Semiparametric Single Index Model Parameter and Bandwidth Selection

Description

npindexbw computes a npindexbw bandwidth specification using the model Y = G(XB) + epsilon. 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,
          ...)

## 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,
          ...)

Arguments

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:

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.

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.

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

# 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)

## Not run: 

# 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)