Data-Driven Uncertainty Set Construction with ARIMA–GARCH Modeling for Robust Portfolio Optimization

Abstract

Portfolio optimization models are highly sensitive to estimation errors in expected returns and covariance matrices, often resulting in unstable allocations. Robust optimization mitigates parameter uncertainty by optimizing against worst-case realizations within a specified uncertainty set, whose construction critically determines the effectiveness of the approach. This paper proposes a data-driven framework for constructing polyhedral uncertainty sets that integrates Gaussian mixture models (GMMs) to identify heterogeneous distributional components and ARIMA-GARCH models to capture time-varying volatility dynamics. The construction proceeds in three stages. First, ARIMA-GARCH filters remove serial dependence and volatility clustering. Second, GMM clustering applied to the standardized residuals identifies latent market regimes. Third, convex hulls of observations lying within a Mahalanobis distance threshold form component-wise polyhedral sets, which are aggregated into a global convex uncertainty set. The robust counterpart is derived via linear programming duality, transforming the robust constraint into a tractable quadratic program that preserves convexity and polyhedrality. We prove that the constructed sets are convex and polyhedral, establish probabilistic coverage guarantees under mild regularity conditions, and analyze the computational complexity of the framework. Empirical analysis of Indonesian equity data confirms heavy tails and volatility clustering, with GMM identifying three distinct regimes of approximately equal proportions. Controlled synthetic experiments show that uncertainty set geometry fundamentally influences portfolio outcomes: overlapping clusters yield stable allocations across regimes, whereas well-separated clusters reveal that convex hull aggregation introduces conservatism that masks regime distinctions. Rolling window backtests demonstrate that the proposed approach produces economically higher Sharpe ratios than standard uncertainty set formulations in the reported out-of-sample period, although statistical significance is limited by the small number of independent rebalancing periods (18 quarterly events). The practical advantage should therefore be interpreted as conditional on moderate transaction costs and manageable turnover. These findings provide a statistically grounded, geometrically faithful, and computationally tractable methodology for practical implementation, contributing to financial resilience and sustainable economic growth with broader implications for stable capital markets.