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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.

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Sandra Addo
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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.
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Michael Chernick
Director of Biostatistical Services
Lankenau Institute for Medical Research
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Sandra,

I just spent the last several months putting the techniques in Wayne Fuller's 1976 time series book into R functions.  In Section 7.4 of the 1976 book, Dr. Fuller gives methods for working with multivariate spectrums, which includes estimating the phase between two time series and testing if the phase is different from zero.  In the book, the variable for the phase is phi. 

If you are interested, I can email you the R function.


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Margot Tollefson
Owner
Vanward Statistical Consulting
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Not that it matters but I think Fuller has since come out with a second edition of the book.  As an aside in 1981 when I taught a graudate seminar in time series I used Fuller's book.
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Michael Chernick
Director of Biostatistical Services
Lankenau Institute for Medical Research
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Original Message:
Sent: 07-08-2011 15:21
From: Margot Tollefson
Subject: Spectral Analysis

Sandra,

I just spent the last several months putting the techniques in Wayne Fuller's 1976 time series book into R functions.  In Section 7.4 of the 1976 book, Dr. Fuller gives methods for working with multivariate spectrums, which includes estimating the phase between two time series and testing if the phase is different from zero.  In the book, the variable for the phase is phi. 

If you are interested, I can email you the R function.


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Margot Tollefson
Owner
Vanward Statistical Consulting
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Michael,

You are correct.  I took time series from Fuller in 1990, I believe, before he had the new edition out, so the old edition is what I have on my shelves.

Margot

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Margot Tollefson
Owner
Vanward Statistical Consulting
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Original Message:
Sent: 07-08-2011 16:06
From: Michael Chernick
Subject: Spectral Analysis


Not that it matters but I think Fuller has since come out with a second edition of the book.  As an aside in 1981 when I taught a graudate seminar in time series I used Fuller's book.
-------------------------------------------
Michael Chernick
Director of Biostatistical Services
Lankenau Institute for Medical Research
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