Kernel Consistent Density Asymmetry Test with Mixed Data Types

npsymtest implements the consistent metric entropy test of asymmetry as described in Maasoumi and Racine (2009).

Usage

npsymtest(data = NULL,
          method = c("integration","summation"),
          boot.num = 399,
          bw = NULL,
          boot.method = c("iid", "geom"),
          random.seed = 42,
          ...)

Arguments

Data, Bandwidth Inputs And Formula Interface

These arguments identify the data, statistic variant, and any supplied bandwidth.

bw: a numeric (scalar) bandwidth. Defaults to plug-in (see details below).
data: a vector containing the variable.
method: a character string used to specify whether to compute the integral version or the summation version of the statistic. Can be set as integration or summation (see below for details). Defaults to integration.

Bootstrap Controls

These arguments control bootstrap execution and reproducibility settings.

boot.method: a character string used to specify the bootstrap method. Can be set as iid or geom (see below for details). Defaults to iid.
boot.num: an integer value specifying the number of bootstrap replications to use. Defaults to 399.
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 specify the bandwidth type, kernel types, and so on. This is used since we specify bw as a numeric scalar and not a bandwidth object, and is of interest if you do not desire the default behaviours. To change the defaults, you may specify any of bwscaling, bwtype, ckertype, ckerorder, ukertype, okertype.

Value

npsymtest returns an object of type symtest with the following components

Srho: the statistic Srho
Srho.bootstrap: contains the bootstrap replications of Srho
P: the P-value of the statistic
boot.num: number of bootstrap replications
data.rotate: the rotated data series
bw: the numeric (scalar) bandwidth

summary supports object of type symtest.

Book And Method Pointers

npsymtest tests symmetry by comparing the density or probability function of the observed data with that of the data reflected around the relevant center. The same entropy-distance logic used by the univariate density tests underlies the statistic.

For book-length background, see Racine (2019), Chapter 2 Continuous Density and Cumulative Distribution Functions and Li and Racine (2007), Chapter 13 Nonsmoothing Tests and Chapter 12 Model Specification Tests where related density-feature tests are treated.

References

Granger, C.W. and E. Maasoumi and J.S. Racine (2004), “A dependence metric for possibly nonlinear processes”, Journal of Time Series Analysis, 25, 649-669.

Maasoumi, E. and J.S. Racine (2009), “A robust entropy-based test of asymmetry for discrete and continuous processes,” Econometric Reviews, 28, 246-261.

Politis, D.N. and J.P. Romano (1994), “The stationary bootstrap,” Journal of the American Statistical Association, 89, 1303-1313.

Author

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

Details

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

npsymtest computes the nonparametric metric entropy (normalized Hellinger of Granger, Maasoumi and Racine (2004)) for testing symmetry using the densities/probabilities of the data and the rotated data, \(D[f(y), f(\tilde y)]\). See Maasoumi and Racine (2009) for details. Default bandwidths are of the plug-in variety (bw.SJ for continuous variables and direct plug-in for discrete variables).

For bootstrapping the null distribution of the statistic, iid conducts simple random resampling, while geom conducts Politis and Romano's (1994) stationary bootstrap using automatic block length selection via the b.star function in the np package. See the boot package for details.

The summation version of this statistic may be numerically unstable when y is sparse (the summation version involves division of densities while the integration version involves differences). Warning messages are produced should this occur (‘integration recommended’) and should be heeded.

Usage Issues

When using data of type factor it is crucial that the variable not be an alphabetic character string (i.e. the factor must be integer-valued). The rotation is conducted about the median after conversion to type numeric which is then converted back to type factor. Failure to do so will have unpredictable results. See the example below for proper usage.

Examples

if (FALSE) { # \dontrun{
set.seed(1234)

n <- 100

## Asymmetric discrete probability distribution function

x <- factor(rbinom(n,2,.8))
npsymtest(x,boot.num=99)

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

## Symmetric discrete probability distribution function

x <- factor(rbinom(n,2,.5))
npsymtest(x,boot.num=99)

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

## Asymmetric continuous distribution function

y <- rchisq(n,df=2)
npsymtest(y,boot.num=99)

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

## Symmetric continuous distribution function

y <- rnorm(n)
npsymtest(y,boot.num=99)

## Time-series bootstrap

ar.series <- function(phi,epsilon) {
  n <- length(epsilon)
  series <- numeric(n)
  series[1] <- epsilon[1]/(1-phi)
  for(i in 2:n) {
    series[i] <- phi*series[i-1] + epsilon[i]
  }
  return(series)
}

## Asymmetric time-series

yt <- ar.series(0.5,rchisq(n,df=3))
npsymtest(yt,boot.num=99,boot.method="geom")

## Symmetric time-series

yt <- ar.series(0.5,rnorm(n))
npsymtest(yt,boot.num=99,boot.method="geom")

} # }