A method has been developed to suss out causality in natural experiments. Joshua Angrist and Guido Imbens were awarded the Nobel Memorial Prize in Economics for their work in developing those methodologies.
I can't begin to understand math on how that is done or if this guy at Columbia used any of those methods, but it is possible to determine causality in natural experiments in some cases.
Also, on the specific claim, I think some more controlled experiments with UBI (Universal Basic Income) have shown that giving money to poor people can significantly reduce anxiety and improve scores related to happiness and well-being. (sorry I can't remember those specific studies).
So at least for poor people there is a causal relationship between money and happiness. (This is common sense if you've ever spoken with someone who is going hungry or who is on the edge homelessness.)
@Pat (This being said, causal inference is very difficult to apply to the question of money and happiness because there is a lot of potential confounds. This doesn't mean that we can't conclude anything but that the answer may not be black and white)
@Pat Yes the mathematical framework of causal inference is very useful (I know it quite wel, as a matter of fact I have been using it on my own data). I'm not claiming that only randomised controlled trials can provide evidence about causality... only that Killingsworth Kahneman and Mellers didn't do their job properly