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I am trying to "decompose" a standard deviation of an economy-wide variable into sectoral components. I have data for the year 2010 on the dispersion (standard deviation) of total economy GDP for 10 countries around the mean, where the mean is the cross-country average GDP. I also have sectoral (3 sectors) data for these same 10 countries, and can compute the standard deviation of the sector-level GDP for these three sectors in the similar fashion (so dispersion of the sector-level GDP for 10 countries, where the mean is the cross-country average of that sector GDP). The sum of the sectors GDP adds up to total economy GDP. What I am interested in finding out is whether there is something similar that can be done with the sector standard deviations, i.e. can they be aggregated in a way such that they equal the standard deviation of total economy GDP? The reason I am interested in this is that I want to be able to say that e.g. sector A accounts for 20% of the dispersion of total economy GDP. Does anyone have an idea whether this is possible? Any help is greatly appreciated.

Best,

Oz

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    $\begingroup$ My suggestion: edit the question and add more background. Be more specific about what sort of problem you are tryint to tackle. $\endgroup$
    – Sina Baghal
    Commented Jul 14, 2021 at 2:49
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    $\begingroup$ With this full generality of your statement, i am not quite sure if anyone can help. $\endgroup$
    – Sina Baghal
    Commented Jul 14, 2021 at 2:50
  • $\begingroup$ Let $Z=(V,W,Y....)$ and $d=(a,b,c...)$, which I assume is unknown, and assume $X$ has mean zero. Then you have the variance $\mathbb{E}[X^2] = \mathbb{E}[(d^TZ)^Td^TZ] = \mathbb{E}[Z^Tdd^TZ]$. You now want to retrieve $d$. But observe that for any orthogonal matrix $U$ you have that $\mathbb{E}[((dU)^TZ)^T(dU)^TZ]= \mathbb{E}[Z^Tdd^TZ] = \mathbb{E}[X^2] $. Thus, there is no unique solution for your $d$ (and thus no meaningful interpretation in the way you want it). $\endgroup$
    – a_student
    Commented Jul 14, 2021 at 14:09
  • $\begingroup$ Dear Sina, sorry for not making my question clear, I will adjust it. Dear user, thank you for your explanation. $\endgroup$
    – user319004
    Commented Jul 15, 2021 at 14:30

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Yes, this is possible and it is a well-researched question, particular in your special situation. You (probably) want to perform "Euler" or "Aumann-Shapley" allocation of the risk measure standard deviation. For $$ S=\sum_{i=1}^n w_i X_i$$ this simply boils down to $$ K_i=w_i\frac{\text{Cov}(S,X_i)}{\sqrt{\text{Var}(S)}}.$$

For details see "Capital Allocation to Business Units and Sub-Portfolios: the Euler Principle" by D. Tasche here in particular Section 3.1.

BTW: I think this question is better suited for quant.stackexchange.

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  • $\begingroup$ I am not sure whether this is what I meant, I rewrote my question, maybe that makes it clearer what I am interested in? Your approach seems to be a bit more theoretical if I am not mistaken. And thanks for pointing that out, I will post it there then. $\endgroup$
    – user319004
    Commented Jul 15, 2021 at 14:42
  • $\begingroup$ @user319004 If you would like a less theoretical answer, it seems pretty likely you will get a better answer on a different website. We're fairly theoretical here... $\endgroup$
    – Will Sawin
    Commented Jul 18, 2021 at 18:33

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