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Abstract:
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Modern time series analysis can involve tenor-valued observations. For example, the portfolios (see Fama-French model) can be formed based on the grouping of Size, Book-to-Market, and Operating Profitability, which lead to a three dimensional tensor of returns for each day or month. We propose an autoregressive model for the tensor-valued time series. Comparing with the vector autoregressive model, it leads to a substantial dimension reduction while preserves the tensor structure and admits corresponding interpretations. The model is extended to have multiple lag-one autoregressive terms to provide more flexibility. Estimation procedures of the coefficient matrices and the choice of the number of lag-one terms are investigated with the corresponding theoretical analysis. The performance of the model is demonstrated with simulated and real examples.
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