Time delay embedding is a common method to estimate hidden structure in the system. You can make short-term predictions about future state of the system if you look at its previous states and find ones that are similar to current one. You can analyse transitional probabilities between these states. You can also study scale-invariant properties of the system such as fractal dimension or log-time scaling. There is a separate discipline called 'chaos control' that looks at certain unstable moments when the system can be manipulated by a small external signal (very relevant for dealing with heart attacks, should be interesting for central banks as well).
So, most categorically, current mainstream economic models are by far not the best we can do with at the time. They are grossly inaccurate representation of reality. There is of course emerging field of 'complexity economics' but it is not even close to being accepted in the mainstream.
The biggest issue is that modern mainstream economics is modelling the world as Gaussian. There is no room for 'fat tails' in econometrics, for example, every time you see 'regression' mentioned somewhere -- this should raise alarm. The world is not-linear.
Same applies to other concepts mentioned above. This one issue of crude inefficient approximation (stochastic world rather than the world as a complex nonlinear system) is enough to undermine most of modern economics. This is because a lot of this so called 'science' consists of many very elaborate and intellectually beautiful rigorous constructs, which are unfortunately built on an obsolete base.
Linear models had their place, maybe, 20 year ago, because there was nothing better. But now we have a very solid body of knowledge on which to build further scientific advances, most of it achieved through experiments (with computers), and not though some cute 'mental experiments'.
So, let's not be too sensitive, there is science that is supported by experiments, and there is a kind of abstract sophism which is modern economics. The sooner the public sees that this emperor has no clothes, the better.