🤖 AI Summary
This study addresses a critical limitation of existing integer-valued matrix autoregressive models, which cannot accommodate negative integers and thus fail to model real-world scenarios involving differenced series or financial tick data. To overcome this, the authors propose the Z-MINAR model—the first matrix autoregressive framework defined over the full set of integers, both positive and negative. The approach introduces a signed matrix sparsity operator and innovation terms based on an extended Poisson distribution, thereby preserving the underlying matrix topology. Parameter estimation is achieved via projected conditional least squares. Theoretical analysis establishes the model’s stationarity, causality, and asymptotic normality. Simulations demonstrate that Z-MINAR substantially outperforms existing methods in estimation accuracy, robustness, and adaptability, while empirical application successfully uncovers the dynamic spatiotemporal dependence structure in urban crime count data.
📝 Abstract
Integer-valued time series are ubiquitous in fields such as finance, economics, and epidemiology. As spatiotemporal data structures in these domains grow increasingly complex and high-dimensional, the matrix integer-valued autoregressive (MINAR) model efficiently captures row-column cross-correlations to reduce dimensionality. However, it fundamentally fails to accommodate negative values, which is a critical flaw for analyzing real-world differenced data or financial tick fluctuations. To bridge this theoretical and practical gap, this paper introduces the Z-MINAR model, a novel matrix autoregressive framework defined on the full integer domain (Z). By pioneering a signed matrix thinning operator and utilizing an extended poisson distribution for the innovations, the Z-MINAR model elegantly handles both positive and negative integers while strictly preserving the crucial topological interactions inherent in matrix data. Furthermore, we employ a projection-based conditional least squares estimation procedure and rigorously establish the model's stationarity, causality, and asymptotic normality. Extensive simulations demonstrate the superior estimation accuracy, robustness, and adaptability of Z-MINAR over existing benchmark models. Finally, an empirical application focusing on crime count variations across different urban regions confirms the model's practical efficacy in uncovering dynamic spatiotemporal dependence structures in Z-valued matrix time series.