npregbw: Kernel Regression Bandwidth Selection with Mixed Data Types

Description

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).

Usage

npregbw(...)
## S3 method for class 'formula':
npregbw(formula, data, subset, na.action, call, \dots)
## S3 method for class 'NULL':
npregbw(xdat = stop("invoked without data 'xdat'"),
        ydat = stop("invoked without data 'ydat'"),
        bws,
        ...)
## S3 method for class 'default':
npregbw(xdat = stop("invoked without data 'xdat'"),
        ydat = stop("invoked without data 'ydat'"),
        bws,
        bandwidth.compute = TRUE,
        nmulti,
        remin,
        itmax,
        ftol,
        tol,
        small,
        regtype,
        bwmethod,
        bwscaling,
        bwtype,
        ckertype,
        ckerorder,
        ukertype,
        okertype,
        ...)
## S3 method for class 'rbandwidth':
npregbw(xdat = stop("invoked without data 'xdat'"),
        ydat = stop("invoked without data 'ydat'"),
        bws,
        bandwidth.compute = TRUE,
        nmulti,
        remin = TRUE,
        itmax = 10000,
        ftol = 1.19209e-07,
        tol = 1.49012e-08,
        small = 2.22045e-16,
        ...)

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

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

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 regressors which bandwidth selection will be performed. The data types may be continuous, discrete (unordered and ordered factors), or some combination thereof.

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 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 eith

...

additional arguments supplied to specify the bandwidth type, kernel types, selection methods, and so on, detailed below.

regtype

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.

bwmethod

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

bwscaling

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

bwtype

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 ba

bandwidth.compute

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

ckertype

character string used to specify the continuous kernel type.
    Can be set as gaussian, epanechnikov, or
    uniform. Defaults to gaussian.

ckerorder

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.

ukertype

character string used to specify the unordered categorical kernel type.
    Can be set as aitchisonaitken or liracine. Defaults to
    aitchisonaitken.

okertype

character string used to specify the ordered categorical kernel type.
    Can be set as wangvanryzin or liracine. Defaults to
    wangvanryzin.

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)).

remin

a logical value which when set as TRUE the search routine
    restarts from located minima for a minor gain in accuracy. Defaults
    to TRUE.

itmax

integer number of iterations before failure in the numerical
    optimization routine. Defaults to 10000.

ftol

tolerance on the value of the cross-validation function 
    evaluated at located minima. Defaults to 1.19e-07
      (FLT_EPSILON).

tol

tolerance on the position of located minima of the 
    cross-validation function. Defaults to 1.49e-08
      (sqrt(DBL_EPSILON)).

small

a small number, at about the precision of the data type
    used. Defaults to 2.22e-16 (DBL_EPSILON).

`Value`

npregbw returns a rbandwidth object, with the
  following components:
bwbandwidth(s), scale factor(s) or nearest neighbours for the
    data, xdat
fvalobjective 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 $k$th 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.

`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 $i$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.

`Details`

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.

`References`

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

`See Also`

npreg

`Examples`

Run this code# 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)
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