# Index as sum of random variables: Reduction of dimension

I have a multivariate time series X, that is, T observations of Dimensions K. These K different factors are summed up to form an index (equally-weighted for simplicity):

$I_t = X_t^1 + X_t^2 + .. + X_t^K$

where $X_t^i$ is the i-th factor of the observation at $t$.

My question would be how to choose a smaller number of factors $\hat{K}<K$ that already explains a large majority of the variation of index $I_t$.

My approach would have been to run step-wise regressions / calculate variance-inflation factors to determine which factors to add and which not, similarly as for a standard regression in which you have a large amount of potential predictors that are highly correlated and you need to decide which to include in your analysis.

Does this approach make sense? PCA also came to mind, but does not seem to fit the problem.

Thanks a lot for any help & best regards noclue

• It would help if you gave some more details about what you plan to do with the index after constructing it. But generally speaking it will not be a good idea to use equal weights. At the very least, you want to make sure that each of the $X_t^k$ is standardized. PCA is the first thing I would reach for in a setting like this: the first PC is the maximum variance linear combination of the $X_t^k$ which sounds like what you're after. Again I would probably want to standardize the variables first to avoid situations where one variable completely determines the first PC. – inhuretnakht Dec 22 '17 at 15:45
• Thanks a lot for your comment and sorry for my very late reply. You are right that my question is perhaps too vague / incomplete. The thing is, I have an index $I_t = \frac{\sum q_t^i p_t^i}{\sum q_0^i p_0^i}$ and I want to find a subset of its constituents which still explains most of its variation. So the question is more like reducing the dimension of the basket of products without removing too much information. What I just noticed is that one would also need to still fit the level of the index. So, products with large weights but low variation in prices would still need to be included.. – noclue Jan 3 '18 at 18:29