Identification of Local Treatment Effects Using a Proxy for an Instrument

The method of indirect least squares (ILS) using a proxy for a discrete instrument is shown to identify a weighted average of local treatment effects. The weights are nonnegative if and only if the proxy is intensity preserving for the instrument. A similar result holds for instrumental variables (IV) methods such as two stage least squares. Thus, one should carefully interpret estimates for causal effects obtained via ILS or IV using an error-laden proxy of an instrument, a proxy for an instrument with missing or imputed observations, or a binary proxy for a multivalued instrument. Favorably, the proxy need not satisfy all the assumptions required for the instrument. Specifically, an individual's proxy can depend on others' instrument and the proxy need not affect the treatment nor be exogenous. In special cases such as with binary instrument, ILS using any suitable proxy for an instrument identifies local average treatment effects.