Testing Missing at Random Using Instrumental Variables

Abstract

This article proposes a test for missing at random (MAR). The MAR assumption is shown to be testable given instrumental variables which are independent of response given potential outcomes. A nonparametric testing procedure based on integrated squared distance is proposed. The statistic’s asymptotic distribution under the MAR hypothesis is derived. In particular, our results can be applied to testing missing completely at random (MCAR). A Monte Carlo study examines finite sample performance of our test statistic. An empirical illustration analyzes the nonresponse mechanism in labor income questions.

KEY WORDS:

• Consistent testing
• Incomplete data
• Instrumental variable
• Missing-data mechanism
• Nonparametric hypothesis testing
• Selection model
• Series estimation

Acknowledgments

The author thanks the joint editor Rajeev Dehejia and two anonymous referees for their comments and suggestions that improved the article. In addition, the article benefited from discussions with Timothy Armstrong, Xiaohong Chen, Stefan Hoderlein, Arthur Lewbel, Enno Mammen, and Peter Robinson and research assistance by Boryana Ilieva. This research was supported by the DFG postdoctoral fellowship BR 4874/1-1. The author is also grateful for support and hospitality of the Cowles Foundation.

Notes

1 Since conditional probabilities/expectations are defined only up to equality a.s., all equalities with conditional probabilities/expectations are understood as equalities a.s., even if we do not say so explicitly.

2 In Germany, a typical age for a graduate student to take up a full-time job is 26 and the average retirement age is close to 63.

3 There are only 11 individuals who do never report their gross labor income level and for those we use average gross labor income for males of the respective profession, in the respective working sector, available on Eurostat for the year 2010. Similarly, we replace nine missing values for actual work time by the average work time with respect to job classification according to ISCO-8 for German male workers in the year 2013.