Abstract

Producing reliable small area estimates remains challenging when survey data are scarce, incomplete, or unevenly distributed across domains. We propose a hierarchical random regression coefficients time two-fold Fay-Herriot model that leverages both temporal correlation and domain-level variability to improve estimation under limited information. The model incorporates random slopes and time-correlated intercepts through an AR(1) structure, enabling more flexible borrowing of strength across areas and years.
Estimation and prediction rely on residual maximum likelihood and empirical best linear unbiased predictors, with practical implementation via Fisher-scoring. Simulation studies demonstrate notable gains in precision and robustness compared with standard two-fold Fay-Herriot models, particularly when domains have small effective sample sizes. We assess three alternative mean squared error estimators, analytic and parametric bootstrap approaches, showing that the analytic estimator offers the best compromise between bias and computational burden.
The methodology is applied to estimate provincial and gender-specific disposable income in Spain (2013-2022), revealing persistent regional disparities and an increasing post-pandemic gender gap. Results underscore the value of methods specifically designed for inference under scarce data.

Acknowledgements
This research is part of the R&D project PID2020-113578RB-I00, funded by MCIN/AEI/10.13039/501100011033/. It has also been supported by the Spanish R&D project PID2023-147127OB-I00 (funded by MICIU/AEI/10.13039/501100011033/ and "FEDER/UE"), and PID2022-136878NB-I00, the Valencian grant CIPROM/2024/34, by the Xunta de Galicia (Competitive Reference Groups ED431C-2024/14) and by CITIC , as a center accredited for excellence within the Galician University System and a member of the CIGUS Network, receives subsidies from the Department of Education, Science, Universities, and Vocational Training of the Xunta de Galicia. Additionally, it is co-financed by the EU through the FEDER Galicia 2021-27 operational program (Ref. ED431G 2023/01). The first author was also sponsored by the Spanish Grant for Predoctoral Research Trainees RD 103/2019 being this work part of grant PRE2021-100857, funded by MCIN/AEI/10.13039/501100011033/ and ESF+. In addition, we thank the Centro de Supercomputación de Galicia (CESGA) for providing their services for part of the simulations in this work.