Large sample properties of GMM estimators under second-order identification
二阶识别下 GMM 估计量的大样本性质AI 解读 ↓
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二阶识别下 GMM 估计量的大样本性质
- 研究问题
- 当参数全局可识别，但雅可比矩阵存在一维秩亏且局部识别依赖二阶信息时，恰好识别与过度识别如何影响 GMM 极限分布？
- 方法与思路
- 作者通过不同阶数的 GMM 目标函数展开区分两类识别情形，分析慢收敛参数的符号分布，并讨论间接推断与最优权重。
- 材料陈述的贡献
- 摘要称，既有分布理论中的条件实际仅适用于过度识别；作者补充恰好识别情形、相关间接推断理论，并推导两组参数的最优权重。
- 阅读关注点
- 应核对全局识别与一维秩亏条件、不同识别情形所需的展开阶数，以及符号选择和最优权重矩阵的定义及推导。
- 阅读边界
- 这是 arXiv 预印本，同行评审或期刊接收未核验。摘要未给出完整正则条件与分布表达式，无法据此独立确认对既有理论的修正。
原文摘要
Dovonon and Hall (Journal of Econometrics, 2018) proposed a limiting distribution theory for GMM estimators for a p - dimensional globally identified parameter vector φ when local identification conditions fail at first-order but hold at second-order. They assumed that the first-order underidentification is due to the expected Jacobian having rank p-1 at the true value φ_{0}, i.e., having a rank deficiency of one. After reparametrizing the model such that the last column of the Jacobian vanishes, they showed that the GMM estimator of the vector comprising the first p-1 parameters, φ_{1}, converges at rate T^{-1/2} and the GMM estimator of the remaining parameter, φ_{p}, converges at rate T^{-1/4}. They also provided a limiting distribution of T^{1/4}(φ_{p}-hat-φ_{0,p}) subject to a (non-transparent) condition which they claimed to be not restrictive in general. However, as we show in this paper, their condition is in fact only satisfied when φ is overidentified and the limiting distribution of T^{1/4}(φ_{p}-hat-φ_{0,p}), which is non-standard, depends on whether φ is exactly identified or overidentified. In particular, the limiting distributions of the sign of T^{1/4}(φ_{p}-hat-φ_{0,p}) for the cases of exact and overidentification, respectively, are different and are obtained by using expansions of the GMM objective function of different orders. Unsurprisingly, we find that the limiting distribution theories of Dovonon and Hall (2018) for Indirect Inference (II) estimation under two different scenarios with second-order identification where the target function is a GMM estimator of the auxiliary parameter vector, are incomplete for similar reasons. We discuss how our results for GMM estimation can be used to complete both theories. We also derive the optimal weight matrices for φ_{1}-hat and φ_{p}-hat, respectively.