ASYMPTOTIC SIZE OF KLEIBERGEN’S LM AND CONDITIONAL LR TESTS FOR MOMENT CONDITION MODELS
矩条件模型中 Kleibergen LM 与条件 LR 检验的渐近尺寸AI 解读 ↓
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矩条件模型中 Kleibergen LM 与条件 LR 检验的渐近尺寸
- 研究问题
- 在参数未识别或弱识别的非线性矩条件模型中，LM 与条件似然比型检验能否一致控制渐近尺寸，且结果是否依赖加权方式？
- 方法与思路
- 作者在选定的零假设分布参数空间内分析一致意义下的尺寸与相似性，并区分参数维数、矩方差加权及雅可比方差加权。
- 材料陈述的贡献
- 摘要称，LM 与单参数 CLR 具有所述性质；多参数 CLR 在矩方差加权结合指定秩统计量时可获尺寸保证，另一加权未获保证。
- 阅读关注点
- 应核对零假设参数空间、一致相似性的定义及秩统计量条件，并理解多参数情形下关键样本量的渐近独立性为何影响结论。
- 阅读边界
- 仅据所给 Crossref 版本摘要，不能判断具体尺寸公式或全部识别情形；未保证雅可比方差加权正确，不等于断言其必然失效。
原文摘要
An influential paper by Kleibergen (2005, Econometrica 73, 1103–1123) introduces Lagrange multiplier (LM) and conditional likelihood ratio-like (CLR) tests for nonlinear moment condition models. These procedures aim to have good size performance even when the parameters are unidentified or poorly identified. However, the asymptotic size and similarity (in a uniform sense) of these procedures have not been determined in the literature. This paper does so. This paper shows that the LM test has correct asymptotic size and is asymptotically similar for a suitably chosen parameter space of null distributions. It shows that the CLR tests also have these properties when the dimension p of the unknown parameter θ equals 1. When p ≥ 2, however, the asymptotic size properties are found to depend on how the conditioning statistic, upon which the CLR tests depend, is weighted. Two weighting methods have been suggested in the literature. The paper shows that the CLR tests are guaranteed to have correct asymptotic size when p ≥ 2 when the weighting is based on an estimator of the variance of the sample moments, i.e., moment-variance weighting, combined with the Robin and Smith (2000, Econometric Theory 16, 151–175) rank statistic. The paper also determines a formula for the asymptotic size of the CLR test when the weighting is based on an estimator of the variance of the sample Jacobian. However, the results of the paper do not guarantee correct asymptotic size when p ≥ 2 with the Jacobian-variance weighting, combined with the Robin and Smith (2000, Econometric Theory 16, 151–175) rank statistic, because two key sample quantities are not necessarily asymptotically independent under some identification scenarios. Analogous results for confidence sets are provided. Even for the special case of a linear instrumental variable regression model with two or more right-hand side endogenous variables, the results of the paper are new to the literature.