npregbw computes a bandwidth object for a
\(p\)-variate kernel regression estimator defined over mixed
continuous and discrete (unordered, ordered) data using expected
Kullback-Leibler cross-validation, or least-squares cross validation
using the method of Racine and Li (2004) and Li and Racine (2004).
npregbw(…)# S3 method for formula
npregbw(formula, data, subset, na.action, call, …)
# S3 method for NULL
npregbw(xdat = stop("invoked without data 'xdat'"),
ydat = stop("invoked without data 'ydat'"),
bws,
…)
# S3 method for default
npregbw(xdat = stop("invoked without data 'xdat'"),
ydat = stop("invoked without data 'ydat'"),
bws,
bandwidth.compute = TRUE,
nmulti,
remin,
itmax,
ftol,
tol,
small,
lbc.dir,
dfc.dir,
cfac.dir,
initc.dir,
lbd.dir,
hbd.dir,
dfac.dir,
initd.dir,
lbc.init,
hbc.init,
cfac.init,
lbd.init,
hbd.init,
dfac.init,
scale.init.categorical.sample,
regtype,
bwmethod,
bwscaling,
bwtype,
ckertype,
ckerorder,
ukertype,
okertype,
…)
# S3 method for rbandwidth
npregbw(xdat = stop("invoked without data 'xdat'"),
ydat = stop("invoked without data 'ydat'"),
bws,
bandwidth.compute = TRUE,
nmulti,
remin = TRUE,
itmax = 10000,
ftol = 1.490116e-07,
tol = 1.490116e-04,
small = 1.490116e-05,
lbc.dir = 0.5,
dfc.dir = 3,
cfac.dir = 2.5*(3.0-sqrt(5)),
initc.dir = 1.0,
lbd.dir = 0.1,
hbd.dir = 1,
dfac.dir = 0.25*(3.0-sqrt(5)),
initd.dir = 1.0,
lbc.init = 0.1,
hbc.init = 2.0,
cfac.init = 0.5,
lbd.init = 0.1,
hbd.init = 0.9,
dfac.init = 0.375,
scale.init.categorical.sample = FALSE,
…)
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 regressors on which bandwidth selection will be performed. The data types may be continuous, discrete (unordered and ordered factors), or some combination thereof.
a one (1) dimensional numeric or integer vector of dependent data, each
element \(i\) corresponding to each observation (row) \(i\) of
xdat.
a bandwidth specification. This can be set as a rbandwidth
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 bandwidth type, kernel types, selection methods, and so on, detailed below.
a character string specifying which type of kernel regression
estimator to use. lc specifies a local-constant estimator
(Nadaraya-Watson) and ll specifies a local-linear
estimator. Defaults to lc.
which method to use to select bandwidths. cv.aic specifies
expected Kullback-Leibler cross-validation (Hurvich, Simonoff, and
Tsai (1998)), and cv.ls specifies least-squares
cross-validation. 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/1.4826, 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 to compute and return in the
bandwidth object. Defaults to fixed. Option
summary:
fixed: compute fixed bandwidths
generalized_nn: compute generalized nearest neighbors
adaptive_nn: compute adaptive nearest neighbors
a logical value which specifies whether to do a numerical search for
bandwidths or not. If set to FALSE, a rbandwidth object
will be returned with bandwidths set to those specified
in bws. Defaults to TRUE.
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)).
a logical value which when set as TRUE the search routine
restarts from located minima for a minor gain in accuracy. Defaults
to TRUE.
integer number of iterations before failure in the numerical
optimization routine. Defaults to 10000.
fractional tolerance on the value of the cross-validation function
evaluated at located minima (of order the machine precision or
perhaps slightly larger so as not to be diddled by
roundoff). Defaults to 1.490116e-07
(1.0e+01*sqrt(.Machine$double.eps)).
tolerance on the position of located minima of the cross-validation
function (tol should generally be no smaller than the square root of
your machine's floating point precision). Defaults to
1.490116e-04 (1.0e+04*sqrt(.Machine$double.eps)).
a small number used to bracket a minimum (it is hopeless to ask for
a bracketing interval of width less than sqrt(epsilon) times its
central value, a fractional width of only about 10-04 (single
precision) or 3x10-8 (double precision)). Defaults to small
= 1.490116e-05 (1.0e+03*sqrt(.Machine$double.eps)).
lower bound, chi-square
degrees of freedom, stretch factor, and initial non-random values
for direction set search for Powell's algorithm for numeric
variables. See Details
lower bound, upper bound, stretch factor, and initial non-random values for direction set search for Powell's algorithm for categorical variables. See Details
lower bound, upper bound, and
non-random initial values for scale factors for numeric
variables for Powell's algorithm. See Details
lower bound, upper bound, and non-random initial values for scale factors for categorical variables for Powell's algorithm. See Details
a logical value that when set
to TRUE scales lbd.dir, hbd.dir,
dfac.dir, and initd.dir by \(n^{-2/(2P+l)}\),
\(n\) the number of observations, \(P\) the order of the
kernel, and \(l\) the number of numeric variables. See
Details
npregbw returns a rbandwidth object, with the
following components:
bandwidth(s), scale factor(s) or nearest neighbours for the
data, xdat
objective function value at minimum
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 kth nearest neighbors are returned for the continuous variables while the discrete kernel bandwidths are returned for the discrete variables. Bandwidths are stored under the component name bw, with each element i corresponding to column i of input data xdat.
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 ftol=.01 and tol=.01 and
conduct multistarting (the default is to restart min(5, ncol(xdat))
times) as is done for a number of examples. 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.
npregbw implements a variety of methods for choosing
bandwidths for multivariate (\(p\)-variate) regression 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.
The cross-validation methods employ multivariate numerical search algorithms (direction set (Powell's) methods in multidimensions).
Bandwidths can (and will) differ for each variable which is, of course, desirable.
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 \(x\).
npregbw 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 and ydat
parameters. Use of these two interfaces is mutually exclusive.
Data contained in the data frame xdat may be a mix of
continuous (default), unordered discrete (to be specified in the data
frame xdat using factor), and ordered discrete
(to be specified in the data frame xdat using
ordered). 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
~ explanatory data,
where dependent data is a univariate response, and
explanatory data is a
series of variables specified by name, separated by
the separation character '+'. For example, y1 ~ x1 + x2
specifies that the bandwidths for the regression of response y1
and
nonparametric regressors x1 and x2 are to be estimated.
See below for further examples.
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.
The use of compactly supported kernels or the occurrence of small bandwidths during cross-validation can lead to numerical problems for the local linear estimator when computing the locally weighted least squares solution. To overcome this problem we rely on a form or ‘ridging’ proposed by Cheng, Hall, and Titterington (1997), modified so that we solve the problem pointwise rather than globally (i.e. only when it is needed).
The optimizer invoked for search is Powell's conjugate direction
method which requires the setting of (non-random) initial values and
search directions for bandwidths, and, when restarting, random values
for successive invocations. Bandwidths for numeric variables
are scaled by robust measures of spread, the sample size, and the
number of numeric variables where appropriate. Two sets of
parameters for bandwidths for numeric can be modified, those
for initial values for the parameters themselves, and those for the
directions taken (Powell's algorithm does not involve explicit
computation of the function's gradient). The default values are set by
considering search performance for a variety of difficult test cases
and simulated cases. We highly recommend restarting search a large
number of times to avoid the presence of local minima (achieved by
modifying nmulti). Further refinement for difficult cases can
be achieved by modifying these sets of parameters. However, these
parameters are intended more for the authors of the package to enable
‘tuning’ for various methods rather than for the user
themselves.
Aitchison, J. and C.G.G. Aitken (1976), “Multivariate binary discrimination by the kernel method,” Biometrika, 63, 413-420.
Cheng, M.-Y. and P. Hall and D.M. Titterington (1997), “On the shrinkage of local linear curve estimators,” Statistics and Computing, 7, 11-17.
Hall, P. and Q. Li and J.S. Racine (2007), “Nonparametric estimation of regression functions in the presence of irrelevant regressors,” The Review of Economics and Statistics, 89, 784-789.
Hurvich, C.M. and J.S. Simonoff and C.L. Tsai (1998), “Smoothing parameter selection in nonparametric regression using an improved Akaike information criterion,” Journal of the Royal Statistical Society B, 60, 271-293.
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.
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): For this example, we compute a
# Bivariate nonparametric regression estimate for Giovanni Baiocchi's
# Italian income panel (see Italy for details)
data("Italy")
attach(Italy)
# Compute the least-squares cross-validated bandwidths for the local
# constant estimator (default)
bw <- npregbw(formula=gdp~ordered(year))
summary(bw)
# Sleep for 5 seconds so that we can examine the output...
Sys.sleep(5)
# Supply your own bandwidth...
bw <- npregbw(formula=gdp~ordered(year), bws=c(0.75),
bandwidth.compute=FALSE)
summary(bw)
# Sleep for 5 seconds so that we can examine the output...
Sys.sleep(5)
# Treat year as continuous and supply your own scaling factor c in
# c sigma n^{-1/(2p+q)}
bw <- npregbw(formula=gdp~year, bws=c(1.06),
bandwidth.compute=FALSE,
bwscaling=TRUE)
summary(bw)
# Note - see also the example for npudensbw() for more extensive
# multiple illustrations of how to change the kernel function, kernel
# order, bandwidth type and so forth.
detach(Italy)
# EXAMPLE 1 (INTERFACE=DATA FRAME): For this example, we compute a
# Bivariate nonparametric regression estimate for Giovanni Baiocchi's
# Italian income panel (see Italy for details)
data("Italy")
attach(Italy)
# Compute the least-squares cross-validated bandwidths for the local
# constant estimator (default)
bw <- npregbw(xdat=ordered(year), ydat=gdp)
summary(bw)
# Sleep for 5 seconds so that we can examine the output...
Sys.sleep(5)
# Supply your own bandwidth...
bw <- npregbw(xdat=ordered(year), ydat=gdp, bws=c(0.75),
bandwidth.compute=FALSE)
summary(bw)
# Sleep for 5 seconds so that we can examine the output...
Sys.sleep(5)
# Treat year as continuous and supply your own scaling factor c in
# c sigma n^{-1/(2p+q)}
bw <- npregbw(xdat=year, ydat=gdp, bws=c(1.06),
bandwidth.compute=FALSE,
bwscaling=TRUE)
summary(bw)
# Note - see also the example for npudensbw() for more extensive
# multiple illustrations of how to change the kernel function, kernel
# order, bandwidth type and so forth.
detach(Italy)
# }
# NOT RUN {
# }
# NOT RUN {
<!-- % enddontrun -->
# }
Run the code above in your browser using DataLab