We have used composite spectra generated from more than 22 000 quasi-stellar objects (QSOs) observed during the course of the 2dF and 6dF QSO Redshift Surveys to investigate the relationship between the velocity width of emission lines and QSO luminosity. We find that the velocity width of the broad emission lines Hβ, Hγ, Mg ii, C iii] and C iv are correlated with the continuum luminosity, with a significance of more than 99 per cent. Of the major narrow emission lines ([O iii]λ5007, [O ii]λ3727, [Ne iii]λ3870 and [Ne v]λ3426) only [O iii] exhibits a significant correlation between linewidth and luminosity. Assuming that the gas is moving in Keplerian orbits and that the radius of the broad line region is related to the QSO continuum luminosity, we use the velocity widths of the broad lines to derive average black hole masses for the QSOs contributing to the composite spectra. The resultant QSO mass–luminosity relationship is consistent with M∝L^0.97±0.16. We find that the correlation between linewidth and redshift, if present, must be weak, and only C iv shows significant evidence of evolution. This enables us to constrain the redshift evolution of the black hole mass–luminosity ratio to be ∼(1 +z)^β with β≲ 1, much less than the ∼(1 +z)³ evolution seen in QSO luminosity evolution. Assuming that the motion of the broad line region gas is Keplerian and that its radius depends on the QSO luminosity, our models indicate that the observed weak redshift dependence is too small for the observed QSO luminosity function to be due to the evolution of a single long-lived population of sources.