Kernel Consistent Quantile Regression Model Specification Test with Mixed Data Types

npqcmstest implements a consistent test for correct specification of parametric quantile regression models (linear or nonlinear) as described in Racine (2006) which extends the work of Zheng (1998).

Usage

npqcmstest(formula,
           data = NULL,
           subset,
           xdat,
           ydat,
           model = stop(paste(sQuote("model")," has not been provided")),
           tau = 0.5,
           distribution = c("bootstrap", "asymptotic"),
           bwydat = c("y","varepsilon"),
           boot.method = c("iid","wild","wild-rademacher"),
           boot.num = 399,
           pivot = TRUE,
           density.weighted = TRUE,
           random.seed = 42,
           ...)

Details

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

Arguments

Data, Bandwidth Inputs And Formula Interface

These arguments identify the model formula/data interface and explicit data inputs.

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(formula), typically the environment from which the function is called.
formula: a symbolic description of the quantile regression model to be tested. If xdat and ydat are omitted, the data are extracted from this formula and data.
model: a model object obtained from a call to rq. Important: the call to rq must have the argument model=TRUE or npqcmstest will not work.
subset: an optional vector specifying a subset of observations to be used.
xdat: a \(p\)-variate data frame of explanatory data (training data) used to calculate the quantile regression estimators.
ydat: a one (1) dimensional numeric or integer vector of dependent data, each element \(i\) corresponding to each observation (row) \(i\) of xdat.

Bootstrap And Test Controls

These arguments control the quantile level, test statistic, bootstrap procedure, and reproducibility settings.

boot.method: a character string used to specify the bootstrap method. iid will generate independent identically distributed draws. wild will use a wild bootstrap. wild-rademacher will use a wild bootstrap with Rademacher variables. Defaults to iid.
boot.num: an integer value specifying the number of bootstrap replications to use. Defaults to 399.
bwydat: a character string used to specify the left hand side variable used in bandwidth selection. "varepsilon" uses \(1-\tau,-\tau\) for ydat while "y" will use \(y\). Defaults to "y".
density.weighted: a logical value specifying whether the statistic should be weighted by the density of xdat. Defaults to TRUE.
distribution: a character string used to specify the method of estimating the distribution of the statistic to be calculated. bootstrap will conduct bootstrapping. asymptotic will use the normal distribution. Defaults to bootstrap.
pivot: a logical value specifying whether the statistic should be normalised such that it approaches \(N(0,1)\) in distribution. Defaults to TRUE.
random.seed: an integer used to seed R's random number generator. This is to ensure replicability. Defaults to 42.
tau: a numeric value specifying the \(\tau\)th quantile is desired

Additional Arguments

Further arguments are passed to the bandwidth-selection routines used by the test.

...: additional arguments supplied to control bandwidth selection on the residuals. One can specify the bandwidth type, kernel types, and so on. To do this, you may specify any of bwscaling, bwtype, ckertype, ckerorder, ukertype, okertype, as described in npregbw. This is necessary if you specify bws as a \(p\)-vector and not a bandwidth object, and you do not desire the default behaviours.

Value

npqcmstest returns an object of type cmstest with the following components. Components will contain information related to Jn or In depending on the value of pivot:

Jn: the statistic Jn
In: the statistic In
Omega.hat: as described in Racine, J.S. (2006).
q.*: the various quantiles of the statistic Jn (or In if pivot=FALSE) are in components q.90, q.95, q.99 (one-sided 1%, 5%, 10% critical values)
P: the P-value of the statistic
Jn.bootstrap: if pivot=TRUE contains the bootstrap replications of Jn
In.bootstrap: if pivot=FALSE contains the bootstrap replications of In

summary supports object of type cmstest.

Book And Method Pointers

npqcmstest tests a parametric conditional-quantile specification against a nonparametric conditional-quantile alternative at the supplied \(\tau\). The test therefore sits at the intersection of conditional CDF/quantile estimation and model specification testing.

For book-length background, see Li and Racine (2007), Chapter 12 Model Specification Tests and Chapter 6 Conditional CDF and Quantile Estimation. For the later workflow treatment of conditional CDFs and quantiles, see Racine (2019), Chapter 4 Conditional Probability Density and Cumulative Distribution Functions.

References

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

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

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

Murphy, K. M. and F. Welch (1990), “Empirical age-earnings profiles,” Journal of Labor Economics, 8, 202-229.

Pagan, A. and A. Ullah (1999), Nonparametric Econometrics, Cambridge University Press.

Racine, J.S. (2006), “Consistent specification testing of heteroskedastic parametric regression quantile models with mixed data,” manuscript.

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

Zheng, J. (1998), “A consistent nonparametric test of parametric regression models under conditional quantile restrictions,” Econometric Theory, 14, 123-138.

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.

Examples

if (FALSE) { # \dontrun{
# EXAMPLE 1: For this example, we conduct a consistent quantile regression
# model specification test for a parametric wage quantile regression
# model that is quadratic in age. The work of Murphy and Welch (1990)
# would suggest that this parametric quantile regression model is
# misspecified.

library("quantreg")

data("cps71")
with(cps71, {

model <- rq(logwage~age+I(age^2), tau=0.5, model=TRUE)

if (interactive()) plot(age, logwage)
lines(age, fitted(model))

X <- data.frame(age)

# Note - this may take a few minutes depending on the speed of your
# computer...

npqcmstest(model = model, xdat = X, ydat = logwage, tau=0.5)

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

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

# Next try Murphy & Welch's (1990) suggested quintic specification.

model <- rq(logwage~age+I(age^2)+I(age^3)+I(age^4)+I(age^5), model=TRUE)

if (interactive()) plot(age, logwage)
lines(age, fitted(model))

X <- data.frame(age)

# Note - this may take a few minutes depending on the speed of your
# computer...

npqcmstest(model = model, xdat = X, ydat = logwage, tau=0.5)

})
} # }