In this article, we consider estimating the innovation variance function when the conditional mean model is characterised by a structural break autoregressive model, which exhibits multiple unit root, explosive and stationary collapse segments, allowing for behaviour often seen in financial data where bubble and crash episodes are present. Estimating the variance function normally proceeds in two steps: estimating the conditional mean model, then using the residuals to estimate the variance function. In this article, a non-parametric approach is proposed to estimate the complicated parametric conditional mean model in the first step. The approach turns out to provide a convenient solution to the problem and achieve robustness to any structural break features in the conditional mean model without the need of estimating them parametrically. In the second step, kernel-smoothed squares of the truncated first-step residuals are shown to consistently estimate the variance function. In Monte Carlo simulations, we show that our proposed method performs very well in the presence of explosive and stationary collapse segments compared with the popular rolling standard deviation estimator that is commonly used in economics and finance. As an empirical illustration of our new approach, we apply the volatility estimator to recent Bitcoin data.

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具有非平稳和爆炸段的结构断裂自回归模型的方差函数估计

在这篇文章中，我们考虑在条件均值模型以结构断裂自回归模型为特征时估计创新方差函数，该模型表现出多个单位根、爆炸性和平稳崩溃段，允许在金融数据中经常出现的泡沫和崩溃事件的行为存在。估计方差函数通常分两步进行：估计条件均值模型，然后使用残差估计方差函数。在本文中，提出了一种非参数方法来估计第一步中复杂的参数条件均值模型。事实证明，该方法为该问题提供了一种方便的解决方案，并实现了对条件均值模型中任何结构断裂特征的鲁棒性，而无需对其进行参数估计。在第二步中，截断的第一步残差的核平滑平方被证明可以一致地估计方差函数。在 Monte Carlo 模拟中，我们表明，与经济学和金融学中常用的流行滚动标准差估计器相比，我们提出的方法在存在爆炸性和静止坍塌段时表现得非常好。作为我们新方法的实证说明，我们将波动率估计器应用于最近的比特币数据。我们表明，与经济学和金融学中常用的流行滚动标准差估计器相比，我们提出的方法在存在爆炸性和静止坍塌段时表现得非常好。作为我们新方法的实证说明，我们将波动率估计器应用于最近的比特币数据。我们表明，与经济学和金融学中常用的流行滚动标准差估计器相比，我们提出的方法在存在爆炸性和静止坍塌段时表现得非常好。作为我们新方法的实证说明，我们将波动率估计器应用于最近的比特币数据。