  This paper proposes a frequency domain approach to test the hypothesis that a
complex-valued vector time series is proper, i.e., for testing whether the
vector time series is uncorrelated with its complex conjugate. If the
hypothesis is rejected, frequency bands causing the rejection will be
identified and might usefully be related to known properties of the physical
processes. The test needs the associated spectral matrix which can be estimated
by multitaper methods using, say, $K$ tapers. Standard asymptotic distributions
for the test statistic are of no use since they would require $K \rightarrow
\infty,$ but, as $K$ increases so does resolution bandwidth which causes
spectral blurring. In many analyses $K$ is necessarily kept small, and hence
our efforts are directed at practical and accurate methodology for hypothesis
testing for small $K.$ Our generalized likelihood ratio statistic combined with
exact cumulant matching gives very accurate rejection percentages and
outperforms other methods. We also prove that the statistic on which the test
is based is comprised of canonical coherencies arising from our complex-valued
vector time series.Our methodology is demonstrated on ocean current data
collected at different depths in the Labrador Sea. Overall this work extends
results on propriety testing for complex-valued vectors to the complex-valued
vector time series setting.
