Abstract:We consider a factor model for high-dimensional time series with regimeswitching dynamics. The switching is assumed to be driven by an unobserved Markov chain; the mean, factor loading matrix, and covariance matrix of the noise process are different among the regimes. The model is an extension of the traditional factor models for time series and provides flexibility in dealing with applications in which underlying states may be changing over time. We propose an iterative approach to estimating the loading space of each regime and clustering the data points, combining eigenanalysis and the Viterbi algorithm. The theoretical properties of the procedure are investigated. Simulation results and the analysis of a data example are presented.
We consider a threshold factor model for high-dimensional time series in which the dynamics of the time series is assumed to switch between different regimes according to the value of a threshold variable. This is an extension of threshold modeling to a high-dimensional time series setting under a factor structure. Specifically, within each threshold regime, the time series is assumed to follow a factor model. The regime switching mechanism creates structural change in the factor loading matrices. It provides flexibility in dealing with situations that the underlying states may be changing over time, as often observed in economic time series and other applications. We develop the procedures for the estimation of the loading spaces, the number of factors and the threshold value, as well as the identification of the threshold variable, which governs the regime change mechanism. The theoretical properties are investigated.
Functional data analysis has became an increasingly popular class of problems in statistical research. However, functional data observed over time with serial dependence remains a less studied area. Motivated by Bosq (2000), who first introduced the functional autoregressive models, we propose a convolutional functional autoregressive model, where the function at time t is a result of the sum of convolutions of the past functions and a set of convolution functions, plus a noise process, mimicking the vector autoregressive process. It provides an intuitive and direct interpretation of the dynamics of a stochastic process. Instead of principal component analysis commonly used in functional data analysis, we adopt a sieve estimation procedure based on B-spline approximation of the convolution functions. We establish convergence rate of the proposed estimator, and investigate its theoretical properties. The model building, model validation, and prediction procedures are also developed. Both simulated and real data examples are presented.
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