Locally stationary time series frequently appears in both finance and environmental sciences (e.g., daily air pollutant concentration or financial returns). However, constructing the multi-step-ahead prediction interval for such time series remains an open question. Hence, we extend the nonparametric regression model with autoregressive errors for equally spaced designs to the time series setup. We propose a B-spline estimator for the trend function and a kernel estimator for the variance function to implement the model. The prediction interval of multi-step-ahead future observations is also constructed after fitting the autoregressive model of errors and obtaining the quantile of prediction residuals. The proposed method is illustrated by various simulation studies and an example of air pollutant data, containing 8 years of daily air pollutant concentrations in Xi'an. Our results demonstrate that our method outperforms others owing to its higher prediction accuracy and versatility.

中文翻译：

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应用于大气污染物浓度数据的局部平稳时间序列多步超前预测区间

局部平稳时间序列经常出现在金融和环境科学中（例如，每日空气污染物浓度或财务回报）。然而，为这样的时间序列构建多步提前预测区间仍然是一个悬而未决的问题。因此，我们将等距设计的具有自回归误差的非参数回归模型扩展到时间序列设置。我们提出了趋势函数的 B 样条估计器和方差函数的核估计器来实现模型。在拟合误差的自回归模型并获得预测残差的分位数后，也构建了多步超前未来观测的预测区间。所提出的方法通过各种模拟研究和空气污染物数据的例子来说明，含西安 8 年大气污染物日浓度。我们的结果表明，我们的方法由于其更高的预测准确性和多功能性而优于其他方法。