A new robust Kalman filter for filtering the microstructure noise

We propose a robust Kalman filter (RKF) to estimate the true but hidden return when microstructure noise is present. Following Zhou's definition, we assume the observed return Yt is the result of adding microstructure noise to the true but hidden return Xt. Microstructure noise is assumed to be independent and identically distributed (i.i.d.); it is also independent of Xt. When Xt is sampled from a geometric Brownian motion process to yield Yt, the Kalman filter can produce optimal estimates of Xt from Yt. However, the covariance matrix of microstructure noise and that of Xt must be known for this claim to hold. In practice, neither covariance matrix is known so they must be estimated. Our RKF, in contrast, does not need the covariance matrices as input. Simulation results show that the RKF gives essentially identical estimates to the Kalman filter, which has access to the covariance matrices. As applications, estimated Xt can be used to estimate the volatility of Xt.