Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Linear Multidimensional Regression with Interactive Fixed-Effects

Hugo Freeman

TL;DR

This paper extends linear regression to multidimensional panel data with interactive fixed-effects, modeled as $\boldsymbol{Y} = \sum_k \boldsymbol{X}_k β_k^0 + \boldsymbol{\mathcal{A}} + \boldsymbol{ε}$, where $\boldsymbol{\mathcal{A}} = \sum_{\ell=1}^L \varphi^{(1)}_\ell \circ \dots \circ \varphi^{(d)}_\ell$ and $L$ is fixed. It shows how to embed the problem into a two-dimensional panel and apply Bai (2009) factor methods for a slow initial convergence, then introduces a kernel-weighted within transformation to remove interactive fixed-effects at the parametric rate when combined with a double debias procedure, yielding asymptotically normal estimates for $β$. Theoretical results establish consistency and asymptotic normality under both homoskedastic and heteroskedastic errors, with a detailed set of regularity conditions, and simulations illustrate finite-sample performance in growing and fixed sample regimes. An empirical beer demand application demonstrates that a kernel-weighted within transformation yields elasticities comparable to IV estimates but with markedly improved precision, while standard two-dimensional factor approaches are sensitive to how the data are organized. Overall, the paper provides a robust toolkit for inference in high-dimensional interactive fixed-effects models.

Abstract

This paper studies a linear model for multidimensional panel data of three or more dimensions with unobserved interactive fixed-effects. The main estimator uses double debias methods, and requires two preliminary steps. First, the model is embedded within a two-dimensional panel framework where factor model methods in Bai (2009) lead to consistent, but slowly converging, estimates. The second step develops a weighted-within transformation that is robust to multidimensional interactive fixed-effects and achieves the parametric rate of consistency. This is combined with a double debias procedure for asymptotically normal estimates. The methods are implemented to estimate the demand elasticity for beer.

Linear Multidimensional Regression with Interactive Fixed-Effects

TL;DR

This paper extends linear regression to multidimensional panel data with interactive fixed-effects, modeled as , where and is fixed. It shows how to embed the problem into a two-dimensional panel and apply Bai (2009) factor methods for a slow initial convergence, then introduces a kernel-weighted within transformation to remove interactive fixed-effects at the parametric rate when combined with a double debias procedure, yielding asymptotically normal estimates for . Theoretical results establish consistency and asymptotic normality under both homoskedastic and heteroskedastic errors, with a detailed set of regularity conditions, and simulations illustrate finite-sample performance in growing and fixed sample regimes. An empirical beer demand application demonstrates that a kernel-weighted within transformation yields elasticities comparable to IV estimates but with markedly improved precision, while standard two-dimensional factor approaches are sensitive to how the data are organized. Overall, the paper provides a robust toolkit for inference in high-dimensional interactive fixed-effects models.

Abstract

This paper studies a linear model for multidimensional panel data of three or more dimensions with unobserved interactive fixed-effects. The main estimator uses double debias methods, and requires two preliminary steps. First, the model is embedded within a two-dimensional panel framework where factor model methods in Bai (2009) lead to consistent, but slowly converging, estimates. The second step develops a weighted-within transformation that is robust to multidimensional interactive fixed-effects and achieves the parametric rate of consistency. This is combined with a double debias procedure for asymptotically normal estimates. The methods are implemented to estimate the demand elasticity for beer.
Paper Structure (19 sections, 9 theorems, 75 equations, 2 figures, 3 tables)

This paper contains 19 sections, 9 theorems, 75 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Model
  3. Notation and preliminaries
  4. Estimation
  5. Matrix low-rank approximation estimator
  6. Weighted-within estimator
  7. Inference Corrected Estimator
  8. Discussion of estimators
  9. Matrix method results
  10. Estimating kernel weight proxies
  11. Simulations
  12. Growing sample exercise
  13. Fixed sample exercise
  14. Empirical application - demand estimation for beer
  15. Conclusion
  16. ...and 4 more sections

Key Result

Proposition 1

Let $\widehat{\beta}_{(n^*)}^{2D}$ be the estimator from Bai2009 after first flattening along dimension $n^*\in\mathcal{L}$. If Assumptions ass:norms-ass:NonSing hold, the subset $\mathcal{L}$ is non-empty, and the estimated number of factors $\widehat{r}_{n^*}\geq r_{n^*}$, then, for each $n^*\in\m

Figures (2)

  • Figure 1: Bias of $\beta$ estimates with 95% empirical bounds for DGP in \ref{['eqn:simdgp1']}
  • Figure 2: Elasticities over number of factors. Top panel het. SEs, bottom panel HAC SEs

Theorems & Definitions (13)

  • Proposition 1
  • Proposition 2: Upper bound on kernel weighted estimator
  • Corollary 1
  • Theorem 1: Asymptotic distribution under homoskedasticity
  • Theorem 2: Asymptotic distribution under heteroskedasticity
  • Theorem 3: Asymptotic distribution under heteroskedasticity and correlation
  • Lemma 1: Theorem 1 from BaiNg2002
  • proof : Proof of Proposition \ref{['lemma:consKernel']}
  • proof : Proof of Theorem \ref{['thm:AsyNormHetACCC']}
  • Lemma A.1: Verification of Assumption \ref{['ass:regCondKer']}.(ii)
  • ...and 3 more