘noise’ around the true technology and it may be better to estimate a lower order expansion. To investigate this possibility, we also use the Chalfant and Gallant 1985 settings of A53 J 51 13
parameters and A56 J 51 22 parameters.
2.2. The AIM cost function ¨
The Barnett and Jonas 1983 AIM model is defined on the basis of the Muntz–Szatz approximation of the cost function:
Mm n
u
a , j
i
C P 5
O
a
P
p
k i
j j 51
i 50
which is homogeneous of degree one in factor prices if the exponent sets sum to one. AIM is dense in the space of continuous functions if all
u , 1. Following Barnett et al. 1991, Terrell 1995 and
a j
Jensen 1997, we use the series 1 2, 1 4, 1 8, . . . to complete the definition of AIM. The resulting cost function is best understood by looking at the three input model at low orders:
AIM 1: Six parameters
1 2 1 2
1 2 1 2
1 2 1 2
C P 5 a p 1 a p 1 a p 1 a p
p 1
a p p
1 a p
p
1 1
1 2
2 3
3 4
1 2
5 1
3 6
2 3
AIM 2: Fifteen parameters
3 4 1 4
3 4 1 4
1 2 1 2
C P 5
a p 1 a p 1 a p 1 a p p
1 a p
p 1
a p p
2 1
1 2
2 3
3 4
1 2
5 1
3 6
1 2
1 2 1 4
1 4 1 2
1 2 1 4
3 4 1 4
1 2 1 4
1 a p
p p
1 a p
p 1
a p p
1 a p
p p
7 1
2 3
8 1
3 9
1 2
10 1
2 3
1 4 1 4
1 2 1 4
3 4 3 4
1 4 1 2
1 2 1 4
3 4
1 a p
p p
1 a p
p 1
a p p
1 a p
p 1
a p p
11 1
2 3
12 1
3 13
2 3
14 2
3 15
2 3
It is apparent that AIM 1 is the generalized Leontief flexible functional form. AIM 2 includes all combinations of prices with exponents 1 4,1 2,3 4 summing to one and has AIM 1 as a special case.
m
At order m, AIM includes combinations of prices raised to the power 1 2 . The system of AIM input demands estimated are derived using Shephard’s lemma.
2.3. The neural network approximation of the cost function
4
The neural network NN approximation of a three-input cost function of a system of demand Eqs. is: NNh: 7h 13 parameters, 1 hidden layer
x p 5 f f pW 1 w W 1 w
1 1
2 2
where, x is a 1 by 3 vector of input demands and p is the associated 1 x 3 input price vector. W , w ,
1 1
W , and w , are matrices or vectors of parameters weights to be estimated. W is dimensioned 3 x h
2 2
1
and w is 1 x h. For conformability, W and w must be h x 3 and 1 x 3, respectively. In general, w
1 2
2 1
4
Note that unlike the AIM and Fourier forms, this neural network fails to incorporate the homogeneity and symmetry restrictions into the cost function. In the results section we discuss attempts to impose symmetry and homogeneity using a
penalty function.
238 A
.R. Fleissig et al. Economics Letters 68 2000 235 –244
and w are analogous to the intercept in conventional econometric models. The number of nodes in
2
this single-hidden-layer model is denoted by h, and must be determined empirically we considered values of h 51, 3, and 5. A larger h more nodes in the hidden layer implies a higher order
5
functional approximation. The function f, operating on its matrix argument on an element-by-element basis, is an arbitrary bounding function whose first derivative is typically a function of f. The function
f is often approximated by the sigmoid function fx 5 1 1 1 exp2x with corresponding first derivative, f 9x 5 fx1 2 fx and is used in this paper.
2.4. The translog In addition to the SNP forms, we estimate two Diewert flexible forms, the generalized Leontief
which is equivalent to AIM1 and the following translog cost function: C y,P 5 exph y,P where h y,P 5 d 1
O
d ln p 1
O O
g ln p ln p
i i
ij i
j i
i j
The parameters satisfy the restrictions of symmetry and linear homogeneity in prices o d 5 1,
i i
g 5 g , and o g 5 0 and the input demands are derived using Shephard’s lemma.
ij ji
i ij
3. Experimental design
