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Kernel Quantile Regression with Mixed Data Types

np.qregression.Rd

npqreg computes a kernel quantile regression estimate of a one (1) dimensional dependent variable on \(p\)-variate explanatory data, given a set of evaluation points, training points (consisting of explanatory data and dependent data), and a bandwidth specification using the methods of Li and Racine (2008) and Li, Lin and Racine (2013). A bandwidth specification can be a condbandwidth object, or a bandwidth vector, bandwidth type and kernel type.

Usage

npqreg(bws, ...)

# S3 method for class 'formula'
npqreg(bws, data = NULL, newdata = NULL, ...)


# S3 method for class 'condbandwidth'
npqreg(bws,
       txdat = stop("training data 'txdat' missing"),
       tydat = stop("training data 'tydat' missing"),
       exdat,
       tau = 0.5,
       gradients = FALSE,
       tol = 1.490116e-04,
       small = 1.490116e-05,
       itmax = 10000,
       ...)

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

Arguments

Data, Bandwidth Inputs And Formula Interface

These arguments identify the bandwidth specification, formula/data interface, and training data.

bws

a bandwidth specification. This can be set as a condbandwidth object returned from an invocation of npcdistbw, or as a vector of bandwidths, with each element \(i\) corresponding to the bandwidth for column \(i\) in txdat. 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.

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 npcdistbw was called.

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.

Evaluation Data And Returned Quantities

These arguments control where the quantile regression is evaluated and which fitted quantities are returned.

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.

gradients

[currently not supported] a logical value indicating that you want gradients computed and returned in the resulting npregression object. Defaults to FALSE.

newdata

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

tau

a numeric value specifying the \(\tau\)th quantile is desired. Defaults to 0.5.

Quantile Solver Controls

These arguments control the one-dimensional numerical quantile extraction step.

itmax

integer maximum number of iterations allowed in the one-dimensional quantile refinement. Defaults to 10000.

small

minimum interval width used by the one-dimensional quantile refinement. Defaults to 1.490116e-05 (approximately 1000*sqrt(.Machine$double.eps)).

tol

tolerance on the one-dimensional quantile location refinement. Defaults to 1.490116e-04 (approximately 10000*sqrt(.Machine$double.eps)).

Additional Arguments

Further arguments are passed to the regression estimator or bandwidth interpretation path as needed.

...

additional arguments supplied to specify the regression type, bandwidth type, kernel types, training data, and so on. To do this, you may specify any of bwmethod, bwscaling, bwtype, cxkertype, cxkerorder, cykertype, cykerorder, uxkertype, uykertype, oxkertype, oykertype, as described in npcdistbw.

Details

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

Given a conditional distribution bandwidth object, npqreg estimates the conditional distribution at candidate response values and extracts the requested conditional quantile by a one-dimensional numerical refinement over the observed support of the dependent variable. The refinement minimizes the squared residual between the estimated conditional distribution and the requested probability tau. The arguments tol, small, and itmax control this one-dimensional refinement.

Value

npqreg returns a npqregression object. The generic functions fitted (or quantile), se, predict (when using predict you must add the argument tau= to generate predictions other than the median), and gradients, extract (or generate) estimated values, asymptotic 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

quantile

estimation of the quantile regression function (conditional quantile) at the evaluation points

quanterr

asymptotic standard errors of the quantile regression estimates, based on the estimated conditional density at the fitted quantile

quantgrad

gradients at each evaluation point

tau

the \(\tau\)th quantile computed

References

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

Hall, P. and J.S. Racine and Q. Li (2004), “Cross-validation and the estimation of conditional probability densities,” Journal of the American Statistical Association, 99, 1015-1026.

Koenker, R. W. and G.W. Bassett (1978), “Regression quantiles,” Econometrica, 46, 33-50.

Koenker, R. (2005), Quantile Regression, Econometric Society Monograph Series, Cambridge University Press.

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

Li, Q. and J.S. Racine (2008), “Nonparametric estimation of conditional CDF and quantile functions with mixed categorical and continuous data,” Journal of Business and Economic Statistics, 26, 423-434.

Li, Q. and J. Lin and J.S. Racine (2013), “Optimal Bandwidth Selection for Nonparametric Conditional Distribution and Quantile Functions”, Journal of Business and Economic Statistics, 31, 57-65.

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

Author

Tristen Hayfield [email protected], Jeffrey S. Racine [email protected]

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.

See also

np.kernels, np.options, plot quantreg

Examples

if (FALSE) { # \dontrun{
# EXAMPLE 1 (INTERFACE=FORMULA): For this example, we compute a
# bivariate nonparametric quantile regression estimate for Giovanni
# Baiocchi's Italian income panel (see Italy for details)

data("Italy")
attach(Italy)

# First, compute the cross-validated bandwidths.  Note - this may take a
# few minutes depending on the speed of your computer...

bw <- npcdistbw(formula=gdp~ordered(year))

# Note - numerical search for computing the quantiles may take a minute
# or so...

model.q0.25 <- npqreg(bws=bw, tau=0.25)
model.q0.50 <- npqreg(bws=bw, tau=0.50)
model.q0.75 <- npqreg(bws=bw, tau=0.75)

# Plot the resulting quantiles manually...

plot(ordered(year), gdp,
     main="CDF Quantile Estimates for the Italian Income Panel",
     xlab="Year",
     ylab="GDP Quantiles")

lines(ordered(year), model.q0.25$quantile, col="red", lty=2)
lines(ordered(year), model.q0.50$quantile, col="blue", lty=3)
lines(ordered(year), model.q0.75$quantile, col="red", lty=2)

legend(ordered(1951), 32, c("tau = 0.25", "tau = 0.50", "tau = 0.75"),
       lty=c(2, 3, 2), col=c("red", "blue", "red"))

detach(Italy)

# EXAMPLE 1 (INTERFACE=DATA FRAME): For this example, we compute a
# bivariate nonparametric quantile regression estimate for Giovanni
# Baiocchi's Italian income panel (see Italy for details)

data("Italy")
attach(Italy)
data <- data.frame(ordered(year), gdp)

# First, compute the likelihood cross-validation bandwidths (default).
# Note - this may take a few minutes depending on the speed of your
# computer...

bw <- npcdistbw(xdat=ordered(year), ydat=gdp)

# Note - numerical search for computing the quantiles will take a
# minute or so...

model.q0.25 <- npqreg(bws=bw, tau=0.25)
model.q0.50 <- npqreg(bws=bw, tau=0.50)
model.q0.75 <- npqreg(bws=bw, tau=0.75)

# Plot the resulting quantiles manually...

plot(ordered(year), gdp,
     main="CDF Quantile Estimates for the Italian Income Panel",
     xlab="Year",
     ylab="GDP Quantiles")

lines(ordered(year), model.q0.25$quantile, col="red", lty=2)
lines(ordered(year), model.q0.50$quantile, col="blue", lty=3)
lines(ordered(year), model.q0.75$quantile, col="red", lty=2)

legend(ordered(1951), 32, c("tau = 0.25", "tau = 0.50", "tau = 0.75"),
       lty=c(2, 3, 2), col=c("red", "blue", "red"))

detach(Italy)
} # }