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,
           B = 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. The sequence must be complete and regularly spaced. Missing values (including NaN) are rejected: omitting them would change the time lags. If necessary, explicitly select a contiguous complete segment; do not delete interior missing observations. For a numeric vector, its order defines time.
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.

B: 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.

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

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 uses deterministic tensor-product, corrected composite trapezoidal quadrature. Along each axis the domain extends eight relevant bandwidths beyond the observed support and uses two evaluation points per smallest relevant bandwidth. A registered native callback evaluates the Gaussian marginal and joint densities in bounded tiles, so package-owned working memory does not grow as a dense sample-by-quadrature array. The integration version is generally more computationally demanding than summation, but avoids the density division that can make the summation statistic unstable.

The summation version fuses both marginal density sums and the joint density sum in one registered native pass with constant auxiliary storage. Bootstrap samples remain ordered index resamples: marginal counts alone cannot represent the lag pairs used by the statistic.

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,B=99,method="summation")

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

## Stationary independent time-series

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

## Stationary persistent time-series

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

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

## Stationary independent time-series

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

} # }