research

Working Papers

1. 

   Efficient Estimation of Continuous Treatment Effects

   Fangzhou Yu

   Feb 2026

   Working Paper

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   Estimating causal effects of continuous treatments is challenging because traditional methods rely on unstable estimation of the score of the conditional treatment density. We address this instability by extending the balancing weights framework to continuous treatments. By shifting the analytical primitive to the outcome weight, we guarantee a bounded statistical functional by design and bypass density derivative estimation. Within this framework, we characterize the nonparametric efficiency bound and derive the optimal continuous balancing weights under homoskedasticity and heteroskedasticity. Under homoskedasticity, the optimally efficient estimand coincides with the population projection coefficient in a Partially Linear Regression. Under heteroskedasticity, the optimal estimands center the treatment around a precision-weighted propensity score, generalizing the overlap-weighted effect of categorical treatments. Finally, we derive the efficient influence functions and develop \sqrtn-consistent estimators based on Double/Debiased Machine Learning, providing a semiparametric toolkit for empirical practice. We illustrate the proposed estimators in an application regarding the effect of winning a lottery prize on labor supply.

   @unpublished{yu2026efficient,
     title = {Efficient Estimation of Continuous Treatment Effects},
     author = {Yu, Fangzhou},
     year = {2026},
     month = feb,
     note = {Working Paper},
   }

2. Sensitivity, Informativeness, and Misspecification in GMM Estimation

   Seojeong Lee and Fangzhou Yu

   Apr 2026

   Working Paper

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   This paper develops a sensitivity and informativeness framework for GMM estimators that remains valid under general misspecification of moment conditions. Sensitivity is defined through the conditional expectation of the estimator given the moments and is characterized using the influence functions of the estimator and the moments. Under misspecification, sensitivity must be evaluated at the estimator’s pseudo-true probability limit, and additional sources of variation arising from the Jacobian and from estimated weight matrices become first-order relevant. We derive misspecification-robust sensitivity measures for one-step, two-step and iterated GMM estimators, and introduce an informativeness measure that quantifies the fraction of an estimator’s asymptotic variance explained by sampling variation in the moments themselves. This measure provides a notion of structural efficiency under misspecification that complements standard specification tests. Applications to the automobile demand model of Berry et al. (1995) and the consumption insurance model of Blundell et al. (2008) illustrate that accounting for misspecification can substantially alter sensitivity rankings and reveal large losses in informativeness.

   @unpublished{lee2026sensitivity,
     title = {Sensitivity, Informativeness, and Misspecification in GMM Estimation},
     author = {Lee, Seojeong and Yu, Fangzhou},
     year = {2026},
     month = apr,
     note = {Working Paper},
   }

3. 

   A Variance-Based Test for Heterogeneous Treatment Effects

   Fangzhou Yu

   Feb 2026

   Working Paper

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   This paper proposes a robust nonparametric hypothesis test for the existence of heterogeneous treatment effects. We focus on the variance of the Conditional Average Treatment Effect (CATE) as a natural omnibus parameter, where a non-zero variance implies the presence of relevant heterogeneity. Standard inference for this parameter faces a fundamental theoretical challenge. On one hand, evaluating variance components on the same sample leads to null degeneracy, where the asymptotic variance collapses to zero under the null hypothesis of homogeneity, invalidating standard Gaussian inference. On the other hand, decoupling the empirical processes via standard sample-splitting breaks the Neyman orthogonality of the doubly robust scores due to their nonlinear squared loss, which prevents the cancellation of first-order regularization biases. To resolve this challenge, we propose a novel Intra-Fold Sample-Splitting algorithm. By evaluating variance components on mutually disjoint subsamples while coupling them to identical out-of-fold nuisance estimators, our procedure achieves algebraic cancellation of the nuisance biases. We prove this restores consistency and asymptotic normality, and ensures Type I error control. Monte Carlo simulations demonstrate that the proposed test achieves superior size control against existing tests while maintaining high power. In an empirical application to the NSW job training program, the test detects significant heterogeneity that traditional nonparametric test fails to uncover.

   @unpublished{yu2026variance,
     title = {A Variance-Based Test for Heterogeneous Treatment Effects},
     author = {Yu, Fangzhou},
     year = {2026},
     month = feb,
     note = {Working Paper},
   }