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英文文献：Estimation in Threshold Autoregressive Models with Nonstationarity-非平稳性阈值自回归模型的估计
英文文献作者：Jiti Gao,Dag Tjostheim,Jiying Yin
英文文献摘要：
This paper proposes a class of new nonlinear threshold autoregressive models with both stationary and nonstationary regimes. Existing literature basically focuses on testing for a unit?–root structure in a threshold autoregressive model. Under the null hypothesis, the model reduces to a simple random walk. Parameter estimation then becomes standard under the null hypothesis. How to estimate parameters involved in an alternative nonstationary model, when the null hypothesis is not true, becomes a nonstandard estimation problem. This is mainly because models under such an alternative are normally null recurrent Markov chains. This paper thus proposes to establish a parameter estimation method for such nonlinear threshold autoregressive models with null recurrent structure. Under certain assumptions, we show that the ordinary least squares (OLS) estimates of the parameters involved are asymptotically consistent. Furthermore, it can be shown that the OLS estimator of the coefficient parameter involved in the stationary regime can still be asymptotically normal while the OLS estimator of the coefficient parameter involved in the nonstationary regime has a nonstandard asymptotic distribution. In the limit, the rate of convergence in the stationary regime is n^(-1/4) , whereas it is n^(-1) in the nonstationary regime. The proposed theory and estimation method is illustrated by both simulated and real data examples.
本文提出了一类新的具有平稳和非平稳状态的非线性阈值自回归模型。现有文献基本上集中在阈值自回归模型中对单位-根结构的检验。在零假设下，模型简化为一个简单的随机游走。在原假设下，参数估计成为标准。在零假设不成立的情况下，如何对非平稳模型中的参数进行估计，成为一个非标准估计问题。这主要是因为在这种替代下的模型通常是空循环马氏链。为此，本文提出了一种基于零回归结构的非线性阈值自回归模型的参数估计方法。在一定的假设条件下，我们证明了所涉及参数的一般最小二乘估计是渐近一致的。进一步地，证明了非平稳区系数参数的OLS估计量仍然可以是渐近正态的，而非平稳区系数参数的OLS估计量具有非标准渐近分布。在极限情况下，平稳区域的收敛速度为n^(-1/4)，而非平稳区域的收敛速度为n^(-1)。通过仿真和实际数据实例说明了所提出的理论和估计方法。
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