A Novel Grouped-Gram-Based Algorithm for Fast and Memory-Efficient Fixed Effects Estimation

Fixed effects models often rely on the within transformation, which constructs demeaned arrays prior to forming cross-products. This paper develops an estimator that avoids the for- mation of demeaned arrays by exploiting grouped summaries built from per-unit sufficient statistics. A complete derivation shows that the grouped Gram representation reproduces the classical estimator exactly. The difference lies in memory access patterns and byte movement. The grouped estimator concentrates operations into unit-level accumulations, avoiding the writes associated with array centering. Gains arise once the panel reaches a scale where mem- ory traffic governs run time. Simulations examine coefficient accuracy, bootstrap dispersion, run time, and memory use.