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

${\displaystyle {\widehat {m}}_{h}(x)={\frac {\sum _{i=1}^{n}K_{h}(x-x_{i})y_{i}}{\sum _{i=1}^{n}K_{h}(x-x_{i})}}}$ 

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 ${\displaystyle \int _{-\infty }^{\infty }uK(u)\,du=0}$ .

Derivation edit

Starting with the definition of conditional expectation,

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

we estimate the joint distributions f(x,y) and f(x) using kernel density estimation with a kernel K:

${\displaystyle {\hat {f}}(x,y)={\frac {1}{n}}\sum _{i=1}^{n}K_{h}(x-x_{i})K_{h}(y-y_{i}),}$ 

${\displaystyle {\hat {f}}(x)={\frac {1}{n}}\sum _{i=1}^{n}K_{h}(x-x_{i}),}$ 

We get:

${\displaystyle {\begin{aligned}\operatorname {\hat {E}} (Y\mid X=x)&=\int y{\frac {{\hat {f}}(x,y)}{{\hat {f}}(x)}}\,dy,\\[6pt]&=\int y{\frac {\sum _{i=1}^{n}K_{h}(x-x_{i})K_{h}(y-y_{i})}{\sum _{j=1}^{n}K_{h}(x-x_{j})}}\,dy,\\[6pt]&={\frac {\sum _{i=1}^{n}K_{h}(x-x_{i})\int y\,K_{h}(y-y_{i})\,dy}{\sum _{j=1}^{n}K_{h}(x-x_{j})}},\\[6pt]&={\frac {\sum _{i=1}^{n}K_{h}(x-x_{i})y_{i}}{\sum _{j=1}^{n}K_{h}(x-x_{j})}},\end{aligned}}}$ 

which is the Nadaraya–Watson estimator.

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

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

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

where ${\displaystyle s_{i}={\frac {x_{i-1}+x_{i}}{2}}.}$ 

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.

Script for example edit

The following commands of the R programming language use the npreg() 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 library
data(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 to David Salsburg, the algorithms used in kernel regression were independently developed and used in fuzzy 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 on these pages (this toolbox is a part of the book ^[6]).
• Python: the KernelReg class for mixed data types in the statsmodels.nonparametric sub-package (includes other kernel density related classes), the package kernel_regression as an extension of scikit-learn (inefficient memory-wise, useful only for small datasets)
• R: the function npreg of the np 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 (with Microsoft Excel).
• An online kernel regression demonstration Requires .NET 3.0 or later.
• Kernel regression with automatic bandwidth selection (with Python)