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This is the third paper in a series describing the spectroscopic propertiesof a sample of 39 AGN at $z \sim 1.5$, selected to cover a large range in blackhole mass ($M_{BH}$) and Eddington ratio ($L/L_{Edd}$). In this paper, wecontinue the analysis of the VLT/X-shooter observations of our sample with theaddition of 9 new sources. We use an improved Bayesian procedure, which takesinto account intrinsic reddening, and improved $M_{BH}$ estimates, to fit thinaccretion disc (AD) models to the observed spectra and constrain the spinparameter ($a_*$) of the central black holes. We can fit 37 out of 39 AGN withthe thin AD model, and for those with satisfactory fits, we obtain constraintson the spin parameter of the BHs, with the constraints becoming generally lesswell defined with decreasing BH mass. Our spin parameter estimates range from$\sim$$-$0.6 to maximum spin for our sample, and our results are consistentwith the "spin-up" scenario of BH spin evolution. We also discuss how theresults of our analysis vary with the inclusion of non-simultaneous GALEXphotometry in our thin AD fitting. Simultaneous spectra covering the rest-frameoptical through far-UV are necessary to definitively test the thin AD theoryand obtain the best constraints on the spin parameter.

Active galactic nuclei at z ~ 1.5: III. Accretion discs and black hole spin