Instatistics,kernel regression is anon-parametric technique to estimate theconditional expectation of arandom variable. The objective is to find a non-linear relation between a pair of random variablesX andY.

In anynonparametric regression, theconditional expectation of a variable${\displaystyle Y}$ relative to a variable${\displaystyle X}$ may be written:

${\displaystyle \mathrm{E}(Y\mid X)=m(X)}$

where${\displaystyle m}$ is an unknown function.

Nadaraya andWatson, both in 1964, proposed to estimate${\displaystyle m}$ as a locally weighted average, using akernel as a weighting function.^[1][2][3] The Nadaraya–Watson estimator is:

${\displaystyle {\hat{m}}_h(x)=\frac{\sum _{i=1}^n{K}_h(x-x_i)y_i}{\sum _{i=1}^n{K}_h(x-x_i)}}$

where${\displaystyle K_h(t)=\frac{1}{h}K\left(\frac{t}{h}\right)}$ is a kernel with a bandwidth${\displaystyle h}$ such that${\displaystyle K(\cdot )}$ is of order at least 1, that is${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}uK(u)du=0}$.

Starting with the definition ofconditional expectation,

${\displaystyle \mathrm{E}(Y\mid X=x)=\int yf(y\mid x)dy=\int y\frac{f(x,y)}{f(x)}dy}$

we estimate the joint distributionsf(x,y) andf(x) usingkernel density estimation with a kernelK:

${\displaystyle \hat{f}(x,y)=\frac{1}{n}\sum _{i=1}^n{K}_h(x-x_i)K_h(y-y_i),}$

${\displaystyle \hat{f}(x)=\frac{1}{n}\sum _{i=1}^n{K}_h(x-x_i),}$

We get:

which is the Nadaraya–Watson estimator.

${\displaystyle {\hat{m}}_{PC}(x)=h^{-1}\sum _{i=2}^n(x_i-x_{i-1})K\left(\frac{x-x_i}{h}\right)y_i}$

where${\displaystyle h}$ is the bandwidth (or smoothing parameter).

${\displaystyle {\hat{m}}_{GM}(x)=h^{-1}\sum _{i=1}^n\left[{\int }_{s_{i-1}}^{s_i}K\left(\frac{x-u}{h}\right)du\right]y_i}$

where${\displaystyle s_i=\frac{x_{i-1}+x_i}{2}.}$^[4]

This example is based upon Canadian cross-section wage data consisting of a random sample taken from the 1971 Canadian Census Public Use Tapes for male individuals having common education (grade 13). There are 205 observations in total.^{[citation needed]}

The figure to the right shows the estimated regression function using a second order Gaussian kernel along with asymptotic variability bounds.

The following commands of theR programming language use thenpreg() function to deliver optimal smoothing and to create the figure given above. These commands can be entered at the command prompt via cut and paste.

install.packages("np")library(np)# non parametric librarydata(cps71)attach(cps71)m<-npreg(logwage~age)plot(m,plot.errors.method="asymptotic",plot.errors.style="band",ylim=c(11,15.2))points(age,logwage,cex=.25)detach(cps71)

According toDavid Salsburg, the algorithms used in kernel regression were independently developed and used infuzzy systems: "Coming up with almost exactly the same computer algorithm, fuzzy systems and kernel density-based regressions appear to have been developed completely independently of one another."^[5]

• GNU Octave mathematical program package
• Julia:KernelEstimator.jl
• MATLAB: A free MATLAB toolbox with implementation of kernel regression, kernel density estimation, kernel estimation of hazard function and many others is available onthese pages (this toolbox is a part of the book^[6]).
• Python: theKernelReg class for mixed data types in thestatsmodels.nonparametric sub-package (includes other kernel density related classes), the packagekernel_regression as an extension ofscikit-learn (inefficient memory-wise, useful only for small datasets)
• R: the functionnpreg of thenp package can perform kernel regression.^[7][8]
• Stata:npregress,kernreg2

• Kernel smoother
• Local regression

• Henderson, Daniel J.; Parmeter, Christopher F. (2015).Applied Nonparametric Econometrics. Cambridge University Press.ISBN 978-1-107-01025-3.
• Li, Qi; Racine, Jeffrey S. (2007).Nonparametric Econometrics: Theory and Practice. Princeton University Press.ISBN 978-0-691-12161-1.
• Pagan, A.; Ullah, A. (1999).Nonparametric Econometrics. Cambridge University Press.ISBN 0-521-35564-8.
• Racine, Jeffrey S. (2019).An Introduction to the Advanced Theory and Practice of Nonparametric Econometrics: A Replicable Approach Using R. Cambridge University Press.ISBN 9781108483407.
• Simonoff, Jeffrey S. (1996).Smoothing Methods in Statistics. Springer.ISBN 0-387-94716-7.

• Scale-adaptive kernel regression (with Matlab software).
• Tutorial of Kernel regression using spreadsheet (withMicrosoft Excel).
• An online kernel regression demonstration Requires .NET 3.0 or later.
• Kernel regression with automatic bandwidth selection (with Python)

• Nonparametric regression

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