Response surface regressions for critical value bounds and approximate p-values in equilibrium correction models

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.