Partially Linear Kernel Regression Bandwidth Selection with Mixed Data Types

Description

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 = XB + 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

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 kth 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 ith column of X on Z. See examples for more information.

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 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.

Author(s)

Tristen Hayfield hayfield@phys.ethz.ch, Jeffrey S. Racine racinej@mcmaster.ca

References

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

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.

Racine, J.S. and L. Liu (2006), “A partially linear kernel estimator for categorical data,” manuscript.

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.

See Also

npregbw, npreg

Examples

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

set.seed(123)

n <- 250
x1 <- rnorm(n)
x2 <- rbinom(n, 5, .3)

z1 <- rbinom(n, 2, .3)
z2 <- rnorm(n)

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

X <- data.frame(x1, ordered(x2))
Z <- data.frame(ordered(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+ordered(x2)|ordered(z1)+z2, tol=.1, ftol=.1)

summary(bw)

## Not run: 

# 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", tol=.1, ftol=.1)

# 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
                           0.99, 0.23), # X[,2] on Z
                ncol = ncol(Z), byrow=TRUE)

bw <- npplregbw(formula=y~x1+ordered(x2)|ordered(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(123)

n <- 250
x1 <- rnorm(n)
x2 <- rbinom(n, 5, .3)

z1 <- rbinom(n, 2, .3)
z2 <- rnorm(n)

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

X <- data.frame(x1, ordered(x2))
Z <- data.frame(ordered(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, tol=.1, ftol=.1)

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", tol=.1, ftol=.1)

# 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
                           0.99, 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)
## End(Not run)