I don't want to try to answer all your questions but I will attempt 1 and 2 and give a comment about 3 and 4.
1) The test for significance that you are using is standard. You should do the test because the periodogram is just an estimate of the spectral density and you can observe spurious peaks in it.
2) There is always the possibility of a type I error with a significance test. As far as proceeding with cross-spectral analysis I am assuming that you are trying to determine that both series have the same period shifted by a time lag the cross-spectral analysis might help you determine that. I am also assuming that you have some apriori reason for believing there is a common period.
If that is the case and both series have a peak at the same frequency with one highly significant and the other marginally non-significance I think common sense would say that there is some evidence of a common period and the cross-spectral analysis might help confirm it. On the other hand if one is marginally significant and the other marginally non-significant I would have more reason to think the commonality is just a coincidence (chance occurrence). I think you could also similarly look at this in the time domain.
3) As with any sample statistic how high a statistic should be to reach a particular conclusion should depend on the sample size and the result of a statistical hypothesis test.
4) If you determine that the two series are cross correlated you won't be able to tell from the data whether or not one series is leading or lagging the other but the magnitude of the lead/lag could be identified by simply looking at the two series. It also might be identified by the lag/lead with the highest cross-correlation. Not sure about the phase or cross-spectrum but I think they also can help identify it.
Sorry that I am a little rusty on my spectral analysis. Perhaps others can give better answers on 3 and 4.
-------------------------------------------
Michael Chernick
Director of Biostatistical Services
Lankenau Institute for Medical Research
-------------------------------------------
Original Message:
Sent: 07-07-2011 17:10
From: Sandra Addo
Subject: Spectral Analysis
Hello all,
Thanks so much for the reply on quantifying seasonality and volatility. I have another question:
I am using spectral analysis to determine cyclic components in some variables and also cross-spectral
to assess association between the variables and other econometric variables. These are my questions:
1) How do I assess statistical significance of the peaks?. Currently, I am using Fisher Test on the variance explained by the periodogram. Must I even bother with significance test of the peaks?
2) If the peak in one of the variable is not statistically significant based on the above test, does it mean I should not proceed with cross-spectral analysis?
3) How high should the coherence value be to conclude that there is a relationship between the time series?
4) How do I determine the lead-lag relationship based on phase function?
I have read a lot of papers and some books but I am getting confused.
Thank you very much.
-------------------------------------------
Sandra Addo
-------------------------------------------