We propose a two-stage least squares (2SLS) estimator whose first stage is the equal-weighted average over a complete subset with k instruments among K available, which we call the complete subset averaging (CSA) 2SLS. The approximate mean squared error (MSE) is derived as a function of the subset size k by the Nagar (1959) expansion. The subset size is chosen by minimising the sample counterpart of the approximate MSE. We show that this method achieves asymptotic optimality among the class of estimators with different subset sizes. To deal with averaging over a growing set of irrelevant instruments, we generalise the approximate MSE to find that the optimal k is larger than otherwise. An extensive simulation experiment shows that the CSA-2SLS estimator outperforms the alternative estimators when instruments are correlated. As an empirical illustration, we estimate the logistic demand function in Berry et al. (1995) and find that the CSA-2SLS estimate is better supported by economic theory than are the alternative estimates.

中文翻译：

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使用多种仪器完成子集平均

我们提出了一个两阶段最小二乘 (2SLS) 估计器，其第一阶段是在K 个可用工具中有k 个工具的完整子集上的等权平均，我们称之为完整子集平均 (CSA) 2SLS。近似均方误差（MSE）推导子集大小的函数ķ由格尔（1959) 扩展。通过最小化近似 MSE 的样本对应物来选择子集大小。我们表明，该方法在具有不同子集大小的估计器类之间实现了渐近最优。为了处理对越来越多的不相关工具进行平均，我们概括了近似 MSE 以发现最佳k大于其他情况。广泛的模拟实验表明，当仪器相关时，CSA-2SLS 估计器的性能优于替代估计器。作为实证说明，我们估计了 Berry 等人的逻辑需求函数。(1995）并发现 CSA-2SLS 估计比替代估计更能得到经济理论的支持。