Causal inference, a field I've been studying for a few months, is often touted as achieving this. That's not quite true -- the causes do not live in the data.

What causal inference does is estimate the parameters of a given causal model, and does so by applying algorithms on the model to determine which variables to control for. Once the researcher to supply directions of causality, you can use the data to validate the model, or ask questions about it that you already know the answer to as a cross check. But there can be multiple valid models, and the data alone cannot help you select the "right" one. Worse, sometimes (often?) validating the "right" model requires data you do not have access to. But you don't need to do the experiment to find that out at least!

Underlying all this appears to be Bayesian methods. Which makes sense as you are often seeking answers to conditional probabilities, Bayes is good for that.

First thing: all claims to have proved a causal relationship are relative to a set of assumptions which may or may not be satisfied (even in experimental sciences).

Taking the experiment as an ideal we cannot reach, there are many settings where there is useful variation in the data which can approximate the random assignment to treatment which is the essence of experiments.

Some canonical research designs which can be used here include differences in differences, regression discontinuity, synthetic control methods, or instrumental variables methods. They are differently appropriate to different settings but permit causal claims to be made. These methods are widely used in contemporary empirical economics (indeed this is a nearly exhaustive list of methods used in applied micro!).

Useful, accessible references here include Angrist and Pischke, “mostly harmless econométrics,” or “causal inference the mixtape” by a guy at Baylor whose name I’m forgetting, and the very recent “the effect” by Huntington-Klein, which I have not yet read.

Other slightly more exotic models are used in (for example) industrial organization which still permit causal claims to be made about the effects of increasing prices or changing product features on demand for a product or set of products.

There is a whole subfield dedicated to the question called causal inference. You can read something like the "causal inference : the mixtape" https://mixtape.scunning.com/

1. Experiments, and natural experiments

2. causal discovery methods + generic assumptions

Economists are really putting effort into sound methodology.