Robust Inference for Convex Pairwise Difference Estimators

Technical Summary: Robust Inference for Convex Pairwise Difference Estimators

Problem Statement
This paper addresses the inference challenges associated with a broad class of convex pairwise difference estimators, defined as minimizers of a kernel-weighted convex objective function over observation pairs with similar covariates. These estimators are widely used to address heterogeneity in nonlinear models (e.g., partially linear regression, logit, and Tobit models).

The central difficulty lies in the dependence of the estimator's large-sample behavior on the bandwidth (localization) parameter, $h_n$. Classical inference relies on restrictive bandwidth conditions: specifically, that $n h_n^d \to \infty$ (ensuring sufficient pairs for asymptotic linearity) and $n h_n^4 \to 0$ (ensuring bias is negligible). Under these conditions, the estimator is asymptotically linear and standard nonparametric bootstrap inference is consistent. However, in practice, selecting a bandwidth that satisfies both conditions simultaneously is difficult, particularly as the dimension $d$ of the covariate vector increases. Existing methods often fail to provide valid inference when the bandwidth is too small (violating the linearity condition) or too large (violating the bias condition).

Methodology
The authors develop a novel inference framework that combines three distinct theoretical ideas to achieve robustness across a wider range of bandwidth choices:

1. Small Bandwidth Asymptotics: Extending the framework of Cattaneo, Crump, and Jansson (2014a), the paper derives Gaussian distributional approximations for the estimator without imposing the condition $n h_n^d \to \infty$. Instead, it establishes validity under the weaker condition $n^2 h_n^d \to \infty$. This approach accounts for the additional variance induced by high localization, resulting in a convergence rate and variance structure that explicitly depend on the degree of localization.
2. Debiasing via Generalized Jackknifing: To address bias when $h_n$ is large (violating $n h_n^4 \to 0$), the authors employ a debiasing procedure based on generalized jackknifing (Schucany and Sommers, 1977). This involves constructing a linear combination of estimators computed with different bandwidths. Crucially, unlike using higher-order kernels (which would introduce negative weights and destroy the convexity of the objective function), the generalized jackknife preserves the convexity of the estimator. This allows for the use of larger bandwidths while maintaining the computational and theoretical benefits of convex optimization.
3. Bootstrap Rescaling: The paper identifies that the standard nonparametric bootstrap is inconsistent under small bandwidth asymptotics because it fails to capture the specific variance inflation caused by localization. The authors propose a corrected bootstrap procedure where the localization parameter used in the bootstrap resampling is rescaled by a factor of $3^{1/d}$. This adjustment aligns the bootstrap variance with the true asymptotic variance, ensuring consistency even when $n h_n^d$ is bounded.

Key Contributions and Results
The paper provides a comprehensive distribution theory and a practical inference method that unifies these three components.

• Generalized Gaussian Approximation: Theorem 1 establishes that under $n^2 h_n^d \to \infty$, the estimator converges to a Gaussian distribution. The asymptotic variance includes a term reflecting the "small bandwidth" uncertainty, which vanishes only if $n h_n^d \to \infty$. This result covers three regimes: asymptotic linearity ($n h_n^d \to \infty$), root-$n$ consistency without linearity ($n h_n^d \to c \in (0, \infty)$), and slower-than-root-$n$ consistency ($n h_n^d \to 0$ but $n^2 h_n^d \to \infty$).
• Debiased Estimator Properties: Theorem 2 extends the Gaussian approximation to the debiased estimator. It shows that by combining estimators with bandwidths $h_n, c_1 h_n, \dots, c_L h_n$ using specific weights, one can eliminate bias terms up to order $h_n^{2L+2}$. This allows for valid inference with bandwidths that violate the standard $n h_n^4 \to 0$ condition, provided $n h_n^{4(L+1)} \to 0$.
• Valid Bootstrap Inference: Theorem 3 and Corollary 1 demonstrate that the standard bootstrap is inconsistent under small bandwidth asymptotics due to variance mismatch. However, by rescaling the bandwidth in the bootstrap step to $3^{1/d} h_n$, the method yields a consistent distributional approximation.
• Robust Coverage: The paper compares four inference procedures: (1) Classical, (2) Classical Debiased, (3) Small Bandwidth, and (4) Small Bandwidth Debiased. Theoretical analysis and simulation evidence (Section 6) show that the combination of debiasing and bandwidth rescaling (Procedure 4) provides the most robust coverage probabilities across the widest range of bandwidth choices. It mitigates the conservativeness of the standard bootstrap at small bandwidths and the undercoverage due to bias at large bandwidths.

Significance
The paper claims to offer a "demonstrably more robust" inference method for convex pairwise difference estimators compared to existing approaches. Its primary significance lies in:

1. Relaxing Bandwidth Restrictions: It removes the necessity of the restrictive $n h_n^d \to \infty$ condition for valid Gaussian approximation, allowing for higher levels of localization.
2. Preserving Convexity: By utilizing generalized jackknifing rather than higher-order kernels, the method retains the convexity of the objective function, which is essential for fast computation and weaker regularity conditions.
3. Practical Robustness: The proposed method (combining debiasing and rescaling) ensures that confidence intervals maintain nominal coverage levels even when the bandwidth choice is uncertain or data-driven, a common scenario in empirical work.

The authors note that their theoretical development relies heavily on the convexity of the objective function and the foundational work of Hjort and Pollard (1993) and Pollard (1991) applied to U-processes. The paper concludes by suggesting future extensions to higher-order stochastic expansions for bandwidth selection, consistent variance estimators, and estimators based on generated regressors.