Least Squares Estimation Of Large Dimensional Threshold Factor Models

Leo Migdal
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least squares estimation of large dimensional threshold factor models

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Tensor Factor Models (TFM) are appealing dimension reduction tools for high-order large-dimensional tensor time series, and have wide applications in economics, finance and medical imaging. In this paper, we propose a projection estimator for the Tucker-decomposition based TFM, and provide its least-square interpretation which parallels to the least-square interpretation of the Principal Component Analysis (PCA) for the vector factor... The projection technique simultaneously reduces the dimensionality of the signal component and the magnitudes of the idiosyncratic component tensor, thus leading to an increase of the signal-to-noise ratio. We derive a convergence rate of the projection estimator of the loadings and the common factor tensor which are faster than that of the naive PCA-based estimator. Our results are obtained under mild conditions which allow the idiosyncratic components to be weakly cross- and auto- correlated. We also provide a novel iterative procedure based on the eigenvalue-ratio principle to determine the factor numbers.

Extensive numerical studies are conducted to investigate the empirical performance of the proposed projection estimators relative to the state-of-the-art ones. Tensors, or high-order arrays, emerge ubiquitously in all applied sciences, such as macroeconomic data ([18, 9]) and financial data ([30, 33]) where typical applications consider higher-order realized moments [43, 12, 6, 36]. They appear also in computer vision data ([44]); neuroimaging and functional MRI data (see e.g. [53, 35, 17]), typically consisting of hundreds of thousands of voxels observed over time; recommender systems (see e.g. [23], and the various references therein); data arising in psychometrics ([13, 21]), and chemometrics (see [46], and also the references in [1]). For a review of further applications we also refer to Kolda and Bader [37] and Bi et al.

[11]. Hence, inference in the context of tensor-valued time series has now become one of the most active research areas in statistics and machine learning. Given that the analysis of tensor-valued data poses significant computational challenges due to high-dimensionality, a natural approach is to model tensor-valued time series through a low-dimensional projection on a space of common factors. Indeed, factor models are one of the most powerful dimension reduction tools in time series for extracting comovements among multi-dimensional features, and, in the past decades, factor models applied to vector-valued time series have... In recent years, the literature has extended the tools developed for the analysis of vector-valued data to the context of matrix-valued data. In particular, we refer to the seminal contribution by [48], who proposed Matrix Factor Model (MFM) exploiting the double low-rank structure of matrix-valued observations, and to the subsequent paper by Chen and Fan [15],...

Furthermore, Yu et al. [49] study a projected version of PCA; Gao et al. [26] study the Maximum Likelihood approach; and Zhang et al. [52] study a hierarchical CP product MFM. In related contributions, [32] propose a strong rule to determine whether there is a factor structure of matrix time series; Chen et al. [16] study the MFM under linear constraints; and Liu and Chen [42] and He et al.

[34] study non-linearities in the form of a threshold regression, or a changepoint model, respectively. In contrast with this plethora of contributions, the statistical analysis of tensor factor models (TFM) is still in its infancy, and most of the recent contributions focus on i.i.d. data, or on models with i.i.d. noise, thus limiting the applicability to tensor-valued time series. In such a context, a typical approach would be to decompose an observed K𝐾Kitalic_K-fold tensor time series {𝒳t∈ℝp1×p2×⋯×pK,t∈ℤ}formulae-sequencesubscript𝒳𝑡superscriptℝsubscript𝑝1subscript𝑝2⋯subscript𝑝𝐾𝑡ℤ\{\mathcal{X}_{t}\in\mathbb{R}^{p_{1}\times p_{2}\times\dots\times p_{K}},\;t% \in\mathbb{Z}\}{ caligraphic_X start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT × italic_p... However, whereas in the vector case the possible structure of the signal component is uniquely defined as a product of a loadings matrix times a vector of factors, in the context of a tensor...

Lettau [41] for a recent discussion and a comparison. The choice between the two relies mainly on the kind of data one is dealing with and the problem one is interested in studying. The Tucker decomposition is easier to estimate, less restrictive and more flexible than the CP decomposition. This flexibility comes at a cost, however, since Tucker tensor factor models, as the vector factors models, are not uniquely identified, while CP factor models, if they exist, are uniquely identified. An advantage of Tucker tensor factor models is that they allow for different numbers of factors for each mode, while the CP tensor factor model restricts the number of factors to be the same...

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