  We investigate the impact of noise processes on high-precision pulsar timing.
Our analysis focuses on the measurability of the second spin frequency
derivative $\ddot{\nu}$. This $\ddot{\nu}$ can be induced by several factors
including the radial velocity of a pulsar. We use Bayesian methods to model the
pulsar times-of-arrival in the presence of red timing noise and dispersion
measure variations, modelling the noise processes as power laws. Using
simulated times-of-arrival that both include red noise, dispersion measure
variations and non-zero $\ddot{\nu}$ values, we find that we are able to
recover the injected $\ddot{\nu}$, even when the noise model used to inject and
recover the input parameters are different. Using simulations, we show that the
measurement uncertainty on $\ddot{\nu}$ decreases with the timing baseline $T$
as $T^\gamma$, where $\gamma=-7/2+\alpha/2$ for power law noise models with
shallow power law indices $\alpha$ ($0<\alpha<4$). For steep power law indices
($\alpha>8$), the measurement uncertainty reduces with $T^{-1/2}$. We applied
this method to times-of-arrival from the European Pulsar Timing Array and the
Parkes Pulsar Timing Array and determined $\ddot{\nu}$ probability density
functions for 49 millisecond pulsars. We find a statistically significant
$\ddot{\nu}$ value for PSR\,B1937+21 and consider possible options for its
origin. Significant (95 per cent C.L.) values for $\ddot{\nu}$ are also
measured for PSRs\,J0621+1002 and J1022+1001, thus future studies should
consider including it in their ephemerides. For binary pulsars with small
orbital eccentricities, like PSR\,J1909$-$3744, extended ELL1 models should be
used to overcome computational issues. The impacts of our results on the
detection of gravitational waves are also discussed.
