How to test statistical significance of coherence coefficients in spectral analysis?

Question

I would be very thankful for advising me how to test statistical significance of coherence coefficients. I have a statistical software that calculates them for me but it does not test their statistical significance. As far as I know spectral analysis results may be biased and high coherence may not prove anything. Therefore, I need a test that compares my results to a results I would get for two white noise series (at lest, this is my idea how this test would look like?).

My second request is whether there is any statistical package that would do it for me. If not, I would be thankful for formulae to calculate it on my own.

And last, but not least, is there any other method that would allow me to search for connection/similarities between two de-trended and cleared with Christiano-Fitzgerald filter series? I would like to prove that long-term US business cycles strongly influence business cycles in Europe (contrary to IMF results) but I do not know what tool to use to receive reliable results. When I used coherence I was told that high coherence can be the result of the method imperfections (spectral analysis) and that I must not even attempt to analyse similarities between UE business cycles and US business cycles using this approach.

Answer 1

I would think that the use of coherence (that is, correlation in the frequency domain) would require high-quality data. You can find some references for inference from http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.177.670

and its references.

Answer 2

Function spectrum in R on a bi-variate time series will produce both coherency and phase with confidence intervals.

Regarding your last paragraph: you want to "prove" that US cycles strongly influence Eusope cycles. You must bear in mind that coherency is a sort of correlation coefficient among frequency components aligned. How large the alignment is, you can see in the phase estimate: but phase is modulo $2 \pi$. In other words, you cannot tell a lead of $\omega$ radians from a lag of $2\pi - \omega$ radians.

Thus, from a bi-variate spectral analysis you may find that US cycles of a certain frequency are quite coherent with European cycles of the same frequency; but which one leads (which I assume is a requisite for "influencing") the other, you will have to decide on other grounds.