Partially Linear Kernel Regression Bandwidth Selection with Mixed Data Types

npplregbw computes a bandwidth object for a partially linear kernel regression estimate of a one (1) dimensional dependent variable on \(p+q\)-variate explanatory data, using the model \(Y = X\beta + \Theta (Z) + \epsilon\) 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.

Usage

npplregbw(...)

# S3 method for class 'formula'
npplregbw(formula, data, subset, na.action, call, ...)

# S3 method for class 'NULL'
npplregbw(xdat = stop("invoked without data `xdat'"),
          ydat = stop("invoked without data `ydat'"),
          zdat = stop("invoked without data `zdat'"),
          bws,
          ...)

# Default S3 method
npplregbw(xdat = stop("invoked without data `xdat'"),
          ydat = stop("invoked without data `ydat'"),
          zdat = stop("invoked without data `zdat'"),
          bws,
          ...,
          bandwidth.compute = TRUE,
          nmulti,
          remin,
          itmax,
          ftol,
          tol,
          small)

# S3 method for class 'plbandwidth'
npplregbw(xdat = stop("invoked without data `xdat'"),
          ydat = stop("invoked without data `ydat'"),
          zdat = stop("invoked without data `zdat'"),
          bws,
          nmulti,
          ...)

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), corresponding to \(X\) in the model equation, whose linear relationship with the dependent data \(Y\) is posited.

ydat

a one (1) dimensional numeric or integer vector of dependent data, each element \(i\) corresponding to each observation (row) \(i\) of xdat.

zdat

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)

bws

a bandwidth specification. This can be set as a plbandwidth object returned from an invocation of npplregbw, or as a matrix of bandwidths, where 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, and so on.

If left unspecified, npplregbw will search for optimal bandwidths using npregbw in the course of calculations. If specified, npplregbw will use the given bandwidths as the starting point for the numerical search for optimal bandwidths, unless you specify bandwidth.compute = FALSE.

...

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.

bandwidth.compute

a logical value which specifies whether to do a numerical search for bandwidths or not. If set to FALSE, a plbandwidth object will be returned with bandwidths set to those specified in bws. Defaults to TRUE.

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(zdat)).

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)

Details

npplregbw 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 \(x\).

npplregbw 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 zdat may be a mix of continuous (default), unordered discrete (to be specified in the data frame zdat using factor), and ordered discrete (to be specified in the data frame zdat 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 ~ 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 partially linear model with response y1, linear parametric regressors x1 and x2, and nonparametric regressor z1 is 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.

Value

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 list under the component name bw. Each element is an rbandwidth object. The first element of the list corresponds to the regression of \(Y\) on \(Z\). Each subsequent element is the bandwidth object corresponding to the regression of the \(i\)th column of \(X\) on \(Z\). See examples for more information.

References

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

Gao, Q. and L. Liu and J.S. Racine (2015), “A partially linear kernel estimator for categorical data,” Econometric Reviews, 34 (6-10), 958-977.

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.

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.

Author

Tristen Hayfield tristen.hayfield@gmail.com, Jeffrey S. Racine racinej@mcmaster.ca

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(zdat)) 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.

Examples

if (FALSE) { # \dontrun{
# EXAMPLE 1 (INTERFACE=FORMULA): For this example, we simulate an
# example for a partially linear model and perform bandwidth selection

set.seed(42)

n <- 250
x1 <- rnorm(n)
x2 <- rbinom(n, 1, .5)

z1 <- rbinom(n, 1, .5)
z2 <- rnorm(n)

y <- 1 + x1 + x2 + z1 + sin(z2) + rnorm(n)

X <- data.frame(x1, factor(x2))
Z <- data.frame(factor(z1), z2)

# Compute data-driven bandwidths... this may take a minute or two
# depending on the speed of your computer...

bw <- npplregbw(formula=y~x1+factor(x2)|factor(z1)+z2)

summary(bw)

# Note - the default is to use the local constant estimator. If you wish
# to use instead a local linear estimator, this is accomplished via
# npplregbw(xdat=X, zdat=Z, ydat=y, regtype="ll")

# Note - see the example for npudensbw() for multiple illustrations
# of how to change the kernel function, kernel order, and so forth.

# You may want to manually specify your bandwidths
bw.mat <- matrix(data =  c(0.19, 0.34,  # y on Z
                           0.00, 0.74,  # X[,1] on Z
                           0.29, 0.23), # X[,2] on Z
                ncol = ncol(Z), byrow=TRUE)

bw <- npplregbw(formula=y~x1+factor(x2)|factor(z1)+z2, 
                          bws=bw.mat, bandwidth.compute=FALSE)
summary(bw)

# Sleep for 5 seconds so that we can examine the output...

Sys.sleep(5)

# You may want to tweak some of the bandwidths
bw$bw[[1]] # y on Z, alternatively bw$bw$yzbw
bw$bw[[1]]$bw <- c(0.17, 0.30)

bw$bw[[2]] # X[,1] on Z
bw$bw[[2]]$bw[1] <- 0.00054

summary(bw)

# EXAMPLE 1 (INTERFACE=DATA FRAME): For this example, we simulate an
# example for a partially linear model and perform bandwidth selection

set.seed(42)

n <- 250
x1 <- rnorm(n)
x2 <- rbinom(n, 1, .5)

z1 <- rbinom(n, 1, .5)
z2 <- rnorm(n)

y <- 1 + x1 + x2 + z1 + sin(z2) + rnorm(n)

X <- data.frame(x1, factor(x2))
Z <- data.frame(factor(z1), z2)

# Compute data-driven bandwidths... this may take a minute or two
# depending on the speed of your computer...

bw <- npplregbw(xdat=X, zdat=Z, ydat=y)

summary(bw)

# Note - the default is to use the local constant estimator. If you wish
# to use instead a local linear estimator, this is accomplished via
# npplregbw(xdat=X, zdat=Z, ydat=y, regtype="ll")

# Note - see the example for npudensbw() for multiple illustrations
# of how to change the kernel function, kernel order, and so forth.

# You may want to manually specify your bandwidths
bw.mat <- matrix(data =  c(0.19, 0.34,  # y on Z
                           0.00, 0.74,  # X[,1] on Z
                           0.29, 0.23), # X[,2] on Z
                ncol = ncol(Z), byrow=TRUE)

bw <- npplregbw(xdat=X, zdat=Z, ydat=y, 
                          bws=bw.mat, bandwidth.compute=FALSE)
summary(bw)

# Sleep for 5 seconds so that we can examine the output...

Sys.sleep(5)

# You may want to tweak some of the bandwidths
bw$bw[[1]] # y on Z, alternatively bw$bw$yzbw
bw$bw[[1]]$bw <- c(0.17, 0.30)

bw$bw[[2]] # X[,1] on Z
bw$bw[[2]]$bw[1] <- 0.00054

summary(bw)
} # }