Benchmarking a HAR(1) Fay-Herriot Model for Estimating Labour Indicators from the Spanish Quarterly Labour Cost Survey

Abstract

[Abstract]: We develop a scalable multivariate small-area framework to produce publication-grade labour-cost indicators from Spain’s Quarterly Labour Cost Survey (QLCS) over fine domains defined by Autonomous Communities x 2-digit NACE divisions x three enterprise-size classes. A bivariate Fay-Herriot model with a heteroscedastic autoregressive HAR(1) structure jointly links each cost component to its natural “base” (labour cost with workers; wage cost with effective hours), exploiting cross-equation dependence while remaining parsimonious and stable. Estimation proceeds via REML; uncertainty is quantified with closed-form MSE expressions specific to the HAR(1) setting, reported as coefficients of variation (CV). To ensure tractability at national scale, we introduce a Divide-et-Impera strategy: models are fitted in parallel by Autonomous Community and optimally recombined using Fisher-information weights, delivering large speed-ups with negligible efficiency loss. For institutional coherence, we implement a fast triple-conformity benchmarking heuristic with positivity constraints, aligning model-based totals with official aggregates across enterprise size, industry and territory; ratios are reconstructed from benchmarked numerators and denominators. Using 2024Q2 data and auxiliary covariates, the approach substantially stabilises estimates: median CVs are ≤ 20% and seldom exceed 40%, meeting international dissemination standards. The pipeline, HAR(1) BFH + Divide-et-Impera + positivity-preserving benchmarking, offers a reproducible, policy-ready solution for releasing coherent labour- and wage-cost indicators at granular levels.