Assessing causality and delay within a frequency band

The frequency-specific Granger causality test is extended to a more general null hypothesis that allows causality testing at unknown frequencies within a pre-specified range of frequencies. This setup corresponds better to empirical situations encountered in applied research and it is easily implemented in vector autoregressive models. Furthermore tools are provided to estimate and determine the sampling uncertainty of the phase shift/delay at some pre-specified frequency or frequency band. In an empirical application dealing with the dynamics of CO2 emissions and US temperatures it is found that emissions cause temperature changes only at very low frequencies with more than 30 years of oscillation. In a business cycle application the causality and leading properties of new orders for German industrial production are analyzed at the interesting frequencies.