This article proposes frequentist multiple-equation least-squares averaging approaches for multistep forecasting with vector autoregressive (VAR) models. The proposed VAR forecast averaging methods are based on the multivariate Mallows model averaging (MMMA) and multivariate leave-h-out cross-validation averaging (MCVAh) criteria (with h denoting the forecast horizon), which are valid for iterative and direct multistep forecast averaging, respectively. Under the framework of stationary VAR processes of infinite order, we provide theoretical justifications by establishing asymptotic unbiasedness and asymptotic optimality of the proposed forecast averaging approaches. Specifically, MMMA exhibits asymptotic optimality for one-step-ahead forecast averaging, whereas for direct multistep forecast averaging, the asymptotically optimal combination weights are determined separately for each forecast horizon based on the MCVAh procedure. To present our methodology, we investigate the finite-sample behavior of the proposed averaging procedures under model misspecification via simulation experiments.

中文翻译：

---

最优多步 VAR 预测平均

本文提出了使用向量自回归 (VAR) 模型进行多步预测的频率主义多方程最小二乘平均方法。所提出的 VAR 预测平均方法基于多元 Mallows 模型平均 (MMMA) 和多元休假H-out 交叉验证平均（MCVAH) 标准（与H表示预测范围），它们分别适用于迭代和直接多步预测平均。在无限阶平稳 VAR 过程的框架下，我们通过建立所提出的预测平均方法的渐近无偏性和渐近​​最优性来提供理论依据。具体来说，MMMA 对于一步提前预测平均表现出渐近最优性，而对于直接多步预测平均，渐近最优组合权重是根据 MCVA 为每个预测范围单独确定的H程序。为了展示我们的方法，我们通过模拟实验研究了在模型错误指定下所提出的平均程序的有限样本行为。