We propose a new class of spatio-temporal models with unknown and banded autoregressive coefficient matrices. The setting represents a sparse structure for highdimensional spatial panel dynamic models when panel members represent economic (or other type) individuals at many different locations. The structure is practically meaningful when the order of panel members is arranged appropriately. Note that the implied autocovariance matrices are unlikely to be banded, and therefore, the proposal is radically different from the existing literature on the inference for high-dimensional banded covariance matrices. Due to the innate endogeneity, we apply the least squares method based on a Yule-Walker equation to estimate autoregressive coefficient matrices. The estimators based on multiple Yule-Walker equations are also studied. A ratio-based method for determining the bandwidth of autoregressive matrices is also proposed. Some asymptotic properties of the inference methods are established. The proposed methodology is further illustrated using both simulated and real data sets.

This article considers a structural‐factor approach to modeling high‐dimensional time series and space‐time data by decomposing individual series into trend, seasonal, and irregular components. For ease in analyzing many time series, we employ a time polynomial for the trend, a linear combination of trigonometric series for the seasonal component, and a new factor model for the irregular components. The new factor model simplifies the modeling process and achieves parsimony in parameterization. We propose a Bayesian information criterion to consistently select the order of the polynomial trend and the number of trigonometric functions, and use a test statistic to determine the number of common factors. The convergence rates for the estimators of the trend and seasonal components and the limiting distribution of the test statistic are established under the setting that the number of time series tends to infinity with the sample size, but at a slower rate. We study the finite‐sample performance of the proposed analysis via simulation, and analyze two real examples. The first example considers modeling weekly PM2.5 data of 15 monitoring stations in the southern region of Taiwan and the second example consists of monthly value‐weighted returns of 12 industrial portfolios.

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Abstract:The threshold autoregressive (TAR) model and the smooth threshold autoregressive (STAR) model have been among the most popular parametric nonlinear time series models for the past three decades or so. However, as yet there is no formal statistical test in the literature for one against the other. The two models are fundamentally different in their autoregressive functions, the TAR model being generally discontinuous while the STAR model being smooth (except in the limit of infinitely fast switching for some cases). Following the approach initiated by Cox (1961Cox ( , 1962, we treat the test problem as one of separate families of hypotheses, thus filling a serious gap in the literature. The test statistic under a STAR model is shown to follow asymptotically a chi-squared distribution, and the one under a TAR model expressed as a functional of a chi-squared process. We present numerical results with both simulated and real data to assess the performance of our procedure.

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