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We have used X-ray data from the Neutron Star Interior Composition Explorer(NICER) to search for long time-scale, correlated variations ("red noise") inthe pulse times of arrival from the millisecond pulsars PSR J1824$-$2452A andPSR B1937+21. These data more closely track intrinsic noise because X-rays areunaffected by the radio-frequency dependent propagation effects of theinterstellar medium. Our Bayesian search methodology yields strong evidence(natural log Bayes factor of $9.634 \pm 0.016$) for red noise in PSRJ1824$-$2452A, but is inconclusive for PSR B1937+21. In the interest of futureX-ray missions, we devise and implement a method to simulate longer and higherprecision X-ray datasets to determine the timing baseline necessary to detectred noise. We find that the red noise in PSR B1937+21 can be reliably detectedin a 5-year mission with a time-of-arrival (TOA) error of 2 microseconds and anobserving cadence of 20 observations per month compared to the 5 microsecondTOA error and 11 observations per month that NICER currently achieves in PSRB1937+21. We investigate detecting red noise in PSR B1937+21 with othercombinations of observing cadences and TOA errors. We also find that aninjected stochastic gravitational wave background (GWB) with an amplitude of$A_{\rm GWB}=2\times10^{-15}$ and spectral index of $\gamma_{\rm GWB}=13/3$ canbe detected in a pulsar with similar TOA precision to PSR B1937+21, but with noadditional red noise, in a 10-year mission that observes the pulsar 15 timesper month and has an average TOA error of 1 microsecond.

A Detection of Red Noise in PSR J1824$-$2452A and Projections for PSR B1937+21 using NICER X-ray Timing Data