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Conditional Likelihood Ratio Test with Many Weak Instruments

Sreevidya Ayyar, Yukitoshi Matsushita, Taisuke Otsu

TL;DR

This work extends Moreira's conditional likelihood ratio (CLR) framework to instrumental-variable models with unknown error variance and many weak instruments. It shows that the conventional CLR, when using a plug-in variance estimator, fails to maintain exact similarity under many weak instruments and proposes a Modified CLR (MCLR) that uses a four-statistic likelihood-ratio representation and a robust, data-dependent critical-value function. The authors prove that the MCLR is asymptotically valid under two growth regimes for the number of instruments and provide non-normal-error results under mild conditions, complemented by a comprehensive simulation demonstrating improved size control and competitive power relative to existing robust tests. The approach offers a practical method for inference in IV settings with large instrument sets, preserving the favorable properties of CLR while achieving robustness to many weak instruments. Overall, the MCLR enhances reliability of conditional inference in IV regressions where instrument count is large and identification can be weak, with clear implications for empirical work in macroeconomics and finance.

Abstract

This paper extends validity of the conditional likelihood ratio (CLR) test developed by Moreira (2003) to instrumental variable regression models with unknown error variance and many weak instruments. In this setting, we argue that the conventional CLR test with estimated error variance loses exact similarity and is asymptotically invalid. We propose a modified critical value function for the likelihood ratio (LR) statistic with estimated error variance, and prove that this modified test achieves asymptotic validity under many weak instrument asymptotics. Our critical value function is constructed by representing the LR using four statistics, instead of two as in Moreira (2003). A simulation study illustrates the desirable properties of our test.

Conditional Likelihood Ratio Test with Many Weak Instruments

TL;DR

This work extends Moreira's conditional likelihood ratio (CLR) framework to instrumental-variable models with unknown error variance and many weak instruments. It shows that the conventional CLR, when using a plug-in variance estimator, fails to maintain exact similarity under many weak instruments and proposes a Modified CLR (MCLR) that uses a four-statistic likelihood-ratio representation and a robust, data-dependent critical-value function. The authors prove that the MCLR is asymptotically valid under two growth regimes for the number of instruments and provide non-normal-error results under mild conditions, complemented by a comprehensive simulation demonstrating improved size control and competitive power relative to existing robust tests. The approach offers a practical method for inference in IV settings with large instrument sets, preserving the favorable properties of CLR while achieving robustness to many weak instruments. Overall, the MCLR enhances reliability of conditional inference in IV regressions where instrument count is large and identification can be weak, with clear implications for empirical work in macroeconomics and finance.

Abstract

This paper extends validity of the conditional likelihood ratio (CLR) test developed by Moreira (2003) to instrumental variable regression models with unknown error variance and many weak instruments. In this setting, we argue that the conventional CLR test with estimated error variance loses exact similarity and is asymptotically invalid. We propose a modified critical value function for the likelihood ratio (LR) statistic with estimated error variance, and prove that this modified test achieves asymptotic validity under many weak instrument asymptotics. Our critical value function is constructed by representing the LR using four statistics, instead of two as in Moreira (2003). A simulation study illustrates the desirable properties of our test.
Paper Structure (20 sections, 5 theorems, 47 equations, 2 figures, 2 tables)

This paper contains 20 sections, 5 theorems, 47 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Setup and test statistics
  3. Setup
  4. Likelihood ratio statistic with known $\Omega$
  5. Likelihood ratio statistic with unknown $\Omega$
  6. Conditional likelihood ratio test with many weak instruments
  7. Numerical illustrations
  8. Critical value function
  9. Simulation
  10. Conclusion
  11. Mathematical appendix
  12. Proof of Proposition \ref{['prop:1']}
  13. Proof of Proposition \ref{['prop:2']}
  14. Proof of (i)
  15. Proof of (ii)
  16. ...and 5 more sections

Key Result

Proposition 1

$LR_{1}$ can be written as a function of $n$, $k$, and $(\bar{S}^{\prime}\bar{S},\bar{S}^{\prime}\bar{T},\bar{T}^{\prime}\bar{T},\tilde{S}^{\prime}\tilde{S},\tilde{S}^{\prime}\tilde{T},\tilde{T}^{\prime}\tilde{T})$:

Figures (2)

  • Figure 1: Critical value function of MCLR test with $k=4$
  • Figure 2: Calibrated Power Curves

Theorems & Definitions (7)

  • Proposition 1
  • Proposition 2
  • Remark 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 2