Assessing causality and delay within a frequency band

We extend the frequency-specific Granger-causality test of Breitung et al. (2006) to a more general null hypothesis that allows causality testing at unknown frequencies within a prespecified range of frequencies. This setup corresponds better to empirical situations encountered in applied research and it is easily implemented in vector autoregressive models. We also provide tools for estimating the phase shift/delay at some prespecified frequency or frequency band. In an empirical application dealing with the dynamics of US temperatures and CO2 emissions we find that emissions cause temperature changes only at very low frequencies with more than 30 years of oscillation. Furthermore we analyze the indicator properties of new orders for German industrial production by assessing the delay at the frequencies of interest.