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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
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We present the results using graphs, tables and response surface estimation. We focus on the average root mean square error fitted values less true values from each function across the three
input demands and a similar average across the six Morishima elasticities. The CES data sets vary the elasticity of substitution between inputs from 0.2 to 4.0. The Box–Cox parameter settings correspond
to design technologies 5 and 7 of Chalfant and Gallant 1985. By construction, all Box–Cox and CES parameter settings lead to technologies satisfying concavity, monotonicity, homogeneity and
symmetry at every price used in the data set and at all forecast prices. Homogeneity and symmetry
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constraints are imposed in the translog, AIM, and Fourier models. Given that the Fourier is estimated in shares, it may approximate the shares slightly better than the AIM and NN. For similar reasons,
AIM and the NN may have an advantage in approximating input demands. Our analysis reveals that the results are very similar for the shares and input demands; thus we only report the results for the
input demands. The functions are estimated using the first 100 observations, with the last 50 observations reserved for out-of-sample forecasts.
4.1. Basic results for simulated data For each technology, we generated data with no error, 10 error, and 20 error. Not surprisingly
all three SNP forms performed well for data generated with no error. Increasing the order of expansion led to a better approximation in all cases, though AIM and the Fourier series tended to
provide a slightly better approximation than the neural networks for most technologies. For data generated with error, fitting the observed data points does not guarantee an accurate
approximation of the unknown technology. Thus, we evaluate how well the SNP functions approximate the true technology using data generated with 10 and 20 error. In most cases the results
show that the SNP forms continue to better approximate the true data generating function than the translog but there is some evidence of overfitting by the SNP forms.
Fig. 1 graphs the average RMSE for all inputs from the SNP functions for the most nonlinear CES s 54 technology having 10 error, divided by the average RMSE for the translog. The
approximation of the true technology improves for each SNP form by increasing the order of the expansion from AIM 1 to AIM 2, Fourier with J 51, A53 to J 51, A56 and a neural network with
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Both the Fourier system of the cost function and the first two shares and the AIM system of three input demands are estimated using maximum likelihood estimation. The likelihood functions often proved quite flat, so convergence criteria
were tightened and results were replicated with numerous starting values. Matlab Numeric Computation Software Math Works Inc., Natick, Massachusetts, 1994 was used to perform the network estimations. For a more comprehensive
description of the neural network functional form and the associated analytical derivatives for partial effects Kastens et al., 1995.
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Homogeneity and symmetry restrictions were imposed on the neural networks using penalty functions. No results are reported because results failed to show any improvements over the unconstrained model. Optimization difficulties, such as
the lack of convergence, also led us to have less confidence in the restricted neural networks.
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Fig. 1. Ratio of RMSE to translog RMSE for input demands.
one to three nodes. All SNP forms better approximate the underlying technology than the translog at these orders confirming the results using data generated without error. However, the approximation
error increases for each SNP form when moving from the second highest order to the highest order considered. Fig. 2 shows that the same general pattern occurs for elasticities as well with 10 error.
Results are similar when using data having 20 error.
In contrast to the results using data generated without error, all SNP forms show violations of concavity. However, for most technologies the violations of concavity occur at less than 5 of the
data points. It may be that violations of concavity serve as a warning of overfitting given that the true technology adheres to microeconomic theory. Our results tend to support this conjecture particularly
when there is a considerable amount of error 20. Also, when the violations of concavity occur, the quality of the approximation decreases when moving to the highest order of the SNP expansion. In
Fig. 2. Ratio of RMSE to translog RMSE for elasticities.
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summary, when many violations of concavity occur, it may be preferable to use a lower order of an
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SNP expansion. Results for out-of-sample forecasts closely mirror the in-sample results. 4.2. Response surface analysis
An alternative method for analyzing the results is to estimate a response surface using different covariates. We focus on the estimates of substitution elasticities which are more difficult to
approximate than input demands. The response surfaces are estimated by regressing the average RMSEs for Morishima elasticities on covariates. Results are reported in Table 2. Model 1 regresses
RMSE on a dummy variable for AIM, Fourier, and neural network models with the translog serving as the excluded group. Thus, the intercept gives an average RMSE of the translog across all simulated
data sets of 0.559. The other coefficients show that on average AIM models lowered the RMSE by 0.453 below the translog, Fourier models lowered the RMSE by 0.359, and neural network models by
0.06.
Model 2 adds the mean elasticity, proportion of the variation attributable to error and the number of parameters used in estimation. As predicted, it becomes more difficult to predict the elasticity when
both the size of the elasticity and amount of error increase. In addition, increasing the number of parameters tended to lower the root mean square error. To evaluate the principle of parsimony, Model
3 includes an interaction term between number of parameters and the size of the random error. The positive coefficient on the interaction supports the principle of parsimony by finding that fewer
parameters should be used in data with more noise.
In all specifications, the results show advantages of choosing the AIM and Fourier models over the translog, but indicate that the neural network models fail to yield the same improvement. One
explanation for this result lies in the fact that the translog, AIM, and Fourier models impose symmetry and homogeneity restrictions by construction while the neural network models fail to incorporate this
restriction. Models 4 and 5 include a dummy variable set equal to one for all forms that impose homogeneity and symmetry restrictions. Results show that the restrictions lower the RMSE especially
Table 2 Response surface estimates standard errors are in parentheses
Variable Model 1
Model 2 Model 3
Model 4 Model 5
Intercept 0.559 0.135
0.136 0.101 0.211 0.108
0.138 0.082 0.168 0.111
AIM 20.453 0.155
20.448 0.112 20.448 0.112
Fourier 20.359 0.155
20.354 0.112 20.354 0.112
Neural net 20.060 0.155
20.055 0.112 20.055 0.112
Elasticity 0.259 0.020
0.259 0.020 0.259 0.021
0.259 0.021 Size of errors
1.077 0.355 0.336 0.540
1.077 0.037 0.769 0.839
Number of parameters 20.0003 0.002
20.005 0.003 20.003 0.002
20.008 0.004 NumberSize of errors
0.048 0.026 0.045 0.028
Theory 20.299 0.066
20.243 0.105 TheoryError
20.567 0.812
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Tables containing the RMSEs for all simulated technologies, both in sample and out-of-sample, are available in pdf form at http: www.bus.lsu.edu economics faculty dterrell tables.pdf.
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for data generated with error. This suggests that there may be benefits for imposing these conditions on neural networks if these properties hold in the underlying data generating process.
5. Conclusion
