Kernel Consistent Serial Dependence Test for Univariate Nonlinear Processes

npsdeptest implements the consistent metric entropy test of nonlinear serial dependence as described in Granger, Maasoumi and Racine (2004).

Usage

npsdeptest(data = NULL,
           lag.num = 1,
           method = c("integration","summation"),
           bootstrap = TRUE,
           boot.num = 399,
           random.seed = 42)

Arguments

Data, Bandwidth Inputs And Formula Interface

These arguments identify the data series, lag count, and statistic variant.

data: a vector containing the variable that can be of type numeric or ts.
lag.num: an integer value specifying the maximum number of lags to use. Defaults to 1.
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.num: an integer value specifying the number of bootstrap replications to use. Defaults to 399.
bootstrap: a logical value which specifies whether to conduct the bootstrap test or not. If set to FALSE, only the statistic will be computed. Defaults to TRUE.
random.seed: an integer used to seed R's random number generator. This is to ensure replicability. Defaults to 42.

Value

npsdeptest returns an object of type deptest with the following components

Srho: the statistic vector Srho
Srho.cumulant: the cumulant statistic vector Srho.cumulant
Srho.bootstrap.mat: contains the bootstrap replications of Srho
Srho.cumulant.bootstrap.mat: contains the bootstrap replications of Srho.cumulant
P: the P-value vector of the Srho statistic vector
P.cumulant: the P-value vector of the cumulant Srho statistic vector
bootstrap: a logical value indicating whether bootstrapping was performed
boot.num: number of bootstrap replications
lag.num: the number of lags
bw.y: the numeric vector of bandwidths for data marginal density at lag num.lag
bw.y.lag: the numeric vector of bandwidths for lagged data marginal density at lag num.lag
bw.joint: the numeric matrix of bandwidths for data and lagged data joint density at lag num.lag

summary supports object of type deptest.

Book And Method Pointers

npsdeptest applies the dependence-test idea to a time series by comparing the joint law of current and lagged values with the product of their marginal laws. The lag and bootstrap choices determine the serial-dependence alternatives being probed and the null calibration.

For book-length background, see Li and Racine (2007), Chapter 13 Nonsmoothing Tests, especially Sections 13.4 and 13.5, and Chapter 12 Model Specification Tests for related distributional tests. For entropy and density context, see Racine (2019), Chapter 2.

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.

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.

npsdeptest computes the nonparametric metric entropy (normalized Hellinger of Granger, Maasoumi and Racine (2004)) for testing for nonlinear serial dependence, \(D[f(y_t, \hat y_{t-k}), f(y_t)\times f(\hat y_{t-k})]\). Default bandwidths are of the Kullback-Leibler variety obtained via likelihood cross-validation.

The test may be applied to a raw data series or to residuals of user estimated models.

The summation version of this statistic may be numerically unstable when data 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

The integration version of the statistic uses multidimensional numerical methods from the cubature package. See adaptIntegrate for details. The integration version of the statistic will be substantially slower than the summation version, however, it will likely be both more accurate and powerful.

Examples

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

## A function to create a time series

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)
}

n <- 100

## Stationary persistent time-series

yt <- ar.series(0.95,rnorm(n))
npsdeptest(yt,lag.num=2,boot.num=99,method="summation")

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

## Stationary independent time-series

yt <- ar.series(0.0,rnorm(n))
npsdeptest(yt,lag.num=2,boot.num=99,method="summation")

## Stationary persistent time-series

yt <- ar.series(0.95,rnorm(n))
npsdeptest(yt,lag.num=2,boot.num=99,method="integration")

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

## Stationary independent time-series

yt <- ar.series(0.0,rnorm(n))
npsdeptest(yt,lag.num=2,boot.num=99,method="integration")

} # }