np.plregression: Partially Linear Kernel Regression with Mixed Data Types

In np: Nonparametric Kernel Smoothing Methods for Mixed Data Types

Partially Linear Kernel Regression with Mixed Data Types

Description

npplreg computes a partially linear kernel regression estimate of a one (1) dimensional dependent variable on p+q-variate explanatory data, using the model Y = X\beta + \Theta (Z) + \epsilon given a set of estimation points, training points (consisting of explanatory data and dependent data), and a bandwidth specification, which can be a rbandwidth object, or a bandwidth vector, bandwidth type and kernel type.

Usage

npplreg(bws, ...)

## S3 method for class 'formula'
npplreg(bws, data = NULL, newdata = NULL, y.eval =
FALSE, ...)

## S3 method for class 'call'
npplreg(bws, ...)

## S3 method for class 'plbandwidth'
npplreg(bws,
        txdat = stop("training data txdat missing"),
        tydat = stop("training data tydat missing"),
        tzdat = stop("training data tzdat missing"),
        exdat,
        eydat,
        ezdat,
        residuals = FALSE,
        ...)

Arguments

bws	a bandwidth specification. This can be set as a plbandwidth object returned from an invocation of npplregbw, or as a matrix of bandwidths, where each row is a set of bandwidths for Z, with a column for each variable Z_i. In the first row are the bandwidths for the regression of Y on Z, the following rows contain the bandwidths for the regressions of the columns of X on Z. If specified as a matrix additional arguments will need to be supplied as necessary to specify the bandwidth type, kernel types, training data, and so on.
...	additional arguments supplied to specify the regression type, bandwidth type, kernel types, selection methods, and so on. To do this, you may specify any of regtype, bwmethod, bwscaling, bwtype, ckertype, ckerorder, ukertype, okertype, as described in npregbw.
data	an optional data frame, list or environment (or object coercible to a data frame by as.data.frame) containing the variables in the model. If not found in data, the variables are taken from environment(bws), typically the environment from which npplregbw was called.
newdata	An optional data frame in which to look for evaluation data. If omitted, the training data are used.
y.eval	If newdata contains dependent data and y.eval = TRUE, np will compute goodness of fit statistics on these data and return them. Defaults to FALSE.
txdat	a p-variate data frame of explanatory data (training data), corresponding to X in the model equation, whose linear relationship with the dependent data Y is posited. Defaults to the training data used to compute the bandwidth object.
tydat	a one (1) dimensional numeric or integer vector of dependent data, each element i corresponding to each observation (row) i of txdat. Defaults to the training data used to compute the bandwidth object.
tzdat	a q-variate data frame of explanatory data (training data), corresponding to Z in the model equation, whose relationship to the dependent variable is unspecified (nonparametric). Defaults to the training data used to compute the bandwidth object.
exdat	a p-variate data frame of points on which the regression will be estimated (evaluation data). By default, evaluation takes place on the data provided by txdat.
eydat	a one (1) dimensional numeric or integer vector of the true values of the dependent variable. Optional, and used only to calculate the true errors. By default, evaluation takes place on the data provided by tydat.
ezdat	a q-variate data frame of points on which the regression will be estimated (evaluation data). By default, evaluation takes place on the data provided by tzdat.
residuals	a logical value indicating that you want residuals computed and returned in the resulting plregression object. Defaults to FALSE.

Details

npplreg uses a combination of OLS and nonparametric regression to estimate the parameter \beta in the model Y = X\beta + \Theta (Z) + \epsilon.

npplreg implements a variety of methods for nonparametric regression on multivariate (q-variate) explanatory data defined over a set of possibly continuous and/or discrete (unordered, ordered) data. The approach is based on Li and Racine (2003) who employ ‘generalized product kernels’ that admit a mix of continuous and discrete data types.

Three classes of kernel estimators for the continuous data types are available: fixed, adaptive nearest-neighbor, and generalized nearest-neighbor. Adaptive nearest-neighbor bandwidths change with each sample realization in the set, x_i, when estimating the density at the point x. Generalized nearest-neighbor bandwidths change with the point at which the density is estimated, x. Fixed bandwidths are constant over the support of x.

Data contained in the data frame tzdat may be a mix of continuous (default), unordered discrete (to be specified in the data frame tzdat using factor), and ordered discrete (to be specified in the data frame tzdat using ordered). Data can be entered in an arbitrary order and data types will be detected automatically by the routine (see np for details).

A variety of kernels may be specified by the user. Kernels implemented for continuous data types include the second, fourth, sixth, and eighth order Gaussian and Epanechnikov kernels, and the uniform kernel. Unordered discrete data types use a variation on Aitchison and Aitken's (1976) kernel, while ordered data types use a variation of the Wang and van Ryzin (1981) kernel.

Value

npplreg returns a plregression object. The generic accessor functions coef, fitted, residuals, predict, and vcov, extract (or estimate) coefficients, estimated values, residuals, predictions, and variance-covariance matrices, respectively, from the returned object. Furthermore, the functions summary and plot support objects of this type. The returned object has the following components:

evalx	evaluation points
evalz	evaluation points
mean	estimation of the regression, or conditional mean, at the evaluation points
xcoef	coefficient(s) corresponding to the components \beta_i in the model
xcoeferr	standard errors of the coefficients
xcoefvcov	covariance matrix of the coefficients
bw	the bandwidths, stored as a plbandwidth object
resid	if residuals = TRUE, in-sample or out-of-sample residuals where appropriate (or possible)
R2	coefficient of determination (Doksum and Samarov (1995))
MSE	mean squared error
MAE	mean absolute error
MAPE	mean absolute percentage error
CORR	absolute value of Pearson's correlation coefficient
SIGN	fraction of observations where fitted and observed values agree in sign

Usage Issues

If you are using data of mixed types, then it is advisable to use the data.frame function to construct your input data and not cbind, since cbind will typically not work as intended on mixed data types and will coerce the data to the same type.

Author(s)

Tristen Hayfield [e-mail], Jeffrey S. Racine [e-mail]

References

Aitchison, J. and C.G.G. Aitken (1976), “Multivariate binary discrimination by the kernel method,” Biometrika, 63, 413-420.

Doksum, K. and A. Samarov (1995), “Nonparametric estimation of global functionals and a measure of the explanatory power of covariates in regression,” The Annals of Statistics, 23 1443-1473.

Gao, Q. and L. Liu and J.S. Racine (2015), “A partially linear kernel estimator for categorical data,” Econometric Reviews, 34 (6-10), 958-977.

Li, Q. and J.S. Racine (2007), Nonparametric Econometrics: Theory and Practice, Princeton University Press.

Li, Q. and J.S. Racine (2004), “Cross-validated local linear nonparametric regression,” Statistica Sinica, 14, 485-512.

Pagan, A. and A. Ullah (1999), Nonparametric Econometrics, Cambridge University Press.

Racine, J.S. and Q. Li (2004), “Nonparametric estimation of regression functions with both categorical and continuous data,” Journal of Econometrics, 119, 99-130.

Robinson, P.M. (1988), “Root-n-consistent semiparametric regression,” Econometrica, 56, 931-954.

Wang, M.C. and J. van Ryzin (1981), “A class of smooth estimators for discrete distributions,” Biometrika, 68, 301-309.

See Also

npregbw, npreg

Examples

## Not run: 
# EXAMPLE 1 (INTERFACE=FORMULA): For this example, we simulate an
# example for a partially linear model and compare the coefficient
# estimates from the partially linear model with those from a correctly
# specified parametric model...

set.seed(42)

n <- 250
x1 <- rnorm(n)
x2 <- rbinom(n, 1, .5)

z1 <- rbinom(n, 1, .5)
z2 <- rnorm(n)

y <- 1 + x1 + x2 + z1 + sin(z2) + rnorm(n)

# First, compute data-driven bandwidths. This may take a few minutes
# depending on the speed of your computer...

bw <- npplregbw(formula=y~x1+factor(x2)|factor(z1)+z2)

# Next, compute the partially linear fit

pl <- npplreg(bws=bw)

# Print a summary of the model...

summary(pl)

# Sleep for 5 seconds so that we can examine the output...

Sys.sleep(5)

# Retrieve the coefficient estimates and their standard errors...

coef(pl)
coef(pl, errors = TRUE)

# Compare the partially linear results to those from a correctly
# specified model's coefficients for x1 and x2

ols <- lm(y~x1+factor(x2)+factor(z1)+I(sin(z2)))

# The intercept is coef()[1], and those for x1 and x2 are coef()[2] and
# coef()[3]. The standard errors are the square root of the diagonal of
# the variance-covariance matrix (elements 2 and 3)

coef(ols)[2:3]
sqrt(diag(vcov(ols)))[2:3]

# Sleep for 5 seconds so that we can examine the output...

Sys.sleep(5)

# Plot the regression surfaces via plot() (i.e., plot the `partial
# regression surface plots').

plot(bw)

# Note - to plot regression surfaces with variability bounds constructed
# from bootstrapped standard errors, use the following (note that this
# may take a minute or two depending on the speed of your computer as
# the bootstrapping is done in real time, and note also that we override
# the default number of bootstrap replications (399) reducing them to 25
# in order to quickly compute standard errors in this instance - don't
# of course do this in general)

plot(bw,
     plot.errors.boot.num=25, 
     plot.errors.method="bootstrap")


# EXAMPLE 1 (INTERFACE=DATA FRAME): For this example, we simulate an
# example for a partially linear model and compare the coefficient
# estimates from the partially linear model with those from a correctly
# specified parametric model...

set.seed(42)

n <- 250
x1 <- rnorm(n)
x2 <- rbinom(n, 1, .5)

z1 <- rbinom(n, 1, .5)
z2 <- rnorm(n)

y <- 1 + x1 + x2 + z1 + sin(z2) + rnorm(n)

X <- data.frame(x1, factor(x2))
Z <- data.frame(factor(z1), z2)

# First, compute data-driven bandwidths. This may take a few minutes
# depending on the speed of your computer...

bw <- npplregbw(xdat=X, zdat=Z, ydat=y)

# Next, compute the partially linear fit

pl <- npplreg(bws=bw)

# Print a summary of the model...

summary(pl)

# Sleep for 5 seconds so that we can examine the output...

Sys.sleep(5)

# Retrieve the coefficient estimates and their standard errors...

coef(pl)
coef(pl, errors = TRUE)

# Compare the partially linear results to those from a correctly
# specified model's coefficients for x1 and x2

ols <- lm(y~x1+factor(x2)+factor(z1)+I(sin(z2)))

# The intercept is coef()[1], and those for x1 and x2 are coef()[2] and
# coef()[3]. The standard errors are the square root of the diagonal of
# the variance-covariance matrix (elements 2 and 3)

coef(ols)[2:3]
sqrt(diag(vcov(ols)))[2:3]

# Sleep for 5 seconds so that we can examine the output...

Sys.sleep(5)

# Plot the regression surfaces via plot() (i.e., plot the `partial
# regression surface plots').

plot(bw)

# Note - to plot regression surfaces with variability bounds constructed
# from bootstrapped standard errors, use the following (note that this
# may take a minute or two depending on the speed of your computer as
# the bootstrapping is done in real time, and note also that we override
# the default number of bootstrap replications (399) reducing them to 25
# in order to quickly compute standard errors in this instance - don't
# of course do this in general)

plot(bw,
     plot.errors.boot.num=25, 
     plot.errors.method="bootstrap")

## End(Not run) 

np documentation built on March 31, 2023, 9:41 p.m.