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I am new to nonparametric regressions. What tests should one perform to choose a non-parametric regression model over a parametric one(Or vice versa)? Let's assume in our analysis we have a continuous dependent variable Y and regressors - X1 which is categorical and X2, which is continuous. We then perform a a simple linear regression for the parametric model and then run the nonparametric regression.

What should be tested if both regressors look significant and with reasonable coefficients?

EDIT: The nonparametric model here in particular is "Local Constant Kernel Regression"

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    $\begingroup$ There are many forms of nonparametric regression and even more forms of semiparametric models. Describe which methods you are referring to. State the over-arching goals of your analysis also. $\endgroup$ – Frank Harrell Dec 29 '16 at 13:54
  • $\begingroup$ Well, one reason to not use OLS parametric regression is due to violations of normality from the residuals. What do your residuals look like? $\endgroup$ – Jon Dec 29 '16 at 20:02
  • $\begingroup$ Normal for both models. $\endgroup$ – user3810441 Dec 29 '16 at 20:07
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    $\begingroup$ it's really important whether by "nonparametric" you mean something like additive models/splines or whether you are referring to classical (rank-based) nonparametric approaches such as Spearman correlations/Kendall's tau. (Or quantile regression or ...) $\endgroup$ – Ben Bolker Dec 29 '16 at 20:55
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    $\begingroup$ @user3810441 If you're asking a specific question about a specific dataset to answer a specific problem, then be specific. I gave you a general answer because you weren't specific. $\endgroup$ – Kodiologist Dec 29 '16 at 22:17
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Answering the question of which model to use for a given problem is an area of statistics unto itself, which is called model selection. Methods for this include AIC, cross-validation, and Bayesian model selection. Most introductory statistics textbooks will include material on model selection.

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    $\begingroup$ I don't think this addresses the OP's question. In my experience at least, model selection assumes we are assessing competing parametric models. $\endgroup$ – Ben Bolker Dec 29 '16 at 20:53
  • $\begingroup$ @BenBolker There's no reason you can't do model selection between parametric and nonparametric models. Some methods may be limited to parametric models (such as AIC), but that's just a limitation of the specific method. My own preference is for cross-validation, which is totally neutral to the parametric form (or lack thereof) of the model. $\endgroup$ – Kodiologist Dec 29 '16 at 22:15
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    $\begingroup$ Hmm. OK (especially since OP says they're doing kernel regression) -- I was thinking about methods like Spearman correlations, which provide p-values but not predictions. Nevertheless ... this isn't very specific advice ... $\endgroup$ – Ben Bolker Dec 30 '16 at 0:12
  • $\begingroup$ @BenBolker To my understanding, Spearman correlation isn't a regression model, or even a model at all. It's just a measure of association. $\endgroup$ – Kodiologist Dec 30 '16 at 0:39
  • $\begingroup$ Fair enough. Again, OP was pretty vague about what they were doing (although the fact that they say "reasonable coefficients" implies a real regression model). $\endgroup$ – Ben Bolker Dec 30 '16 at 0:49

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