Slope Heterogeneity and TWFE estimator

[Frank Giaquinto]

Dear Professors,

I have panel data with 110 countries and 10 time periods, so N=110 and T=10.
The equation I need to estimate comes from economic theory and it is:

y_it = β₁ x_1,it + β₂ x_2,it + β₃(x_{2,it *} δ_it) + c_i + ξ_t + u_it,

where δ_it is a dummy variable taking value 1 if x_2,it ≥ K and zero otherwise, with K a scalar (a given threshold level).
There are some complications here, and what I’m gonna say comes from economic theory:
x_1,it is endogenous;
x_2,it is predetermined.
In your opinion, what's the best way to deal with endogeneity, slope heterogeneity and for the presence of country fixed effects?

Still, from theory, I cannot place a random effect assumption and I cannot assume that c_i is uncorrelated with my regressors.
The TWFE estimator will be inconsistent, but maybe it can still be a decent benchmark to compare point estimates using other estimators since the TWFE estimator is inconsistent of the order 1/T.

Since I don't have external instruments, the only way I know to deal simultaneously with country fixed effects and endogeneity is the Arellano and Bond estimator.
Is there any better option to consider to estimate the equation above and still exploit internal instruments?

Should I consider a multivariate time series analysis and rely on some of the recent work by Pesaran et al.? My actual time dimension is T=50 but I took 5-year averages to smooth out cyclical fluctuations in the data

Thanks a lot

[Jeff Wooldridge]

Frank: You have to be very clear about what you mean by endogeneity. If x_1,it is contemporaneously correlated with u_it then the bias of TWFE is not on the order of 1/T; it is essentially constant. Without external IVs, you can try AB, but then you're assuming the model is truly contemporaneous on no lags of x_1,it matter. If x_2,it is only predetermined and not strictly exogenous, you might have to instrument for it (and the interaction) in the differenced equation.

[Frank Giaquinto]

Dear Jeff,

Thank you very much for the insights. If x_1,it is assumed to respond to current and past shocks (it is endogenous), the bias is constant. Thank you for pointing this out.

On the other hand, x_2,it is assumed to respond only to past shocks (it is predetermined).

The point estimates of AB and TWFE are very close—does this indicate anything significant?

Should I also try using SYS-GMM to exploit additional moment conditions on the level part of the system of equations?

Thanks a lot for your help!!