  Observational studies often use linear regression to assess the effect of
ambient air pollution on outcomes of interest, such as human health indicators
or crop yields. Yet pollution datasets are typically noisy and include only a
subset of the potentially relevant pollutants, giving rise to both measurement
error bias (MEB) and omitted variable bias (OVB). While it is well understood
that these biases exist, less is understood about whether these biases tend to
be positive or negative, even though it is sometimes falsely claimed that
measurement error simply biases regression coefficient estimates towards zero.
In this paper, we study the direction of these biases under the realistic
assumptions that the concentrations of different types of air pollutants are
positively correlated with each other and that each type of pollutant has a
nonpositive association with the outcome variable. We demonstrate both
theoretically and using simulations that under these two assumptions, the OVB
will typically be negative and that more often than not the MEB for null
pollutants or for pollutants that are perfectly measured will be negative. We
also use a crop yield and air pollution dataset to show that these biases tend
to be negative in the setting of our motivating application. We do this by
introducing a validation scheme that does not require knowing the true
coefficients. While this paper is motivated by studies assessing the effect of
air pollutants on crop yields, the findings are also relevant to
regression-based studies assessing the effect of air pollutants on human health
outcomes. The validation scheme can also be used to empirically study OVB or
MEB in other contexts.
