Exploring the role of covariance matrix: from finance to hierarchical forecast reconciliation

The covariance matrix is a key tool in many fields. This thesis proposes novelties in three areas. In systemic risk detection in financial market, we introduce an early warning indicator of systemic portfolio risk based on the eigenvalues of returns' factors correlation matrix. The indicator is constructed using the Marchenko-Pastur theorem and extracting factors via an autoencoder; the resulting investment strategy outperforms standard benchmarks. Second, we address the propagation of risks in financial markets. A covariance-based approach is limited, as it captures correlations but not causal relationships. We use Granger causality tests on volatility time series and then formulate the problem as a Quadratic Unconstrained Binary Optimization (QUBO) to estimate a causal network for credit spread markets, represented by a Directed Acyclic Graph (DAG). This causal approach yields more stable and effective portfolio strategies than correlation-based methods. Finally, we improve the treatment of covariance matrix in hierarchical time series forecasting. We propose a novel shrinkage estimator that incorporates conditional dependencies suggested by the hierarchical structure, yielding more accurate reconciled forecasts compared to classical approaches. We then develop a Bayesian model that explicitly accounts for the uncertainty in covariance matrix estimation, resulting in more reliable prediction intervals of the reconciled forecasts.