Semiparametric Single Index Model

npindex computes a semiparametric single index model for a dependent variable and \(p\)-variate explanatory data using the model \(Y = G(X\beta) + \epsilon\), given a set of evaluation points, training points (consisting of explanatory data and dependent data), and a npindexbw bandwidth specification. Note that for this semiparametric estimator, the bandwidth object contains parameters for the single index model and the (scalar) bandwidth for the index function.

Usage

npindex(bws, ...)

# S3 method for class 'formula'
npindex(bws,
        data = NULL,
        newdata = NULL,
        y.eval = FALSE,
        ...)

# S3 method for class 'call'
npindex(bws,
        ...)

# Default S3 method
npindex(bws,
        txdat,
        tydat,
        ...) 

# S3 method for class 'sibandwidth'
npindex(bws,
        txdat = stop("training data 'txdat' missing"),
        tydat = stop("training data 'tydat' missing"),
        exdat,
        eydat,
        gradients = FALSE,
        residuals = FALSE,
        errors = FALSE,
        boot.num = 399,
        ...)

Arguments

bws

a bandwidth specification. This can be set as a sibandwidth 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 txdat where the first element is normalized to 1, and a scalar bandwidth (h).

gradients

a logical value indicating that you want gradients and the asymptotic covariance matrix for beta computed and returned in the resulting singleindex object. Defaults to FALSE.

residuals

a logical value indicating that you want residuals computed and returned in the resulting singleindex object. Defaults to FALSE.

errors

a logical value indicating that you want (bootstrapped) standard errors for the conditional mean, gradients (when gradients=TRUE is set), and average gradients (when gradients=TRUE is set), computed and returned in the resulting singleindex object. Defaults to FALSE.

boot.num

an integer specifying the number of bootstrap replications to use when performing standard error calculations. Defaults to 399.

...

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

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(bws), typically the environment from which npindexbw was called.

newdata

An optional data frame in which to look for evaluation data. If omitted, the training data are used.

y.eval

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

txdat

a \(p\)-variate data frame of explanatory data (training data) used to calculate the regression estimators. Defaults to the training data used to compute the bandwidth object.

tydat

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

exdat

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

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.

Details

A matrix of gradients along with average derivatives are computed and returned if gradients=TRUE is used.

Value

npindex returns a npsingleindex object. The generic functions fitted, residuals, coef, vcov, se, predict, and gradients, extract (or generate) estimated values, residuals, coefficients, variance-covariance matrix, bootstrapped standard errors on estimates, predictions, and gradients, respectively, from the returned object. Furthermore, the functions summary and plot support objects of this type. The returned object has the following components:

eval

evaluation points

mean

estimates of the regression function (conditional mean) at the evaluation points

beta

the model coefficients

betavcov

the asymptotic covariance matrix for the model coefficients

merr

standard errors of the regression function estimates

grad

estimates of the gradients at each evaluation point

gerr

standard errors of the gradient estimates

mean.grad

mean (average) gradient over the evaluation points

mean.gerr

bootstrapped standard error of the mean gradient estimates

R2

if method="ichimura", coefficient of determination (Doksum and Samarov (1995))

MSE

if method="ichimura", mean squared error

MAE

if method="ichimura", mean absolute error

MAPE

if method="ichimura", mean absolute percentage error

CORR

if method="ichimura", absolute value of Pearson's correlation coefficient

SIGN

if method="ichimura", fraction of observations where fitted and observed values agree in sign

confusion.matrix

if method="kleinspady", the confusion matrix or NA if outcomes are not available

CCR.overall

if method="kleinspady", the overall correct classification ratio, or NA if outcomes are not available

CCR.byoutcome

if method="kleinspady", a numeric vector containing the correct classification ratio by outcome, or NA if outcomes are not available

fit.mcfadden

if method="kleinspady", the McFadden-Puig-Kerschner performance measure or NA if outcomes are not available

References

Aitchison, J. and C.G.G. Aitken (1976), “Multivariate binary discrimination by the kernel method,” Biometrika, 63, 413-420.

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

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.

McFadden, D. and C. Puig and D. Kerschner (1977), “Determinants of the long-run demand for electricity,” Proceedings of the American Statistical Association (Business and Economics Section), 109-117.

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.

vcov requires that gradients=TRUE be set.

Examples

if (FALSE) { # \dontrun{
# EXAMPLE 1 (INTERFACE=FORMULA): Generate a simple linear model then
# estimate it using a semiparametric single index specification and
# Ichimura's nonlinear least squares coefficients and bandwidth
# (default). Also compute the matrix of gradients and average derivative
# estimates.

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)

summary(bw)

model <- npindex(bws=bw, gradients=TRUE)

summary(model)

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

Sys.sleep(5)

# Or you can visualize the input with plot.

plot(bw)

Sys.sleep(5)

# EXAMPLE 1 (INTERFACE=DATA FRAME): Generate a simple linear model then
# estimate it using a semiparametric single index specification and
# Ichimura's nonlinear least squares coefficients and bandwidth
# (default). Also compute the matrix of gradients and average derivative
# estimates.

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)

summary(bw)

model <- npindex(bws=bw, gradients=TRUE)

summary(model)

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

Sys.sleep(5)

# Or you can visualize the input with plot.

plot(bw)

Sys.sleep(5)

# EXAMPLE 2 (INTERFACE=FORMULA): Generate a simple binary outcome linear
# model then estimate it using a semiparametric single index
# specification and Klein and Spady's likelihood-based coefficients and
# bandwidth (default). Also compute the matrix of gradients and average
# derivative estimates.

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)

model <- npindex(bws=bw, gradients=TRUE)

# Note that, since the outcome is binary, we can assess model
# performance using methods appropriate for binary outcomes. We look at
# the confusion matrix, various classification ratios, and McFadden et
# al's measure of predictive performance.

summary(model)

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

Sys.sleep(5)

# EXAMPLE 2 (INTERFACE=DATA FRAME): Generate a simple binary outcome
# linear model then estimate it using a semiparametric single index
# specification and Klein and Spady's likelihood-based coefficients and
# bandwidth (default). Also compute the matrix of gradients and average
# derivative estimates.

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)

model <- npindex(bws=bw, gradients=TRUE)

# Note that, since the outcome is binary, we can assess model
# performance using methods appropriate for binary outcomes. We look at
# the confusion matrix, various classification ratios, and McFadden et
# al's measure of predictive performance.

summary(model)

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

Sys.sleep(5)

# EXAMPLE 3 (INTERFACE=FORMULA): Replicate the DGP of Klein & Spady
# (1993) (see their description on page 405, pay careful attention to
# footnote 6 on page 405).

set.seed(123)

n <- 1000

# x1 is chi-squared having 3 df truncated at 6 standardized by
# subtracting 2.348 and dividing by 1.511

x <- rchisq(n, df=3)
x1 <- (ifelse(x < 6, x, 6) - 2.348)/1.511

# x2 is normal (0, 1) truncated at +- 2 divided by 0.8796

x <- rnorm(n)
x2 <- ifelse(abs(x) < 2 , x, 2) / 0.8796

# y is 1 if y* > 0, 0 otherwise.

y <- ifelse(x1 + x2 + rnorm(n) > 0, 1, 0)

# Compute the parameter vector and bandwidth. 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")

# Next, create the evaluation data in order to generate a perspective
# plot

# Create an evaluation data matrix

x1.seq <- seq(min(x1), max(x1), length=50)
x2.seq <- seq(min(x2), max(x2), length=50)
X.eval <- expand.grid(x1=x1.seq, x2=x2.seq)

# Now evaluate the single index model on the evaluation data

fit <- fitted(npindex(exdat=X.eval, 
               eydat=rep(1, nrow(X.eval)), 
               bws=bw))

# Finally, coerce the fitted model into a matrix suitable for 3D
# plotting via persp()

fit.mat <- matrix(fit, 50, 50)

# Generate a perspective plot similar to Figure 2 b of Klein and Spady
# (1993)

persp(x1.seq, 
      x2.seq, 
      fit.mat, 
      col="white", 
      ticktype="detailed", 
      expand=0.5, 
      axes=FALSE, 
      box=FALSE, 
      main="Estimated Semiparametric Probability Perspective", 
      theta=310, 
      phi=25)

# EXAMPLE 3 (INTERFACE=DATA FRAME): Replicate the DGP of Klein & Spady
# (1993) (see their description on page 405, pay careful attention to
# footnote 6 on page 405).

set.seed(123)

n <- 1000

# x1 is chi-squared having 3 df truncated at 6 standardized by
# subtracting 2.348 and dividing by 1.511

x <- rchisq(n, df=3)
x1 <- (ifelse(x < 6, x, 6) - 2.348)/1.511

# x2 is normal (0, 1) truncated at +- 2 divided by 0.8796

x <- rnorm(n)
x2 <- ifelse(abs(x) < 2 , x, 2) / 0.8796

# y is 1 if y* > 0, 0 otherwise.

y <- ifelse(x1 + x2 + rnorm(n) > 0, 1, 0)

# Create the X matrix

X <- cbind(x1, x2)

# Compute the parameter vector and bandwidth. 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")

# Next, create the evaluation data in order to generate a perspective
# plot

# Create an evaluation data matrix

x1.seq <- seq(min(x1), max(x1), length=50)
x2.seq <- seq(min(x2), max(x2), length=50)
X.eval <- expand.grid(x1=x1.seq, x2=x2.seq)

# Now evaluate the single index model on the evaluation data

fit <- fitted(npindex(exdat=X.eval, 
               eydat=rep(1, nrow(X.eval)), 
               bws=bw))

# Finally, coerce the fitted model into a matrix suitable for 3D
# plotting via persp()

fit.mat <- matrix(fit, 50, 50)

# Generate a perspective plot similar to Figure 2 b of Klein and Spady
# (1993)

persp(x1.seq, 
      x2.seq, 
      fit.mat, 
      col="white", 
      ticktype="detailed", 
      expand=0.5, 
      axes=FALSE, 
      box=FALSE, 
      main="Estimated Semiparametric Probability Perspective", 
      theta=310, 
      phi=25)
} # }