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

当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。她可以拒绝真实的零假设，也可以接受错误的假设。为了实施测试，应用研究人员选择一个显著水平，这是研究人员愿意犯第一类错误的最大概率。这种发生第一错误的最大概率也称为测试的大小。相同(大样本)大小的比较测试过程通常根据它们的相对功率属性来排序，其中功率表示1减去第二类型错误的概率。然而，注意到支撑计量经济学模型的关键假设在实践中往往是值得怀疑的，PI建议根据测试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来对测试进行替代排名。PI包括对比较的测试，当关键模型假设成立时，大样本大小等于标称大小，并且当模型被点识别时，相对于固定备选方案是一致的。作为一个更雄心勃勃的目标--除了根据新的度量对现有测试进行排序之外--PI打算调查是否存在最优测试，即对于给定程度的模型假设的局部违反，当模型假设为真时具有正确的渐近大小并且当模型被点识别时具有一致的测试类中具有最小大样本大小失真的测试。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。当样本大小n增加到无穷大时，相关性以n^(-1/2)的速率逐渐消失。PI考虑当相关性实际上为零时具有正确的大样本大小的测试，以及当IV较强且与误差项不相关时一致的测试。在所考虑的各种检验中，PI发现在IVs和误差项的局部相关性下，Anderson-and-Rubin类型的检验是最不失真的。其次，PI考虑了由矩不等定义的部分辨识模型中未知参数向量的检验。当矩不等被局部违反时，PI根据测试的大样本大小失真对测试进行排序，比率n^(-1/2)。PI发现，在考虑的测试中，基于插件渐近临界值的测试是在局部错误指定下扭曲的最小大小。两个例子的最优性理论都在研究中。Borader影响：如果应用研究人员基于其有利的功率特性选择测试，那么当模型假设略有违反时，她的推断可能会受到严重的大小扭曲。不幸的是，违反模型似乎在经验应用中无处不在。相反，新标准建议使用限制尺寸失真的测试，同时仍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。预计这项研究提供的方法将被学术界和公共部门的社会科学应用研究人员频繁使用。

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.