R: Semiparametric Single Index Model Parameter and Bandwidth...
npindexbw
R Documentation
Semiparametric Single Index Model Parameter and Bandwidth Selection
Description
npindexbw computes a npindexbw bandwidth specification
using the model Y = G(XB) +
epsilon. For continuous Y, the approach is that of Hardle, Hall
and Ichimura (1993) which jointly minimizes a least-squares
cross-validation function with respect to the parameters and
bandwidth. For binary Y, a likelihood-based cross-validation
approach is employed which jointly maximizes a likelihood
cross-validation function with respect to the parameters and
bandwidth. The bandwidth object contains parameters for the single
index model and the (scalar) bandwidth for the index function.
Usage
npindexbw(...)
## S3 method for class 'formula'
npindexbw(formula, data, subset, na.action, call, ...)
## S3 method for class 'NULL'
npindexbw(xdat = stop("training data xdat missing"),
ydat = stop("training data ydat missing"),
bws,
...)
## Default S3 method:
npindexbw(xdat = stop("training data xdat missing"),
ydat = stop("training data ydat missing"),
bws,
bandwidth.compute = TRUE,
nmulti,
random.seed,
optim.method,
optim.maxattempts,
optim.reltol,
optim.abstol,
optim.maxit,
only.optimize.beta,
...)
## S3 method for class 'sibandwidth'
npindexbw(xdat = stop("training data xdat missing"),
ydat = stop("training data ydat missing"),
bws,
bandwidth.compute = TRUE,
nmulti,
random.seed = 42,
optim.method = c("Nelder-Mead", "BFGS", "CG"),
optim.maxattempts = 10,
optim.reltol = sqrt(.Machine$double.eps),
optim.abstol = .Machine$double.eps,
optim.maxit = 500,
only.optimize.beta = FALSE,
...)
Arguments
formula
a symbolic description of variables on which bandwidth selection is
to be performed. The details of constructing a formula are
described below.
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(formula), typically the environment from which the
function is called.
subset
an optional vector specifying a subset of observations to be used in
the fitting process.
na.action
a function which indicates what should happen when the data contain
NAs. The default is set by the
na.action setting of options, and is
na.fail if that is unset. The (recommended) default is
na.omit.
call
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.
xdat
a p-variate data frame of explanatory data (training data) used to
calculate the regression estimators.
ydat
a one (1) dimensional numeric or integer vector of dependent data, each
element i corresponding to each observation (row) i of
xdat.
bws
a bandwidth specification. This can be set as a
singleindexbandwidth
object returned from an invocation of npindexbw, or
as a vector of parameters (beta) with each element i
corresponding to the coefficient for column i in xdat
where the first element is normalized to 1, and a scalar bandwidth
(h). If specified as a vector, then additional arguments will need
to be supplied as necessary to specify the bandwidth type, kernel
types, and so on.
method
the single index model method, one of either “ichimura”
(Ichimura (1993)) or “kleinspady” (Klein and Spady
(1993)). Defaults to
ichimura.
nmulti
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)).
random.seed
an integer used to seed R's random number generator. This ensures
replicability of the numerical search. Defaults to 42.
bandwidth.compute
a logical value which specifies whether to do a numerical search for
bandwidths or not. If set to FALSE, a bandwidth object
will be returned with bandwidths set to those specified
in bws. Defaults to TRUE.
optim.method
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.
optim.maxattempts
maximum number of attempts taken trying to achieve successful
convergence in optim. Defaults to 100.
optim.abstol
the absolute convergence tolerance used by optim. Only useful
for non-negative functions, as a tolerance for reaching
zero. Defaults to .Machine$double.eps.
optim.reltol
relative convergence tolerance used by optim. The algorithm
stops if it is unable to reduce the value by a factor of 'reltol *
(abs(val) + reltol)' at a step. Defaults to
sqrt(.Machine$double.eps), typically about 1e-8.
optim.maxit
maximum number of iterations used by optim. Defaults
to 500.
only.optimize.beta
signals the routine to only minimize the objective function with
respect to beta
...
additional arguments supplied to specify the parameters to the
sibandwidth S3 method, which is called during the numerical
search.
Details
We implement Ichimura's (1993) method via
joint estimation of the bandwidth and coefficient vector using
leave-one-out nonlinear least squares. We implement Klein and Spady's
(1993) method maximizing the leave-one-out log likelihood function
jointly with respect to the bandwidth and coefficient vector. Note
that Klein and Spady's (1993) method is for binary outcomes
only, while Ichimura's (1993) method can be applied for any outcome
data type (i.e., continuous or discrete).
We impose the identification condition that the first element of the
coefficient vector beta is equal to one, while identification also
requires that the explanatory variables contain at least one
continuous variable.
npindexbw 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.
Note that, unlike most other bandwidth methods in the np
package, this implementation uses the R optim nonlinear
minimization routines and npksum. We have implemented
multistarting and strongly encourage its use in practice. 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(xdat)) times) as is done
for a number of examples.
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 bandwidth object for
the regression of response y1 and semiparametric regressors
x1 and x2 are to be estimated. See below for further
examples.
Value
npindexbw returns a sibandwidth object, with the
following components:
bw
bandwidth(s), scale factor(s) or nearest neighbours for the
data, xdat
beta
coefficients of the model
fval
objective function value at minimum
If bwtype is set to fixed, an object containing a scalar
bandwidth for the function G(XB) and an estimate of
the parameter vector B is returned.
If bwtype is set to generalized_nn or
adaptive_nn, then instead the scalar kth nearest neighbor
is returned.
The functions coef, predict,
summary, and plot support
objects of this class.
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.
Caution: multivariate data-driven bandwidth selection methods are, by
their nature, computationally intensive. Virtually all methods
require dropping the ith 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(xdat)) 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.
Aitchison, J. and C.G.G. Aitken (1976), “Multivariate binary
discrimination by the kernel method,” Biometrika, 63, 413-420.
Hardle, W. and P. Hall and H. Ichimura (1993), “Optimal
Smoothing in Single-Index Models,” The Annals of Statistics, 21,
157-178.
Ichimura, H., (1993), “Semiparametric least squares (SLS) and
weighted SLS estimation of single-index models,” Journal of
Econometrics, 58, 71-120.
Klein, R. W. and R. H. Spady (1993), “An efficient semiparametric
estimator for binary response models,” Econometrica, 61, 387-421.
Li, Q. and J.S. Racine (2007), Nonparametric Econometrics:
Theory and Practice, Princeton University Press.
Wang, M.C. and J. van Ryzin (1981), “A class of smooth
estimators for discrete distributions,” Biometrika, 68, 301-309.
Examples
## Not run:
# EXAMPLE 1 (INTERFACE=FORMULA): Generate a simple linear model then
# compute coefficients and the bandwidth using Ichimura's nonlinear
# least squares approach.
set.seed(12345)
n <- 100
x1 <- runif(n, min=-1, max=1)
x2 <- runif(n, min=-1, max=1)
y <- x1 - x2 + rnorm(n)
# Note - this may take a minute or two depending on the speed of your
# computer. Note also that the first element of the vector beta is
# normalized to one for identification purposes, and that X must contain
# at least one continuous variable.
bw <- npindexbw(formula=y~x1+x2, method="ichimura")
summary(bw)
# Sleep for 5 seconds so that we can examine the output...
Sys.sleep(5)
# EXAMPLE 1 (INTERFACE=DATA FRAME): Generate a simple linear model then
# compute coefficients and the bandwidth using Ichimura's nonlinear
# least squares approach.
set.seed(12345)
n <- 100
x1 <- runif(n, min=-1, max=1)
x2 <- runif(n, min=-1, max=1)
y <- x1 - x2 + rnorm(n)
X <- cbind(x1, x2)
# Note - this may take a minute or two depending on the speed of your
# computer. Note also that the first element of the vector beta is
# normalized to one for identification purposes, and that X must contain
# at least one continuous variable.
bw <- npindexbw(xdat=X, ydat=y, method="ichimura")
summary(bw)
# Sleep for 5 seconds so that we can examine the output...
Sys.sleep(5)
# EXAMPLE 2 (INTERFACE=DATA FRAME): Generate a simple binary outcome
# model then compute coefficients and the bandwidth using Klein and
# Spady's likelihood-based approach.
n <- 100
x1 <- runif(n, min=-1, max=1)
x2 <- runif(n, min=-1, max=1)
y <- ifelse(x1 + x2 + rnorm(n) > 0, 1, 0)
# Note that the first element of the vector beta is normalized to one
# for identification purposes, and that X must contain at least one
# continuous variable.
bw <- npindexbw(formula=y~x1+x2, method="kleinspady")
summary(bw)
# EXAMPLE 2 (INTERFACE=DATA FRAME): Generate a simple binary outcome
# model then compute coefficients and the bandwidth using Klein and
# Spady's likelihood-based approach.
n <- 100
x1 <- runif(n, min=-1, max=1)
x2 <- runif(n, min=-1, max=1)
y <- ifelse(x1 + x2 + rnorm(n) > 0, 1, 0)
X <- cbind(x1, x2)
# Note that the first element of the vector beta is normalized to one
# for identification purposes, and that X must contain at least one
# continuous variable.
bw <- npindexbw(xdat=X, ydat=y, method="kleinspady")
summary(bw)
## End(Not run)