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Title: Fully data-driven non-parametric estimation of Toeplitz covariance matrices Authors:  Karolina Klockmann - University of Vienna (Austria) [presenting]
Tatyana Krivobokova - University of Vienna (Austria)
Abstract: Estimating the Toeplitz covariance matrix of a single realization of a stationary stochastic process is a central problem in time series econometrics. It is well known that the sample auto-covariance matrix is inconsistent in the spectral norm, so regularized versions, such as the tapered or banded covariance estimators, have been proposed. However, such estimators are not guaranteed to be positive definite, and data-driven choices of the regularization parameters are not available. We present an estimator for the Toeplitz covariance matrix and its inverse, which overcome these drawbacks. First, we derive an alternative version of the Whittle likelihood based on the Discrete Cosine Transform matrix, which is shown to asymptotically diagonalize Toeplitz matrices. Using variance stabilizing transforms, we transform the resulting Gamma regression problem into an approximate Gaussian regression setting for the log-spectral density. The resulting estimators for the Toeplitz covariance matrix and its inverse are positive definite and all regularization parameters are data-driven. As our main result, we show that our estimators attain the minimax optimal convergence rate under the spectral norm for Gaussian stationary time series. The performance of our estimators is demonstrated in simulations and real data analysis.