Testing Error Distribution by Kernelized Stein Discrepancy in Multivariate Time Series Models

Goodness‐of‐fit tests for the multivariate Student‐tdistribution based on i.i.d. data, and for GARCH observations

“…These models correspond to the non-diagonal version of the CCC-GARCH, i.e. the ECCC-GARCH, with dimensions p = 2 and p = 5, and the precise values of the GARCH parameters may be found in section 4.3 of Luo et al (2023). The results are presented in Table III.…”

“…Following Luo et al (2023), we now look whether the null distributions with highest p-values from Table IV give us better portfolio VaR forecasts in comparison with others. Let X t be the bivariate time series of the aforementioned monthly log returns.…”

“…case is apparently not affine invariant, which implies potential dependence of the finite-sample distribution on the actual location vector and scatter matrix. These remarks notwithstanding, the test of Luo et al (2023) remains a formidable competitor, and further comparison with our test will be discussed by means of Monte Carlo simulations in Section 5.…”

“…This method has the advantage of not being tailored, but applicable to any distribution under test provided of course that the underlying density (assumed to exist) is not too heavy-tailed, and that both density and score functions are sufficiently smooth, while for our test we only need the CF, which always exists, and moment assumptions are incidental in our case and due to the estimation step preceding the test. Moreover, and unlike our method, the test of Luo et al (2023) requires data subsampling, and in the i.i.d. case is apparently not affine invariant, which implies potential dependence of the finite-sample distribution on the actual location vector and scatter matrix.…”

“…We present a study of the power performance of our test as compared to the following competitors: the test based on the Wasserstein metric proposed in Hallin et al (2021) (denoted as W), the kernel Stein discrepancy-(KSD) based test proposed in Luo et al (2023) (denoted as L), and the tests based on Rosenblatt transform for three different weight functions, denoted as (R k , k = 1, 2, 3), which correspond to the statistics denoted as (T k , k = 1, 2, 3), in the original paper of Meintanis et al (2014). The test statistics are available in the Appendix S1.…”