%
\name{npplreg}
\alias{npplreg}
\alias{npplreg.default}
\alias{npplreg.formula}
\alias{npplreg.plbandwidth}

\title{Partially Linear Kernel Regression with Mixed Data Types}

\description{
  \code{npplreg} computes a partially linear kernel regression estimate
  of a one (1) dimensional dependent variable on \eqn{p+q}-variate
  explanatory data, using the model \eqn{Y = X\beta + \Theta (Z) +
  \epsilon}{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 \code{rbandwidth}
  object, or a bandwidth vector, bandwidth type and kernel type.
}

\usage{
npplreg(bws, 
        \dots)

\method{npplreg}{formula}(bws, 
       data = NULL, 
       newdata = NULL, 
       y.eval = FALSE, 
       \dots)

\method{npplreg}{default}(bws,
        txdat,
        tydat,
        tzdat,
        nomad = FALSE,
        \dots)

\method{npplreg}{plbandwidth}(bws,
        txdat = stop("training data txdat missing"),
        tydat = stop("training data tydat missing"),
        tzdat = stop("training data tzdat missing"),
        exdat,
        eydat,
        ezdat,
        residuals = FALSE,
        \dots)
}
\arguments{
\subsection{Data, Bandwidth Inputs And Formula Interface}{
These arguments identify the bandwidth specification, formula/data interface, and partially linear training data.
}

  \item{bws}{
  a bandwidth specification. This can be set as a \code{plbandwidth}
  object returned from an invocation of \code{\link{npplregbw}}, or as
  a matrix of bandwidths, where each row is a set of bandwidths for \eqn{Z},
  with a column for each variable \eqn{Z_i}{Z[i]}. In the first row are
  the bandwidths for the regression of \eqn{Y} on \eqn{Z}, the following
  rows contain the bandwidths for the regressions of the columns of
  \eqn{X} on \eqn{Z}.  If specified as a matrix
  additional arguments will need to be supplied as necessary to
  specify the bandwidth type, kernel types, training data, and so on.

  }



  \item{data}{
    an optional data frame, list or environment (or object
    coercible to a data frame by \code{\link{as.data.frame}}) containing the variables
    in the model. If not found in data, the variables are taken from
    \code{environment(bws)}, typically the environment from which
   \code{\link{npplregbw}} was called.
  }


  \item{txdat}{
  a \eqn{p}-variate data frame of explanatory data (training data),
  corresponding to \eqn{X} in the model equation, whose linear
  relationship with the dependent data \eqn{Y} is posited. Defaults to
  the training data used to
    compute the bandwidth object.
  }


  \item{tydat}{
  a one (1) dimensional numeric or integer vector of dependent data, each
  element \eqn{i} corresponding to each observation (row) \eqn{i} of
  \code{txdat}. Defaults to the training data used to
    compute the bandwidth object.
  }


  \item{tzdat}{
  a \eqn{q}-variate data frame of explanatory data (training data),
  corresponding to \eqn{Z} in the model equation, whose relationship
  to the dependent variable is unspecified (nonparametric). Defaults to
  the training data used to
    compute the bandwidth object.
  }


\subsection{Local-Polynomial Degree And Bandwidth Search}{
This argument controls the recommended automatic local-polynomial NOMAD route, which jointly selects continuous polynomial degree and bandwidths when these are computed inside \code{npplreg}.
}

  \item{nomad}{
    logical shortcut passed through to \code{\link{npplregbw}} when
    bandwidths are computed inside \code{npplreg}. When \code{TRUE},
    the partially linear bandwidth route fills any missing values among
    \code{regtype}, \code{search.engine}, \code{degree.select},
    \code{bernstein.basis}, \code{degree.min}, \code{degree.max},
    \code{degree.verify}, and \code{bwtype} with the recommended
    automatic local-polynomial degree-and-bandwidth NOMAD preset documented in \code{\link{npplregbw}}.
    The preset selects local-polynomial degrees and bandwidths
    separately for the nuisance regressions \eqn{E[Y \mid Z]} and
    \eqn{E[X_j \mid Z]} before the final partially linear solve.
    Additional NOMAD tuning arguments such as \code{nomad.nmulti} may
    also be supplied through \code{\dots}; \code{nmulti} remains the
    outer restart count while \code{nomad.nmulti} controls inner
    \code{crs::snomadr()} multistarts within each outer restart.
    After fitting, inspect \code{fit$bws$nomad.shortcut} on the
    returned object \code{fit} to see the normalized shortcut metadata.
  }


\subsection{Evaluation Data And Returned Quantities}{
These arguments control where the partially linear regression is evaluated and which fitted quantities are returned.
}

  \item{exdat}{
  a \eqn{p}-variate data frame of points on which the regression will be
  estimated (evaluation data).  By default,
    evaluation takes place on the data provided by \code{txdat}.
  }


  \item{eydat}{
  a one (1) dimensional numeric or integer vector of the true values
  of the dependent variable. Optional, and used only to calculate the
  true errors.  By default,
    evaluation takes place on the data provided by \code{tydat}.
  }


  \item{ezdat}{
    a \eqn{q}-variate data frame of points on which the regression will
    be estimated (evaluation data).  By default,
    evaluation takes place on the data provided by \code{tzdat}.
  }


  \item{newdata}{
    An optional data frame in which to look for evaluation data. If
    omitted, the training data are used.
  }


  \item{residuals}{
    a logical value indicating that you want residuals computed and
    returned in the resulting \code{plregression} object. Defaults to
    \code{FALSE}.
  }


  \item{y.eval}{
    If \code{newdata} contains dependent data and \code{y.eval = TRUE},
    \code{\link{np}} will compute goodness of fit statistics on these
    data and return them. Defaults to \code{FALSE}.
  }


\subsection{Additional Arguments}{
Further arguments are passed to \code{\link{npplregbw}} and its component \code{\link{npregbw}} searches when bandwidths are computed internally.
}

  \item{\dots}{
    additional arguments supplied to \code{\link{npplregbw}} when
    \code{npplreg} computes bandwidths internally, or arguments needed
    to interpret a numeric or matrix \code{bws} specification. This is
    where bandwidth selection controls such as \code{bwmethod},
    \code{bwtype}, and \code{bwscaling}, kernel/support controls such
    as \code{ckertype}, \code{ckerorder}, and \code{ckerbound},
    categorical kernel controls such as \code{ukertype} and
    \code{okertype}, search controls such as \code{nmulti} and
    \code{scale.factor.search.lower}, and local-polynomial/NOMAD
    controls such as \code{regtype}, \code{degree},
    \code{bernstein.basis}, \code{degree.select}, and
    \code{nomad.nmulti} are supplied. See \code{\link{npplregbw}} and
    \code{\link{npregbw}} for the complete bandwidth-selection
    argument surface, including the child-indexed matrix/list
    \code{degree} form for fixed local-polynomial partially linear
    fits.
  }
}
\details{
  Documentation guide: see \code{\link{npplregbw}} for partially linear
  bandwidth selection, \code{\link{npregbw}} for the component
  nonparametric regression search controls, \code{\link{np.kernels}}
  for kernels, \code{\link{np.options}} for global options, and
  \code{\link{plot}}, \code{\link{plot.np}} for plotting options.

  When \code{bws} is omitted, the formula and default methods call
  \code{\link{npplregbw}} first and pass bandwidth-selection arguments
  from \code{\dots} to that call. When \code{bws} is already a
  \code{plbandwidth} object, \code{npplreg} estimates with the stored
  bandwidth metadata in that object.

  Argument groups for bandwidth selection are documented on
  \code{\link{npplregbw}} and, for the component nonparametric
  regressions, \code{\link{npregbw}}. The most common workflow is to
  choose the linear \code{X} variables and nonparametric \code{Z}
  variables first, then bandwidth/search controls for the
  \code{Z}-side nonparametric regressions, and finally
  local-polynomial/NOMAD controls when using polynomial-adaptive fits.

  For S3 plotting help, see \code{\link{plot.np}}. You can list
  available plot methods with \code{methods("plot")}.

 \code{npplreg} uses a combination of OLS and nonparametric
 regression to estimate the parameter \eqn{\beta}{B} in the model
 \eqn{Y = X\beta + \Theta (Z) + \epsilon}{Y = XB + theta(Z) + epsilon}.

 \code{npplreg} implements a variety of methods for
 nonparametric regression on multivariate (\eqn{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 \sQuote{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, \eqn{x_i}{x[i]}, when estimating the
 density at the point \eqn{x}. Generalized nearest-neighbor bandwidths change
 with the point at which the density is estimated, \eqn{x}. Fixed bandwidths
 are constant over the support of \eqn{x}.

 Data contained in the data frame \code{tzdat} may be a mix of
 continuous (default), unordered discrete (to be specified in the data
 frame \code{tzdat} using \code{\link{factor}}), and ordered discrete
 (to be specified in the data frame \code{tzdat} using
 \code{\link{ordered}}). Data can be entered in an arbitrary order and
 data types will be detected automatically by the routine (see
 \code{\link{np}} for details).

 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.

 For practitioners who want the recommended automatic local-polynomial degree-and-bandwidth NOMAD route
 without spelling out all LP tuning arguments,
 \code{npplreg(..., nomad=TRUE)} and \code{npplregbw(..., nomad=TRUE)}
 expand missing settings to the same documented preset. Explicit
 incompatible settings fail fast rather than being silently rewritten.
}

\value{
  \code{npplreg} returns a \code{plregression} object. The generic
  accessor functions \code{\link{coef}}, \code{\link{fitted}},
  \code{\link{residuals}}, \code{\link{predict}}, and
  \code{\link{vcov}}, extract (or
  estimate) coefficients, estimated values, residuals,
  predictions, and variance-covariance matrices,
  respectively, from
  the returned object. Furthermore, the functions \code{summary}
  and \code{plot} support objects of this type. The returned object
  has the following components:

  For \code{plregression} objects, \code{predict(object, se.fit = TRUE)}
  returns a list with components \code{fit} and \code{se.fit}. The
  \code{se.fit} component contains asymptotic prediction standard errors
  computed from the partially linear fitted-value linear operator and
  the squared residuals from the training fit.

  \item{evalx}{ evaluation points }
  \item{evalz}{ evaluation points }
  \item{mean}{ estimation of the regression, or conditional mean, at the
    evaluation points }
  \item{xcoef}{ coefficient(s) corresponding to the components
    \eqn{\beta_i}{B[i]} in the model }
  \item{xcoeferr}{ standard errors of the coefficients }
  \item{xcoefvcov}{ covariance matrix of the coefficients }
  \item{bws}{ the canonical bandwidth object, stored as a
    \code{plbandwidth} object }
  \item{bw}{ backward-compatible alias for \code{bws} }
  \item{resid}{ if \code{residuals = TRUE}, in-sample or out-of-sample
    residuals where appropriate (or possible) }
  \item{R2}{ coefficient of determination (Doksum and Samarov (1995))}
  \item{MSE}{ mean squared error }
  \item{MAE}{ mean absolute error }
  \item{MAPE}{ mean absolute percentage error }
  \item{CORR}{ absolute value of Pearson's correlation coefficient }
  \item{SIGN}{ fraction of observations where fitted and observed values
    agree in sign }

}

\section{Book And Method Pointers}{
  The partially linear model is
  \eqn{Y=X\beta+\Theta(Z)+\epsilon}. The Robinson-style construction
  residualizes \eqn{Y} and each column of \eqn{X} on \eqn{Z}, then
  solves the final least-squares problem using those residualized
  variables. Thus the nonparametric part of the problem is a sequence
  of conditional-mean regressions \eqn{E[Y\mid Z]} and
  \eqn{E[X_j\mid Z]}, followed by the parametric solve for
  \eqn{\beta}. In local-polynomial/NOMAD fits, the nuisance regressions
  may select child-specific continuous polynomial degrees.

  For book-length derivations, see Li and Racine (2007), Chapter 7
  \emph{Semiparametric Partially Linear Models}, especially Sections 7.1,
  7.2, and 7.4, and Racine (2019), Chapter 8 \emph{Semiparametric
  Conditional Mean Function Estimation}, especially Robinson's
  partially linear model.
}

\references{

 Aitchison, J. and C.G.G. Aitken (1976), \dQuote{Multivariate binary
   discrimination by the kernel method,} Biometrika, 63, 413-420.

 Doksum, K. and A. Samarov (1995), \dQuote{Nonparametric estimation of
 global functionals and a measure of the explanatory power of covariates
 in regression,} The Annals of Statistics, 23 1443-1473.

 Gao, Q. and L. Liu and J.S. Racine (2015), \dQuote{A partially linear
 kernel estimator for categorical data,} Econometric Reviews, 34 (6-10),
 958-977.

 Li, Q. and J.S. Racine (2007), \emph{Nonparametric Econometrics: Theory
 and Practice,} Princeton University Press.

 Li, Q. and J.S. Racine (2004), \dQuote{Cross-validated local linear
   nonparametric regression,} Statistica Sinica, 14, 485-512.

 Pagan, A. and A. Ullah (1999), \emph{Nonparametric Econometrics,}
 Cambridge University Press.

 Racine, J.S. and Q. Li (2004), \dQuote{Nonparametric estimation of regression
 functions with both categorical and continuous data,} Journal of
 Econometrics, 119, 99-130.

 Robinson, P.M. (1988), \dQuote{Root-n-consistent semiparametric regression,}
 Econometrica, 56, 931-954.

 Wang, M.C. and J. van Ryzin (1981), \dQuote{A class of smooth estimators
 for discrete distributions,}  Biometrika, 68, 301-309.

}
\author{
    Tristen Hayfield \email{tristen.hayfield@gmail.com}, Jeffrey S. Racine
    \email{racinej@mcmaster.ca}
}

\section{Usage Issues}{
  If you are using data of mixed types, then it is advisable to use the
  \code{\link{data.frame}} function to construct your input data and not
  \code{\link{cbind}}, since \code{\link{cbind}} will typically not work as
  intended on mixed data types and will coerce the data to the same
  type.
}

\seealso{
  \code{\link{np.kernels}}, \code{\link{np.options}}, \code{\link{plot}}, \code{\link{plot.np}}
  \code{\link{npregbw}}, \code{\link{npreg}} }
\examples{
\dontrun{
# EXAMPLE 1 (INTERFACE=FORMULA): For this example, we simulate an
# example for a partially linear model and compare the coefficient
# estimates from the partially linear model with those from a correctly
# specified parametric model...

set.seed(42)

n <- 250
x1 <- rnorm(n)
x2 <- rbinom(n, 1, .5)

z1 <- rbinom(n, 1, .5)
z2 <- rnorm(n)

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

# First, compute data-driven bandwidths. This may take a few minutes
# depending on the speed of your computer...

bw <- npplregbw(formula=y~x1+factor(x2)|factor(z1)+z2)

# Next, compute the partially linear fit

pl <- npplreg(bws=bw)

# Print a summary of the model...

summary(pl)

# Sleep for 5 seconds so that we can examine the output...

if (interactive()) Sys.sleep(5)

# Retrieve the coefficient estimates and their standard errors...

coef(pl)
coef(pl, errors = TRUE)

# Compare the partially linear results to those from a correctly
# specified model's coefficients for x1 and x2

ols <- lm(y~x1+factor(x2)+factor(z1)+I(sin(z2)))

# The intercept is coef()[1], and those for x1 and x2 are coef()[2] and
# coef()[3]. The standard errors are the square root of the diagonal of
# the variance-covariance matrix (elements 2 and 3)

coef(ols)[2:3]
sqrt(diag(vcov(ols)))[2:3]

# Sleep for 5 seconds so that we can examine the output...

if (interactive()) Sys.sleep(5)

# Plot the regression surfaces via plot() (i.e., plot the `partial
# regression surface plots').

if (interactive()) plot(bw)

# Note - to plot regression surfaces with variability bounds constructed
# from bootstrapped standard errors, use the following (note that this
# may take a minute or two depending on the speed of your computer as
# the bootstrapping is done in real time, and note also that we override
# the default number of bootstrap replications (399) reducing them to 25
# in order to quickly compute standard errors in this instance - don't
# of course do this in general)

plot(bw,
     B=25,
     errors="bootstrap")


# EXAMPLE 1 (INTERFACE=DATA FRAME): For this example, we simulate an
# example for a partially linear model and compare the coefficient
# estimates from the partially linear model with those from a correctly
# specified parametric model...

set.seed(42)

n <- 250
x1 <- rnorm(n)
x2 <- rbinom(n, 1, .5)

z1 <- rbinom(n, 1, .5)
z2 <- rnorm(n)

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

X <- data.frame(x1, factor(x2))
Z <- data.frame(factor(z1), z2)

# First, compute data-driven bandwidths. This may take a few minutes
# depending on the speed of your computer...

bw <- npplregbw(xdat=X, zdat=Z, ydat=y)

# Next, compute the partially linear fit

pl <- npplreg(bws=bw)

# Print a summary of the model...

summary(pl)

# Sleep for 5 seconds so that we can examine the output...

if (interactive()) Sys.sleep(5)

# Retrieve the coefficient estimates and their standard errors...

coef(pl)
coef(pl, errors = TRUE)

# Compare the partially linear results to those from a correctly
# specified model's coefficients for x1 and x2

ols <- lm(y~x1+factor(x2)+factor(z1)+I(sin(z2)))

# The intercept is coef()[1], and those for x1 and x2 are coef()[2] and
# coef()[3]. The standard errors are the square root of the diagonal of
# the variance-covariance matrix (elements 2 and 3)

coef(ols)[2:3]
sqrt(diag(vcov(ols)))[2:3]

# Sleep for 5 seconds so that we can examine the output...

if (interactive()) Sys.sleep(5)

# Plot the regression surfaces via plot() (i.e., plot the `partial
# regression surface plots').

if (interactive()) plot(bw)

# Note - to plot regression surfaces with variability bounds constructed
# from bootstrapped standard errors, use the following (note that this
# may take a minute or two depending on the speed of your computer as
# the bootstrapping is done in real time, and note also that we override
# the default number of bootstrap replications (399) reducing them to 25
# in order to quickly compute standard errors in this instance - don't
# of course do this in general)

plot(bw,
     B=25,
     errors="bootstrap")
} % enddontrun
}
\keyword{nonparametric}
