fCH_test: Test for Conditional Heteroscedasticity of Functional Time...

In FTSgof: White Noise and Goodness-of-Fit Tests for Functional Time Series

Test for Conditional Heteroscedasticity of Functional Time Series

Description

It tests the null hypothesis that the objective functional curve data is not conditionally heteroscedastic. If a small p-value rejects the null hypothesis, the curves exhibit conditional heteroscedasticity.

Usage

fCH_test(f_data, H = 10, stat_Method = "functional", pplot = FALSE)

Arguments

f_data	A J \times N matrix of functional time series data, where J is the number of discrete points in a grid and N is the sample size.
H	A positive integer specifying the maximum lag for which test statistic is computed.
stat_Method	A string specifying the test method to be used in the "ch" test. Options include: "norm" Uses V_{N,H}. "functional" Uses M_{N,H}.
pplot	A Boolean value. If TRUE, the function will produce a plot of p-values of the test as a function of maximum lag H, ranging from H=1 to H=20, which may increase the computation time.

Details

Given the objective curve data X_i(t), for 1\leq i \leq N, t\in[0,1], the test aims at distinguishing the hypotheses:

H_0: the sequence X_i(t) is IID;
H_1: the sequence X_i(t) is conditionally heteroscedastic.

Two portmanteau type statistics are applied:

1. the norm-based statistic: V_{N,H}=N\sum_{h=1}^H\hat{\gamma}^2_{X^2}(h), where \hat{\gamma}^2_{X^2}(h) is the sample autocorrelation of the time series ||X_1||^2,\dots,||X_N||^2, and H is a pre-set maximum lag length.

2. the fully functional statistic M_{N,H}=N\sum_{h=1}^H||\hat{\gamma}_{X^2,N,h}||^2, where the autocovariance kernel \hat{\gamma}_{X^2,N,h}(t,s)=N^{-1}\sum_{i=1}^{N-h}[X_i^2(t)-\bar{X}^2(t)][X^2_{i+h}(s)-\bar{X}(s)], for ||\cdot || is the L^2 norm, and \bar{X}^2(t)=N^{-1}\sum_{i=1}^N X^2_i(t).

Value

A list that includes the test statistic and the p-value will be returned.

References

Rice, G., Wirjanto, T., Zhao, Y. (2020). Tests for conditional heteroscedasticity of functional data. Journal of Time Series Analysis. 41(6), 733-758. <doi:10.1111/jtsa.12532>.

Examples

# generate discrete evaluations of the iid curves under the null hypothesis.
yd_ou = dgp.ou(50, 100)

# test the conditional heteroscedasticity.
fCH_test(yd_ou, H=5, stat_Method="functional")

FTSgof documentation built on Oct. 4, 2024, 1:06 a.m.