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, 
          ...)

# Default S3 method
npindexbw(xdat = stop("training data xdat missing"),
          ydat = stop("training data ydat missing"),
          bws,
          bandwidth.compute = TRUE,
          basis = c("glp", "additive", "tensor"),
          bernstein.basis = FALSE,
          degree = NULL,
          degree.select = c("manual", "coordinate", "exhaustive"),
          search.engine = c("nomad+powell", "cell", "nomad"),
          nomad = FALSE,
          nomad.nmulti = 0L,
          degree.min = NULL,
          degree.max = NULL,
          degree.start = NULL,
          degree.restarts = 0L,
          degree.max.cycles = 20L,
          degree.verify = FALSE,
          nmulti,
          nomad.remin = FALSE,
          powell.remin = TRUE,
          only.optimize.beta,
          optim.abstol,
          optim.maxattempts,
          optim.maxit,
          optim.method,
          optim.reltol,
          random.seed,
          regtype = c("lc", "ll", "lp"),
          scale.factor.init.lower = 0.1,
          scale.factor.init.upper = 2.0,
          scale.factor.init = 0.5,
          scale.factor.search.lower = NULL,
          ...,
          nomad.opts = list())

# S3 method for class 'sibandwidth'
npindexbw(xdat = stop("training data xdat missing"),
          ydat = stop("training data ydat missing"),
          bws,
          bandwidth.compute = TRUE,
          .certify.selected = TRUE,
          nmulti,
          only.optimize.beta = FALSE,
          optim.abstol = .Machine$double.eps,
          optim.maxattempts = 10,
          optim.maxit = 500,
          optim.method = c("Nelder-Mead", "BFGS", "CG"),
          optim.reltol = sqrt(.Machine$double.eps),
          random.seed = 42,
          scale.factor.init.lower = 0.1,
          scale.factor.init.upper = 2.0,
          scale.factor.init = 0.5,
          scale.factor.search.lower = NULL,
          .restore.first.start = TRUE,
          ...,
          .fixed.h.lower = NULL)

Arguments

Data, Bandwidth Inputs And Formula Interface

These arguments identify the data, formula interface, method label, and whether bandwidths are supplied or computed.

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.

.certify.selected

internal logical control requiring one raw-objective replay of the selected search point. This argument is used by package search owners and is not intended for direct user control.

.restore.first.start

internal logical control enabling bounded restoration of eligible automatically generated raw-invalid first optimizer evaluations. It is used by package search owners and is not intended for direct user control.

.fixed.h.lower

internal physical-bandwidth floor passed from a NOMAD search to its fixed-degree refinement. The default retains the direct-call scale-factor domain. This argument is not intended for direct user control.

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 design column $i$ (see Factors and Index Coordinates) 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.

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.

data

This argument belongs to the formula interface (including supported formula-bearing bandwidth-object refits). Native calls must supply their vectors, matrices or data frames directly, or use with(data, ...). Supplying data, including explicit data = NULL, to a native route that does not consume it is an error; see np-data-interfaces.

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.

formula

a symbolic description of variables on which bandwidth selection is to be performed. The details of constructing a formula are described below.

method

the single index model method, one of either “ichimura” (Ichimura (1993)) or “kleinspady” (Klein and Spady (1993)). Defaults to ichimura.

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.

subset

an optional vector specifying a subset of observations to be used in the fitting process.

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. For method="kleinspady", the response must be numeric 0/1 with both values present after jointly omitting rows with missing predictors or responses. Missing outcomes are permitted; a complete sample containing only one response class is rejected. Factor responses are rejected rather than converted to internal category codes; explanatory variables may still be factors.

Automatic Degree Search Controls

These arguments control automatic local-polynomial degree search.

degree.max: optional scalar or integer vector giving upper bounds for automatic degree search when degree.select != "manual".
degree.max.cycles: positive integer giving the maximum number of coordinate-search sweeps over the degree vector. Ignored for "manual" and "exhaustive" degree selection.
degree.min: optional scalar or integer vector giving lower bounds for automatic degree search when degree.select != "manual".
degree.restarts: non-negative integer giving the number of additional deterministic coordinate-search restarts. Ignored for "manual" and "exhaustive" degree selection.
degree.select: character string controlling local-polynomial degree handling when regtype="lp". "manual" (default) treats degree as fixed. "coordinate" performs cached coordinate-wise search over admissible degree values for the index smoother. "exhaustive" evaluates the full admissible degree grid when search.engine="cell". For NOMAD-based search engines, any non-"manual" value requests direct joint search over the degree and bandwidth coordinates.
degree.start: optional starting degree vector for automatic coordinate search. If omitted, the search starts from the degree-zero local-constant baseline for the index smoother.
degree.verify: logical value indicating whether an automatic degree-search solution should be exhaustively certified over the admissible degree grid after the heuristic phase completes. With search.engine="cell", this verifies a coordinate-search solution using repeated fixed-degree solves. With NOMAD-based search engines, this first runs the joint NOMAD search over bandwidth and degree coordinates, then verifies the result over the requested degree grid using repeated fixed-degree solves. The default is FALSE because certification can be substantially slower than the heuristic search. Set degree.verify=TRUE when degree-selection certification or sensitivity checking is more important than runtime.

Continuous Scale-Factor Search Initialization

These controls define deterministic and random continuous scale-factor starts and the lower admissibility floor for fixed-bandwidth search.

scale.factor.init: deterministic initial scale factor for continuous fixed-bandwidth search. Defaults to 0.5. The value supplied by the user is not rewritten, but the effective first start passed to the optimizer is max(scale.factor.init, scale.factor.search.lower). See Details.
scale.factor.init.lower: lower endpoint for random continuous scale-factor starts. Defaults to 0.1. The value supplied by the user is not rewritten, but the effective random-start lower endpoint is max(scale.factor.init.lower, scale.factor.search.lower). See Details. Random fixed-bandwidth restarts use the index scale at their own drawn beta. After conversion to search coordinates, an automatically generated bandwidth outside the existing search bounds is set to the nearest bound without drawing a new random value. The first start and explicitly supplied or held bandwidths are not changed by this step.
scale.factor.init.upper: upper endpoint for random continuous scale-factor starts. Defaults to 2.0. It must be greater than or equal to the effective lower endpoint, max(scale.factor.init.lower, scale.factor.search.lower); otherwise bandwidth search errors rather than silently expanding the interval. See Details.
scale.factor.search.lower: optional nonnegative scalar giving the hard lower admissibility bound for continuous fixed-bandwidth search candidates. Defaults to NULL. If NULL, an existing bandwidth object's stored value is inherited when available; otherwise the package default 0.1 is used. This floor applies to computed/search bandwidth candidates and to effective search starts only. It does not rewrite explicit bandwidths supplied for storage with bandwidth.compute = FALSE. Final fixed-bandwidth search candidates must also have a finite valid raw objective value. A supplied positive fixed bandwidth held by a only.optimize.beta=TRUE refit is not a bandwidth-search candidate and is not restricted by this control.

Local-Polynomial Model Specification

These arguments control the index smoother, local-polynomial basis, and fixed degree specification.

basis: local polynomial basis selector used when regtype="lp": one of "glp", "additive", or "tensor". Ignored for "lc" and "ll".
bernstein.basis: logical flag used when regtype="lp". For basis="glp", TRUE selects the complete degree-graded, orthonormal shifted-Legendre representation defined in npreg; additive and tensor bases retain their fixed-coordinate-degree Bernstein representation. When automatic degree search is requested and bernstein.basis is not explicitly supplied, the search route defaults to TRUE for numerical stability. Explicit bernstein.basis=FALSE is honored, but raw-polynomial search can be poorly conditioned at higher degrees.
degree: integer degree vector for continuous predictors when regtype="lp". When degree.select="manual", this must be supplied explicitly. For single-index regression this is typically length one.
regtype: a character string specifying local smoothing type for the nonparametric index regression fit used downstream in npindex. Supported values are "lc", "ll", and "lp".

NOMAD Search Controls

These arguments control the optional NOMAD direct-search route for local-polynomial degree and bandwidth search.

nomad: logical or character shortcut for the recommended automatic local-polynomial NOMAD route. When TRUE, any missing values among regtype, search.engine, degree.select, bernstein.basis, degree.min, degree.max, degree.verify, and bwtype are filled with regtype="lp", search.engine="nomad+powell", degree.select="coordinate", bernstein.basis=TRUE, degree.min=0L, degree.max=10L, degree.verify=FALSE, and bwtype="fixed". Explicit incompatible settings error immediately; in particular, explicit nomad=TRUE currently requires regtype="lp", bwtype="fixed", automatic degree search, no explicit degree, and search.engine %in% c("nomad", "nomad+powell"). This shortcut does not change the meaning of nmulti or nomad.nmulti: nmulti remains the outer restart count, while nomad.nmulti controls inner native crs NOMAD API multistarts within each outer restart. Returned bandwidth objects retain this normalized preset metadata in bw$nomad.shortcut for a returned object bw; when available, nomad.time records the direct-search component. The legacy field powell.time records the subsequent family-native R optim refinement selected by optim.method.
nomad.nmulti: non-negative integer controlling the inner the native crs NOMAD C API multistart count used within each outer NOMAD restart when regtype="lp" and automatic degree search uses search.engine="nomad" or "nomad+powell". Defaults to 0L, which preserves the current one-start-per- restart behavior. This does not replace nmulti: nmulti controls outer restarts, while nomad.nmulti controls inner NOMAD multistarts within each outer restart.
nomad.opts: optional named list of native NOMAD parameters forwarded to the crs NOMAD C API for NOMAD-backed bandwidth searches. NULL is treated as no extra NOMAD options. Non-list values and invalid or unsupported parameters are rejected before solver entry rather than silently ignored. Routes that evaluate R callbacks require serial callback evaluation and reject parallel callback settings such as NB_THREADS_PARALLEL_EVAL > 1.
nomad.remin: logical flag controlling the optional second NOMAD hot start. When TRUE, NOMAD is restarted once from the best full candidate found, including both bandwidth and degree coordinates. Defaults to FALSE; current simulation evidence favors the one-pass NOMAD default for routine use, while leaving this switch available for sensitivity checks.
search.engine: character string controlling the automatic local-polynomial search backend when regtype="lp" and degree.select != "manual". The compatibility value "nomad+powell" (default) performs direct joint search over the index-smoother bandwidth and degree using the native crs NOMAD C API, then applies one family-native R optim hot start from the NOMAD solution using optim.method. The value is retained for API compatibility; "nomad" omits this R-optimizer refinement. "cell" profiles the criterion over the admissible degree grid using repeated fixed-degree solves. Direct exhaustive degree-grid search is available with degree.select="exhaustive" and search.engine="cell". NOMAD-based search is intended to avoid exhaustive degree enumeration; users who want post-search certification over the requested degree grid can set degree.verify=TRUE. NOMAD-based search requires the suggested package crs to be installed.

Numerical Search Controls

These controls set search restart behavior.

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(2,ncol(xdat)).
powell.remin: compatibility argument shared with bandwidth families that use the Powell direction-set solver. Single-index fixed-degree and post-NOMAD refinement use R optim with optim.method; consequently, this argument does not alter their refinement behavior.

Optimization Controls

These arguments control outer optimization behavior for the semiparametric search.

only.optimize.beta

minimizes the objective function with respect to beta while retaining the existing polynomial degree and bandwidth (or nearest-neighbor count). Automatic degree search, including NOMAD and cell search, is not run; nomad, degree.select and other degree-search controls do not replace the retained model specification. A bandwidth object carries its selected degree; with a numeric bws vector and regtype="lp", supply degree explicitly. The public scale.factor.search.lower control does not restrict a supplied positive fixed bandwidth held in this way, whether inherited or explicitly supplied. The control is retained for later bandwidth searches. Positivity, finiteness and raw-objective validity checks still apply. When no positive bandwidth is supplied, the usual initialization rules apply. A supplied normalized beta vector is used as the first starting point, including when all its free coefficients are zero. With nmulti=1, this provides a single hot start; the usual convergence checks and recovery rules still apply.

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.maxattempts

maximum number of attempts taken trying to achieve successful convergence in optim. Defaults to 10.

optim.maxit

maximum number of iterations used by optim. Defaults to 500.

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.

The default "Nelder-Mead" method is a reasonable, robust derivative-free default when the search dimension is small, such as one scalar bandwidth plus only a few index coefficients. As the number of predictors grows, the $\beta$ search becomes higher dimensional and optim.method="BFGS" can be a useful performance and sensitivity comparison. Because the single-index objective can be nonconvex and route-sensitive, compare the selected objective value, convergence behavior, and resulting bandwidth object against the default search before relying on the result.

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.

random.seed

an integer used to seed R's random number generator. This ensures replicability of the numerical search. Defaults to 42.

Additional Arguments

These arguments collect remaining controls passed through S3 methods.

...: additional arguments supplied to specify the parameters to the sibandwidth S3 method, which is called during the numerical search.

Details

Factors and Index Coordinates

For newly constructed bandwidth objects, factors enter the linear index through R model-matrix contrasts. An unordered factor with $k$ observed levels ordinarily contributes $k-1$ coefficients. Ordered factors use their R contrasts (polynomial contrasts by default), including when their level labels look numeric. To fit a single coefficient on specified scores instead, supply those scores as a numeric predictor.

For supported additive formulas, coefficient names and predictor columns follow model.matrix(). The intercept is absorbed into the unknown link and is not an index coefficient. The first retained design coefficient is normalized to one; the package does not reorder predictors to choose a different normalization. Thus a factor listed first can supply the normalized contrast coefficient. These are relative index coefficients, not linear-regression slopes. This representation does not add support for otherwise unsupported formula syntax.

Native data-frame calls use the corresponding additive design. A manually supplied bws vector contains one beta for each expanded design column, in model-matrix order, followed by the scalar bandwidth. Automatic start vectors use the same coordinates. Numeric-only inputs keep their existing representation. A new factor design with aliased columns or a constant direction is rejected: for example, y ~ 0 + x + group supplies all group indicators and has an unidentified constant direction. Use the ordinary contrast formula y ~ x + group or independent numeric columns.

Factor levels and the actual contrast matrices are retained at construction. Prediction uses this design even if global contrast options change or an evaluation sample omits some training levels. New evaluation levels are an error. Levels absent from the complete training sample do not identify coefficients. Custom contrasts must remain identifiable on that sample.

Older saved bandwidth objects without retained design metadata preserve their historical single-score factor representation. Reconstruct a bandwidth object to adopt the contrast design; the package does not silently reinterpret an existing coefficient vector. The index.design component records the raw predictor names, retained levels/contrasts, and expanded column mapping; coefficient names and dimensions refer to the expanded design.

Formula bandwidth objects retain the prepared training sample (after alignment, subsetting and missing-value handling) and the resolved missing-value policy. Later refits use that sample even if constructor variables are rebound. Supplying explicit training data to the estimator replaces the training data; evaluation via newdata does not. Retention requires storage proportional to the sample size times the number of variables, not a pairwise weight matrix. Older saved objects without this retained state still rely on their original data bindings; rebuild their bandwidth object from the intended data before changing those bindings. User-defined transformation and missing-value functions should not rely on subsequently modified closure state.

The scale.factor.* controls are dimensionless search controls. The package converts scale factors to bandwidths using the estimator-specific scaling encoded in the bandwidth object, including kernel order and the number of continuous variables relevant for the estimator. Users should not pre-multiply these controls by sample-size or standard-deviation factors.

scale.factor.init controls the deterministic first search start when that control is exposed. scale.factor.init.lower and scale.factor.init.upper define the random multistart interval when exposed. scale.factor.search.lower is the lower admissibility bound for continuous fixed-bandwidth search candidates. The effective first start is max(scale.factor.init, scale.factor.search.lower) when both controls are present, and the effective random-start lower endpoint is max(scale.factor.init.lower, scale.factor.search.lower). scale.factor.init.upper must be at least that effective lower endpoint; the package errors rather than silently expanding the user's interval.

When scale.factor.search.lower is NULL, an existing bandwidth object's stored floor is inherited when available; otherwise the package default 0.1 is used. Explicit bandwidths supplied for storage with bandwidth.compute = FALSE are not rewritten by the search floor.

Categorical search-start controls such as dfac.init, lbd.init, and hbd.init have separate semantics and are not affected by scale.factor.search.lower.

Documentation guide: see np.kernels for kernels, np.options for global options, and plot, plot.np for plotting options.

For S3 plotting help, see plot.np. You can list available plot methods with methods("plot").

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).

Invalid outer-search guidance uses the exact declared leave-one-out null: the constant-model least-squares null for Ichimura and the clipped Bernoulli likelihood null for Klein–Spady. Inner regression evaluation remains raw; only terminal invalidity is mapped at the outer boundary. The selected point is replayed once against the raw objective and is rejected if it is not finite and valid.

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.

Interpreting $\beta$ Coefficients

The scale of $X^\prime\beta$ and the scale of the unknown link function are not separately identified, so npindexbw normalizes the first coefficient to one. The remaining coefficients should therefore be read relative to the first explanatory variable, not as standalone linear-model slopes. Holding other variables fixed, a one-unit increase in $X_j$ changes the index by the same amount as a $\beta_j$-unit increase in $X_1$. Where the link derivative exists and is nonzero, marginal-effect ratios satisfy $\partial g(X^\prime\beta)/\partial X_j \; /\; \partial g(X^\prime\beta)/\partial X_1 = \beta_j$. Signs and magnitudes are therefore relative to the chosen first variable and its units; choose the first continuous regressor deliberately when interpretation is important.

For fixed-bandwidth R optim searches, an automatic raw-invalid starting bandwidth is doubled while keeping its starting coefficients and degree fixed. This applies to all supported regression types, kernel orders and support options, including generated starts in automatic degree-search children. Each multistart uses its own coefficients' index scale. Only the first scalar evaluation already requested by optim is checked. A valid original start continues in that same optimizer invocation without an additional objective evaluation. A rejected start does not incur a numerical-gradient stencil.

Restoration permits at most eight doublings (nine starting scalar evaluations) for each multistart or existing convergence-retry start and respects the existing search bounds. Each rejected start adds one scalar evaluation, which can be costly. The random-draw control scale.factor.init.upper is not a search upper bound, so restoration may exceed it. Explicit initial bandwidths and bandwidths held by only.optimize.beta=TRUE are never expanded; a raw-invalid first evaluation fails with the method, start/retry, attempted bandwidth range, count and reason. Budget exhaustion also fails the complete call.

This rule does not change subsequent optimizer trials, finite differences, penalties, convergence-retry controls or final exact-point certification. A valid start does not guarantee convergence or a valid final point. Increasing the bandwidth does not guarantee feasibility for higher-order kernels or local-polynomial fits; every attempted start must pass the same raw-objective check. Nearest-neighbour bandwidths and NOMAD's own direct search retain their existing feasibility mechanisms. An explicit NOMAD refinement start is checked but not expanded. No alternative solver or earlier multistart winner is substituted when restoration fails.

Choosing `optim.method`

The default "Nelder-Mead" search is often a good starting point for low-dimensional single-index problems because it uses only objective values and is comparatively tolerant of rough objective surfaces. When $p$ is larger, especially when many coefficients enter $\beta$, "BFGS" may reduce runtime for some problems, but it is more dependent on local smoothness and on the quality of numerical gradient information. For important fits, try both methods with adequate multistarts and compare the final objective values, convergence messages, and fitted objects.

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(2, ncol(xdat)) times) as is done for a number of examples. See the optimizer guidance above when choosing between the default "Nelder-Mead" method and optim.method="BFGS" for higher-dimensional index searches.

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.

When regtype="lp" and degree.select != "manual", npindexbw can jointly determine the local-polynomial degree for the index smoother together with its bandwidth coordinate. With search.engine="cell", the criterion is profiled over the admissible degree grid using cached coordinate-wise or exhaustive search. With search.engine="nomad" or "nomad+powell", the criterion is optimized directly over the joint degree/bandwidth space using the native crs NOMAD C API. For this family, the compatibility value "nomad+powell" then performs one family-native R optim hot start using optim.method and retains the better of the direct NOMAD and refined solutions. For the index-smoother local-polynomial component, this polynomial-adaptive joint-search route follows Hall and Racine (2015).

Setting nomad=TRUE is a convenience preset for this automatic LP route, not a generic optimizer alias. For single-index bandwidth selection it expands any missing values to the equivalent long-form call


    npindexbw(...,
              regtype = "lp",
              search.engine = "nomad+powell",
              degree.select = "coordinate",
              bernstein.basis = TRUE,
              degree.min = 0L,
              degree.max = 10L,
              degree.verify = FALSE,
              bwtype = "fixed")

Compatible explicit tuning arguments are respected. Incompatible explicit settings fail fast so the shortcut never silently changes user-selected semantics. The character value nomad="auto" applies the same LP shortcut but leaves search.engine and degree.select eligible for data-driven resolution when both were not supplied explicitly: scalar continuous degree searches use the exhaustive degree-grid route, while higher-dimensional degree searches keep the compatibility "nomad+powell" route. Explicit search.engine or degree.select choices are honored.

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.

Book And Method Pointers

Local-polynomial index search inherits the numerical-ridging policies described in npreg; see Seifert and Gasser (2000) there.

npindexbw selects the index direction, bandwidth, and when requested local-polynomial degree/search metadata for the single-index model. The continuous-outcome target is typically written $E[Y\mid X]=g(X^\prime\beta)$, with $\beta$ normalized for identification; binary outcomes use the Klein-Spady route.

For book-length derivations, see Li and Racine (2007), Chapter 8 Semiparametric Single Index Models, especially Sections 8.1, 8.2.1, 8.4, 8.5, and 8.10, and Racine (2019), Chapter 8.

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.

Hall, P. and J.S. Racine (2015), “Infinite Order Cross-Validated Local Polynomial Regression,” Journal of Econometrics, 185, 510-525.

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(2, 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...

if (interactive()) 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...

if (interactive()) 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)
} # }