On the Relative Robustness of the Size of Tests to Local Model Violations

当一个应用研究人员检验一个假设时，她可能会犯两个错误中的一个。她可以拒绝一个正确的零假设，也可以接受一个错误的零假设。为了实施检验，应用研究者选择一个显著性水平，这是研究者愿意犯第一种错误的最大概率。出现第一个错误的最大概率也称为测试的大小。相等（大样本）规模的竞争测试程序通常根据其相对功率属性进行排名，其中功率表示1减去第二类错误的概率。然而，注意到计量经济学模型的关键假设在实践中经常受到质疑，PI根据在局部违反某些模型假设的情况下相对较大的样本量失真，提出了另一种测试排名。PI包括在关键模型假设成立时具有等于名义大小的大样本量的比较测试，以及在确定模型时与固定替代方案一致的比较测试。除了根据新度量对现有测试进行排名之外，PI还打算研究是否存在最优测试，即在模型假设为真时具有正确渐近大小的测试类别中，对于给定程度的局部违反模型假设的测试，具有最小的大样本量失真的测试，并且在模型为点识别时具有一致性。在众多例子中，PI重点关注两个主要例子。首先，PI考虑了线性工具变量（IVs）模型中涉及结构参数向量的假设检验，其中IVs和结构误差项可能相关。当样本量n增加到无穷大时，相关性以n^（-1/2）的速率逐渐消失。当相关性实际上为零时，PI考虑具有正确的大样本量的测试，当iv很强且与误差项不相关时，PI考虑一致性的测试。在考虑的各种测试中，PI发现anderson -and- rubin型测试在IVs和误差项的局部相关下失真最小。其次，PI考虑对由矩不等式定义的部分识别模型中未知参数向量的测试。当矩不等式以n^（-1/2）的速率局部违反时，PI根据其大样本量失真对测试进行排名。PI发现，在考虑的测试中，基于插件渐近临界值的测试在局部错误规范下的尺寸扭曲最小。对于这两个例子，最优化理论正在研究中。边界影响：如果应用研究人员根据其有利的功率特性选择测试，那么当模型假设稍微违反时，她的推断可能会遭受严重的尺寸扭曲。不幸的是，模型违反似乎在经验应用中普遍存在。新标准建议使用限制尺寸扭曲的测试，同时在标准假设下保持一致。所提出的方法将具有广泛的经验影响，并具有改进推理的潜力。预计本研究提供的方法将经常被学术界和公共部门的社会科学应用研究人员使用。

When an applied researcher tests a hypothesis, she can make one of two mistakes. She can reject a true null hypothesis or accept a false one. To implement the test the applied researcher picks a significance level, which is the maximal probability at which the researcher is willing to commit the first type of error. This maximal probability at which the first error occurs is also called the size of the test.Competing testing procedures of equal (large sample) size are typically ranked according to their relative power properties, where power denotes 1 minus the probability of the second type of error. However, noting that key assumptions underlying Econometric models are often questionable in practice, the PI proposes an alternative ranking of tests according to their relative large sample size distortion under local violations of certain model assumptions. The PI includes tests into the comparison that have large sample size equal to nominal size when the key model assumptions hold true and that are consistent against fixed alternatives when the model is point identified. As a more ambitious goal--beyond ranking existing tests according to the new measure--the PI intends to investigate whether there exists an optimal test, that is, a test that has smallest large sample size distortion for a given degree of local violations of the model assumptions in the class of tests that have correct asymptotic size when the model assumptions are true and are consistent when the model is point identified.Out of many examples, the PI focuses on two lead examples. First, the PI considers hypothesis tests involving the structural parameter vector in the linear instrumental variables (IVs) model where the IVs and the structural error term may be correlated. The correlation fades away at rate n^(-1/2) as the sample size n increases to infinity. The PI considers tests that have correct large sample size when the correlation is in fact zero and that are consistent when the IVs are strong and uncorrelated with the error term. Of the various tests considered the PI finds that Anderson-and-Rubin-type tests are the least distorted under local correlation of the IVs and the error term.Second, the PI considers tests for the unknown parameter vector in partially identified models defined by moment inequalities. The PI ranks the tests with respect to their large sample size distortion when the moment inequalities are locally violated at rate n^(-1/2). The PI finds that among the tests considered those based on plug-in asymptotic critical values are the least size distorted under local misspecification. An optimality theory is under investigation for both examples.Borader Impact: If an applied researcher chooses a test based on its favorable power properties, then her inference may suffer from severe size distortion when the model assumptions are slightly violated. Unfortunately, model violations seem to be pervasive in empirical applications. The new criterion instead suggests using tests that limit the size distortion while still being consistent under standard assumptions. The proposed methods will have broad empirical impact and has the potential to improve inference. It is expected that the methods provided by this research will find frequent use by applied researchers in social sciences within academia and the public sector.