Soumia Kharfouchi

We propose a new class of spatio-temporal GARCH models designed to capture volatility dynamics that propagate jointly across time and space. Existing spatio-temporal GARCH formulations typically account for either lagged spatial spillovers or contemporaneous interactions separately, and therefore fail to capture the combined effect of instantaneous spatial volatility feedback and its propagation over time. To address this gap, we introduce a unified framework that incorporates both contemporaneous and lagged spatial volatility interactions within a single coherent model. At each time point, conditional variances evolve according to a temporal GARCH recursion combined with both contemporaneous and lagged spatial volatility interactions defined on a lattice. This structure allows volatility shocks to diffuse instantaneously across neighboring locations and persist over time through spatially structured feedback mechanisms, extending existing spatial and spatio-temporal GARCH formulations. We establish sufficient conditions for the existence of a unique strictly stationary and ergodic solution based on contraction properties of a combined spatial–temporal operator. Statistical inference is conducted via Gaussian quasi-maximum likelihood estimation (QMLE). We derive consistency and asymptotic normality of the QMLE under two asymptotic regimes: (i) increasing temporal domain with fixed spatial size, and (ii) joint asymptotics where both the number of time periods and spatial locations diverge. In both cases, the asymptotic covariance matrix admits a standard sandwich form and can be consistently estimated. An extensive Monte Carlo study confirms the theoretical results. The simulations show that the QMLE performs well even under strong spatial and temporal persistence and remains robust to heavy-tailed innovations. In particular, increasing the spatial domain substantially improves estimation accuracy, highlighting the efficiency gains induced by spatial information. The proposed model provides a flexible and tractable framework for analyzing volatility processes evolving jointly in time and space.

Recursive constructions derived from finite projective geometries generate scalable families of resolvable block designs exhibiting strong algebraic regularity and intrinsic symmetry. In this work, we investigate the structural optimization of recursion depth in such constructions and demonstrate that the combinatorial growth of recursive chains is governed by a quadratic scaling law originating from the asymptotic expansion of Gaussian binomial coefficients. We show that the resulting exponent is strictly concave, which guarantees the existence and uniqueness of an optimal recursion depth. This optimum occurs at the midpoint of the projective dimension and reflects the dual symmetry of the lattice of projective subspaces. To analyze this behavior, we introduce a normalized objective function that compares recursion depths and reveals a unique maximum corresponding to the midpoint of the projective dimension. Theoretical analysis is supported by exact enumeration and asymptotic validation, confirming that the optimal depth is robust to lower-order perturbations and remains invariant under the natural duality of projective geometry. The proposed framework establishes a direct connection between symmetry properties of discrete geometric structures and optimality in nonlinear discrete systems. These results provide a unified perspective on recursive design constructions, revealing that symmetry not only governs combinatorial structure but also induces a mathematically inevitable optimal configuration. The approach opens new directions for studying symmetry-induced optimality in combinatorial geometry, discrete optimization, and related nonlinear mathematical models.

This paper introduces a rigorous class of two-dimensional Markov-switching autoregressive moving-average (2D MS-ARMA) models for spatial lattice data exhibiting regime-dependent dynamics. The switching mechanism is governed by a latent causal Markov random field that drives spatial transitions between regime-specific autoregressive and moving-average structures. We provide sufficient conditions for the existence of a strictly stationary solution through the top Lyapunov exponent associated with a sequence of random matrices obtained from a state-space representation constructed along the lexicographic order. For the first-order bidirectional specification, we derive explicit spectral conditions linking stationarity to the regime-dependent spectral radii. Sufficient conditions ensuring the existence of finite second-order moments are also provided. Parameter estimation is carried out using a variational expectation–maximization (VEM) algorithm based on a mean-field approximation of the posterior distribution of the hidden regimes. The E-step yields closed-form coordinate ascent updates, while the M-step relies on gradient-based numerical optimization with derivatives computed via recursive differentiation. Under increasing-domain asymptotics, we discuss the consistency and asymptotic behavior of the variational estimator. The proposed framework fills a methodological gap between classical one-dimensional Markov-switching ARMA models and spatial autoregressive structures by extending regime-switching theory to multi-indexed processes with rigorous probabilistic foundations. It provides a comprehensive basis for statistical inference, model diagnostics, and prediction in spatially heterogeneous environments.

This paper proposes a recursive method for constructing intra-resolvable balanced incomplete block designs (BIBDs). The approach exploits the algebraic and geometric structure of finite projective geometries over Galois fields to generate resolvable designs with improved efficiency in terms of the number of blocks and treatment replications. The recursive procedure produces symmetric and uniform designs that are particularly suitable for high-dimensional settings. By systematically nesting resolvable blocks, we derive a new class of balanced n-ary designs that are both economical and scalable. These designs hold significant value for the statistical community, offering broad applicability in resource-constrained experimental environments such as precision agriculture, high-throughput drug screening, and computer-based simulation studies. We provide theoretical foundations through explicit constructions and comparative evaluations, demonstrating the advantages of our method over classical approaches.