np (version 0.40-13)

npqreg: 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 method of Li and Racine (2008). A bandwidth specification can be a conbandwidth object, or a bandwidth vector, bandwidth type and kernel type.

Usage

npqreg(bws, ...)
## S3 method for class 'formula':
npqreg(bws, data = NULL, newdata = NULL, \dots)
## S3 method for class 'call':
npqreg(bws, \dots)
## S3 method for class 'conbandwidth':
npqreg(bws,
       txdat = stop("training data 'txdat' missing"),
       tydat = stop("training data 'tydat' missing"),
       exdat,
       tau = 0.5,
       gradients = FALSE,
       ftol = 1.19209e-07,
       tol = 1.49012e-08,
       small = 2.22045e-16,
       itmax = 10000,
       ...)
## S3 method for class 'default':
npqreg(bws, txdat, tydat, \dots)

Arguments

bws

a bandwidth specification. This can be set as a conbandwidth object returned from an invocation of npcdensbw, or as a vector of bandwidths, with each element $i$ corresponding to

tau

a numeric value specifying the $\tau$th 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

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

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

tolerance on the value of the objective function evaluated at located minima. Defaults to 1.19e-07 (FLT_EPSILON).

tol

tolerance on the position of located minima of the objective function. Defaults to 1.49e-08 (sqrt(DBL_EPSILON)).

small

a small number, at about the precision of the data type used. Defaults to 2.22e-16 (DBL_EPSILON).

Value

• npqreg returns a npqregression object. The generic functions fitted (or quantile), se, predict, 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 $\tau$th 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.

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

See Also

quantreg

Examples

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

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

# 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)

# 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 <- npcdensbw(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)