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Perspectives on nonparametric and semiparametric modeling.

Article Excerpt
Nonparametric regression techniques hold out the promise of more flexible modeling of data in many areas of physical, biological and social sciences. However, their use is hampered by the "curse of dimensionality" which imposes enormous data requirements as the number of explanatory variables increases. After summarizing two of the most commonly used methods for mitigating the "curse", this paper outlines a new approach which exploits data on derivatives. In economics, such circumstances arise in the joint estimation of cost and factor demand functions, or when production function data are combined with data on factor prices. The ideas are illustrated using empirical examples from energy economics.

PREFATORY REMARKS

I had the distinct honor and privilege of working with Campbell Watkins for many years when we jointly edited The Energy Journal. Our backgrounds and interests were complementary, our interactions always collegial and friendly. I recall the day Campbell called to inform me of his illness. His voice never faltered, though he was aware of the gravity of his circumstances. In the few months that remained for him, he fulfilled his editorial responsibilities to the very highest standards, and remarkably, continued his research program. (I know this because I assigned our best graduate student--Brian McCaig--to assist him.) Campbell and I had long talked about collaborating on a research paper. Alas, that never came to fruition. As part of his legacy, Campbell assigned to me the subject area "nonparametric demand functions" as a topic for this volume. This paper will attempt to offer some perspectives on the use of flexible modeling techniques such as nonparametric and semi-parametric estimation. Though all the examples are drawn from energy economics, the tools have much broader applicability in economics and in other fields.

1. INTRODUCTION

Economists frequently rely upon optimization problems to model the behavior of economic agents. Textbook examples include utility maximization in the theory of the consumer and cost minimization or profit maximization in the theory of the firm. Indeed, the expression "to economize" connotes some kind of constrained optimization process.

These same economic theories rarely provide sufficient structure to dictate specific functional forms. Thus, the modeler faces competing objectives. On the one hand, simple specifications will lead to more precise results: standard errors of parameter estimates will be generally smaller and prediction ranges will be narrower. On the other hand, the model may be an over-simplification that ignores not just statistically significant features of the data, but elements that may materially impact upon policy decisions. In that event, one would prefer a richer specification which yields more robust if less precise predictions.

To illustrate, consider an electricity distribution industry populated by firms which are identical with respect to all exogenous variables except their "size", which we will measure by the number of customers served. It is reasonable to suppose that there are scale economies in this business so that unit costs (e.g., costs per customer) should decline, at least initially. Once minimum efficient scale is reached, one might expect unit costs to level off. Unfortunately, standard linear, quadratic or polynomial models lack the capacity to approach an asymptote. An alternative approach is to produce a scatter-plot of the data, then to "smooth" the observations to estimate the effects of size on unit costs. "Smoothing" can be accomplished by taking "local averages". For example, to estimate unit costs for a utility serving say 50,000 customers, one might average data for firms serving 40,000 to 60,000 customers. This idea of local averaging is the essence of nonparametric regression techniques.

The graph in Figure 1 illustrates the results of such a procedure applied to 81 distributors of varying sizes in Ontario, Canada. The data have been standardized (using methods described below) to remove the effects of other differences across these utilities. Unit costs (measured in logarithms) fall steadily as the (log of the) number of customers increases. Beyond a certain point, they appear to remain approximately flat with the exception of the largest firm which has among the highest unit costs.

2. THE CURSE OF DIMENSIONALITY

The essence of the smoothing procedure described above is this--to estimate the effect of firm size, one calculates an average over firms of similar size. If there are many such firms, the estimate should be accurate, that is, it will have a small standard error. If not, the standard error will be large.

Why would one not always rely upon nonparametric...