SPECTRAL ANALYSIS OF LARGE DIMENSIONAL SAMPLE AUTOCORRELATION AND AUTOCOVARIANCE MATRICES

Determining the number of common factors is an important problem in high dimensional factor model which is widely concerned in financial econometrics. Most existing methods are based on a sample covariance matrix or a sample auto­covariance matrix. While this thesis tries to construct estimators of the total number of factors based on sample auto­correlation and sample weighted covariance matrices. As a first step, we focus on the spectral analysis of sample auto­correlation and auto­covariance matrices under the high­-dimensional context when both dimension and sample size are large. These theoretical results play important roles in parameter estimation and inference in the high dimensional factor model. As the first part of this thesis we consider the sample auto­correlation matrix, we study the limiting behaviors of singular values of a lag­ged sample auto­correlation matrix of error process in the high­dimensional factor model. Specifically, we derive the limiting spectral distribution (LSD) which characterizes the global spectrum, and find out the limit of its largest singular value. Under mild assumptions, we show that the LSD is the same as that of the lag­ged sample auto­covariance matrix. Based on this asymptotic equivalence, we additionally show that the largest singular value of converges almost surely to the right end point of the support of its LSD. Our results take the first step to identifying the number of factors in factor analysis. Our theoretical results are fully supported by numerical experiments as well. As the second part of this thesis, we propose a sample weighted covariance matrix combining sample covariance matrix and sample auto­covariance matrix, in order to make use of more information from sample covariance and sample auto­covariance matrices. We derive the LSD of sample weighted covariance matrix of error process in the high dimensional factor model. Then we propose a new estimator of the total number of factors based on the sample weighted covariance matrix. We also provide numerical experiments to check the performance of the estimator. We find out that the new estimator demonstrates its estimation consistency under the scenarios we consider.

因子个数的估计是金融计量经济学中广泛关注的高维因子模型的核心问题， 大多数现有的研究方法都是基于样本协方差矩阵或者样本自协方差矩阵。而这篇 论文重点研究高维数据背景下，在维数与样本量呈比例趋近于无穷大的理论框架 下，样本自相关和加权自协方差矩阵的渐近谱理论，并应用到高维因子模型里的 估计和统计推断问题中。 在论文的第一部分我们首先考虑的是样本自相关矩阵，我们研究了高维因子 模型中误差项 𝜖 的样本自相关矩阵的奇异值的极限性质。特别地，我们推导出R_𝜖^𝜏的极限谱分布，并且找到其最大奇异值的极限。在特定的假设下，我们证明了 样本自协方差矩阵的极限谱分布和样本自相关矩阵的极限谱分布是相同的。基于 这种渐近等价性，我们还证明了R_𝜖^𝜏的最大奇异值收敛到其极限谱分布的支撑集 的右边界。我们的结果为高维因子模型中因子个数估计的问题奠定了理论的基础。 我们的理论结果也得到了数值实验的充分支持。 而本论文的第二部分，为了更多地利用样本协方差和样本自协方差矩阵的信 息，我们构造了一个结合样本协方差矩阵和样本自协方差矩阵的加权样本协方差 矩阵。我们推导出了加权样本协方差矩阵的极限谱分布。并基于加权样本协方差 矩阵提出了新的因子个数的估计量，并且通过数值模拟来检验我们的估计量的效 果。基于数值模拟结果，新的估计量对于我们数值模拟中设置的实例有相合性。

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