| npplreg {np} | R Documentation |
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 = XB + 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.
npplreg(bws, ...)
## S3 method for class 'formula':
npplreg(bws, data = NULL, newdata = NULL, ...)
## 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,
...)
bws |
a bandwidth specification. This can be set as a plbandwidth
object returned from an invocation of npplregbw, or as
a matrix of bandwidths, 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. |
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.
|
npplreg uses a combination of OLS and nonparametric
regression to estimate the parameter B in the model
Y = XB + 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.
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 B[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 |
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 |
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.
Tristen Hayfield hayfield@phys.ethz.ch, Jeffrey S. Racine racinej@mcmaster.ca
Aitchison, J. and C.G.G. Aitken (1976), “Multivariate binary discrimination by the kernel method,” Biometrika, 63, 413-420.
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.
Racine, J.S. and L. Liu (2006), “A partially linear kernel estimator for categorical data,” manuscript.
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.
# 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(123)
n <- 250
x1 <- rnorm(n)
x2 <- rbinom(n, 5, .3)
z1 <- rbinom(n, 2, .3)
z2 <- rnorm(n)
y <- 1 + x1 + x2 + z1 + sin(z2) + rnorm(n)
# First, compute data-driven bandwidths. We override the default
# tolerances for the search method as the objective function is
# well-behaved (don't of course do this in general). This may take a few
# minutes depending on the speed of your computer...
bw <- npplregbw(formula=y~x1+ordered(x2)|ordered(z1)+z2, tol=.1, ftol=.1)
# Next, compute the partially linear fit
pl <- npplreg(bws=bw)
# Print a summary of the model...
summary(pl)
## Not run:
# 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+x2+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 npplot() (i.e., plot the `partial
# regression surface plots').
npplot(bws=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)
npplot(bws=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(123)
n <- 250
x1 <- rnorm(n)
x2 <- rbinom(n, 5, .3)
z1 <- rbinom(n, 2, .3)
z2 <- rnorm(n)
y <- 1 + x1 + x2 + z1 + sin(z2) + rnorm(n)
X <- data.frame(x1, ordered(x2))
Z <- data.frame(ordered(z1), z2)
# First, compute data-driven bandwidths. We override the default
# tolerances for the search method as the objective function is
# well-behaved (don't of course do this in general). This may take a few
# minutes depending on the speed of your computer...
bw <- npplregbw(xdat=X, zdat=Z, ydat=y, tol=.1, ftol=.1)
# 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+x2+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 npplot() (i.e., plot the `partial
# regression surface plots').
npplot(bws=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)
npplot(bws=bw,
plot.errors.boot.num=25,
plot.errors.method="bootstrap")
## End(Not run)