Many economists uncritically accept the appropriateness of a functional form on the basis of convention or convenience; others try several forms for their relations but report only the one that in some sense “looks best” a posteriori.
One is often interested not only in the curve itself but also in special qualitative characteristics of the smooth. The regression function may be constrained to simple shape characteristics, for example, and the smooth should preferably have the same qualitative characteristics. A quite common shape characteristic is a monotonic or unimodal relationship between the predictor variable and the response variable. This a priori knowledge about the qualitative form of the curve should be built into the estimation technique. Such qualitative features do not necessarily lead to better rates of convergence but help the experimenter in interpretation of the obtained curves.
In economic applications involving demand, supply and price, functions with prescribed shape (monotonicity, convexity, etc.) are common. Lipsey, Sparks and Steiner (1976, chapter 5) present a number of convex decreasing demand curves and convex increasing supply curves (in both cases, price as a function of quality). They also give an example for quantity demanded as a function of household income. A more complex procedure could be applied to the potato Engel curve in Figure 1.2. The nonparametric fit shows a partially increasing and a decreasing segment. This curve could be estimated by a unimodal regression technique.
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