Single-equation conditional equilibrium correction models can be used to test for the existence of a level relationship among the variables of interest. The distributions of the respective test statistics are nonstandard under the null hypothesis of no such relationship and critical values need to be obtained with stochastic simulations. We compute more than 95 billion F -statistics and 57 billion t-statistics for a large number of specifications of the Pesaran, Shin, and Smith (2001, Journal of Applied Econometrics 16: 289Ð326) bounds test. Our large-scale simulations enable us to draw smooth density functions and to estimate response surface models that improve upon and substantially extend the set of available critical values for the bounds test. Besides covering the full range of possible sample sizes and lag orders, our approach notably allows for any number of variables in the long-run level relationship by exploiting the diminishing effect on the distributions of adding another variable to the model. The computation of approximate p-values enables a fine-grained statistical inference and allows us to quantify the finite-sample distortions from using asymptotic critical values. We find that the bounds test can be easily oversized by more than 5 percentage points in small samples.

中文翻译：

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平衡校正模型中临界值界限和近似 p 值的响应面回归

单方程条件均衡校正模型可用于检验相关变量之间是否存在水平关系。在没有这种关系的零假设下，各个检验统计量的分布是非标准的，需要通过随机模拟获得临界值。我们为 Pesaran、Shin 和 Smith (2001, Journal of Applied Econometrics 16: 289Ð326) 边界检验的大量规范计算了超过 950 亿个 F 统计量和 570 亿个 t 统计量。我们的大规模模拟使我们能够绘制平滑的密度函数并估计响应面模型，这些模型改进并大大扩展了边界测试的可用临界值集。除了涵盖所有可能的样本量和滞后订单外，我们的方法通过利用将另一个变量添加到模型中对分布的递减影响，特别允许长期水平关系中的任意数量的变量。近似 p 值的计算可以进行细粒度的统计推断，并允许我们通过使用渐近临界值来量化有限样本失真。我们发现边界测试在小样本中很容易被放大超过 5 个百分点。