We derive a nonparametric test for constant (continuous) beta over a fixed interval of time. Continuous beta is defined as the ratio of the continuous covariation between an asset and observable risk factor (e.g., the market return) and the continuous variation of the latter. Our test is based on discrete observations of a bivariate Itsemimartingale with mesh of the observation grid shrinking to zero. We rst form a consistent and asymptotically mixed normal estimate of beta using all the observations within the time interval under the null hypothesis that beta is constant. Using it we form an estimate of the residual component of the asset returns that is orthogonal (in martingale sense) to the risk factor. Our test is then based on the distinctive asymptotic behavior, under the null and alternative hypothesis, of the sample covariation between the risk factor and the estimated residual component of the asset returns over blocks with asymptotically shrinking time span. Optimality of the test is considered as well. We document satisfactory finite sample properties of the test on simulated data. In an empirical application based on 10-minute data we analyze the time variation in market betas of four assets over the period 2006-2012. The results suggest that (for likely structural reasons) for one of the assets there is statistically nontrivial variation in market beta even for a period as short as a week. On the other hand, for the rest of the assets in our analysis we find evidence that a window of constant beta of one week to one month is statistically plausible.
nonparametric tests time-varying beta stochastic volatility high-frequency data