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Statistica Sinica 32 (2022), 755-786

TIME SERIES MODELS FOR REALIZED COVARIANCE
MATRICES BASED ON THE MATRIX-F DISTRIBUTION

Jiayuan Zhou1 , Feiyu Jiang2 , Ke Zhu3 and Wai Keung Li3;4

1University of Florida, 2Fudan University,
3The University of Hong Kong and 4The Education University of Hong Kong

Abstract: We propose a new Conditional BEKK matrix-F (CBF) model for time-varying realized covariance (RCOV) matrices. This CBF model is capable of capturing a heavy-tailed RCOV, which is an important stylized fact, but is not handled adequately by Wishart-based models. To further mimic the long-memory feature of an RCOV, we introduce a special CBF model with a conditional heterogeneous autoregressive structure. Moreover, we provide a systematic study of the probabilistic properties and statistical inferences of the CBF model, including exploring its stationarity, establishing the asymptotics of its maximum likelihood estimator, and giving new inner-product-based tests for model checking. In order to handle a large-dimensional RCOV matrix, we construct two reduced CBF models: the variance-target CBF model (for a moderate but fixed-dimensional RCOV matrix), and the factor CBF model (for a high-dimensional RCOV matrix). For both reduced models, the asymptotic theory of the estimated parameters is derived. The importance of our methodology is illustrated by means of simulations and two real examples.

Key words and phrases: Factor model, heavy-tailed innovation, long memory, matrix-F distribution, matrix time series model, model checking, realized covariance matrix, variance target.

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