What is a non-parametric regression?

[Jun Park]

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 STATA different from the normal xreg?

[Jared Greathouse]

To be non technical, it's where no functional form is assumed about the distribution of the outcome. For poisson, for example, we assume a count distribution. For logit, we assume a functional form of 0 1, a logit.

[Jun Park]

Jared Greathouse Thanks for your reply! So by looking at the equation above, is there no assumed distribution? Because to me it just looks like a normal regression equation with covariates and fixed effects..

[Jared Greathouse]

I don't know how the Lagrangian multiplier is being used in this case. I'd need to look and read about that specific method.

[Jun Park]

Jared Greathouse
This is the excerpt from the book prior to that equation. And it's the only information that explains that equation:

In Denmark, I study the effect of involuntary unemployment on indebtedness in an event study framework, which is well-suited for administrative records with a long time-frame. This model estimates the effect of each year before and after unemployment in year t ̄i = 0 by defining an event-dummy vector for each individual i ranging from L years before and F years after unem- ployment. I code individuals as unemployed in a given year if they were unemployed for 40 weeks or longer.11 I consider household heads who were employed for at least two consecutive years before unemployment to cleanly identify the effect. Since individuals may experience more than one spell of unemployment during their time in the panel, I select the first year the head be- comes unemployment as the beginning of the spell and construct a balanced panel of individuals who are observed for at least five years before and after unemployment began. I then estimate the following non-parametric regression model: