fun_hetero: Test for Conditional Heteroscedasticity of Functional Data

In yzhao7322/CurVol: Curve Data Volatility Analysis

Test for Conditional Heteroscedasticity of Functional Data

Description

fun_hetero function 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

fun_hetero(yd, K = NULL, stat_Method, pplot = NULL)

Arguments

yd	A (grid_point) x (number of observations) matrix drawn from N intra-day return curves.
K	The lag autocorrelation coefficients served for the conditional heteroscedasticity test. If it is missing, a default value "K=20" is used.
stat_Method	A string to indicate which test will be implemented: "norm" - V_{N,K}; "functional" - M_{N,K}. Or if it is missing, the "functional"-type method is used.
pplot	an optional argument to compute and plot the P-values (either "norm" or "functional") as a function of K, K=1,2,...,20. If pplot=1, the p-values will be computed and figured; if it is missing, this step will be skipped.

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:
the norm-based statistic: V_{N,K}=N\sum_{h=1}^K\hat{\rho}_h^2, where \hat{\rho}_h is the sample autocorrelation of the time series ||x_1||^2,\dots,||x_N||^2, and K is a pre-set maximum lag length.
the fully functional statistic M_{N,K}=N\sum_{h=1}^K||\hat{\gamma}_h||^2, where the autocovariance kernel \hat{\gamma}_{h}(t,s)=\frac{1}{N}\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}(t)=1/N\sum_{i=1}^N x^2_i(t).

Value

List of objects:

stats: the test statistic.

p_value: the P_value.

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>.

See Also

sample_data

Examples

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

# test the conditional heteroscedasticity.
fun_hetero(yd, K=5, "functional")

yzhao7322/CurVol documentation built on June 11, 2025, 8:30 p.m.