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We have been spectroscopically monitoring 88 quasars selected to have broadH$\beta$ emission lines offset from their systemic redshift by thousands of kms$^{-1}$. By analogy with single-lined spectroscopic binary stars, we considerthese quasars to be candidates for hosting supermassive black hole binaries(SBHBs). In this work we present new radial velocity measurements, typically3-4 per object over a time period of up to 12 years in the observer's frame. In29/88 of the SBHB candidates no variability of the shape of the broad H$\beta$profile is observed, which allows us to make reliable measurements of radialvelocity changes. Among these, we identify three objects that have displayedsystematic and monotonic velocity changes by several hundred km s$^{-1}$ andare prime targets for further monitoring. Because the periods of thehypothetical binaries are expected to be long, we cannot hope to observe manyorbital cycles during our lifetimes. Instead, we seek to evaluate thecredentials of the SBHB candidates by attempting to rule out the SBHBhypothesis. In this spirit, we present a method for placing a lower limit onthe period, and thus the mass, of the SBHBs under the assumption that thevelocity changes we observe are due to orbital motion. Given the duration ofour monitoring campaign and the uncertainties in the radial velocities, we wereable to place a lower limit on the total mass in the range$4.7\times10^4-3.8\times10^8$ $M_{\scriptscriptstyle \odot}$, which does notyet allow us to rule out the SBHB hypothesis for any candidates.

A Large Systematic Search for Close Supermassive Binary and Rapidly Recoiling Black Holes - III. Radial Velocity Variations