Spatial modelling and volatility matrix estimation in high dimension statistics with financial applications

Qian, Cheng (2018) Spatial modelling and volatility matrix estimation in high dimension statistics with financial applications. PhD thesis, London School of Economics and Political Science.

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Abstract

High dimension modelling is an important area in modern statistics. For example, a large number of problems that arise in finance are also inspired by more and more available high dimensional data. The main objective of this thesis is to investigate three methodologies in high dimension statistics with the application in finance. The subsequent chapters are organized as follow. The first two chapters are about spatial modeling and its inference respectively. The third chapter tackles a different problem about the estimation of large integrated volatility matrix of high frequency data. In the first chapter, a dynamic spatial model with different weight matrices for different time-lagged spatial effects is proposed. Unlike assuming a known spatial weight matrix, the proposed method estimates each spatial weight matrix for corresponding spatial effect by a linear combination of a set of specified spatial weight matrices to avoid misspecification. To estimate the coefficients for linear combinations and covariates, the pro- filed least square estimation is used with instrumental-like variables. A further selection on spatial weight matrices is introduced by adding an adaptive LASSO penalty on the coefficients of linear combination. All theoretical results are built on the scenario when the sample size T and panel dimension N go to infinity. The functional dependence in time series proposed by Wu (2005) is applied for the asymptotic normality of the estimated parameters. The oracle properties for model selection are developed including the asymptotic normality and sign consistency. Apart from a simulated data used to illustrate the performance of the proposed model, we also apply the proposed model to 32 important stocks from the Euro Stoxx 50 and S&P 500 in 2015 to invest the spatial interaction of them. The second chapter discusses the inference for the spatial dynamic model.To estimate the spatial weight matrices for contemporaneous and timelagged spatial effects, two linear combinations of a set of the specified spatial weight matrices are adopted respectively. We extend the quasimaximum likelihood estimation for the linear combination cofficients in our model and their consistency and asymptotic normality are established when both N and T are large. Using the asymptotic normality of the quasi-maximum likelihood estimators, a Wald test can be employed on the coefficients of the linear combination. Then, a diagnostic test proposed in Chang et al. (2017) is applied to test whether the fitted residuals perform like a white noise vector in our large N and large T setting. Simulated and real data are used to demonstrate the performance of the proposed quasi-maximum likelihood estimation and all above tests. The third chapter is about the estimation of large integrated volatility matrix for high frequency data. Besides the microstructure noises and non-synchronous trading times for high frequency data analysis should be fixed, the bias in the extreme eigenvalues coming from the high dimensionality are also not negligible. A nonparametric eigenvalue regularization proposed in Lam (2016) is applied on three existing volatility matrix estimators, such as multi-scale, kernel and pre-averaging realized volatility matrix estimators. One advantage for the proposed estimators is no need for implicit assumptions on the structure of the true integrated volatility matrix. It can be proved that the bias in the extreme eigenvalues can be shrunk and the regularized volatility estimators are positive definite in probability. Incidentally, the bias-corrected versions of kernel and pre-averaging estimators, which have faster rate of convergence at n??1=4 but are not guaranteed to be positive definite in Barndorff-Nielsen et al. (2011) and Christensen et al. (2010) respectively, are now regularized to be positive definite in probability, and we prove their rates of convergence to an \ideal" estimator under the spectral norm are also at n??1=4 under p=n ! c > 0. Jump and its removal by wavelet method in Fan and Wang (2007) are also included and all theoretical results are still hold. All proposed methods are applied on the simulated data. We also test the performance of the proposed methods on the stocks from the list "Fifty Most Active Stocks on NYSE" and "Fifty Most Active Stocks by Dollar Volume on NYSE".

Item Type:	Thesis (PhD)
Additional Information:	© 2018 Cheng Qian
Library of Congress subject classification:	H Social Sciences > HG Finance Q Science > QA Mathematics
Sets:	Departments > Statistics
Supervisor:	Lam, Clifford
URI:	http://etheses.lse.ac.uk/id/eprint/3901

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