Tractable Estimation of Nonlinear Panels with Interactive Fixed Effects

Interactive fixed effects are routinely controlled for in linear panel models. While an analogous fixed effects (FE) estimator for nonlinear models has been available in the literature (Chen, Fernandez-Val and Weidner, 2021), it sees much more limited use in applied research because its implementation involves solving a high-dimensional non-convex problem. In this paper, we complement the theoretical analysis of Chen, Fernandez-Val and Weidner (2021) by providing a new computationally efficient estimator that is asymptotically equivalent to their estimator. Unlike the previously proposed FE estimator, our estimator avoids solving a high-dimensional optimization problem and can be feasibly computed in large nonlinear panels. Our proposed method involves two steps. In the first step, we convexify the optimization problem using nuclear norm regularization (NNR) and obtain preliminary NNR estimators of the parameters, including the fixed effects. Then, we find the global solution of the original optimization problem using a standard gradient descent method initialized at these preliminary estimates. Thus, in practice, one can simply combine our computationally efficient estimator with the inferential theory provided in Chen, Fernandez-Val and Weidner (2021) to construct confidence intervals and perform hypothesis testing; we also provide an R package for empirical implementation.

💡 Research Summary

The paper addresses a major computational bottleneck in estimating nonlinear panel models with interactive fixed effects (IFE). While Chen, Fernández‑Val and Weidner (2021) introduced a fixed‑effects (FE) maximum‑likelihood estimator that is theoretically sound—consistent and asymptotically normal—it requires solving a high‑dimensional non‑convex optimization problem over the common parameters β and the low‑rank matrices of unit loadings (λ_i) and time factors (γ_t). As N (units) and T (time periods) grow, the dimensionality of (λ,γ) becomes prohibitive, and standard algorithms struggle to find the global optimum.

The authors propose a two‑step estimator that is computationally tractable yet asymptotically equivalent to the original FE estimator.

Step 1 – Convexification via Nuclear‑Norm Regularization (NNR).
The low‑rank constraint ΛΓ′ is relaxed by adding a nuclear‑norm penalty ∥ΛΓ′∥_nuc to the likelihood. This yields a convex problem that can be solved efficiently with proximal‑gradient or ADMM methods. The solution provides preliminary estimates (β̂_NNR, Λ̂_NNR, Γ̂_NNR). Crucially, the authors derive a new error bound for these NNR estimates in the nonlinear IFE setting, relying on a Restricted Strong Convexity (RSC) condition. They translate the abstract RSC into primitive, verifiable assumptions (eigenvalue conditions on covariates, boundedness of errors, etc.) and show that the NNR estimator converges at a rate fast enough to fall inside a shrinking neighbourhood of the true parameters with probability approaching one.

Step 2 – Local Optimization of the Original Non‑Convex Objective.
Using the NNR estimates as starting values, the original log‑likelihood (still non‑convex) is optimized with a standard gradient‑descent algorithm. The authors prove that, within a neighbourhood whose radius shrinks at a specific rate as N,T→∞, the objective function becomes locally convex in all parameters (β, Λ, Γ). Because the NNR estimates lie in this neighbourhood with high probability, the gradient descent converges to the global solution of the original FE problem. Hence the two‑step estimator inherits the same asymptotic distribution as the FE estimator.

Theoretical Contributions.

1. An improved error bound for nuclear‑norm regularized estimators in nonlinear panel models, extending prior results that were limited to linear or single‑index settings.
2. A rigorous demonstration of local convexity in a shrinking neighbourhood, a novel technical device required because the Hessian dimension grows with N and T.
3. Proof of asymptotic equivalence between the two‑step estimator and the FE estimator, implying that all inferential tools (standard errors, confidence intervals, average partial effects) developed in Chen et al. (2021) remain valid.

Practical Implementation.
The paper supplies concrete algorithms for both steps, data‑driven choices of the nuclear‑norm tuning parameter, and a method to estimate the unknown number of factors R (e.g., information‑criterion based). An R package, NNRPanel, implements the full procedure, handling panels up to N,T≈1000–2000 with modest computational resources.

Monte‑Carlo and Empirical Evidence.
Simulation studies across a range of N,T, factor dimensions, and link functions (binary probit/logit, Poisson) show that the two‑step estimator matches the FE estimator in bias, RMSE, and coverage, while reducing computation time by an order of magnitude or more. An empirical re‑analysis of the gravity model of trade from Chen et al. (2021) demonstrates that the new method can estimate a panel with 150 countries over 30 years in minutes, whereas the original FE approach would be infeasible without massive computing power.

Conclusion.
By replacing the high‑dimensional non‑convex optimization with a convex nuclear‑norm relaxation followed by a locally convergent gradient descent, the authors deliver a scalable, theoretically justified estimator for nonlinear panels with interactive fixed effects. This bridges the gap between the sophisticated asymptotic theory of Chen et al. (2021) and the practical needs of applied researchers working with large‑scale panel data.