  We develop a nonparametric test for deciding whether volatility of an asset
follows a standard semimartingale process, with paths of finite quadratic
variation, or a rough process with paths of infinite quadratic variation. The
test utilizes the fact that volatility is rough if and only if volatility
increments are negatively autocorrelated at high frequencies. It is based on
the sample autocovariance of increments of spot volatility estimates computed
from high-frequency asset return data. By showing a feasible CLT for this
statistic under the null hypothesis of semimartingale volatility paths, we
construct a test with fixed asymptotic size and an asymptotic power equal to
one. The test is derived under very general conditions for the data-generating
process. In particular, it is robust to jumps with arbitrary activity and to
the presence of market microstructure noise. In an application of the test to
SPY high-frequency data, we find evidence for rough volatility.
