The ability to identify a causal relationship in a controlled experiment has nothing to do with the statistical model used—it comes from the logic of the experimental design. Only if the experiment is designed properly will statistical analysis of the results provide any insights into cause and effect.
Unfortunately, for some scientific questions, experimentation is hard, or even impossible. Climate change is a good example. Even though it’s possible to manipulate the climate (as indeed we are currently doing, by adding more greenhouse gases), we can’t set up a carefully controlled experiment, because we only have one planet to work with. Instead, we use numerical models, which simulate the causal factors—a kind of virtual experiment. An experiment conducted in a causal model won’t necessarily tell us what will happen in the real world, but it often gives a very useful clue. If we run the virtual experiment many times in our causal model, under slightly varied conditions, we can then turn back to a statistical model to help analyze the results. But without the causal model to set up the experiment, a statistical analysis won’t tell us much.
