Jackknife Inference for Fixed Effects Models

Abstract: This paper develops a general method of inference for fixed effects models which is (i) automatic, (ii) computationally inexpensive, and (iii) highly model agnostic. Specifically, we show how to combine a collection of subsample estimators into a self-normalised jackknife $t$-statistic, from which hypothesis tests, confidence intervals, and $p$-values are readily obtained.

• The paper introduces a jackknife method that corrects bias in fixed effects models by combining full-sample and structured subsample estimators.
• It constructs a minimum variance unbiased jackknife estimator and a self-normalized t-statistic achieving asymptotically pivotal distributions.
• Simulation studies show superior bias correction and robust coverage compared to traditional analytic and bootstrap methods in complex panels.

Jackknife Inference for Fixed Effects Models: Formal Summary

Motivation and Context

The paper "Jackknife Inference for Fixed Effects Models" (2602.21903) introduces a general framework for statistical inference in fixed effects models, with particular emphasis on procedures that are automatic, computationally efficient, and model-agnostic. The challenge addressed arises from the incidental parameters problem (IPP) endemic to fixed effects models with many nuisance parameters, especially in contemporary settings with multiple dimensions and complex data dependence structures. Classical analytic bias corrections are model-specific, often cumbersome, and do not readily generalize to nonlinear or high-dimensional panels. Bootstrap solutions, while more automatic, require design and computational complexity that can be prohibitive.

Methodological Contributions

The core advancement is the development of a general jackknife method that achieves bias correction and inference for an estimator of interest by systematically combining full-sample and structured subsample estimators into a self-normalized jackknife $t$-statistic. The framework is predicated on two high-level assumptions:

1. The existence of an estimator that, centered and scaled, is asymptotically normal up to unknown deterministic bias terms and variance.
2. The existence of structured subsample estimators whose joint distributions encode the same leading bias terms with known loadings by design.

Given these, the methodology proceeds as follows:

1. Subsample Construction and Bias Representation:

Panel data are partitioned (by cross-section, time, or higher dimensions) into a small number of large, possibly overlapping blocks. Each block produces a subsample estimator. The arrangement is such that the leading bias terms enter estimators in predictable, linear combinations. This enables algebraic identification of bias loadings and facilitates the elimination of bias via recombination.

2. Minimum Variance Unbiased Jackknife (MVUJ) Estimator:

The main estimator is formed as a linear combination of the full-sample and subsample estimators, with weights chosen to annihilate all leading bias terms and to minimize asymptotic variance under the covariance structure implied by the subsample design. In many standard designs, explicit closed-form expressions for these weights can be derived analytically (examples in the paper show this for one-way and two-way fixed effects panels).

3. Self-Normalization and Jackknife $t$-Statistic:

A self-normalized $t$-statistic is constructed using the MVUJ estimator and an appropriately weighted (possibly multi-dimensional) contrast of the subsample estimators. This yields a $t$-distribution in the asymptotic limit, obviating the need for variance estimation or knowledge of the bias parameters.

4. Generalization (Jackknife $t_q$-Statistic):

When designs allow it, multiple orthogonal variance weights can be constructed, leading to self-normalized statistics that converge to $t_q$ distributions with $q > 1$—this directly translates to improved power and shorter confidence intervals.

Theoretical Properties

• The asymptotic theory establishes that the jackknife $t$-statistic is exactly pivotal under broad regularity: it converges to a $t$-distribution with a finite number of degrees of freedom determined by the subsample construction.
• The framework is completely agnostic to model and estimator-specific details beyond the high-level bias and variance structure. All necessary quantities are encoded by the subsample-design matrices, which are deterministic and computable given the partitioning protocol.
• The procedure can be systematically extended to panels of arbitrary dimension, and to estimators with higher-order bias expansions, by recursively constructing subsamples tailored to the bias structure.

Numerical Results

Simulation studies in the paper show that the jackknife estimators achieve:

• Superior bias correction in small to moderate samples compared to both analytic and bootstrap-based approaches. In direct comparison, analytic and bootstrap procedures show sensitivity to tuning parameters, and their coverage is less stable.
• Robust coverage properties for confidence intervals constructed from the proposed jackknife $t$-statistics, especially as sample size grows.
• Competitive or improved efficiency: While jackknife-corrected estimators exhibit modest inflation in standard errors compared to uncorrected estimators (as expected from bias-variance trade-offs), this inflation diminishes with larger samples. The use of multiple variance weights ($t_q$-statistics with larger $q$) enables sharper inference.

These results hold across both linear and nonlinear panel contexts, including models with interactive effects and predetermined regressors.

Technical Implications and Extensions

The framework unifies several disparate "split-panel" and "leave-one-out" methods under a single algebraic structure, grounded in explicit bias and covariance matrices, and generalizes them to higher dimensional, nonlinear, and nonstandard settings. The method is shown to be particularly advantageous in high-dimensional and complex panel environments, where analytic bias corrections are either unknown or infeasible, and where bootstrap reliability is suspect or computationally expensive.

By reducing inference to linear algebraic calculations with a fixed, small number of subsample estimators, the approach remains computationally trivial relative to bootstrap or simulation-based alternatives. Furthermore, the methodology provides detailed recipes for embedding higher-order bias corrections and guiding subsample construction in multi-way panel designs, supporting systematic application in empirical contexts.

Future Directions

The author identifies several avenues for further development:

• Systematic principles for the construction of optimal subsample partitions, potentially minimizing variance subject to model-specific dependence structures.
• Simultaneous inference on multiple restrictions and extension to functionals of parameters (e.g., average partial effects, distributional quantities).
• Exploration of optimal design choices in the presence of complex dependencies (network data, cross-sectional/time-series dependence).

Conclusion

This work establishes a rigorous, flexible, and computationally light approach to valid inference in fixed effects models. It synthesizes recent advances in jackknife theory, generalizes them to modern high-dimensional contexts, and provides both practical implementation strategies and strong theoretical guarantees. For practitioners working with large, complex panels—especially in settings with nonlinear models, interactive or multi-way effects, and nontrivial dependence structures—the techniques of this paper supply a principled alternative to conventional analytic or bootstrap inference.

Reference:

"Jackknife Inference for Fixed Effects Models" (2602.21903).