DEA-Inspired Constrained Log-Log Quantile Frontier: Smooth Bench-marking, Calibration, and Dynamic Interpretation

Abstract

This paper proposes a Data Envelopment Analysis (DEA)-inspired smooth benchmarking approach based on a constrained log-log quantile production frontier. The frontier is estimated by quantile regression under economically motivated inequality restrictions—monotonicity in inputs and a non-increasing-returns (concavity-compatible) restriction within the Cobb-Douglas class—yielding a continuously differentiable benchmark with elasticity-based interpretation and tractable inference via constrained quantile regression theory. Our contribution is threefold: we (i) formalize large-sample inference for constrained quantile frontiers under economically interpretable inequality restrictions, (ii) introduce a calibration perspective for probabilistic frontiers via exceedance/coverage diagnostics (and a simple non-crossing adjustment for multiple quantiles), and (iii) derive closed-form links between output- and input-oriented quantile efficiency indices through the returns-to-scale parameter, together with a high-quantile interpretation within a one-sided stochastic production model. We emphasize the probabilistic nature of quantile frontiers: for any fixed quantile level τ ∈ (0, 1), the fitted frontier is a coverage benchmark rather than a deterministic envelopment surface, so a non-negligible share of observations may lie above it. Empirically, we apply the framework to firm-level panel data from the SPARK-Interfax information system covering 1,035 Russian manufacturing firms over 2019–2023 (5,175 firm-year observations). We benchmark the proposed smooth quantile frontier against classical DEA and a parametric Stochastic Frontier Analysis (SFA), and we report internal validation (pinball loss, coverage) together with sensitivity diagnostics across τ. The resulting efficiency measures exhibit significant associations with profitability (net Return on Assets (ROA)), changes in profitability, and sales growth, indicating economic relevance. Overall, the proposed approach complements deterministic envelopment by providing smooth differentiability, robustness to noise, and calibration-driven interpretability for heterogeneous datasets, while retaining economically meaningful shape discipline and enabling a dynamic decomposition of frontier shifts and relative performance.