This paper introduces a new framework for high-dimensional covariance matrix estimation, the Blockwise Exponential Covariance Model (BECM), which extends the traditional block-partitioned representation to the log-covariance domain. By exploiting the block-preserving properties of the matrix logarithm and exponential transformations, the proposed model guarantees strict positive definiteness while substantially reducing the number of parameters to be estimated through a blockwise log-covariance parameterization, without imposing any rank constraint. Within each block, intra- and inter-group dependencies are parameterized through interpretable coefficients and kernel-based similarity measures of factor loadings, enabling a data-driven representation of nonlinear groupwise associations. Using monthly stock return data from the U.S. stock market, we conduct extensive rolling-window tests to evaluate the empirical performance of the BECM in minimum-variance portfolio construction. The results reveal three main findings. First, the BECM consistently outperforms the Canonical Block Representation Model (CBRM) and the native 1/N benchmark in terms of out-of-sample Sharpe ratios and risk-adjusted returns. Second, adaptive determination of the number of clusters through cross-validation effectively balances structural flexibility and estimation stability. Third, the model maintains numerical robustness under fine-grained partitions, avoiding the loss of positive definiteness common in high-dimensional covariance estimators. Overall, the BECM offers a theoretically grounded and empirically effective approach to modeling complex covariance structures in high-dimensional financial applications.
Fan et al. (Fri,) studied this question.