Kernel Consistent Pairwise Nonlinear Dependence Test for Univariate Processes

npdeptest implements the consistent metric entropy test of pairwise independence as described in Maasoumi and Racine (2002).

Usage

npdeptest(data.x = NULL,
          data.y = NULL,
          method = c("integration","summation"),
          bootstrap = TRUE,
          B = 399,
          random.seed = 42)

Arguments

Data, Bandwidth Inputs And Formula Interface

These arguments identify the paired samples being tested.

data.x, data.y

two univariate vectors containing two variables that are of type numeric.

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

npdeptest returns an object of type deptest with the following components

Srho: the statistic Srho
Srho.bootstrap.vec: contains the bootstrap replications of Srho
P: the P-value of the Srho statistic
bootstrap: a logical value indicating whether bootstrapping was performed
boot.num: number of bootstrap replications
bw.data.x: the numeric bandwidth for data.x marginal density
bw.data.y: the numeric bandwidth for data.y marginal density
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

npdeptest compares the joint density or probability law of two variables with the product of their marginals. Departures from the product form indicate dependence, lack of fit, or predictability depending on how the two inputs are constructed.

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, especially Sections 12.4.1 and 12.4.3. 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.

Maasoumi, E. and J.S. Racine (2002), “Entropy and Predictability of Stock Market Returns,” Journal of Econometrics, 107, 2, pp 291-312.

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 pairwise nonlinear dependence between the densities of two data series. See Maasoumi and Racine (2002) for details. Default bandwidths are of the Kullback-Leibler variety obtained via likelihood cross-validation. The null distribution is obtained via bootstrap resampling under the null of pairwise independence.

npdeptest computes the distance between the joint distribution and the product of marginals (i.e. the joint distribution under the null), \(D[f(y, \hat y), f(y)\times f(\hat y)]\). Examples include, (a) a measure/test of “fit”, for in-sample values of a variable \(y\) and its fitted values, \(\hat y\), and (b) a measure of “predictability” for a variable \(y\) and its predicted values \(\hat y\) (from a user implemented model).

The summation version of this statistic will be numerically unstable when data.x and data.y lack common support or are 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. During bootstrapping, bounded groups of sampled data.x indices share the invariant data.y marginal work while retaining every draw's positionwise pairing, reduction order, and output order. Workspace grows only with the sample size times the bounded group size; no dense sample-by-sample kernel matrix is retained. Counts alone are not used because they would discard the required pairing with data.y.

Examples

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

## Test/measure lack of fit between y and its fitted value from a
## regression model when x is relevant using the `summation' version.

n <- 100
x <- rnorm(n)
y <- 1 + x + rnorm(n)
model <- lm(y~x)
y.fit <- fitted(model)

npdeptest(y,y.fit,B=99,method="summation")

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

## Test/measure lack of fit between y and its fitted value from a
## regression model when x is irrelevant using the `summation' version.

n <- 100
x <- runif(n,-2,2)
y <- 1 + rnorm(n)
model <- lm(y~x)
y.fit <- fitted(model)

npdeptest(y,y.fit,B=99,method="summation")

## Test/measure lack of fit between y and its fitted value from a
## regression model when x is relevant using the `integration'
## version (default, slower than summation version).

n <- 100
x <- rnorm(n)
y <- 1 + x + rnorm(n)
model <- lm(y~x)
y.fit <- fitted(model)

npdeptest(y,y.fit,B=99)

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

## Test/measure lack of fit between y and its fitted value from a
## regression model when x is irrelevant using the `integration'
## version (default, slower than summation version).

n <- 100
x <- runif(n,-2,2)
y <- 1 + rnorm(n)
model <- lm(y~x)
y.fit <- fitted(model)

npdeptest(y,y.fit,B=99)
} # }