Confidence intervals for intentionally biased estimators

Abstract.

We propose and study three confidence intervals (CIs) centered at an estimator that is intentionally biased to reduce mean squared error. The first CI simply uses an unbiased estimator’s standard error; compared to centering at the unbiased estimator, this CI has higher coverage probability for confidence levels above 91. 7%, even if the biased and unbiased estimators have equal mean squared error. The second CI trades some of this “excess” coverage for shorter length. The third CI is centered at a convex combination of the two estimators to further reduce length. Practically, these CIs apply broadly and are simple to compute.

Keywords:

• Bias–variance tradeoff
• coverage probability
• mean squared error
• smoothing

JEL:

• classification: C13

Acknowledgments

We thank Editor Esfandiar Maasoumi and the anonymous associate editor and referees for their help improving this paper, as well as Alyssa Carlson for feedback on multiple drafts.

Disclosure statement

We (the authors) have no competing interests to declare.

Notes

1. There are multiple algorithms for computing ${\hat{\theta }}_1$, but they all aim to solve for the same unsmoothed solution ${\hat{\theta }}_1$ (up to some smaller-order terms) and are consequently all asymptotically normal; for example, see Chernozhukov and Hansen (2006, Rmk. 3), Chen and Lee (2018, eq. (13)), and Kaido and Wüthrich (2021, Thm. 1, Cor. 2).

2. Thanks to an anonymous reviewer for making this connection and suggesting we consider convex combinations.

3. https://kaplandm.github.io/

4. https://kaplandm.github.io/