The screenshot below is from a paper that I am reading and the author says it is a non-parametric regression. The explanation below just seems like a normal OLS with some covariate, fixed effects.. etc. What exactly is a non-parametric regression and how do we see it from the equation below? When do we use it? The only noticeable difference from standard OLS seems to be the L function, which I don't understand.. Also, when running a non-parametric regression, is the function at R different from the normal lm function?
In general, this is an interesting question that comes up a lot.
I'll be the first to say "non-parametric" regression is not well-defined. You might be referred to Wasserman's text "All of Non-Parametric Statistics" which was the first seminal reference of its kind, attempting to broach the concept. The text wasn't without its issues, and I recall several of the professors in my department being deeply agitated by the material - actual mistakes, not just epistemological disagreements.
In general, to refer to something as "parametric" means that the terms in the regression model index a probability model. In Poisson regression, for instance, it's quite easy to take the design of $X$ and the estimated coefficients, and simulate responses from the results. The same is true of ordinary linear regression when it's treated like maximum likelihood of a normally distributed error term. But linear regression does not actually require normal errors. So, when we perform asymptotic inference, relying on the CLT to give us asymptotically correct CIs for the regression coefficients, we cannot say that linear regression is a parametric routine because our estimates do not, in fact, index a probability model. Whether or not "asymptotic" OLS is semi-parametric or non-parametric was an issue that not even my professors could agree on; but I'm in the non-parametric camp, if we are willing to make minimal assumptions about the existence of first and second moments.
So in my opinion there's nothing fundamentally wrong with writing down what looks like an ordinary least squares model and saying, "This is a non-parametric regression". Recall, a coefficient is only a parameter - necessitating parametric regression - if we claim to believe there's a probability model beneath it - an estimable probability model, that we know to be true, and for which our reg ression model provides reliable estimates of all the actual components.
In your example, the model description confirms that, while these are panel data, the authors are confident in the robustness and surfeit of data to assure us of the reliability of estimates for what appear to be a large number of fixed effects, whereas the error term has no descriptor other than being "idiosyncratic". One may only hope that this at least means these errors are independent or identically distributed - even if not, the OLS can be motivated, but I argue as a semiparametric estimator.