How to combine several time series into a useful average time series? Let's assume we have four time series a, b, c and d with 10 measurments.
a(1), ..., a(10)
b(1), ..., b(10)
c(1), ..., c(10)

d(1), ..., d(10)

a, b and c are assumed to show the same trend and periodicity.
The question is how can I compare d to a combination of a, b and c in order to test whether d is differing from the assumed trend and periodicity.
The problem is that a, b and c have different ranges, so an average
X(i) := ( a(i) + b(i) + c(i) ) / 3

is not useful.
My question is what would be a good way to reach a meaningful combination?
Would it make sense to normalize the mean of all series a, b, c, d to 1 and then compare d to the average of a, b and c? Or would I also have to normalize the standard deviation of all four series to 1 first?
 A: I am not particularly familiar with them, but it seems to me that this would be a reasonable case to use a VAR (vector autoregression) model for.
In particular, a VAR with a linear time trend or period-specific deterministic component would give you a nice summary statistic for the overall trend in a given period.  See for example equation (11.4) here:
http://faculty.washington.edu/ezivot/econ584/notes/varModels.pdf
You could then consider your series d as analagous to the exogenous variable in impulse response modeling (or the X in the equation listed above) and see its joint effect on the vector of Y's (your series a-c).  The F-test seems to be the standard way to do this.
Example
Here's an example that (I think) shows what you are looking to test:
set.seed(1)
n <- 10
dat <- data.frame( a = runif(n), b = runif(n), c = runif(n) )
dat$d <- apply( dat[ c(1,1:9), ], 1, function(x) x[1] + x[2]*3 + x[3]*4 + runif(1) )
mdl <- VAR( dat, p=1, type="const" )
causality(mdl,cause="d")

$Granger

    Granger causality H0: d do not Granger-cause a b c

data:  VAR object mdl
F-Test = 3.8527, df1 = 3, df2 = 16, p-value = 0.02994


$Instant

    H0: No instantaneous causality between: d and a b c

data:  VAR object mdl
Chi-squared = 2.6609, df = 3, p-value = 0.4469

The null Granger hypothesis is rejected at p<.05.  So a shock in d is useful in predicting future values of (a,b,c).
