FTSgof (version 1.0.0)

fCH_test: 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)

Value

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

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 \(\bar{X}^2(t)=N^{-1}\sum_{i=1}^N X^2_i(t)\).

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

# \donttest{
# 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")
# }