The Confidence Interval Method for Selecting Valid Instrumental Variables

Frank Windmeijer*, Xiaoran Liang, Fernando P Hartwig, Jack Bowden

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

4 Citations (Scopus)
151 Downloads (Pure)


We propose a new method, the confidence interval (CI) method,
to select valid instruments from a larger set of potential instruments
for instrumental variables (IV) estimation of the causal effect of
an exposure on an outcome. Invalid instruments are such that they
fail the exclusion conditions and enter the model as explanatory variables.
The CI method is based on the confidence intervals of the per instrument
causal effects estimates and selects the largest group with all confidence
intervals overlapping with each other as the set of valid instruments.
Under a plurality rule, we show that the resulting standard IV, or
two-stage least squares (2SLS) estimator has oracle properties. This
result is the same as for the hard thresholding with voting (HT) method
of Guo et al. (2018). Unlike the HT method, the number of
instruments selected as valid by the CI method is guaranteed to be
monotonically decreasing for decreasing values of the tuning parameter.
For the CI method, we can therefore use a downward testing procedure
based on the Sargan (1958) test for overidentifying restrictions
and a main advantage of the CI downward testing method is that it
selects the model with the largest number of instruments selected
as valid that passes the Sargan test.
Original languageEnglish
Pages (from-to)752-776
Number of pages25
JournalJournal of the Royal Statistical Society: Series B
Issue number4
Early online date4 Aug 2021
Publication statusPublished - Sept 2021

Bibliographical note

Funding Information:
Jack Bowden acknowledges support from the Medical Research Council, MC_UU_00011/2, and Xiaoran Liang from the Economic and Social Research Council, ES/P000630/1. The authors thank two referees, an associate editor and the editors, Aurore Delaigle and Simon Wood for their useful comments, which helped to improve the paper.

Publisher Copyright:
© 2021 The Authors. Journal of the Royal Statistical Society: Series B (Statistical Methodology) published by John Wiley & Sons Ltd on behalf of Royal Statistical Society


  • causal inference
  • instrumental variables
  • invalid instruments


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