Semiparametric Single Index Model Parameter and Bandwidth Selection

npindexbw computes a npindexbw bandwidth specification using the model \(Y = G(X\beta) + \epsilon\). 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 \(G(X\beta)\) and an estimate of the parameter vector \(\beta\) is returned.

If bwtype is set to generalized_nn or adaptive_nn, then instead the scalar \(k\)th nearest neighbor is returned.

The functions coef, predict, summary, and plot support objects of this class.

References

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.

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 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.

Examples

if (FALSE) { # \dontrun{
# 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)
} # }