The problem of estimating the spectral density matrix $f(w)$ of a multivariate time series is revisited with special focus on the frequencies $w=0$ and $w=\pi$. Recognizing that the entries of the spectral density matrix at these two boundary points are real-valued, we propose a new estimator constructed from a local polynomial regression of the real portion of the multivariate periodogram. The case $w=0$ is of particular importance, since $f(0)$ is associated with the large-sample covariance matrix of the sample mean; hence, estimating $f(0)$ is crucial in order to conduct any sort of statistical inference on the mean. We explore the properties of the local polynomial estimator through theory and simulations, and discuss an application to inflation and unemployment.
翻译:由于认识到这两个边界点的光谱密度矩阵条目是真实估价的,我们提议用多变量时间序列实际部分的局部多数值回归制成一个新的估算器。案例为w=0, 特别重要,因为f(0)美元与样本平均值的大型分布式共变量矩阵有关;因此,估算值f(0)美元对于就平均值进行任何统计推论至关重要。我们通过理论和模拟探讨本地多数值估算器的特性,并讨论通货膨胀和失业问题。