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Power Bounds and Efficiency Loss for Asymptotically Optimal Tests in IV Regression

Marcelo J. Moreira, Geert Ridder, Mahrad Sharifvaghefi

Abstract

We characterize the maximal attainable power-size gap in overidentified instrumental variables models with heteroskedastic or autocorrelated (HAC) errors. Using total variation distance and Kraft's theorem, we define the decision theoretic frontier of the testing problem. We show that Lagrange multiplier and conditional quasi likelihood ratio tests can have power arbitrarily close to size even when the null and alternative are well separated, because they do not fully exploit the reduced-form likelihood. In contrast, the conditional likelihood ratio (CLR) test uses the full reduced-form likelihood. We prove that the power-size gap of CLR converges to one if and only if the testing problem becomes trivial in total variation distance, so that CLR attains the decision theoretic frontier whenever any test can. An empirical illustration based on Yogo (2004) shows that these failures arise in empirically relevant configurations.

Power Bounds and Efficiency Loss for Asymptotically Optimal Tests in IV Regression

Abstract

We characterize the maximal attainable power-size gap in overidentified instrumental variables models with heteroskedastic or autocorrelated (HAC) errors. Using total variation distance and Kraft's theorem, we define the decision theoretic frontier of the testing problem. We show that Lagrange multiplier and conditional quasi likelihood ratio tests can have power arbitrarily close to size even when the null and alternative are well separated, because they do not fully exploit the reduced-form likelihood. In contrast, the conditional likelihood ratio (CLR) test uses the full reduced-form likelihood. We prove that the power-size gap of CLR converges to one if and only if the testing problem becomes trivial in total variation distance, so that CLR attains the decision theoretic frontier whenever any test can. An empirical illustration based on Yogo (2004) shows that these failures arise in empirically relevant configurations.
Paper Structure (10 sections, 9 theorems, 69 equations, 3 figures, 1 table)

This paper contains 10 sections, 9 theorems, 69 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Model and test statistics
  3. CLR test
  4. LM-based tests and failure under impossibility designs
  5. Benchmark: strong instruments and local alternatives
  6. Fixed alternatives
  7. Impossibility designs
  8. Implications for CQLR
  9. Empirical Application
  10. Conclusion

Key Result

Lemma 1

Let $\mathcal{C}_{0}$, as defined in eq:null_hypothesis, be the convex hull of distributions under the null hypothesis, and let $\mathcal{C}_{1}$, as defined in eq:alt_hypothesis, be the convex hull of distributions under the alternative hypothesis. Consider the total variation distance between $f(\

Figures (3)

  • Figure 1: Power curves for the impossibility design with $\alpha=0.001$, $k=10$, and $\lambda=100$.
  • Figure 2: Power plot for Japan considering $\mu\in\{\widetilde{\mu },0.1309\,\widetilde{\mu}\}$
  • Figure 3: Projections of intersection of $95\%$-level set and impossibility design (Japan)

Theorems & Definitions (9)

  • Lemma 1
  • Lemma 2
  • Proposition 1
  • Theorem 1
  • Proposition 2
  • Theorem 2
  • Proposition 3
  • Corollary 1
  • Proposition 4