This is the third paper in a series describing the spectroscopic properties of a sample of 39 AGN at $z sim 1.5$, selected to cover a large range in black hole mass ($M_{BH}$) and Eddington ratio ($L/L_{Edd}$). In this paper, we continue the analysis of the VLT/X-shooter observations of our sample with the addition of 9 new sources. We use an improved Bayesian procedure, which takes into account intrinsic reddening, and improved $M_{BH}$ estimates, to fit thin accretion disc (AD) models to the observed spectra and constrain the spin parameter ($a_*$) of the central black holes. We can fit 37 out of 39 AGN with the thin AD model, and for those with satisfactory fits, we obtain constraints on the spin parameter of the BHs, with the constraints becoming generally less well defined with decreasing BH mass. Our spin parameter estimates range from $sim$$-$0.6 to maximum spin for our sample, and our results are consistent with the spin-up scenario of BH spin evolution. We also discuss how the results of our analysis vary with the inclusion of non-simultaneous GALEX photometry in our thin AD fitting. Simultaneous spectra covering the rest-frame optical through far-UV are necessary to definitively test the thin AD theory and obtain the best constraints on the spin parameter.
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