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Abstract

Three chapters are included in this work. The first chapter introduces the remaining chapters. The description of the research chapters (Chapters 2 and 3) can be found in the following paragraphs. In Chapter 2, we consider the problem of estimating high-frequency covariance (quadratic covariation) of two arbitrary assets observed asynchronously. Simple assumptions, such as independence, are usually imposed on the relationship between the prices process and the observation times. In Chapter 2, we introduce a general endogenous two-dimensional nonparametric model. Because an observation is generated whenever an auxiliary process called observation time process hits one of the two boundary processes, it is called the hitting boundary process with time process (HBT) model. We establish a central limit theorem for the Hayashi-Yoshida estimator under HBT in the case where the price process and the observation price process follow a continuous Ito process. We obtain an asymptotic bias. We provide an estimator of the latter as well as a bias-corrected estimator of the high-frequency covariance. In addition, we give a consistent estimator of the associated standard error. In Chapter 3, we show that the techniques used to solve the high-frequency covariance problem can actually be applied to other problems in the high-frequency literature. We give a general time-varying parameter model, where the multidimensional parameter follows a continuous local martingale. As such, we call it the locally parametric model (LPM). The quantity of interest is defined as the integrated value over time of the parameter process.We provide an estimator of this value based on a parametric estimator in the original (non time-varying) parametric model and conditions under which we can show consistency and the corresponding central limit theorem. Since the estimator is obtained by chopping the data into small blocks, then estimating the parameter on each block while pretending it is constant locally and finally taking a block length weighted mean of the estimates on each block, we call it the local parametric estimator (LPE). The class of estimators is very broad, and can contain estimators that are (not too) biased, such as the bias-corrected MLE. We show that the LPM class contains some models that come from popular problems in the high-frequency financial econometrics literature (estimating volatility, high-frequency covariance, integrated betas, leverage effect, volatility of volatility), as well as a new general asset-price diffusion model which allows for endogenous observations and time-varying noise which can be auto-correlated and correlated with the efficient price. Finally, as another example of how to apply the limit theory provided in Chapter 3, we build a time-varying friction parameter extension of the (semi-parametric) model with uncertainty zones (Robert and Rosenbaum (2012)) and we show that we can easily verify the conditions for the estimation of integrated volatility.

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