This paper develops methodology for semiparametric panel data models in a setting where both the time series and the cross section are large. Such settings are common in finance and other areas. We allow for heterogeneous nonparametric covariate effects as well as unobserved time and firm specific fixed effects that may depend on the covariates in an arbitrary way. We use a fixed effect transformation to eliminate the nuisance parameters and then estimate the heterogeneous covariate effects using time series nonparametric regressions. We propose a dimensionality reducing common component structure that allows us to model the covariate effect parsimoniously. We obtain the asymptotic theory of our proposed procedures. We apply our methodology to a specific application, that has been the subject of recent policy interest, that is, the effect of trading venue fragmentation on market quality, such as liquidity and volatility. We use a unique dataset that reports at the weekly frequency since 2008 the location and volume of trading on the FTSE350 companies until 2011. We find that the effect of competition, as measured by the Herfindahl index of concentration, on market quality is nonlinear and non monotonic. The implied quality of the market under perfect competition is superior to that under monopoly provision, but the transition between the two is complicated.