Yazhen Wang

Multi-Scale Jump and Volatility Analysis for High-Frequency Financial Data

The wide availability of high-frequency data for many financial instruments stimulates an upsurge interest in statistical research on the estimation of volatility. Jump-diffusion processes observed with market microstructure noise are frequently used to model high-frequency financial data. Yet existing methods are developed for either noisy data from a continuous-diffusion price model or data from a jump-diffusion price model without noise. We propose methods to cope with both jumps in the price and market microstructure noise in the observed data. These methods allow us to estimate both integrated volatility and jump variation from the data sampled from jump-diffusion price processes, contaminated with the market microstructure noise. Our approach is to first remove jumps from the data and then apply noise-resistant methods to estimate the integrated volatility. The asymptotic analysis and the simulation study reveal that the proposed wavelet methods can successfully remove the jumps in the price processes and the integrated volatility can be estimated as accurately as in the case with no presence of jumps in the price processes. In addition, they have outstanding statistical efficiency. The methods are illustrated by applications to two high-frequency exchange rate data sets.

Sparse linear discriminant analysis by thresholding for high dimensional data

In many social, economical, biological and medical studies, one objective is to classify a subject into one of several classes based on a set of variables observed from the subject. Because the probability distribution of the variables is usually unknown, the rule of classification is constructed using a training sample. The well-known linear discriminant analysis (LDA) works well for the situation where the number of variables used for classification is much smaller than the training sample size. Because of the advance in technologies, modern statistical studies often face classification problems with the number of variables much larger than the sample size, and the LDA may perform poorly. We explore when and why the LDA has poor performance and propose a sparse LDA that is asymptotically optimal under some sparsity conditions on the unknown parameters. For illustration of application, we discuss an example of classifying human cancer into two classes of leukemia based on a set of 7,129 genes and a training sample of size 72. A simulation is also conducted to check the performance of the proposed method.

Vast volatility matrix estimation for high-frequency financial data

High-frequency data observed on the prices of financial assets are commonly modeled by diffusion processes with micro-structure noise, and realized volatility-based methods are often used to estimate integrated volatility. For problems involving a large number of assets, the estimation objects we face are volatility matrices of large size. The existing volatility estimators work well for a small number of assets but perform poorly when the number of assets is very large. In fact, they are inconsistent when both the number, p, of the assets and the average sample size, n, of the price data on the p assets go to infinity. This paper proposes a new type of estimators for the integrated volatility matrix and establishes asymptotic theory for the proposed estimators in the framework that allows both n and p to approach to infinity. The theory shows that the proposed estimators achieve high convergence rates under a sparsity assumption on the integrated volatility matrix. The numerical studies demonstrate that the proposed estimators perform well for large p and complex price and volatility models. The proposed method is applied to real high-frequency financial data.

A Likelihood Ratio Test against Stochastic Ordering in Several Populations

Abstract The likelihood ratio test is often used to test hypotheses involving a stochastic ordering. Distribution theory for the likelihood ratio test has been developed only for two stochastically ordered distributions. For testing equality of distributions against a stochastic ordering in several populations, this paper derives the null asymptotic distribution of the likelihood ratio test statistic, which is characterized by minimization problems and has no closed form. A Monte Carlo simulation is conducted to study the limiting distribution. Because the limiting distribution depends on the specific values of the unknown distributions under the null hypothesis, asymptotic and bootstrap approaches are proposed to overcome practical difficulties and implement tests based on the likelihood principle. Asymptotic validities for these tests are established and simulations are carried out to check their performances for finite sample sizes. The tests are applied to an example involving data for survival time f...

Large Volatility Matrix Inference via Combining Low-Frequency and High-Frequency Approaches

It is increasingly important in financial economics to estimate volatilities of asset returns. However, most of the available methods are not directly applicable when the number of assets involved is large, due to the lack of accuracy in estimating high-dimensional matrices. Therefore it is pertinent to reduce the effective size of volatility matrices in order to produce adequate estimates and forecasts. Furthermore, since high-frequency financial data for different assets are typically not recorded at the same time points, conventional dimension-reduction techniques are not directly applicable. To overcome those difficulties we explore a novel approach that combines high-frequency volatility matrix estimation together with low-frequency dynamic models. The proposed methodology consists of three steps: (i) estimate daily realized covolatility matrices directly based on high-frequency data, (ii) fit a matrix factor model to the estimated daily covolatility matrices, and (iii) fit a vector autoregressive model to the estimated volatility factors. We establish the asymptotic theory for the proposed methodology in the framework that allows sample size, number of assets, and number of days go to infinity together. Our theory shows that the relevant eigenvalues and eigenvectors can be consistently estimated. We illustrate the methodology with the high-frequency price data on several hundreds of stocks traded in Shenzhen and Shanghai Stock Exchanges over a period of 177 days in 2003. Our approach pools together the strengths of modeling and estimation at both intra-daily (high-frequency) and inter-daily (low-frequency) levels.

Change Curve Estimation via Wavelets

Abstract The recently developed theory of wavelets has a remarkable ability to “zoom in” on very short-lived frequency phenomena, such as transients in signals and singularities in functions, and hence provides an ideal tool to study localized changes. This article proposes a wavelet method for estimating jump and sharp cusp curves of a function in the plane. The method involves first computing wavelet transformation of data and then estimating jump and sharp cusp curves by wavelet transformation across fine scales. Asymptotic theory is established, and simulations are carried out to lend some credence to the asymptotic theory. The wavelet estimate is nearly optimal and can be computed by fast algorithms. The method is applied to a real image.

FAST CONVERGENCE RATES IN ESTIMATING LARGE VOLATILITY MATRICES USING HIGH-FREQUENCY FINANCIAL DATA

Financial practices often need to estimate an integrated volatility matrix of a large number of assets using noisy high-frequency data. Many existing estimators of a volatility matrix of small dimensions become inconsistent when the size of the matrix is close to or larger than the sample size. This paper introduces a new type of large volatility matrix estimator based on nonsynchronized high-frequency data, allowing for the presence of microstructure noise. When both the number of assets and the sample size go to infinity, we show that our new estimator is consistent and achieves a fast convergence rate, where the rate is optimal with respect to the sample size. A simulation study is conducted to check the finite sample performance of the proposed estimator.

Optimal sparse volatility matrix estimation for high-dimensional Itô processes with measurement errors

Stochastic processes are often used to model complex scientiflc problems in flelds ranging from biology and flnance to engineering and physical science. This paper investigates rate-optimal estimation of the volatility matrix of a high dimensional It^o process observed with measurement errors at discrete time points. The minimax rate of convergence is established for estimating sparse volatility matrices. By combining the multi-scale and threshold approaches we construct a volatility matrix estimator to achieve the optimal convergence rate. The minimax lower bound is derived by considering a subclass of It^o processes for which the minimax lower bound is obtained through a novel equivalent model of covariance matrix estimation for independent but non-identically distributed observations and through a delicate construction of the least favorable parameters. In addition, a simulation study was conducted to test the flnite sample performance of the optimal estimator, and the simulation results were found to support the established asymptotic theory.

Self-similarity index estimation via wavelets for locally self-similar processes

Many naturally occurring phenomena can be effectively modeled using self-similar processes. In such applications, accurate estimation of the scaling exponent is vital, since it is this index which characterizes the nature of the self-similarity. Although estimation of the scaling exponent has been extensively studied, previous work has generally assumed that this parameter is constant. Such an assumption may be unrealistic in settings where it is evident that the nature of the self-similarity changes as the phenomenon evolves. For such applications, the scaling exponent must be allowed to vary as a function of time, and a procedure must be available which provides a statistical characterization of this progression. In what follows, we propose and describe such a procedure. Our method uses wavelets to construct local estimates of time-varying scaling exponents for locally self-similar processes. We establish a consistency result for these estimates. We investigate the effectiveness of our procedure in a simulation study, and demonstrate its applicability in the analyses of a hydrological and a geophysical time series, each of which exhibit locally self-similar behavior.

Minimax estimation via wavelets for indirect long-memory data

Abstract In this paper, we model linear inverse problems with long-range dependence by a fractional Gaussian noise model and study function estimation based on observations from the model. By using two wavelet-vaguelette decompositions, one for the inverse problem which simultaneously quasi-diagonalizes both the operator and the prior information, and one for long-range dependence which decorrelates fractional Gaussian noise, we establish asymptotics for minimax risks, and show that the wavelet shrinkage estimate can be tuned to achieve the minimax convergence rate and significantly outperform linear estimates.