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We revisit the index leverage effect, that can be decomposed into avolatility effect and a correlation effect. We investigate the latter using amatrix regression analysis, that we call `Principal Regression Analysis' (PRA)and for which we provide some analytical (using Random Matrix Theory) andnumerical benchmarks. We find that downward index trends increase the averagecorrelation between stocks (as measured by the most negative eigenvalue of theconditional correlation matrix), and makes the market mode more uniform. Upwardtrends, on the other hand, also increase the average correlation between stocksbut rotates the corresponding market mode {\it away} from uniformity. There aretwo time scales associated to these effects, a short one on the order of amonth (20 trading days), and a longer time scale on the order of a year. Wealso find indications of a leverage effect for sectorial correlations as well,which reveals itself in the second and third mode of the PRA.

Principal Regression Analysis and the index leverage effect