question on Decomposition

[hardmann]

Dear Tom:


In the trend cycle decomposition of logarithmic time series multiplied by 100, why is the cycle exactly as a percentage ? Is there a certain allocation mechanism for the variance of trends and cyclic components within it? Why does the cyclic component happen to be a percentage of the trend, rather than a tenfold percentage, or a thousandth or even a ten thousandth ratio. When decomposition of seasonally unadjusted time series, does the seasonal component also show a percentage? Why?

Best Regard
Hardmann

[TomDoan]

I'm not sure I understand what you are asking. If you do an additive decomposition of data in 100 x logs, any "zero mean" component will have an interpretation of percentage differences.

[hardmann]

Dear Tom:

What I mean is why in trend/cycle decomposition, the amplitude of the cycle is mostly between positive or negative 5, rather than positive or negative 50, or positive or negative 0.5.


Best Regard
Hardmann

[TomDoan]

While it would certainly depend upon the series in question, for most macro aggregates a cycle that is even +/- 10 percentage points would be rather extreme. Again, if you are modeling a series at 100 x log, any of the components which aren't capturing the overall log level are going to be percentage deviations. If you have a cycle that is straying way off of zero (particularly if it's very persistent) then it's probably picking up something that maybe should be in the trend.