Inference on the Realized Joint Laplace Transform of Volatility From High‐Frequency Data With Microstructure Noise

ABSTRACT In this article, we introduce a new measure for quantifying the joint variational intensity between two assets, termed the realized joint Laplace transform of volatilities. Taking into account jumps and microstructure noise, we first employ the pre‐averaging method to mitigate the impact of noise. Subsequently, we construct an estimator based on the cosine function of the pre‐averaged increments. The proposed estimator is robust against both jump and microstructure noise; under suitable conditions, we establish its consistency and asymptotic normality. The estimator achieves the optimal convergence rate for estimating integrated volatility functionals in the presence of microstructure noise. Furthermore, based on the established results, we extend our theoretical analysis to independent tests for the volatility processes of different assets. We conduct extensive simulation studies to assess the finite sample performance of our estimator and apply the measure to a real high‐frequency dataset.