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A two-step approach to production frontier estimation and the Matsuoka's distribution

Danilo Hiroshi Matsuoka, Guilherme Pumi, Hudson da Silva Torrent, Marcio valk

TL;DR

This work develops a deterministic production frontier model in which efficiency follows the one-parameter Matsuoka distribution on $(0,1)$. It introduces a two-step semiparametric estimator: a nonparametric regression component to recover the frontier and a parametric step to estimate the efficiency parameter $p$ via a moments-based approach, followed by a plug-in frontier reconstruction. The authors establish convergence rates, asymptotic normality, and an oracle property for the parametric estimator, with extensive simulations and an empirical application to Brazilian temporary crops showing CES-like isoquants and a significant regional efficiency difference. The framework offers a parsimonious yet flexible route to frontier estimation, with practical relevance for regional productivity analysis and policy evaluation.

Abstract

In this work, we introduce a deterministic frontier model in which efficiency is governed by the Matsuoka distribution, a parsimonious one-parameter specification on $(0,1)$ designed to reflect patterns typically observed in efficiency data. Based on this formulation, we develop a two-step semiparametric estimation procedure: a nonparametric smoothing for the regression component, followed by a feasible method of moments estimation for the efficiency parameter with plug-in reconstruction of the frontier. Theoretical results establish convergence rates, asymptotic normality, and an oracle property for the parametric estimator of the efficiency parameter. A Monte Carlo study demonstrates that the procedure performs consistently with the theoretical results and improves upon a fully nonparametric alternative. Applying the method to Brazilian temporary crops with land and agrochemicals as inputs, we find that both regions exhibit isoquants close to the constant elasticity substitution form, but differ in the relative productivity of inputs. Most notably, statistical tests provide evidence that the South is relatively more efficient than the Center-West, highlighting the empirical relevance of the proposed approach.

A two-step approach to production frontier estimation and the Matsuoka's distribution

TL;DR

This work develops a deterministic production frontier model in which efficiency follows the one-parameter Matsuoka distribution on . It introduces a two-step semiparametric estimator: a nonparametric regression component to recover the frontier and a parametric step to estimate the efficiency parameter via a moments-based approach, followed by a plug-in frontier reconstruction. The authors establish convergence rates, asymptotic normality, and an oracle property for the parametric estimator, with extensive simulations and an empirical application to Brazilian temporary crops showing CES-like isoquants and a significant regional efficiency difference. The framework offers a parsimonious yet flexible route to frontier estimation, with practical relevance for regional productivity analysis and policy evaluation.

Abstract

In this work, we introduce a deterministic frontier model in which efficiency is governed by the Matsuoka distribution, a parsimonious one-parameter specification on designed to reflect patterns typically observed in efficiency data. Based on this formulation, we develop a two-step semiparametric estimation procedure: a nonparametric smoothing for the regression component, followed by a feasible method of moments estimation for the efficiency parameter with plug-in reconstruction of the frontier. Theoretical results establish convergence rates, asymptotic normality, and an oracle property for the parametric estimator of the efficiency parameter. A Monte Carlo study demonstrates that the procedure performs consistently with the theoretical results and improves upon a fully nonparametric alternative. Applying the method to Brazilian temporary crops with land and agrochemicals as inputs, we find that both regions exhibit isoquants close to the constant elasticity substitution form, but differ in the relative productivity of inputs. Most notably, statistical tests provide evidence that the South is relatively more efficient than the Center-West, highlighting the empirical relevance of the proposed approach.
Paper Structure (12 sections, 8 theorems, 167 equations, 6 figures, 3 tables)

This paper contains 12 sections, 8 theorems, 167 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Production Frontier Model
  3. Probability Distribution of the Efficiency
  4. Proposed two-step estimation
  5. Step 1: estimation of the regression function $g$
  6. Step 2: estimation of the parameter $p$ and plug-in frontier
  7. Plug-in frontier reconstruction.
  8. Asymptotic theory
  9. Estimating the efficiency with order statistics
  10. Simulations
  11. Application to agricultural data
  12. Conclusion and discussion

Key Result

Theorem 3.1

Suppose that A1- A5 hold. Then If in addition the second derivative $g"(\cdot)$ is a continuous function and $nh^3\to\infty$, then for all points $x$ in the interior of $\mathrm{supp}(f_X)$ for which the function $H(y)\coloneqq \mathds{E}(Z^4|X=y)$ is bounded on a neighborhood of $x$, we have that where

Figures (6)

  • Figure 1: Simulated data for $p=1$ and $p=8$.
  • Figure 2: Boxplots of the estimated values $\hat{p}$ for the univariate case.
  • Figure 3: Boxplots of the values $\mathrm{ASE}_r(\hat{g})$ and $\mathrm{ASE}_r(\hat{f})$ for the univariate case.
  • Figure 4: Contour lines of the estimated production functions $\hat{f}_s$ and $f_c$.
  • Figure 5: Observed Chi-square statistics for different values for the parameter $p$.
  • ...and 1 more figures

Theorems & Definitions (8)

  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Theorem 3.5
  • Lemma 3.1
  • Theorem 3.6
  • Theorem 3.7