Test for the Relevance of Noise in Observed Data

Abstract

We show that the Truncated Realized Variance (TRV) of a semimartingale asset price converges to zero when observations are contaminated by microstructure noises. Under the additive iid noise assumption, a central limit theorem is also proved. In consequence it is possible to construct a feasible test allowing us to measure the relevance of the noise in the data of a given asset price at a given observation step. For a given observed price path we thus can optimally select the observation frequency at which we can safely use TRV to estimate the efficient price integrated variance IV. The local size of our test is investigated and its performance is veri¯ed on simulated data. A comparison conducted with the Bandi and Russel (2008) and Ait-Sahalia, Mykland and Zhang (2005) mean square error criterions shows that, in order to estimate IV, in many cases we can rely on TRV for lower observation frequencies than previously indicated when using Realized Variance. The advantages of our method are at least two: on the one hand the underlying model for the efficient asset price is less restrictive in that any kind of Ito semimartingale (SM) jump component is allowed. On the other hand our criterion is pathwise, rather than based on an average estimation error, allowing for a more precise estimation of IV because the choice of the optimal frequency is based on the observed path. Further analysis on both simulated and empirical data is conducted in [Lorenzini, 2012].