Kernel Consistent Model Specification Test with Mixed Data Types

npcmstest implements a consistent test for correct specification of parametric regression models (linear or nonlinear) as described in Hsiao, Li, and Racine (2007).

Usage

npcmstest(formula,
          data = NULL,
          subset,
          xdat,
          ydat,
          model = stop(paste(sQuote("model")," has not been provided")),
          distribution = c("bootstrap", "asymptotic"),
          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 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 lm (or glm). Important: the call to either glm or lm must have the arguments x=TRUE and y=TRUE or npcmstest will not work. Also, the test is based on residual bootstrapping hence the outcome must be continuous (which rules out Logit, Probit, and Count models).
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 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 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.
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.

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

npcmstest 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 Hsiao, C. and Q. Li and J.S. Racine.
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

npcmstest tests a parametric conditional-mean specification against a nonparametric alternative, using residual-based bootstrap or asymptotic calibration as selected by distribution. The statistic is designed for continuous outcomes and fitted lm or glm objects whose model frame, model matrix, and response are available.

For book-length background, see Li and Racine (2007), Chapter 12 Model Specification Tests, especially Section 12.1 and Section 12.1.2. For the conditional-mean estimation context used by the test, see Racine (2019), Chapter 6 Conditional Mean Function Estimation.

References

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

Hsiao, C. and Q. Li and J.S. Racine (2007), “A consistent model specification test with mixed categorical and continuous data,” Journal of Econometrics, 140, 802-826.

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

Maasoumi, E. and J.S. Racine and T. Stengos (2007), “Growth and convergence: a profile of distribution dynamics and mobility,” Journal of Econometrics, 136, 483-508.

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.

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

Author

Tristen Hayfield tristen.hayfield@gmail.com, Jeffrey S. Racine racinej@mcmaster.ca

Usage Issues

npcmstest supports regression objects generated by lm and uses features specific to objects of type lm hence if you attempt to pass objects of a different type the function cannot be expected to work.

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 model
# specification test for a parametric wage regression model that is
# quadratic in age. The work of Murphy and Welch (1990) would suggest
# that this parametric regression model is misspecified.

data("cps71")
with(cps71, {

model <- lm(logwage~age+I(age^2), x=TRUE, y=TRUE)

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

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

npcmstest(model = model, xdat = age, ydat = logwage, boot.num=9, nmulti = 1)

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

model <- lm(logwage~age+I(age^2)+I(age^3)+I(age^4)+I(age^5), x=TRUE, y=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...

npcmstest(model = model, xdat = age, ydat = logwage, boot.num=9, nmulti = 1)

# Note - you can pass in multiple arguments to this function. For
# instance, to use local linear rather than local constant regression,
# you would use npcmstest(model, X, regtype="ll"), while you could also
# change the kernel type (default is second order Gaussian), numerical
# search tolerance, or feed in your own vector of bandwidths and so
# forth.

})

# EXAMPLE 2: For this example, we replicate the application in Maasoumi,
# Racine, and Stengos (2007) (see oecdpanel for details). We
# estimate a parametric model that is used in the literature, then
# subject it to the model specification test.

data("oecdpanel")
with(oecdpanel, {

model <- lm(growth ~ oecd +
            factor(year) +
            initgdp +
            I(initgdp^2) +
            I(initgdp^3) +
            I(initgdp^4) +
            popgro +
            inv +
            humancap +
            I(humancap^2) +
            I(humancap^3) - 1,
            x=TRUE,
            y=TRUE)

X <- data.frame(factor(oecd), factor(year), initgdp, popgro, inv, humancap)

npcmstest(model = model, xdat = X, ydat = growth, boot.num=9, nmulti = 1)

})
} # }