Title: When Moving Average Models Meet High Frequency Data: Uniform Inference on Volatility.
Abstract: In this paper, we propose a general framework of volatility inference with noisy high-frequency data. The observed transaction price follows a continuous-time Ito-semimartingale, contaminated by a discrete-time MA(inf) noise associated with the arrival of trades. Our estimator is obtained by maximizing the likelihood of a misspecified MA model with homoscedastic innovations. We show that this quasi-likelihood estimator is consistent with respect to the quadratic variation of the semimartingale, and that the estimator is asymptotically mixed normal. We propose a AIC/BIC type criterion for order selection, and establish uniformly valid inference on volatility, while allowing for model selection mistakes. In addition, our estimator is adaptive to the presence of the noise, and its convergence rate varies from n^-1/4 to n^-1/2, depending on the magnitude of the noise. We thereby provide uniform inference on volatility over small and large noises. Finally, we present the semiparametric efficiency bound on volatility estimation, from which our estimator deviates slightly. To implement our likelihood estimator, we adopt Kalman filter and a state-space representation, which is tuning-free and warrants a positive estimate in finite sample. In contrast, we show that the classical Whittle approximation is inconsistent under in-fill asymptotics.