How to establish/describe a model where all variables are correlated and can all be viewed as independent variables to each other?

Confusing as it might sounds, allow me to make an example:

given, (time-series) variables A B C D E F, the correlations between any two variables are greater than 0.85. Is it possible to design a model in which we can use the other 5 variable to predict the 6th variable? Is there some non-Machine-Learning way? If not, how may I design it?

(In real life, such example sounds like some elemtary school homework. Give you some patterns and you can have a reasonable guess of whats missed in the blank.)

(I tried Mulitple Linear Regression and the result seem meaningless since the correlation was too high.)

[dpb]

I don't have much real practice/experience but the one general technique of which I am aware is VAR (Vector AutoRegression). The <Wikipedia Vector Autoregression Article> seems a reasonable overview.

If it weren't for the time series part, principal component analysis could be another trick...

[Xiaoxiao Du]

Thanks a lot! It's been very inspiring!

I've been consulting the VAR and PCA you mentioned along with other methodologies from my friends today.

I found PCA is quite suitable for this problem, although the result might be hard for human mind to comprehend. In my understanding, the problem is quite 'static', in which we use a period of historical data of all variables to conclude a 'relationship matrix' among them. Assume the relation could hold for 1 period (skipping the necessarity of doing 'forecasts'), we can derive the value we need from 'all other variables + relationship matrix'.

The other methodology on my list is 'Anomaly Detection' using Machine Learning. The description of 'Anomaly Detection' accurately describe my need, but I wonder if there might exist some other computation-easy, human-friendly methodology to solve this 'Anomaly Detection' Problem?