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Estimation and exclusion restrictions in clustered linear models

Anna Mikusheva, Mikkel Sølvsten, Baiyun Jing

Abstract

We study linear regression models with clustered data, high-dimensional controls, and intricate exclusion restrictions. We propose a correctly centered internal instrument IV estimator that accommodates a broad class of exclusion restrictions and allows within-cluster dependence. The estimator admits a simple leave-out interpretation and is computationally tractable. We derive a central limit theorem for the associated quadratic form and propose a robust variance estimator. We also develop identification-robust inference procedures. Our framework extends dynamic panel methods to general clustered settings. We illustrate the approach in a large-scale fiscal intervention in rural Kenya, where spatial interference generates the exclusion-restriction pattern.

Estimation and exclusion restrictions in clustered linear models

Abstract

We study linear regression models with clustered data, high-dimensional controls, and intricate exclusion restrictions. We propose a correctly centered internal instrument IV estimator that accommodates a broad class of exclusion restrictions and allows within-cluster dependence. The estimator admits a simple leave-out interpretation and is computationally tractable. We derive a central limit theorem for the associated quadratic form and propose a robust variance estimator. We also develop identification-robust inference procedures. Our framework extends dynamic panel methods to general clustered settings. We illustrate the approach in a large-scale fiscal intervention in rural Kenya, where spatial interference generates the exclusion-restriction pattern.

Paper Structure

This paper contains 27 sections, 11 theorems, 84 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Notation.
  3. Exclusion restrictions with clustered data
  4. Data structure
  5. Exclusion restrictions
  6. Asymptotic bias of OLS
  7. Design based modeling
  8. Correctly centered estimators
  9. Class of estimators
  10. Asymptotic efficiency considerations
  11. Solution to the optimization problem
  12. Price of robustness.
  13. Quantification of uncertainty
  14. Importance of cross-dependence
  15. Special case: block-diagonal structure.
  16. ...and 12 more sections

Key Result

Lemma 1

Suppose assumptions ass: panel model and ass: technical (stated in the Appendix) hold. Then, where $M_{\tilde{\ell}\ell}$ are entries of the $n\times n$ projection matrix $M$, and $Q=\mathop{\mathrm{plim}}\nolimits_{n \rightarrow \infty} \frac{1}{n}\sum_{\ell=1}^n\tilde{x}_{\ell}^2$.

Figures (3)

  • Figure 1: Bias of $\hat{\beta}^{\mathrm{LS}}$ in simulations
  • Figure 2: Estimated effects on household consumption
  • Figure 3: Non-zero structure of $A^*$

Theorems & Definitions (37)

  • Example 1: Unbalanced panel data
  • Example 2: Spatial data
  • Example 3: Network data
  • Example \ref{ex: ex 2}: continued
  • Example \ref{ex: ex 1new}: continued
  • Example \ref{ex: new ex- network}: continued
  • Lemma 1
  • Example 4: Dynamic panel data
  • Example \ref{ex: new ex- network}: continued
  • Lemma 2
  • ...and 27 more