This paper is concerned with the findings related to the robust first-stage F-statistic in the Monte Carlo analysis of Andrews (2018), who found in a heteroskedastic grouped-data design that even for very large values of the robust F-statistic, the standard 2SLS confidence intervals had large coverage distortions. This finding appears to discredit the robust F-statistic as a test for underidentification. However, it is shown here that large values of the robust F-statistic do imply that there is first-stage information, but this may not be utilized well by the 2SLS estimator, or the standard GMM estimator. An estimator that corrects for this is a robust GMM estimator, denoted GMMf, with the robust weight matrix not based on the structural residuals, but on the first-stage residuals. For the grouped-data setting of Andrews (2018), this GMMf estimator gives the weights to the group specific estimators according to the group specific concentration parameters in the same way as 2SLS does under homoskedasticity, which is formally shown using weak instrument asymptotics. The GMMf estimator is much better behaved than the 2SLS estimator in the Andrews (2018) design, behaving well in terms of relative bias and Wald-test size distortion at more standard values of the robust F-statistic. We show that the same patterns can occur in a dynamic panel data model when the error variance is heteroskedastic over time. We further derive the conditions under which the Stock and Yogo (2005) weak instruments critical values apply to the robust F-statistic in relation to the behaviour of the GMMf estimator.

中文翻译：

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弱工具、第一阶段异方差、稳健 F 检验和具有基于第一阶段残差的权重矩阵的 GMM 估计器

本文关注 Andrews (2018) 的蒙特卡洛分析中与稳健的第一阶段 F 统计量相关的发现，他在异方差分组数据设计中发现，即使对于稳健 F 统计量的非常大的值，标准的 2SLS 置信区间具有较大的覆盖失真。这一发现似乎否定了作为识别不足检验的稳健 F 统计量。然而，这里表明，鲁棒 F 统计量的大值确实意味着存在第一阶段信息，但这可能不会被 2SLS 估计器或标准 GMM 估计器很好地利用。对此进行校正的估计器是稳健的 GMM 估计器，表示为 GMMf，其稳健的权重矩阵不是基于结构残差，而是基于第一阶段残差。对于 Andrews (2018) 的分组数据设置，该 GMMf 估计量根据组特定浓度参数为组特定估计量赋予权重，其方式与 2SLS 在同方差下所做的相同，使用弱仪器渐近线正式显示。GMMf 估计器比 Andrews (2018) 设计中的 2SLS 估计器表现得更好，在稳健 F 统计量的更多标准值下，在相对偏差和 Wald-test 尺寸失真方面表现良好。我们表明，当误差方差随时间异方差时，动态面板数据模型中可能会出现相同的模式。我们进一步推导了 Stock and Yogo (2005) 弱工具临界值适用于与 GMMf 估计器行为相关的稳健 F 统计量的条件。