Suppose that I have three different time-series for per capita GDPs of countries. Let $X^j_{it}$ denote the GDP estimate of country $i$ in year $t$, made by organization $j \in {1,2,3}$ (so there are three organizations that publish estimates every year).
The estimates don't always agree with each other and for some countries the disagreements are particularly large. I would like to quantify which countries have the most "uncertain" estimates.
To that end, for each country $i$ and for each year $t$, I'll have an estimate from organization 1, organization 2 and organization 3: $(X^1_{it}, X^2_{it}, X^3_{it})$. I can calculate $\sigma_{it}$, which is just the standard deviation of the triplet.
Now for each country I can compute the average the standard deviation over time: $\mu_{\sigma_i} = \frac{1}{T}\sum_{t=1}^T \sigma_{it}$.
I can also get a "standard deviation of the standard deviations" as: $\sigma_{\sigma_{i}} = \sqrt{\frac{\sum_{t=1}^T(\sigma_{it}-\sigma_i)^2}{T-1}}$.
Now let's compare country 1 and 2.
Suppose that $\mu_{\sigma_1} > \mu_{\sigma_2}$, and that $\sigma_{\sigma_1} < \sigma_{\sigma_2}$. Which country has more "uncertain" GDP estimates?
On the one hand the average standard deviation is higher for country 1, but on the other hand the average standard deviation for country 2 is determined with less precision.
How can I quantify the uncertainty? And perhaps more importantly: am I even going about this in the right way?
