This thesis studies testing and scoring high-dimensional covariance matrices when data exhibit heteroskedasticity. The observations are modeled as Y_i = ω_iZ_i, where Z_i's are i.i.d. p-dimensional random vectors with mean 0 and covariance Σ, and ω_i's are positive random scalars reflecting heteroskedasticity. The model is an extension of the elliptical distribution and can capture several stylized facts of financial data including heteroskedasticity, heavy-tailedness and asymmetry, etc..

Firstly, we aim to test H₀ : Σ ∝ Σ₀, in the high-dimensional setting where both the dimension p and the sample size n grow to infinity proportionally. We remove the heteroskedasticity by self-normalizing the observations, and establish a CLT for the linear spectral statistic (LSS) of S̃_n := p/nΣ...[ Read more ]

This thesis studies testing and scoring high-dimensional covariance matrices when data exhibit heteroskedasticity. The observations are modeled as Y_i = ω_iZ_i, where Z_i's are i.i.d. p-dimensional random vectors with mean 0 and covariance Σ, and ω_i's are positive random scalars reflecting heteroskedasticity. The model is an extension of the elliptical distribution and can capture several stylized facts of financial data including heteroskedasticity, heavy-tailedness and asymmetry, etc..

Firstly, we aim to test H₀ : Σ ∝ Σ₀, in the high-dimensional setting where both the dimension p and the sample size n grow to infinity proportionally. We remove the heteroskedasticity by self-normalizing the observations, and establish a CLT for the linear spectral statistic (LSS) of S̃_n := p/nΣ_i=1ⁿY_iY_i^T/ │Y _i│² = p/nΣ_i=1ⁿZ_iZ_i^T/│Z _i│². The CLT is different from the existing ones for the LSS of the usual sample covariance matrix S_n := 1/nΣ_i=1ⁿZ_iZ_i^T ([1], [2]). Our tests based on the new CLT neither assume a specic parametric distribution nor involve the fourth moment of Z_i. Numerical studies show that our tests work well even when Z_i's are heavy-tailed.

Secondly, to evaluate the performance of different covariance matrix predictors, we propose a scoring method in the heteroskedastic setting. Empirically, we use our proposed scores to evaluate different covariance matrix predictors for the returns of 72 stocks in the S&P 500 financials sector. The results show that: 1) self-normalizing the observations can improve the prediction; 2) compared with sparsity, an approximate factor model is more suitable for stock returns. These results can be used to build better minimum-variance portfolios.