Structural shrinkage of nonparametric spectral estimators for multivariate time series

TL;DR

This paper introduces a structural shrinkage method for nonparametric spectral estimators of multivariate time series, improving stability and accuracy by leveraging factor-induced spectral structure without explicit factor modeling.

Contribution

It proposes a model-free, asymptotically optimal linear combination of smoothed periodogram and a parametric estimator based on underfitting factor models.

Findings

Shrinkage improves numerical stability of spectral estimators.

Eigenvalue shrinkage reduces mean squared error.

Structural shrinkage outperforms traditional methods in simulations.

Abstract

In this paper we investigate the performance of periodogram based estimators of the spectral density matrix of possibly high-dimensional time series. We suggest and study shrinkage as a remedy against numerical instabilities due to deteriorating condition numbers of (kernel) smoothed periodogram matrices. Moreover, shrinking the empirical eigenvalues in the frequency domain towards one another also improves at the same time the Mean Squared Error (MSE) of these widely used nonparametric spectral estimators. Compared to some existing time domain approaches, restricted to i.i.d. data, in the frequency domain it is necessary to take the size of the smoothing span as "effective or local sample size" into account. While B\"{o}hm and von Sachs (2007) proposes a multiple of the identity matrix as optimal shrinkage target in the absence of knowledge about the multidimensional structure of the data, here we consider "structural" shrinkage. We assume that the spectral structure of the data is induced by underlying factors. However, in contrast to actual factor modelling suffering from the need to choose the number of factors, we suggest a model-free approach. Our final estimator is the asymptotically MSE-optimal linear combination of the smoothed periodogram and the parametric estimator based on an underfitting (and hence deliberately misspecified) factor model. We complete our theoretical considerations by some extensive simulation studies. In the situation of data generated from a higher-order factor model, we compare all four types of involved estimators (including the one of B\"{o}hm and von Sachs (2007)).