Outlier robust inference in the instrumental variable model with applications to causal effects

Abstract

SummaryThe Anderson‐Rubin (AR) test is an important method that allows for reliable inference in the instrumental variable model when the instruments are weak. Yet, the robustness properties of this test have not been formally studied. As it turns out that the AR test is not robust to outliers, we show how to construct an outlier robust alternative—the robust AR test. We investigate the robustness properties of the robust AR test and show that the robust AR statistic asymptotically follows a chi‐square distribution. The theoretical results are illustrated by a simulation study. Finally, we apply the robust AR test to three different case studies that are affected by different types of outliers.

Outlier Robust Inference in the Instrumental Variable Model With Applications to Causal Effects

“…For more information, data and the construction of the instrument we refer to Ananat (2011). Klooster and Zhelonkin (2024) show that an outlier in the control variable used in the main results of Ananat (2011) inflates the first-stage F -statistic from 1.83 to 19.32. As the outlier was not taken into account in the original study, it was assumed that the instrument was strong.…”

“…In the two-step procedure, the F -test is not robust against outliers (Ronchetti, 1982). Hence, even one outlier is enough to inflate the first-stage F leading to a false impression that the instrument is strong, while it is weak (Klooster and Zhelonkin, 2024), which eventually results in incorrect inference in the second stage. On the other hand, an outlier could also deflate the F -statistic, i.e., the instrument is strong, but due to an outlier it seems weak.…”

“…If the data is cleaned, then one has to correct the inference for the data cleaning step, otherwise the classical asymptotic theory becomes invalid (Chen and Bien, 2020). A typical consequence is underestimation of the (asymptotic) variance which leads to overrejection of the null hypothesis, see Klooster and Zhelonkin (2024) for an example in the IV model. Moreover, in modern applications involving complex structural models outlier detection and its justification are themselves not trivial.…”

“…In the context of IV, several studies showed that classical IV estimators, such as the 2SLS and Limited Information Maximum Likelihood (LIML) estimators, are not robust to outliers and, consequently, introduced robust IV estimators (Cohen Freue et al, 2013;Jiao, 2022;Sølvsten, 2020;Zhelonkin et al, 2012). Klooster and Zhelonkin (2024) developed a robust version of the AR test. However, a general treatment of the problem of (outlier) robust inference in combination with weak instruments is missing.…”