Why aren’t non-parametric statistics more popular in economics?

Abel, a loyal MR reader from Valencia, asks:

I've recently been reading an introductory book on nonparametric models (Nonparametric and Semiparametric Models: an Introduction – Härdle et al) and the apparent flexibility of the approach makes me wonder why aren't those models more used in empirical economics.

¿Are their drawbacks too big? (The so-called "curse of dimension")
¿Perhaps it's just economist's community lack of knowledge or willingness to learn?
¿Are they perceived as a threat to conventional or more established estimation methods?

I would be really glad to hear your opinion and also the feedback in the comments.

He could have added "signalling" to the list, since many non-parametric methods are relatively easy and thus do not demonstrate the skill of the researcher.  But the fundamental reason I think has to do with the nature of economics: non-parametric methods are most likely when you don't have a well-defined, formal structural model in mind.  But many MR readers know more about this than I do, so please offer us your opinions…