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Kernel Consistent Quantile Regression Model Specification Test with Mixed Data Types

np.qcmstest.Rd

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"),
           B = 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.

For distribution = "bootstrap", each quantile model is still refitted separately using the selected bootstrap law. The resulting score vectors are collected in bounded groups, and the fixed-design kernel contractions for \(I_n\) and, when requested, \(\hat\Omega\) are evaluated as multi-response kernel sums. This reuses kernel geometry without forming a dense observation-by-observation weight matrix or changing score order, tau, bwydat, pivoting, density weighting, or random draws.

The selected bandwidth's effective kernel metadata are shared by the observed and bootstrap statistics. Manual bandwidth inputs and selection controls are not forwarded a second time as kernel-sum arguments.

Bootstrap refits preserve the model's observation weights. Weighted quantile models supply their raw design and response through the retained model frame; zero observation weights need not be inverted.

Model residuals, fitted values, response and design use the model's compact complete-observation sample, including when it was fitted with na.exclude. The supplied test data must represent those same complete observations in the same order. Row counts and response values are checked; when those agree, the caller remains responsible for supplying the matching covariate rows. The test does not infer a new pairing or reorder observations. The original model and its missing-value restoration policy are not modified.

Arguments

Data, Bandwidth Inputs And Formula Interface

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

data

This argument belongs to the formula interface (including supported formula-bearing bandwidth-object refits). Native calls must supply their vectors, matrices or data frames directly, or use with(data, ...). Supplying data, including explicit data = NULL, to a native route that does not consume it is an error; see np-data-interfaces.

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 model must retain its model, design and response components, normally requested with model=TRUE. Availability is checked from those components, not the spelling of the original call.

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.

B

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 finite scalar in (0,1) specifying the desired \(\tau\)th quantile. It must equal the single quantile fitted by model; the default remains 0.5, so pass tau explicitly for other fitted quantiles.

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.

Bootstrap P-Values

The bootstrap upper-tail p-value is the proportion of bootstrap statistics greater than or equal to the observed statistic. Ties count as at least as extreme, giving the conservative tie convention for discrete statistics. This is the ordinary bootstrap proportion, without an add-one correction; a reported zero is possible with a finite number of replications.

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.

See also

np.kernels, np.options, plot, npregbw.

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)

})
} # }