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I need to fit a bivariate data using kernel regression (local polynomial regression). It should satisfies two conditions.

  • $\frac{dy}{dx_1} \geq 0$ for all $x_2$

  • $\frac{dy}{dx_2} \geq 0$ for all $x_1$

How can I incorporate these two conditions into local polynomial regression?

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Shape restrictions on the kernel regression function can be imposed using various approach.

One general method was developed in Du, Parmeter and Racine (2013). The basic idea is to impose the constraints (here monotonicity) through weights in a generalized kernel estimator in the spirit of Hall and Huang (2001). The procedure involves solving a quadratic program with linear inequality constraints.

Another method was recently proposed in Horowitz and Lee (2015). The idea is to estimate the unrestricted kernel function, then evaluate what constraints are binding, and finally re-estimate the kernel function under the restrictions found and ignore all others.

Other recent papers study shape restrictions in econometrics. Among others, Chernozukov, Newey and Santos (2015) for conditional moment restriction models although I find it to be a difficult read.

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