We propose a bootstrap test for unconditional and conditional Granger-causality spectra in the frequency domain. Our test aims to detect if the causality at a particular frequency is systematically different from zero. In particular, we consider a stochastic process derived applying independently the stationary bootstrap to the original series. At each frequency, we test the sample causality against the distribution of the median causality across frequencies estimated for that process. Via our procedure, we infer about the relationship between money stock and GDP in the Euro Area during the period 1999–2017. We point out that the money stock aggregate M1 had a significant impact on economic output at all frequencies, while the opposite relationship is significant only at low frequencies.

我们建议对频域中的无条件和条件格兰杰因果关系谱进行自举测试。我们的测试旨在检测特定频率的因果关系是否系统地不同于零。特别是，我们考虑了一个随机过程，该过程将固定自举独立应用于原始序列。在每个频率上，我们根据为该过程估计的频率之间的中值因果关系分布来测试样本因果关系。通过我们的程序，我们推断出 1999 年至 2017 年期间欧元区货币存量与 GDP 之间的关系。我们指出，货币存量总量 M1 在所有频率下都对经济产出产生显着影响，而相反的关系仅在低频率下显着。