  Given observations from a stationary time series, permutation tests allow one
to construct exactly level $\alpha$ tests under the null hypothesis of an
i.i.d. (or, more generally, exchangeable) distribution. On the other hand, when
the null hypothesis of interest is that the underlying process is an
uncorrelated sequence, permutation tests are not necessarily level $\alpha$,
nor are they approximately level $\alpha$ in large samples. In addition,
permutation tests may have large Type 3, or directional, errors, in which a
two-sided test rejects the null hypothesis and concludes that the first-order
autocorrelation is larger than 0, when in fact it is less than 0. In this
paper, under weak assumptions on the mixing coefficients and moments of the
sequence, we provide a test procedure for which the asymptotic validity of the
permutation test holds, while retaining the exact rejection probability
$\alpha$ in finite samples when the observations are independent and
identically distributed. A Monte Carlo simulation study, comparing the
permutation test to other tests of autocorrelation, is also performed, along
with an empirical example of application to financial data.
