(a) Data and sensitivity analysis.
We use Eurostat data. This dataset is often argued to contain a number
of errors, and one might be tempted to think this is a reason for our much
lower estimates of real exchange rate persistence.
However, measurement error does not affect our results: The aggregation
bias is pervasive. The two notes here
and here go through the details of the analysis.
Here is also a direct answer to a note
posted by Julian di Giovanni on this issue.
Technically: One needs to be careful with the econometric specification.
The Random Coefficient estimator is a generalization of Random Effects,
the Mean Group estimator generalizes Fixed Effects. They are equivalent
to each other asymptotically. Just as in standard panels, appropriate tests
should be implemented when deciding which estimator to implement. Longer
time series call for a Mean Group estimator, which implies a large and
positive aggregation bias. We also implement a variety of corrections to
the Eurostat data (inclusive of some suggested by Charles Engel himself).
Our results stand in all cases.
(b) A generalization of the proof for the existence of an aggregation
bias.
The analytical proof in the paper assumed cross-sectional independence
of the errors of relative prices. As shown in this
note however, the same result obtains even when allowing for correlation
in the errors. It is only under extreme and unrealistic assumptions on
the cross-sectional dependence that the sign of the bias can reverse. Thanks
are due to Charles Engel for forcing us to develop this more general and
elegant proof.
(c) Is the aggregation bias empirically important?
Estimating the persistence of autoregressive processes in panel data
is always related to two types of biases: The aggregation bias that we
discuss and the Nickell "small-sample" attenuating bias. The question is
which dominates. This first note uses
Monte-Carlo simulations to show the aggregation bias dominates in most
relevant cases. In this more recent note
, we dedicate some time to the treatment of explosive roots at the sectoral
level, and whether their inclusion influences the magnitude of the aggregation
bias. We show they are innocuous: estimates of the aggregation bias when
explosive roots are excluded are almost identical to our original results.
We also point to existing research showing that, in Monte-Carlo simulations,
a (fixed effects) intercept should be included in both the Data Generating
Process (DGP) and the simulated estimates. Including it only in the latter
severely magnifies the small-sample bias, as shown in this
gauss program.
Technically: Explosive processes are innocuous because we estimate
directly the autoregressive parameters, rather than the half lives which
would indeed be infinite. It is important to deal correctly with the intercept
when deriving the properties of the Fixed Effects estimator. The correct
exercise is to allow for fixed effects in both the DGP and the empirical
model. In that case the aggregation bias dominates. Without fixed effects
in the DGP, the Nickell attenuating bias is aggravated and can potentially
dominate. It is also important to note that the aggregation bias survives
even with a large time-series dimension (as in our paper), while the small
sample attenuating bias disappears as the time-series grows.