  When fitting a particular Economic model on a sample of data, the model may
turn out to be heavily misspecified for some observations. This can happen
because of unmodelled idiosyncratic events, such as an abrupt but short-lived
change in policy. These outliers can significantly alter estimates and
inferences. A robust estimation is desirable to limit their influence. For
skewed data, this induces another bias which can also invalidate the estimation
and inferences. This paper proposes a robust GMM estimator with a simple bias
correction that does not degrade robustness significantly. The paper provides
finite-sample robustness bounds, and asymptotic uniform equivalence with an
oracle that discards all outliers. Consistency and asymptotic normality ensue
from that result. An application to the "Price-Puzzle," which finds inflation
increases when monetary policy tightens, illustrates the concerns and the
method. The proposed estimator finds the intuitive result: tighter monetary
policy leads to a decline in inflation.
