Hello everyone:
I'm trying to estimate production functions for a panel data of manufacturing with 2 identifiers (province, sector) so that each sector will have observations of the different provinces. The first thing I do is to egen a new ID by group(province sector), but it leads to ignoring the unobservable common trend within each province, or sector apparently.

I was considering a fixed effect (LSDV) or a semi-parametric (e.g. Levinsohn and Petrin). The problem is:
(1)for the former fashion, how to correctly set factor variables;
(2)for the latter, how to correctly get betas of K and L for every province-sector section.

The attachment dataex.txt is a part of my data file.The models I thought were:
(1) reg lnYL_go lnKL i.prov_sec_id i.prov_sec_id#c.lnKL i.actual_year, vce(cluster prov_sec_id) (lnYL and lnKL not included, they are simply ln(Y\L), etc, assuming CRS.)
(2) prodest lnY_va, free(lnL) state(lnK) proxy(lnInt) met(lp) va acf id(prov_sec_id) t(actual_year)

I'm not trying to be a free rider, it's just that related references are rare. Any opinion or suggestion would be appreciated, and happy new year!