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We derive a nonparametric test for constant beta over a fixed time intervalfrom high-frequency observations of a bivariate \Ito semimartingale. Beta isdefined as the ratio of the spot continuous covariation between an asset and arisk factor and the spot continuous variation of the latter. The test is basedon the asymptotic behavior of the covariation between the risk factor and anestimate of the residual component of the asset, that is orthogonal (inmartingale sense) to the risk factor, over blocks with asymptotically shrinkingtime span. Rate optimality of the test over smoothness classes is derived.

Nonparametric test for a constant beta between \Ito semi-martingales based on high-frequency data