np.qregression: Kernel Quantile Regression with Mixed Data Types

In np: Nonparametric Kernel Smoothing Methods for Mixed Data Types

Kernel Quantile Regression with Mixed Data Types

Description

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 'call'
npqreg(bws, ...)

## 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,
       ftol = 1.490116e-07,
       tol = 1.490116e-04,
       small = 1.490116e-05,
       itmax = 10000,
       lbc.dir = 0.5,
       dfc.dir = 3,
       cfac.dir = 2.5*(3.0-sqrt(5)),
       initc.dir = 1.0,
       lbd.dir = 0.1,
       hbd.dir = 1,
       dfac.dir = 0.25*(3.0-sqrt(5)),
       initd.dir = 1.0,
       ...)

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

Arguments

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.
tau	a numeric value specifying the \tauth quantile is desired. Defaults to 0.5.
...	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.
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.
newdata	An optional data frame in which to look for evaluation data. If omitted, the training data are used.
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.
gradients	[currently not supported] a logical value indicating that you want gradients computed and returned in the resulting npregression object. Defaults to FALSE.
itmax	integer number of iterations before failure in the numerical optimization routine. Defaults to 10000.
ftol	fractional tolerance on the value of the cross-validation function evaluated at located minima (of order the machine precision or perhaps slightly larger so as not to be diddled by roundoff). Defaults to 1.490116e-07 (1.0e+01*sqrt(.Machine$double.eps)).
tol	tolerance on the position of located minima of the cross-validation function (tol should generally be no smaller than the square root of your machine's floating point precision). Defaults to 1.490116e-04 (1.0e+04*sqrt(.Machine$double.eps)).
small	a small number used to bracket a minimum (it is hopeless to ask for a bracketing interval of width less than sqrt(epsilon) times its central value, a fractional width of only about 10-04 (single precision) or 3x10-8 (double precision)). Defaults to small = 1.490116e-05 (1.0e+03*sqrt(.Machine$double.eps)).
lbc.dir,dfc.dir,cfac.dir,initc.dir	lower bound, chi-square degrees of freedom, stretch factor, and initial non-random values for direction set search for Powell's algorithm for numeric variables. See Details
lbd.dir,hbd.dir,dfac.dir,initd.dir	lower bound, upper bound, stretch factor, and initial non-random values for direction set search for Powell's algorithm for categorical variables. See Details

Details

The optimizer invoked for search is Powell's conjugate direction method which requires the setting of (non-random) initial values and search directions for bandwidths, and, when restarting, random values for successive invocations. Bandwidths for numeric variables are scaled by robust measures of spread, the sample size, and the number of numeric variables where appropriate. Two sets of parameters for bandwidths for numeric can be modified, those for initial values for the parameters themselves, and those for the directions taken (Powell's algorithm does not involve explicit computation of the function's gradient). The default values are set by considering search performance for a variety of difficult test cases and simulated cases. We highly recommend restarting search a large number of times to avoid the presence of local minima (achieved by modifying nmulti). Further refinement for difficult cases can be achieved by modifying these sets of parameters. However, these parameters are intended more for the authors of the package to enable ‘tuning’ for various methods rather than for the user themselves.

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	standard errors of the quantile regression estimates
quantgrad	gradients at each evaluation point
tau	the \tauth quantile computed

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.

Author(s)

Tristen Hayfield [e-mail], Jeffrey S. Racine [e-mail]

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.

See Also

quantreg

Examples

## Not run: 
# 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)

## End(Not run) 

np documentation built on March 31, 2023, 9:41 p.m.