  In this paper, we are interested in testing if the volatility process is
constant or not during a given time span by using high-frequency data with the
presence of jumps and microstructure noise. Based on estimators of integrated
volatility and spot volatility, we propose a nonparametric way to depict the
discrepancy between local variation and global variation. We show that our
proposed test estimator converges to a standard normal distribution if the
volatility is constant, otherwise it diverges to infinity. Simulation studies
verify the theoretical results and show a good finite sample performance of the
test procedure. We also apply our test procedure to do the heteroscedasticity
test for some real high-frequency financial data. We observe that in almost
half of the days tested, the assumption of constant volatility within a day is
violated. And this is due to that the stock prices during opening and closing
periods are highly volatile and account for a relative large proportion of
intraday variation.
