Identifying the effect of a mis-classified, binary, endogenous regressor

Abstract

This paper studies identification of the effect of a mis-classified, binary, endogenous regressor when a discrete-valued instrumental variable is available. We begin by showing that the only existing point identification result for this model is incorrect. We go on to derive the sharp identified set under mean independence assumptions for the instrument and measurement error. The resulting bounds are novel and informative, but fail to point identify the effect of interest. This motivates us to consider alternative and slightly stronger assumptions: we show that adding second and third moment independence assumptions suffices to identify the model.

LATE With Missing or Mismeasured Treatment

“…Battistin et al (2014) use two measures of the misclassified treatment to point-identify LATE, but require re-survey data (i.e., multiple observations of the same individuals). DiTraglia and García-Jimeno (2019) and Yanagi (2018) also obtain point-identification of LATE with mismeasured treatment. The former, however, requires that treatment effects be homogeneous, while the latter requires the availability of two instrumental variables with specific properties, one for the endogenous treatment and the other to deal with the measurement error.…”

Identification of Regression Models With a Misclassified and Endogenous Binary Regressor

“…Assumption 1(f) requires that T * changes the mean of T and corresponds to Assumption 2 in Mahajan (2006) and Assumption 2.2(ii) in DiTraglia and García-Jimeno (2019). Assumption 1(f) holds when Pr(T = 1|T * = 1,Z) > 1/2 and Pr(T = 0|T * = 0) > 1/2, i.e., the value of T * is informative on the value of T.…”

Correcting for Misclassied Binary Regressors Using Instrumental Variables

“…Assumption 1.ii is the usual instrumental variable exclusion condition and Assumption 1.iii is the usual instrumental variable relevance condition. The last two parts of Assumption 1 are standard in the prior literature (e.g., Frazis and Loewenstein 2003;Mahajan 2006;Lewbel 2007;DiTraglia and Garcia-Jimeno 2019). Assumption 1.iv states that there is no additional independent information contained in the mis-measured regressor once the true value of the regressor is known, which is often referred to as "non-differential" measurement error.…”

“…To address this issue, a number of empirical strategies have been developed to consistently estimate the impact of a misclassified binary regressor, many of which make use of an instrumental variable (e.g. , Card 1996;Kane, Rouse, and Staiger 1999;Black, Berger, and Scott 2000;Frazis and Lowenstein 2003;Mahajan 2006;Lewbel 2007;Hu 2008;Chen, Hu, and Lewbel 2008a,b;Battistin, De Nadai, and Sianesi 2014;DiTraglia and Garcia-Jimeno 2019;Calvi, Lewbel, and Tomassi 2017;Yanagi 2019).…”

“…Recent papers require both an instrumental variable and at least one additional variable which provides information on the measurement error in order to achieve identification (Calvi, Lewbel, and Tomassi 2017;Yanagi 2019), whereas our approach only requires a single instrumental variable. Several other papers have utilized restrictions on higher order moments to achieve identification (e.g., Chen, Hu, and Lewbel 2008a,b;DiTraglia and Garcia-Jimeno 2019), but these approaches introduce arguably strong assumptions on the distribution of the errors which we are able to circumvent. 5 Finally, we contribute a new lower bound on the parameter of interest, following a long tradition in this literature (Aigner 1973;Bollinger 1996;Black, Berger, and Scott 2000;Kreider et al 2012;Ura 2018).…”