Value at risk models are concerned with the estimation of conditional quantiles of a time series. Formally, these quantities are a function of conditional volatility and the respective quantile of the innovation distribution. The former is often subject to asymmetric dynamic behaviour, e.g., with respect to past shocks. In this paper, we propose a model in which conditional quantiles follow a generalised autoregressive process governed by two parameter regimes with their weights determined by a smooth transition function. We develop a two-step estimation procedure based on a sieve estimator, approximating conditional volatility by using composite quantile regression, which is then used in the generalised autoregressive conditional quantile estimation. We show that the estimator is consistent and asymptotically normal, and we complement the results with a simulation study. In our empirical application, we consider daily returns of the German equity index (DAX) and the USD/GBP exchange rate. Although only the latter follows a two-regime model, we find that our model performs well in terms of out-of-sample prediction in both cases.

中文翻译：

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风险估计中基于分位数的平滑过渡值

风险价值模型与时间序列的条件分位数的估计有关。从形式上来说，这些数量是条件波动率和创新分布各自分位数的函数。前者经常遭受不对称的动态行为，例如关于过去的冲击。在本文中，我们提出了一个模型，其中条件分位数遵循由两个参数体制控制的广义自回归过程，其权重由平滑过渡函数确定。我们开发了一种基于筛分估计器的两步估计程序，通过使用复合分位数回归来近似条件波动率，然后将其用于广义自回归条件分位数估计中。我们证明了估计量是一致且渐近正态的，我们通过模拟研究对结果进行补充。在我们的经验应用中，我们考虑了德国股票指数（DAX）和美元/英镑汇率的每日收益。尽管只有后者遵循两制度模型，但我们发现我们的模型在两种情况下的样本外预测方面均表现出色。