This paper considers a continuous three-phase polynomial regression model with two threshold points for dependent data with heteroscedasticity. We assume the model is polynomial of order zero in the middle regime, and is polynomial of higher orders elsewhere. We denote this model by ${ℳ}_2$, which includes models with one or no threshold points, denoted by ${ℳ}_1$ and ${ℳ}_0$, respectively, as special cases. We provide an ordered iterative least squares (OiLS) method when estimating ${ℳ}_2$ and establish the consistency of the OiLS estimators under mild conditions. When the underlying model is ${ℳ}_1$ and is $(d_0-1)$th-order differentiable but not $d_0$th-order differentiable at the threshold point, we further show the $O_p(N^{\text{◂+▸}-1/(d_0+2)})$ convergence rate of the OiLS estimators, which can be faster than the $O_p(N^{\text{◂+▸}-1/(2d_0)})$ convergence rate given in Feder when $d_0\ge 3$. We also apply a model-selection procedure for selecting ${ℳ}_{\kappa }$; $\text{◂,▸}\kappa =0,1,2$. When the underlying model exists, we establish the selection consistency under the aforementioned conditions. Finally, we conduct simulation experiments to demonstrate the finite-sample performance of our asymptotic results.

中文翻译：

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中间状态下均值恒定的连续三相多项式回归模型的阈值估计

本文考虑了具有异方差性的相关数据的具有两个阈值点的连续三相多项式回归模型。我们假设模型在中间区域是零阶多项式，在其他地方是高阶多项式。我们将这个模型表示为${ℳ}_{\mathrm{2个}}$，其中包括具有一个或没有阈值点的模型，表示为${ℳ}_{\mathrm{1个}}$和${ℳ}_0$，分别作为特例。我们在估计时提供有序迭代最小二乘 (OiLS) 方法${ℳ}_{\mathrm{2个}}$并在温和条件下建立 OiLS 估计量的一致性。当底层模型是${ℳ}_{\mathrm{1个}}$并且是$(d_0-\mathrm{1个})$th阶可微但不可$d_0$在阈值点的 th 阶可微分，我们进一步证明${欧}_p({否}^{\text{◂+▸}-\mathrm{1个}/(d_0+\mathrm{2个})})$OiLS 估计器的收敛速度，可以比${欧}_p({否}^{\text{◂+▸}-\mathrm{1个}/(\mathrm{2个}{d}_0)})$Feder 给出的收敛率$d_0\ge \mathrm{3个}$. 我们还应用模型选择程序来选择${ℳ}_{\kappa }$;$\text{◂,▸}\kappa =0,\mathrm{1个},\mathrm{2个}$. 当底层模型存在时，我们在上述条件下建立选择一致性。最后，我们进行了模拟实验来证明我们的渐近结果的有限样本性能。