It is standard practice in applied work to study the effect of a binary variable ("treatment") on an outcome of interest using linear models with additive effects. In this paper I study the interpretation of the ordinary and two-stage least squares estimands in such models when treatment effects are in fact heterogeneous. I show that in both cases the coefficient on treatment is identical to a convex combination of two other parameters(different for OLS and 2SLS), which can be interpreted as the average treatment effects on the treated and controls under additional assumptions. Importantly, the OLS and 2SLS weights on these parameters are inversely related to the proportion of  each group. The more units get treatment, the less weight is placed on the effect on the treated. What follows, the reliance on these implicit weights can have serious consequences for applied work . I illustrate some of these issues in four empirical applicationsfrom different fields of economics. I also develop a weighted least squares correction and simple diagnostic tools that applied researchers can use to avoid potential biases  . In an important special case, my diagnostics only require the knowledge of the proportion of treated units.
Applied work often studies the effect of a binary variable ("treatment") using linear models with additive effects. I study the interpretation of the OLS estimands in such models when treatment effects are heterogeneous. I show that the treatment coefficient is a convex combination of two parameters, which under certain conditions can be interpreted as the average treatment effects on the treated and untreated. The weights on these parameters are inversely related to the proportion of observations in each group. Reliance on these implicit weights can have serious consequences for applied work , as I illustrate with two well-known applications. I develop simple diagnostic tools that empirical researchers can use to avoid potential biases . Software for implementing these methods is available in R and Stata . In an important special case, my diagnostics only require the knowledge of the proportion of treated units.
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It is standard practice in applied work to study
Applied work often studies
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on an outcome of interest
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In this paper
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ordinary and two-stage least squares
OLS
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in fact
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in both cases the coefficient on treatment is identical to
the treatment coefficient is
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other parameters(different for OLS and 2SLS), which
parameters, which under certain conditions
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controls under additional assumptions. Importantly, the OLS and 2SLS
untreated. The
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observations in
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The more units get treatment, the less weight is placed on the effect on the treated. What follows, the reliance on
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. I illustrate some of these issues in four empirical applicationsfrom different fields of economics. I also develop a weighted least squares correction and
, as I illustrate with two well-known applications. I develop
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applied
empirical
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. Software for implementing these methods is available in R and Stata