npscoefbw computes a bandwidth object for a smooth
coefficient kernel regression estimate of a one (1) dimensional
dependent variable on
\(p+q\)-variate explanatory data, using the model
\(Y_i = W_{i}^{\prime} \gamma (Z_i) + u_i\) where \(W_i'=(1,X_i')\)
given 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.
npscoefbw(…)# S3 method for formula
npscoefbw(formula, data, subset, na.action, call, …)
# S3 method for NULL
npscoefbw(xdat = stop("invoked without data 'xdat'"),
ydat = stop("invoked without data 'ydat'"),
zdat = NULL,
bws,
…)
# S3 method for default
npscoefbw(xdat = stop("invoked without data 'xdat'"),
ydat = stop("invoked without data 'ydat'"),
zdat = NULL,
bws,
nmulti,
random.seed,
cv.iterate,
cv.num.iterations,
backfit.iterate,
backfit.maxiter,
backfit.tol,
bandwidth.compute = TRUE,
bwmethod,
bwscaling,
bwtype,
ckertype,
ckerorder,
ukertype,
okertype,
optim.method,
optim.maxattempts,
optim.reltol,
optim.abstol,
optim.maxit,
…)
# S3 method for scbandwidth
npscoefbw(xdat = stop("invoked without data 'xdat'"),
ydat = stop("invoked without data 'ydat'"),
zdat = NULL,
bws,
nmulti,
random.seed = 42,
cv.iterate = FALSE,
cv.num.iterations = 1,
backfit.iterate = FALSE,
backfit.maxiter = 100,
backfit.tol = .Machine$double.eps,
bandwidth.compute = TRUE,
optim.method = c("Nelder-Mead", "BFGS", "CG"),
optim.maxattempts = 10,
optim.reltol = sqrt(.Machine$double.eps),
optim.abstol = .Machine$double.eps,
optim.maxit = 500,
…)
a symbolic description of variables on which bandwidth selection is to be performed. The details of constructing a formula are described below.
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(formula), typically the environment from which the
function is called.
an optional vector specifying a subset of observations to be used in the fitting process.
the original function call. This is passed internally by
np when a bandwidth search has been implied by a call to
another function. It is not recommended that the user set this.
a \(p\)-variate data frame of explanatory data (training data),
which, by default, populates the columns \(2\) through \(p+1\)
of \(W\) in the model equation, and in the
absence of zdat, will also correspond to
\(Z\) from the model equation.
a one (1) dimensional numeric or integer vector of dependent data, each
element \(i\) corresponding to each observation (row) \(i\) of
xdat.
an optionally specified \(q\)-variate data frame of explanatory
data (training data), which corresponds to \(Z\)
in the model equation. Defaults to be the same as xdat.
a bandwidth specification. This can be set as a scbandwidth
object returned from a previous invocation, or as a vector of
bandwidths, with each element \(i\) corresponding to the bandwidth
for column \(i\) in xdat. In either case, the bandwidth
supplied will serve as a starting point in the numerical search for
optimal bandwidths. If specified as a vector, then additional
arguments will need to be supplied as necessary to specify the
bandwidth type, kernel types, selection methods, and so on. This can
be left unset.
additional arguments supplied to specify the regression type, bandwidth type, kernel types, selection methods, and so on, detailed below.
a logical value which specifies whether to do a numerical search for
bandwidths or not. If set to FALSE, a scbandwidth object
will be returned with bandwidths set to those specified
in bws. Defaults to TRUE.
which method was used to select bandwidths. cv.ls
specifies least-squares cross-validation, which is all that is
currently supported. Defaults to cv.ls.
a logical value that when set to TRUE the
supplied bandwidths are interpreted as `scale factors'
(\(c_j\)), otherwise when the value is FALSE they are
interpreted as `raw bandwidths' (\(h_j\) for continuous data
types, \(\lambda_j\) for discrete data types). For
continuous data types, \(c_j\) and \(h_j\) are
related by the formula \(h_j = c_j \sigma_j n^{-1/(2P+l)}\), where \(\sigma_j\) is an
adaptive measure of spread of continuous variable \(j\) defined as
min(standard deviation, mean absolute deviation, interquartile
range/1.349), \(n\) the number of observations, \(P\) the
order of the kernel, and \(l\) the number of continuous
variables. For discrete data types, \(c_j\) and
\(h_j\) are related by the formula \(h_j =
c_jn^{-2/(2P+l)}\), where here
\(j\) denotes discrete variable \(j\). Defaults to
FALSE.
character string used for the continuous variable bandwidth type,
specifying the type of bandwidth provided. Defaults to
fixed. Option summary:
fixed: fixed bandwidths or scale factors
generalized_nn: generalized nearest neighbors
adaptive_nn: adaptive nearest neighbors
character string used to specify the continuous kernel type.
Can be set as gaussian, epanechnikov, or
uniform. Defaults to gaussian.
numeric value specifying kernel order (one of
(2,4,6,8)). Kernel order specified along with a
uniform continuous kernel type will be ignored. Defaults to
2.
character string used to specify the unordered categorical kernel type.
Can be set as aitchisonaitken or liracine. Defaults to
aitchisonaitken.
character string used to specify the ordered categorical kernel type.
Can be set as wangvanryzin or liracine. Defaults to
liracine.
integer number of times to restart the process of finding extrema of
the cross-validation function from different (random) initial
points. Defaults to min(5,ncol(xdat)).
an integer used to seed R's random number generator. This ensures replicability of the numerical search. Defaults to 42.
method used by optim for minimization of
the objective function. See ?optim for references. Defaults
to "Nelder-Mead".
the default method is an implementation of that of Nelder and Mead (1965), that uses only function values and is robust but relatively slow. It will work reasonably well for non-differentiable functions.
method "BFGS" is a quasi-Newton method (also known as a
variable metric algorithm), specifically that published
simultaneously in 1970 by Broyden, Fletcher, Goldfarb and Shanno.
This uses function values and gradients to build up a picture of the
surface to be optimized.
method "CG" is a conjugate gradients method based
on that by Fletcher and Reeves (1964) (but with the option of
Polak-Ribiere or Beale-Sorenson updates). Conjugate gradient
methods will generally be more fragile than the BFGS method, but as
they do not store a matrix they may be successful in much larger
optimization problems.
maximum number of attempts taken trying to achieve successful
convergence in optim. Defaults to 100.
the absolute convergence tolerance used by optim. Only useful
for non-negative functions, as a tolerance for reaching
zero. Defaults to .Machine$double.eps.
relative convergence tolerance used by optim. The algorithm
stops if it is unable to reduce the value by a factor of 'reltol *
(abs(val) + reltol)' at a step. Defaults to
sqrt(.Machine$double.eps), typically about 1e-8.
maximum number of iterations used by optim. Defaults
to 500.
boolean value specifying whether or not to perform iterative,
cross-validated backfitting on the data. See details for limitations
of the backfitting procedure. Defaults to FALSE.
integer specifying the number of times to iterate the backfitting
process over all covariates. Defaults to 1.
boolean value specifying whether or not to iterate evaluations of
the smooth coefficient estimator, for extra accuracy, during the
cross-validated backfitting procedure. Defaults to FALSE.
integer specifying the maximum number of times to iterate the
evaluation of the smooth coefficient estimator in the attempt to
obtain the desired accuracy. Defaults to 100.
tolerance to determine convergence of iterated evaluations of the
smooth coefficient estimator. Defaults to .Machine$double.eps.
if bwtype is set to fixed, an object containing
bandwidths (or scale factors if bwscaling = TRUE) is
returned. If it is set to generalized_nn or adaptive_nn,
then instead the \(k\)th nearest neighbors are returned for the
continuous variables while the discrete kernel bandwidths are returned
for the discrete variables. Bandwidths are stored in a vector under the
component name bw. Backfitted bandwidths are stored under the
component name bw.fitted.
The functions predict, summary, and
plot support
objects of this class.
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.
Caution: multivariate data-driven bandwidth selection methods are, by
their nature, computationally intensive. Virtually all methods
require dropping the \(i\)th observation from the data set,
computing an object, repeating this for all observations in the
sample, then averaging each of these leave-one-out estimates for a
given value of the bandwidth vector, and only then repeating
this a large number of times in order to conduct multivariate
numerical minimization/maximization. Furthermore, due to the potential
for local minima/maxima, restarting this procedure a large
number of times may often be necessary. This can be frustrating for
users possessing large datasets. For exploratory purposes, you may
wish to override the default search tolerances, say, setting
optim.reltol=.1 and conduct multistarting (the default is to restart
min(5,ncol(zdat)) times). Once the procedure terminates, you can restart
search with default tolerances using those bandwidths obtained from
the less rigorous search (i.e., set bws=bw on subsequent calls
to this routine where bw is the initial bandwidth object). A
version of this package using the Rmpi wrapper is under
development that allows one to deploy this software in a clustered
computing environment to facilitate computation involving large
datasets.
Support for backfitted bandwidths is experimental and is limited in functionality. The code does not support asymptotic standard errors or out of sample estimates with backfitting.
npscoefbw implements a variety of methods for semiparametric
regression on multivariate (\(p+q\)-variate) explanatory data defined
over a set of possibly continuous 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 \(x\).
npscoefbw may be invoked either with a formula-like
symbolic description of variables on which bandwidth selection is to be
performed or through a simpler interface whereby data is passed
directly to the function via the xdat, ydat, and
zdat parameters. Use of these two interfaces is mutually
exclusive.
Data contained in the data frame xdat may be continuous and in
zdat may be of mixed type. Data can be entered in an arbitrary
order and data types will be detected automatically by the routine (see
np for details).
Data for which bandwidths are to be estimated may be specified
symbolically. A typical description has the form dependent
data ~ parametric explanatory data
| nonparametric explanatory data, where
dependent data is a univariate response, and
parametric explanatory data and
nonparametric explanatory data are both series of
variables specified by name, separated by the separation character
'+'. For example, y1 ~ x1 + x2 | z1 specifies that the
bandwidth object for the smooth coefficient model with response
y1, linear parametric regressors x1 and x2, and
nonparametric regressor (that is, the slope-changing variable)
z1 is to be estimated. See below for further examples. In the
case where the nonparametric (slope-changing) variable is not
specified, it is assumed to be the same as the parametric variable.
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.
Aitchison, J. and C.G.G. Aitken (1976), “Multivariate binary discrimination by the kernel method,” Biometrika, 63, 413-420.
Cai Z. (2007), “Trending time-varying coefficient time series models with serially correlated errors,” Journal of Econometrics, 136, 163-188.
Hastie, T. and R. Tibshirani (1993), “Varying-coefficient models,” Journal of the Royal Statistical Society, B 55, 757-796.
Li, Q. and J.S. Racine (2007), Nonparametric Econometrics: Theory and Practice, Princeton University Press.
Li, Q. and J.S. Racine (2010), “Smooth varying-coefficient estimation and inference for qualitative and quantitative data,” Econometric Theory, 26, 1-31.
Pagan, A. and A. Ullah (1999), Nonparametric Econometrics, Cambridge University Press.
Li, Q. and D. Ouyang and J.S. Racine (2013), “Categorical semiparametric varying-coefficient models,” Journal of Applied Econometrics, 28, 551-589.
Wang, M.C. and J. van Ryzin (1981), “A class of smooth estimators for discrete distributions,” Biometrika, 68, 301-309.
# NOT RUN {
# EXAMPLE 1 (INTERFACE=FORMULA):
set.seed(42)
n <- 100
x <- runif(n)
z <- runif(n, min=-2, max=2)
y <- x*exp(z)*(1.0+rnorm(n,sd = 0.2))
bw <- npscoefbw(formula=y~x|z)
summary(bw)
# EXAMPLE 1 (INTERFACE=DATA FRAME):
n <- 100
x <- runif(n)
z <- runif(n, min=-2, max=2)
y <- x*exp(z)*(1.0+rnorm(n,sd = 0.2))
bw <- npscoefbw(xdat=x, ydat=y, zdat=z)
summary(bw)
# }
# NOT RUN {
# }
# NOT RUN {
<!-- % enddontrun -->
# }
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