Estimation of the Generalized Prediction Error Variance of a Multiple Time Series
针对多元平稳时间序列，提出一种基于多变量Szegö-Kolmogorov公式的非参数估计器来估计广义预测误差方差，证明了其渐近正态性，并通过模拟和实际数据验证了其有效性。
Abstract For a multivariate stationary time series, we propose a nonparametric estimator for its generalized prediction error variance using a multivariate analog of the Szegö-Kolmogorov formula, replacing the integral by a sum and replacing the unknown spectral density matrix by a consistent estimator. Asymptotic normality of this estimator is established, and its small-sample behavior is assessed through simulation and application to two real data sets. These examples show that the proposed method works reasonably well in comparison with parametric models. In contrast to the univariate case where smoothing the periodogram is optional and generally not recommended, in the multivariate case this smoothing is a necessity, because the raw periodogram is a matrix of rank one. To obtain a consistent estimator of a full-rank spectral density matrix, one must necessarily choose larger bandwidths for higher-dimensional time series. A bias-correction factor is computed using the asymptotic properties of our proposed estimator, and a simulation study indicates its important role in reducing the bias.