Bootstrap Inference in Nonlinear Panel Data Models with Interactive Fixed Effects

1. Problem Statement

The paper addresses the incidental parameter problem in nonlinear panel data models featuring interactive fixed effects.

• Context: In models where unobserved heterogeneity is captured by a factor structure (individual effects $\alpha_i$ and time effects $\gamma_t$ interacting as $\alpha_i'\gamma_t$), the Maximum Likelihood Estimator (MLE) for common parameters ($\beta_0$) is biased in finite samples. This bias persists asymptotically as $N, T \to \infty$ (specifically when $T/N \to \kappa$), rendering standard inference invalid.
• Limitations of Existing Methods:
  • Analytical Corrections: Require deriving complex, high-order derivatives of the likelihood function (up to third order), which can be numerically unstable and difficult to implement.
  • Jackknife Methods: While easier to implement, they often suffer from efficiency losses in finite samples and still rely on normal approximations for confidence intervals.
  • Optimization Issues: The log-likelihood function in nonlinear factor models is generally non-concave, making it difficult to guarantee convergence to the global maximizer using standard algorithms.

2. Methodology

The authors propose a parametric bootstrap framework combined with a transformation-based approach to achieve valid inference.

A. Estimation Strategy (Two-Step MLE)

To address the non-concavity of the likelihood function, the authors adopt the nuclear-norm penalized estimator proposed by Zeleneev and Zhang (2026) as a starting point:

1. Step 1: Solve a convex optimization problem using a nuclear-norm penalty to obtain a consistent initial estimator that converges to the global maximizer with probability one.
2. Step 2: Use this initial estimator to start a gradient descent algorithm, ensuring convergence to the global MLE ($\hat{\theta}$) at a low computational cost.

B. Parametric Bootstrap Procedure

The core of the methodology involves generating $B$ bootstrap samples based on the estimated parameters:

1. Data Generating Process (DGP): Generate bootstrap outcomes $Y^*_{it}$ conditional on the original covariates $X_{it}$ and the estimated parameters $\hat{\theta}$:
   $$Y^*_{it} | X_{it}, \hat{\beta}, \hat{\alpha}, \hat{\gamma} \sim f(\cdot | X'_{it}\hat{\beta} + \hat{\alpha}'_i \hat{\gamma}_t)$$
2. Bootstrap Estimation: Compute the MLE ($\hat{\theta}^*$) for each bootstrap sample.
3. Bias Correction: The bias is approximated by the difference between the average of bootstrap estimates and the original estimate:
   $$\hat{\beta}_{boot} = 2\hat{\beta} - \frac{1}{B}\sum_{b=1}^B \hat{\beta}^*_b$$
4. Confidence Intervals: Construct intervals directly from the quantiles of the bootstrap distribution ($\hat{\beta}^* - \hat{\beta}$) rather than relying on asymptotic normality.

C. Transformation-Based Skewness Correction

The authors observe that standard bootstrap intervals can be overly conservative (over-covering) in finite samples due to skewness in the estimator's distribution. To mitigate this:

• They apply a monotone transformation $\phi(\cdot)$ (e.g., Log, Box-Cox, or Yeo-Johnson) to the estimator before constructing the interval.
• The interval is constructed on the transformed scale and mapped back:
  $$[\phi^{-1}(2\phi(\hat{\beta}) - Q^*_{\phi(\hat{\beta}^*)}(1-\alpha/2)), \dots]$$
• This reduces skewness, yielding tighter intervals with coverage probabilities closer to the nominal level.

3. Key Theoretical Contributions

The paper establishes rigorous asymptotic theory under a regime where $N, T \to \infty$ with $T/N \to \kappa \in (0, \infty)$:

• Uniform Asymptotic Normality: The authors prove that the MLE admits a uniform asymptotic expansion in a neighborhood of the true parameter, extending previous results by Chen et al. (2021) and Higgins and Jochmans (2024).
• Bootstrap Consistency: They demonstrate that the parametric bootstrap distribution of $\sqrt{NT}(\hat{\beta}^* - \hat{\beta})$ converges weakly to the true sampling distribution of $\sqrt{NT}(\hat{\beta} - \beta_0)$.
• Validity without Explicit Bias Correction: Crucially, the bootstrap yields asymptotically valid confidence intervals even without explicit bias correction, as the bootstrap distribution inherently replicates the bias structure of the original estimator.
• Extension to Average Partial Effects (APEs): The theory is extended to cover APEs, showing that the bootstrap consistently approximates their distribution.

4. Simulation Results

The authors conduct extensive Monte Carlo simulations (Logit and Probit models) across various scenarios (covariates independent/correlated with factors, outliers, sparse spikes).

• Bias Reduction: The bootstrap method (using mean or median correction) consistently outperforms analytical and split-panel jackknife corrections in reducing bias, particularly when $T$ is small (e.g., $T=20$).
• Coverage Accuracy:
  • Standard bootstrap intervals tend to be conservative (coverage > 95%).
  • Transformation-based intervals (specifically Yeo-Johnson and Box-Cox) significantly improve performance by reducing interval length and correcting the over-coverage, achieving coverage rates much closer to the nominal 95% level.
• Robustness: The method performs well even when covariates are correlated with unobserved factors, a scenario where analytical corrections often fail.

5. Empirical Application

The method is applied to a dataset of 328 U.S. firms (1981–2001) to study innovation behavior (patenting) in the presence of latent heterogeneity.

• Objective: Estimate the effects of technological spillovers (knowledge diffusion) and product market spillovers (business stealing) on patenting.
• Findings:
  • Technological Spillovers: Positive and statistically significant across all methods. However, the analytical correction overestimates the magnitude compared to the bootstrap.
  • Product Market Spillovers: Statistically insignificant, suggesting competitive pressures do not directly drive patenting in this context.
  • Robustness: The bootstrap results confirm the qualitative conclusions of previous studies (e.g., Burda and Harding, 2013) while providing more reliable standard errors and bias-corrected estimates.

6. Significance

• Practical Utility: Provides a computationally feasible and robust alternative to complex analytical bias corrections for nonlinear panel models with interactive fixed effects.
• Finite-Sample Performance: Demonstrates superior performance in "small $T$" settings common in microeconomic data, where traditional asymptotic approximations fail.
• Innovation in Inference: The introduction of monotone transformations to correct bootstrap skewness offers a practical solution to the issue of conservative inference in finite samples.
• Computational Efficiency: By integrating the nuclear-norm penalized estimator, the paper solves the global maximization problem in non-concave likelihoods, making the bootstrap procedure viable for large-scale empirical applications.