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Estimation of the covariance matrix of asset returns is crucial to portfolioconstruction. As suggested by economic theories, the correlation structureamong assets differs between emerging markets and developed countries. It istherefore imperative to make rigorous statistical inference on correlationmatrix equality between the two groups of countries. However, if thetraditional vector-valued approach is undertaken, such inference is eitherinfeasible due to limited number of countries comparing to the relativelyabundant assets, or invalid due to the violations of temporal independenceassumption. This highlights the necessity of treating the observations asmatrix-valued rather than vector-valued. With matrix-valued observations, ourproblem of interest can be formulated as statistical inference on covariancestructures under sub-Gaussian distributions, i.e., testing non-correlation andcorrelation equality, as well as the corresponding support estimations. Wedevelop procedures that are asymptotically optimal under some regularityconditions. Simulation results demonstrate the computational and statisticaladvantages of our procedures over certain existing state-of-the-art methods forboth normal and non-normal distributions. Application of our procedures tostock market data reveals interesting patterns and validates several economicpropositions via rigorous statistical testing.

Testing and Support Recovery of Correlation Structures for Matrix-Valued Observations with an Application to Stock Market Data