  A number of pulsar timing arrays have recently reported preliminary evidence
for the existence of a nanohertz frequency gravitational-wave background. These
analyses rely on detailed noise analyses, which are inherently complex due to
the many astrophysical and instrumental factors that contribute to the pulsar
noise budget. We investigate whether realistic systematic errors, stemming from
misspecified noise models that fail to capture salient features of the pulsar
timing noise, could bias the evidence for gravitational waves. We consider two
plausible forms of misspecification: small unmodeled jumps and unmodeled
chromatic noise. Using simulated data, we calculate the distribution of the
commonly used optimal statistic with no signal present and using plausibly
misspecified noise models. By comparing the optimal statistic distribution with
the distribution created using ``quasi-resampling'' techniques (such as sky
scrambles and phase shifts), we endeavor to determine the extent to which
plausible misspecification might lead to a false positive. The results are
reassuring: we find that quasi-resampling techniques tend to underestimate the
significance of pure-noise datasets. We conclude that recent reported evidence
for a nanohertz gravitational-wave background is likely robust to the most
obvious sources of systematic errors; if anything, the significance of the
signal is potentially underestimated.
