Spectral analysis of large auto-covariance matrices with application to high dimensional time series analysis

Abstract

Sample auto-covariance matrix plays a crucial role in high dimensional times series analysis. In this thesis we study the spectral properties of large sample auto-covariance matrix, the explicit form of the density function of limiting singular value distribution is derived under the so called {\em Mar\v{c}enko-Pastur regime} where the dimension and sample size tend to infinity proportionally.

Based on this result, we look into the problem of identifying number of factors in factor models. Actually a thresholding ratio estimator is proposed based on singular values of sample auto-covariance matrix of the observed time series. We fully characterize the limits of both the factor and noise eigenvalues and establish a phase transition phenomenon which shows how detectable factors are distinguished from the noise. The proposed estimator is capable of detecting all the significant factors including those ones with multi-level strength and exhibits excellent performance in both simulation and empirical studies. The technical tools employed come from recent advances in random matrix theory, including spiked population models and finite-rank perturbations of large random matrices.

A second problem studied in this thesis is to test high dimensional white noise, which is a classical and important tool for diagnostic checking in time series modeling. For vector time series where the dimension is large compared to the sample size, traditional omnibus portmanteau tests such as the multivariate Hosking and Li-McLeod tests become extremely conservative, losing their size and power dramatically. There is thus an urgent need to develop new tests for testing a high-dimensional white noise. Several new tests are proposed to fill in this gap. One of the proposed test statistic is a scalar which encapsulates the serial correlations within and across all components. Precisely, the statistic equals to the sum of squares of the eigenvalues in a symmetrized sample auto-covariance matrix at a certain lag. Two other multiple-lags based tests are also proposed to complement the single-lag based one. We develop adequate high dimensional limiting theory for these test statistics using tools from random matrix theory. Asymptotic normality for the test statistics is derived under different asymptotic regimes when both the dimension $p$ and the sample size $T$ diverge to infinity. We provide evidence for the validity of such high-dimensional limits in a significant range of $(p,T)$ combinations, therefore ensuring broad applications in practice. Extensive simulation experiments confirm an excellent behavior of these high-dimensional tests in finite samples with accurate size and satisfactory power.