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.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
The data used to evaluate the flexible forms are simulated from the CES and generalized Box–Cox functions. These data generating functions are used because they give substitution elasticities of the
magnitudes often found in empirical applications. In addition, Terrell 1995 uses the CES function to evaluate the AIM whereas Chalfant and Gallant 1985 use the generalized Box–Cox technology to
evaluate the Fourier. Therefore, by considering these technologies, it is unlikely that the SNP results are biased toward one functional form because of the choice of the data generating function.
The CES cost function and input demands used by Terrell 1995 to evaluate the AIM are:
n r 21 r
r r 21 1 12
r 1
r 21
C y,P 5 y
O
p and x 5 C y,P
p
S D
i i
i i 51
The generalized Box–Cox and input demands used by Chalfant and Gallant 1985 with symmetry and homogeneity imposed, are:
5
The nesting in the first expression may also continue at additional levels adding more hidden layers. However, the model depicted is a universal functional approximator, given a sufficient number of nodes within the single hidden layer
Hornik et al., 1989.
A .R. Fleissig et al. Economics Letters 68 2000 235 –244
239
n n
1 v
v 2 v 2
12 v
v 221 v 2
C p 5 2 v
O O
u P P
and x 5 2 vC
P
O
u P
F G
ij i
j i
i ij
j i 51 j 51
j
The elasticities are calculated using the Morishima elasticity of substitution: p
p
i i
] ]
s 5 C ? 2 C ?
ij ij
ii
x x
j i
because, as Blackorby and Russell 1989 show, the Allen–Uzawa measure is incorrect when there are more than two variables. Note that the Morishima elasticity is non-symmetric unless the data
generating function is a member of the constant elasticity of substitution CES family. For the CES function, the Morishima elasticities of substitution are symmetric and constant whereas the
generalized Box–Cox function yields variable non-symmetric Morishima elasticities.
The trivariate data generating process of Chalfant and Gallant 1985 is used to construct price series similar to those in the Berndt and Wood 1975 data set. Input prices are deflated by the price of
output and then fitted by a first order autoregression ln P 5 b 1 e with e 5 Re
1 e giving
t t
t t 21
t 6
estimates of R and b, where the elements of b correspond to capital, labor and other inputs. We then
generate a set of 150 observations for each input price by combining normally distributed innovations with the estimates from the autoregression. The input demands with a corresponding wide range of
Morsihima elasticities are generated from the CES and generalized Box–Cox forms using the parameter settings from Table 1 and price series.
Data with 10 and 20 measurement error are generated by adding normally distributed errors
2 2
N0, s
and N0, s
to the input demands. It is important to avoid choosing an unrealistically
10 20
large or small innovation for the stochastic error component of the simulated data. We address this issue by setting the variance of the stochastic component to be a proportion 10 or 20 of the
variance of the nonstochastic of input demands. For example, a 10 error for input i requires setting
2
s 5 0.10 ?Varx , which ensures that approximately 90 of the variation of input i is attributable
i,10 i
Table 1 Parameter settings for simulated data
1. CES r 50.75
Large elasticity 4.00 2. CES
r 5 20.5 Medium elasticity 2 3
3. CES r 5 24.00
Small elasticity 0.20 4. Box–Cox 1
v 51.70711, u 50.2699, u 50.1302, Small elasticity
11 12
u 50.0939, u 50.0405, u 50.0142, u 50.0211
13 22
23 33
5. Box–Cox 3 v 50.29289, u 5 21.0095, u 50.8086,
Large elasticity
11 12
u 50.7645, u 50.0401, u 5 20.6248, u 50.0358
13 22
23 33
6
The Berndt–Wood data have capital, labor, energy, and materials. We construct a share-weighted aggregate of energy and materials inputs defined as other inputs to reduce the number of inputs to three. The estimate of
b was 0.09761, 20.00721, 0.37572.
240 A
.R. Fleissig et al. Economics Letters 68 2000 235 –244
to variation in prices. This provides a total of 15 data sets which have either, no error, 10 error, or 20 error for each technology.
4. Results
