Goodness-of-fit test for Continuous Time Stochastic Volatility Models

• Main Authors: Liang-Ching Lin, 林良靖
• Other Authors: Meihui Guo
• Format: Others
• Language: en_US
• Published: 2013
• Online Access: http://ndltd.ncl.edu.tw/handle/86397692008041741207

博士 === 國立中山大學 === 應用數學系研究所 === 101 === A goodness-of-fit test for stationary distributions of continuous time stochastic processes plays an important role in building up stochastic differential equation (SDE) models. In the first part of this dissertation, we propose two types of goodness-of-fit tests for continuous time stochastic volatility models (SVMs) based on discretely sampled observations. The first type of test is constructed by measuring deviations between the empirical and true characteristic functions obtained from the hypothesized stochastic volatility model. It is shown that under the null, the first proposed test statistics asymptotically follow a weighted sum of products of centered normal random variables. The second type of test is the Bickel-Rosenblatt test which is constructed by measuring integrated squared deviations between the nonparametric kernel density estimate from the observations and a parametric fit of the density. It is shown that under the null hypothesis, the Bickel-Rosenblatt test statistic is asymptotically normal. We also developed the Bickel-Rosenblatt test for the multivariate SVMs with a copula link. Its asymptotic null distribution is derived and bivariate examples are given. Simulation studies and real data analysis are conducted for both proposed tests. In the second part of this thesis, we consider inference for the SVMs with market microstructure noises which are often used to model high frequency financial data. Estimation of the integrated volatility is an important problem for high frequency financial data analysis. We consider the minimum variance unbiased estimator (MVUE) of the integrated volatility proposed by Lin (2007). The MVUE minimizes the finite sample variance in the class of unbiased estimators which are linear combinations of the sample autocovariance functions. The variance of the MVUE converges at a rate of Op(n−1/4). In particular, the MVUE achieves the maximum likelihood estimator efficiency for the constant volatility case. A recursive algorithm is developed to compute the optimal weights of the MVUE. Improved estimators of the microstructure noise variance and the quarticity are also proposed to facilitate the estimation procedure. Simulation results show our proposed estimator attains higher efficiency than state-of-the-art methods for the finite samples. Finally, a real data analysis is conducted for illustration. We also consider the goodness-of-fit test for the SVMs with microstructure noises. Moment estimators of the model parameters are proposed. A goodness-of-fit test based on the characteristic function is proposed. Simulation results of sizes and powers of the proposed test are given.