In the era of big data, leveraging information from multiple clients while preserving data privacy has emerged as a critical challenge in modern statistical modeling and forecasting. This paper introduces a privacy-preserving federated learning framework for high-dimensional vector autoregressive models, where each client's dynamics are characterized by a common low-rank structure augmented with sparse client-specific deviations. We develop a two-stage estimation procedure that integrates differentially private representation learning for the shared component with local personalization for client-specific adjustments, enabling effective information pooling under selective privacy constraints. Non-asymptotic error bounds are established for both the single-client and federated estimators to characterize the inherent privacy-utility trade-off, and consistency of a ridge-type rank selection criterion is proved. Simulation studies demonstrate that federation substantially improves estimation accuracy when local sample sizes are limited. Two empirical applications to analyzing electricity-economy linkages across U.S. states and conducting multi-task macroeconomic forecasting across countries, highlight the superior predictive accuracy of the proposed method over existing single-client benchmarks.

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