CoSmo: A Constrained Scatterplot Smoother for Estimating Convex, Monotonic Transformations

In many of the applied sciences, it is common that the forms of empirical relationships are almost completely unknown prior to study. Scatterplot smoothers used in nonparametric regression methods have considerable potential to ease the burden of model specification that a researcher would otherwise face in this situation. Occasionally the researcher will know the sign of the first or second derivatives, or both. This article develops a smoothing method that can incorporate this kind of information. I show that cubic regression splines with bounds on the coefficients offer a simple and effective approximation to monotonic, convex or concave transformations. I also discuss methods for testing whether the constraints should be imposed. Monte Carlo results indicate that this method, dubbed CoSmo, has a lower approximation error than either locally weighted regression or two other constrained smoothing methods. CoSmo has many potential applications and should be especially useful in applied econometrics. As an illustration, I apply CoSmo in a multivariate context to estimate a hedonic price function and to test for concavity in one of the variables.