Time series LOESS prediction performs better on aggregate data? I'm using the stl function in R (https://stat.ethz.ch/R-manual/R-devel/library/stats/html/stl.html) that uses loess to decompose time series and make predictions.
When the time unit is one month, the error is about 30%. However when I sum over 12 months, (and compare to the actual sum of the variable I'm trying to predict), the error drops to 4%.
My question is: is this a realistic result? My knowledge of variance, and variance of a sum seems to suggest that the error should be higher when tested on the sum over 12 months.
 A: You are expressing error as a percentage, suggesting you're probably dividing by something like the total for that time period or some mean  ($\hat{\sigma_i}/\hat{\mu_i} = 0.3$, say) -- so you appear to be talking about something like a (sample estimate of) coefficient of variation.
Let's say that the annual total, $T = \sum_i Y_i$, where $Y_i$ are monthly variables.
While it's true that the variance of a sum will be larger than the variance of the individual elements ($\sigma^2_T>\sigma^2_i$), if you divide by something that has an expectation that grows like $n\mu_i$, then the variance of the result will be divided by $n^2$. So if you had independence, for example, the $n$ times the variance in the numerator would be divided by $n^2$ in the denominator, and the resulting coefficient of variation $\sigma_T/\mu_T$ would decrease as $n$ increases, as would an estimate of it based on data.
Of course you probably don't have independence and the $\mu_i$ and $\sigma_i$ aren't equal, but under fairly weak assumptions the overall conclusion (that the coefficients of variation will tend to go down as you add terms) should still tend to hold.
So while $\sigma_T>\sigma_i$ (in whatever the original units are), typically $\frac{\sigma_T}{\mu_T} <\frac{\sigma_i}{\mu_i}$.
That may not account for the entirety of the effect you observe, but the fact that you see a decrease should not of itself be at all surprising.
