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An Adversarial Approach to Structural Estimation

Tetsuya Kaji, Elena Manresa, Guillaume Pouliot

TL;DR

The paper introduces adversarial estimation, a GAN-inspired, simulation-based method for structural econometric models, casting parameter estimation as a minimax problem between a generator (the structural model) and a discriminator (a classifier distinguishing real versus simulated data). By using a cross-entropy objective and a sufficiently rich discriminator, the estimator achieves parametric efficiency under correct specification and the parametric rate under misspecification, bridging simulated method of moments and maximum likelihood. The authors demonstrate the method on simple models (logistic location and Roy model) and on a substantive empirical application—elderly saving motives—showing that bequest motives matter across wealth levels and that the approach yields tighter inference than traditional SMM. They also discuss practical challenges, including computational cost, loss-surface roughness with neural discriminators, and the role of discrimination richness for efficiency, offering guidance on implementation and bootstrap-based variance estimation. Overall, adversarial estimation provides a flexible, efficient alternative for complex structural models where likelihoods are intractable but simulation is feasible, with direct applicability to policy-relevant questions like saving behavior in retirement.

Abstract

We propose a new simulation-based estimation method, adversarial estimation, for structural models. The estimator is formulated as the solution to a minimax problem between a generator (which generates simulated observations using the structural model) and a discriminator (which classifies whether an observation is simulated). The discriminator maximizes the accuracy of its classification while the generator minimizes it. We show that, with a sufficiently rich discriminator, the adversarial estimator attains parametric efficiency under correct specification and the parametric rate under misspecification. We advocate the use of a neural network as a discriminator that can exploit adaptivity properties and attain fast rates of convergence. We apply our method to the elderly's saving decision model and show that our estimator uncovers the bequest motive as an important source of saving across the wealth distribution, not only for the rich.

An Adversarial Approach to Structural Estimation

TL;DR

The paper introduces adversarial estimation, a GAN-inspired, simulation-based method for structural econometric models, casting parameter estimation as a minimax problem between a generator (the structural model) and a discriminator (a classifier distinguishing real versus simulated data). By using a cross-entropy objective and a sufficiently rich discriminator, the estimator achieves parametric efficiency under correct specification and the parametric rate under misspecification, bridging simulated method of moments and maximum likelihood. The authors demonstrate the method on simple models (logistic location and Roy model) and on a substantive empirical application—elderly saving motives—showing that bequest motives matter across wealth levels and that the approach yields tighter inference than traditional SMM. They also discuss practical challenges, including computational cost, loss-surface roughness with neural discriminators, and the role of discrimination richness for efficiency, offering guidance on implementation and bootstrap-based variance estimation. Overall, adversarial estimation provides a flexible, efficient alternative for complex structural models where likelihoods are intractable but simulation is feasible, with direct applicability to policy-relevant questions like saving behavior in retirement.

Abstract

We propose a new simulation-based estimation method, adversarial estimation, for structural models. The estimator is formulated as the solution to a minimax problem between a generator (which generates simulated observations using the structural model) and a discriminator (which classifies whether an observation is simulated). The discriminator maximizes the accuracy of its classification while the generator minimizes it. We show that, with a sufficiently rich discriminator, the adversarial estimator attains parametric efficiency under correct specification and the parametric rate under misspecification. We advocate the use of a neural network as a discriminator that can exploit adaptivity properties and attain fast rates of convergence. We apply our method to the elderly's saving decision model and show that our estimator uncovers the bequest motive as an important source of saving across the wealth distribution, not only for the rich.

Paper Structure

This paper contains 33 sections, 13 theorems, 88 equations, 11 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Adversarial Estimation Framework
  3. Illustration with Simple Examples
  4. Logistic Location Model
  5. Efficiency
  6. Normality under Misspecification
  7. Comparison with SMM
  8. The Roy Model
  9. Comparison with MLE
  10. Case with Intractable Likelihood
  11. Challenges of the Adversarial Estimator
  12. Statistical Properties
  13. Consistency
  14. Rate of Convergence
  15. On \ref{['asm:neyman']}
  16. ...and 18 more sections

Key Result

Theorem 1

Suppose that for every open $G\subset\Theta$ containing $\theta_0$, we have $\inf_{\theta\notin G}M_\theta(D_\theta)>M_{\theta_0}(D_{\theta_0})$, that $\{\log D_\theta:\theta\in\Theta\}$ and $\{\log(1-D_\theta)\circ T_\theta:\theta\in\Theta\}$ are $P_0$- and $\tilde{P}_0$-Glivenko--Cantelli respecti

Figures (11)

  • Figure 1: The logistic location model. The curvature of oracle and estimated cross-entropy losses matches the log likelihood (\ref{['fig:logistic:ortho']}). This makes the adversarial estimator comparable with MLE (\ref{['fig:logistic:2']}) and as good as the oracle estimator (\ref{['fig:logistic:1']}). The standard errors (se) are multiplied by $\sqrt{n}$. The vertical dots indicate the true parameter $\theta_0$.
  • Figure 2: Use of a neural network discriminator on the logistic location model.
  • Figure 3: The normally-misspecified logistic location model. The adversarial estimator is comparable with quasi-MLE.
  • Figure 4: The logistic location model with increasing numbers of inputs. The curvature of the cross-entropy loss is very close to the log likelihood up to 7 moments and is still good for 11 moments.
  • Figure 5: The logistic location model with increasing numbers of inputs. Precision of the optimally-weighted SMM rapidly deteriorates as the number of moments increases. The adversarial estimator is much less sensitive. The standard errors (se) are multiplied by $\sqrt{n}$.
  • ...and 6 more figures

Theorems & Definitions (33)

  • Theorem 1: Consistency of generator
  • Theorem 2: Rate of convergence of generator
  • Remark
  • Theorem 3: Asymptotic distribution of generator
  • Corollary 4: Efficiency of generator
  • proof : Proof of \ref{['thm:theta:consistency']}
  • proof : Proof of \ref{['thm:5']}
  • proof : Proof of \ref{['thm:theta:dist']}
  • Remark
  • Definition : Bracketing number and bracketing entropy integral
  • ...and 23 more