  Gravitational wave astronomy using a pulsar timing array requires
high-quality millisecond pulsars, correctable interstellar propagation delays,
and high-precision measurements of pulse times of arrival. Here we identify
noise in timing residuals that exceeds that predicted for arrival time
estimation for millisecond pulsars observed by the North American Nanohertz
Observatory for Gravitational Waves. We characterize the excess noise using
variance and structure function analyses. We find that 26 out of 37 pulsars
show inconsistencies with a white-noise-only model based on the short timescale
analysis of each pulsar and we demonstrate that the excess noise has a red
power spectrum for 15 pulsars. We also decompose the excess noise into
chromatic (radio-frequency-dependent) and achromatic components. Associating
the achromatic red-noise component with spin noise and including additional
power-spectrum-based estimates from the literature, we estimate a scaling law
in terms of spin parameters (frequency and frequency derivative) and data-span
length and compare it to the scaling law of Shannon \& Cordes (2010). We
briefly discuss our results in terms of detection of gravitational waves at
nanohertz frequencies.
