Instruments with Heterogeneous Effects: Bias, Monotonicity, and Localness

Abstract In Instrumental Variables (IV) estimation, the effect of an instrument on an endogenous variable may vary across the sample. In this case, IV produces a local average treatment effect (LATE), and if monotonicity does not hold, then no effect of interest is identified. In this paper, I calculate the weighted average of treatment effects that is identified under general first-stage effect heterogeneity, which is generally not the average treatment effect among those affected by the instrument. I then describe a simple set of data-driven approaches to modeling variation in the effect of the instrument. These approaches identify a Super-Local Average Treatment Effect (SLATE) that weights treatment effects by the corresponding instrument effect more heavily than LATE. Even when first-stage heterogeneity is poorly modeled, these approaches considerably reduce the impact of small-sample bias compared to standard IV and unbiased weak-instrument IV methods, and can also make results more robust to violations of monotonicity. In application to a published study with a strong instrument, the preferred approach reduces error by about 19% in small ( N ≈ 1, 000) subsamples, and by about 13% in larger ( N ≈ 33, 000) subsamples.