npindexbw {np} | R Documentation

## Semiparametric Single Index Model Parameter and Bandwidth Selection

### Description

`npindexbw` computes a `npindexbw` bandwidth specification using the model Y = G(XB) + 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 `NA`s. 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(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 tristen.hayfield@gmail.com, 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.

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.

### Examples

    ## Not run:
    # 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)

    ## End(Not run)

[Package np version 0.60-9 Index]