Let's suppose we have 3 vectors $U_1$, $U_2$ and $U_3$ we construct another vector $V=U_1+U_2+U_3$.
If $X$, $Y$ and $Z$ and independent, we can expect that $VAR(V)=\sum_{i=1}^3 VAR(U_i)$, right?
Now let's look at a more practical example involving time series. We can decompose time series into trend, seasonality and error using STL, so in this case, we can say that the variables $U_i$, are the components extracted via STL decomposition and the $V$ variable represents the original series.
Consider the following example in R:
set.seed(123)
x <- ts(rnorm(144, sd=1), frequency=12)
a <- stl(x, s.window="periodic")
sum(apply(a$time.series, 2, var) / var(x))
# [1] 0.9621605
Can we then conclude that the components (trend, seasonality and errors) are not independent? If so, then why?
In the above example, the value is pretty close to 1, but I have seen instances where this value is approximately 0.7, but I'm afraid I cannot share this data.
