Statistical inference for autoregressive time series models

Abstract

Autoregression is the fundamental and most popular class of time series models and has been widely used for forecasting and structural analysis in a broad variety of fields. Modern technology advances enable researchers to collect an enormous amount of time series and pursue more complicated structures in the data, leading to the emerging interests in both high-dimensional and nonlinear autoregressive modeling.

In the first part of this thesis, a multilinear low-rank vector autoregressive model is proposed for high-dimensional time series, by rearranging parameter matrices into a tensor and then applying tensor decomposition to restrict the parameter space. Compared with the reduced-rank regression model, the low-rank tensor structure substantially expands the capacity of vector autoregression in modeling high-dimensional time series. Moreover, sparsity is imposed on factor matrices to improve interpretability and estimation efficiency. Asymptotic properties of the least squares estimator and non-asymptotic properties of the sparsity-inducing regularized estimator are established, in low- and high-dimensional settings, respectively. Simulation experiments and macroeconomic data analysis demonstrate the advantages of the proposed approach over various existing methods.

Second, this thesis introduces a high-dimensional autoregressive model for matrix- and tensor-valued time series data. In the proposed model, the parameters are folded into a higher-order tensor and a low-Tucker-rank structure is considered to achieve substantial dimension reduction. Under high- dimensional scaling, a novel convex regularization approach, based on the sum of nuclear norms of square matricizations, is proposed for estimation, and its non-asymptotic properties are established. A truncation method is further proposed to consistently select tensor ranks. Extensive simulation experiments and real data analysis in finance demonstrate the effectiveness of the proposed methods.

Third, this thesis proposes a novel compact autoregressive neural network for large-scale nonlinear time series modeling. The proposed neural network has a separable convolutional architecture and can be applied to sequences with long-range dependence since the dimension along the sequential order is reduced. Theoretical studies show that the network architecture improves the learning efficiency, and hence requires much fewer samples for model training. Experiments on synthetic and real-world datasets demonstrate the promising performance of the proposed compact network over the state-of-the-art benchmarks.

Last, this thesis studies the asymptotic statistical inferences for the two-regime buffered autoregressive model with autoregressive unit roots. A sup-likelihood-ratio test is proposed for the nonlinear buffered effect in the possible presence of unit roots, and a class of unit root tests are considered to identify the number of nonstationary regimes in buffered autoregression. The wild bootstrap method is suggested to approximate the critical values of the two tests. Two macroeconomic data, U.S. unemployment rates and real exchange rates, are analyzed to demonstrate the usefulness of the proposed methodology.