  Measurement error can often be harmful when estimating causal effects. Two
scenarios in which this is the case are in the estimation of (a) the average
treatment effect when confounders are measured with error and (b) the natural
indirect effect when the exposure and/or confounders are measured with error.
Methods adjusting for measurement error typically require external data or
knowledge about the measurement error distribution. Here, we propose
methodology not requiring any such information. Instead, we show that when the
outcome regression is linear in the error-prone variables, consistent
estimation of these causal effects can be recovered using constructed
instrumental variables under certain conditions. These variables, which are
functions of only the observed data, behave like instrumental variables for the
error-prone variables. Using data from a study of the effects of prenatal
exposure to heavy metals on growth and neurodevelopment in Bangladeshi
mother-infant pairs, we apply our methodology to estimate (a) the effect of
lead exposure on birth length while controlling for maternal protein intake,
and (b) lead exposure's role in mediating the effect of maternal protein intake
on birth length. Protein intake is calculated from food journal entries, and is
suspected to be highly prone to measurement error.
