As WOOL:10;textualplm, Sec. 10.6.3 observes, if the
idiosyncratic errors in the model in levels are uncorrelated (which
we label hypothesis "fe"), then the errors of the model in first
differences (FD) must be serially correlated with
\(cor(\hat{e}_{it}, \hat{e}_{is}) = -0.5\) for each \(t,s\). If
on the contrary the levels model's errors are a random walk, then
there must be no serial correlation in the FD errors (hypothesis
"fd"). Both the fixed effects (FE) and the first--differenced
(FD) estimators remain consistent under either assumption, but the
relative efficiency changes: FE is more efficient under "fe", FD
under "fd".
Wooldridge (ibid.) suggests basing a test for either hypothesis on
a pooled regression of FD residuals on their first lag:
\(\hat{e}_{i,t}=\alpha + \rho \hat{e}_{i,t-1} +
\eta_{i,t}\). Rejecting the restriction \(\rho = -0.5\) makes us
conclude against the null of no serial correlation in errors of the
levels equation ("fe"). The null hypothesis of no serial
correlation in differenced errors ("fd") is tested in a similar
way, but based on the zero restriction on \(\rho\) (\(\rho =
0\)). Rejecting "fe" favours the use of the first--differences
estimator and the contrary, although it is possible that both be
rejected.
pwfdtest estimates the fd model (or takes an fd model as
input for the panelmodel interface) and retrieves its residuals,
then estimates an AR(1) pooling model on them. The test statistic
is obtained by applying a F test to the latter model to test the
relevant restriction on \(\rho\), setting the covariance matrix
to vcovHC with the option method="arellano" to control for
serial correlation.
Unlike the pbgtest and pdwtest, this test does not rely on
large--T asymptotics and has therefore good properties in ''short''
panels. Furthermore, it is robust to general
heteroskedasticity. The "fe" version can be used to test for
error autocorrelation regardless of whether the maintained
specification has fixed or random effects
@see @DRUK:03plm.