Vector Autoregressions and Causality
推导了水平向量自回归和误差修正模型中格兰杰因果检验的极限理论，发现这些检验在存在随机趋势和协整时往往依赖于讨厌参数且分布非标准，对实际应用有警示作用。
A limit theory for Wald tests of Granger causality in levels vector autoregressions (VAR's) and error correction models (ECM's) is developed, which allows for stochastic trends and cointegration. Earlier work is extended to the general case, thereby characterizing when these Wald tests are asymptotically valid as 'x'(superscript 2) criteria. Our results for inference from unrestricted levels VAR are not encouraging: the limit theory often involves nuisance parameters and nonstandard distributions, a situation offering no satisfactory statistical basis for these tests. Granger causality tests in ECM's also suffer from nuisance parameter dependencies asymptotically and in some cases nonstandard limit theory. Both these results are somewhat surprising in light of earlier research. Copyright 1993 by The Econometric Society.