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


  • Dole, David


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.

Suggested Citation

  • Dole, David, 1999. "CoSmo: A Constrained Scatterplot Smoother for Estimating Convex, Monotonic Transformations," Journal of Business & Economic Statistics, American Statistical Association, vol. 17(4), pages 444-455, October.
  • Handle: RePEc:bes:jnlbes:v:17:y:1999:i:4:p:444-55

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    Cited by:

    1. Ait-Sahalia, Yacine & Duarte, Jefferson, 2003. "Nonparametric option pricing under shape restrictions," Journal of Econometrics, Elsevier, vol. 116(1-2), pages 9-47.
    2. Stefano A. Gattone & Tonio Di Battista, 2009. "A functional approach to diversity profiles," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 58(2), pages 267-284, May.
    3. Yao Luo, 2023. "Bundling and nonlinear pricing in telecommunications," RAND Journal of Economics, RAND Corporation, vol. 54(2), pages 268-298, June.
    4. Pascal Albert & Michael Herold & Matthias Muck, 2023. "Estimation of rare disaster concerns from option prices—An arbitrage‐free RND‐based smile construction approach," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 43(12), pages 1807-1835, December.
    5. Fengler, Matthias & Hin, Lin-Yee, 2011. "Semi-nonparametric estimation of the call price surface under strike and time-to-expiry no-arbitrage constraints," Economics Working Paper Series 1136, University of St. Gallen, School of Economics and Political Science, revised May 2013.
    6. Fengler, Matthias R. & Hin, Lin-Yee, 2015. "Semi-nonparametric estimation of the call-option price surface under strike and time-to-expiry no-arbitrage constraints," Journal of Econometrics, Elsevier, vol. 184(2), pages 242-261.
    7. Yao Luo, 2011. "Nonlinear Pricing with Product Customization in Mobile Service Industry," Working Papers 11-28, NET Institute.

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