Paper # Author Title
This paper studies the use of a discrete instrumental variable to identify the causal effect of an endogenous, mis-measured, binary treatment. We begin by showing that the only existing identification result for this case, which appears in Mahajan (2006), is incorrect. As such, identification in this model remains an open question. We first prove that the treatment effect is unidentified based on conditional first-moment information, regardless of the number of values that the instrument may take. We go on to derive a novel partial identification result based on conditional second moments that can be used to test for the presence of mis-classification and to construct simple and informative bounds for the treatment effect. In certain special cases, we can in fact obtain point identification of the treatment effect based on second moment information alone. When this is not possible, we show that adding conditional third moment information point identifies the treatment effect and the measurement error process. Download Paper
This paper studies the use of a discrete instrumental variable to identify the causal effect of a endogenous, mis-measured, binary treatment. We begin by showing that the only existing identification result for this case, which appears in Mahajan (2006), is incorrect. As such, identification in this model remains an open question. We begin by proving that the treatment effect is unidentified based on conditional first-moment information, regardless of the number of values that the instrument may take. We go on to derive a novel partial identification result based on conditional second moments that can be used to test for the presence of mis-classification and to construct simple and in-formative bounds for the treatment effect. In certain special cases, we can in fact obtain point identification of the treatment effect based on second moment information alone. When this is not possible, we show that adding conditional third moment information point identifies the treatment effect and the measurement error process. Keywords: Instrumental variables, Measurement error, Endogeneity, Binary regressor, Partial Identification JEL Codes: C10, C18, C25, C26 Download Paper
This Version: November 2, 2015, First Version: October 31, 2015 This paper studies the use of a discrete instrumental variable to identify the causal effect of a endogenous, mis-measured, binary treatment in a homogeneous effects model with additively separable errors. We begin by showing that the only existing identification result for this case, which appears in Mahajan (2006), is incorrect. As such, identification in this model remains an open question. We provide a convenient notational framework to address this question and use it to derive a number of results. First, we prove that the treatment effect is unidentified based on conditional first-moment information, regardless of the number of values that the instrument may take. Second, we derive a novel partial identification result based on conditional second moments that can be used to test for the presence of mis-classification and to construct bounds for the treatment effect. In certain special cases, we can in fact obtain point identification of the treatment effect based on second moment information alone. When this is not possible, we show that adding conditional third moment information point identifies the treatment effect and completely characterizes the measurement error process. Keywords: Instrumental variables, Measurement error, Endogeneity, Binary regressor, Partial Identification JEL Codes: C10, C18, C25, C26 Download Paper
The identification of causal effects in linear models relies, explicitly and implicitly, on the imposition of researcher beliefs along several dimensions. Assumptions about measurement error, regressor endogeneity, and instrument validity are three key components of any such empirical exercise. Although in practice researchers reason about these three dimensions independently, we show that measurement error, regressor endogeneity and instrument invalidity are mutually constrained by each other and the data in a manner that is only apparent by writing down the full identified set for the model. The nature of this set makes it clear that researcher beliefs over these objects cannot and indeed should not be independent: there are fewer degrees of freedom than parameters. By failing to take this into account, applied researchers both leave money on the table - by failing to incorporate relevant information in estimation - and more importantly risk reasoning to a contradiction by expressing mutually incompatible beliefs. We propose a Bayesian framework to help researchers elicit their beliefs, explicitly incorporate them into estimation and ensure that they are mutually coherent. We illustrate the practical usefulness of our method by applying it to several well-known papers from the empirical microeconomics literature. Download Paper
The identification of causal effects in linear models relies, explicitly and implicitly, on the imposition of researcher beliefs along several dimensions. Assumptions about measurement error, regressor endogeneity, and instrument validity are three key components of any such empirical exercise. Although in practice researchers reason about these three dimensions independently, we show that measurement error, regressor endogeneity and instrument invalidity are mutually constrained by each other and the data in a manner that is only apparent by writing down the full identified set for the model. The nature of this set makes it clear that researcher beliefs over these objects cannot and indeed should not be independent: there are fewer degrees of freedom than parameters. By failing to take this into account, applied researchers both leave money on the table - by failing to incorporate relevant information in estimation - and more importantly risk reasoning to a contradiction by expressing mutually incompatible beliefs. We propose a Bayesian framework to help researchers elicit their beliefs, explicitly incorporate them into estimation and ensure that they are mutually coherent. We illustrate the practical usefulness of our method by applying it to several well-known papers from the empirical microeconomics literature. Download Paper
Infinite samples, the use of a slightly endogenous but highly relevant instrument can reduce mean-squared error (MSE). Building on this observation, I propose a moment selection criterion for GMM in which moment conditions are chosen based on the MSE of their associated estimators rather than their validity: the focused moment selection criterion (FMSC). I then show how the framework used to derive the FMSC can address the problem of inference post-moment selection. Treating post-selection estimators as a special case of moment-averaging, in which estimators based on different moment sets are given data-dependent weights, I propose a simulation-based procedure to construct valid confidence intervals for a variety of formal and informal moment-selection and averaging procedures. Both the FMSC and confidence interval procedure perform well in simulations. I conclude with an empirical example examining the effect of instrument selection on the estimated relationship between malaria transmission and income. Download Paper
In finite samples, the use of a slightly endogenous but highly relevant instrument can reduce mean-squared error (MSE). Building on this observation, I propose a moment selection criterion for GMM in which moment conditions are chosen based on the MSE of their associated estimators rather than their validity: the focused moment selection criterion (FMSC). I then show how the framework used to derive the FMSC can address the problem of inference post-moment selection. Treating post-selection estimators as a special case of moment-averaging, in which estimators based on different moment sets are given data-dependent weights, I propose a simulation-based procedure to construct valid confidence intervals for a variety of formal and informal moment-selection and averaging procedures. Both the FMSC and confidence interval procedure perform well in simulations. I conclude with an empirical example examining the effect of instrument selection on the estimated relationship between malaria transmission and income. Download Paper
In finite samples, the use of a slightly endogenous but highly relevant instrument can reduce mean-squared error (MSE). Building on this observation, I propose a moment selection criterion for GMM in which moment conditions are chosen based on the MSE of their associated estimators rather than their validity: the focused moment selection criterion (FMSC). I then show how the framework used to derive the FMSC can address the problem of inference post-moment selection. Treating post-selection estimators as a special case of moment-averaging, in which estimators based on different moment sets are given data-dependent weights, I propose a simulation-based procedure to construct valid confidence intervals for a variety of formal and informal moment-selection procedures. Both the FMSC and confidence interval procedure perform well in simulations. I conclude with an empirical example examining the effect of instrument selection on the estimated relationship between malaria transmission and per-capita income. Download Paper