Estimation of Longrun Variance of Continuous Time Stochastic Process Using Discrete Sample

This paper develops the methodology and asymptotic theory for the estimation of longrun variance of continuous time process. We analyze the asymptotic bias and variance of the longrun variance estimator in continuous time, and provide the optimal bandwidth balancing them o↵ and minimizing the asymptotic mean squared error. In the paper, we present not only how to consistently estimate the longrun variance of continuous time process, but also how to choose bandwidth optimally with data dependent procedures, using discrete samples. Our framework is also useful to analyze the high frequency behaviors of usual longrun variance estimators for discrete time series. In particular, we show that they all diverge to infinity as the sampling frequency increases. The relevance and usefulness of our continuous time framework and asymptotic theory are demonstrated by illustration and simulation. JEL Classification: C13, C22