The Statistical Inference Of Some Characteristics On High Frequency Data

In today’s world, many fields are confronted with increasingly large amounts of data. There have been considerable developments of the statistics of processes observed at very high frequency or ultra high frequency, due to the recent availability of such data. This is particularly the case for market prices of stocks, currencies and other financial instruments. The most hot issues of problems are the question of how to estimate daily volatility and covariance matrix for financial prices.For continuous Ito process, a commonly used estimator is the realized volatility, and it is simple with accuracy rate. However, the estimation of integrated volatility becomes tricky when the underlying price process exists the microstructure noise, including, but not limited to, the existence of the bid-ask spread. When prices are sampled at finer intervals, microstructure issues become more pronounced. The empirical finance then suggests that the bias induced by market microstructure effects makes the most finely sampled data unusable, and many authors prefer to sample over longer time horizons to obtain more reasonable estimates. The sampling length of the typical choices in the literature is ad hoc and usually ranges from5to30minutes for exchange rate data, for instance. If the original data are sampled once every second, say, then retaining an observation every5minutes amounts to discarding299out of every300data points. It is difficult to accept that throwing away data from the statistical standpoint. For high frequency, when the sampling data is increasing, the processes of asset price are realized at discrete, but asynchronous, times.To eliminate the microstructure noise, pre-averaging is now becoming available. To synchronize the observed times, HY is becoming available. Meanwhile, many papers consider the process of the asset price with jump, and to deal with the jumps, one of the approaches is so-called threshold estimator.In this paper, we consider the estimation of the integrated volatility, the co-volatility matrix and the self-weighted cross-volatility, and the test about the driving force of an asset price.(1) We consider the estimation of covariation of two asset prices which contain jumps and microstructure noise, based on high frequency data. We propose a realized covariance estimator, which combines pre-averaging method to remove the mi-crostructure noise and the threshold method to reduce the jumps effect.(2) We attempt to develop a procedure that gives a consistent estimator of the integrated volatility in the presence of jumps. We use the threshold method to get rid of jumps, and the re-sulting estimator turns out to work for infinite activity jumps as well as finite ones.(3) We are concerned with the inference of the integrated self-weighted cross volatility, when two asset prices are sampled with microstructure noise and in an asynchronous way. First, we use the HY method to the asynchronous data. The second, we use the "Pre-Averaging" technique which essentially tries to "clean" the contaminated data by smoothing first and then to apply the usual statistical procedures.(4) There has been extensive literatures in using Ito’s semimartingale driven by a Brownian motion to model the asset prices, interest rates and exchange rates. However, the assumption of Brown-ian motion as a driving force of the underlying asset price processes is rarely contested statistically. The purpose of this part is to develop the test to see whether the driving force is a Brownian motion, in terms of the asymptotic normality of the ratio of two realized power variations with different sampling frequencies. Finally, for all our works, simulations are also included to illustrate the performance of the proposed procedure, and, we also adopted the real data sets to analyze the test.