Can sombedoy explain the assumption for the basic linear unobserved effects panel data model? (explanatory variables and unobserved effect)

[Lucas Bordure]

Good morning.
I would liike to know the assumption that concerns the explanatory variables and unobserved effects in a basic linear unobserved effects panel data model.

[Joro Kolev]

The linear panel data model is

(1) Yit = b*Xit + Ui + Eit,

where Ui is the scalar unobserved time constant effect/error that varies only across units, and Eit is the scalar idiosyncratic error that varies both across units and time; Yit is a scalar and Xit is a vector and both are observed, and b is a vector of unknown parameters to be estimated. Take here * to mean the inner product between the vector b and the vector Xit.

Then

(A1) The data is iid in the cross section, that is (Yit, Xit, t=1,2..T) is independent of (Yjt, Xjt, t=1,2..T) and both are unconditionally identically distributed.

(A2 fixed) E(Xis*Eit) = 0, for all t,s=1,2,..T, that is, the idiosyncratic error at t is uncorrelated with any Xis for s=1,2...T, some might say that Eit is strongly exogenous in eq.(1). The idiosyncratic error also has to be uncorrelated with the unobserved time constant effect E(Ui*Eit) = 0.
(A2 random) E[Xis*(Ui+Eit)]=0 for all t,s=1,2,..T, that is the composite error at t is uncorrelated with any Xis for s=1,2...T. Some might say that (Ui+Eit) is strongly exogenous in eq.(1).

Under (A1) and (A2 fixed) the fixed effects estimator is consistent for b.

Under (A1) and (A2 random) the random effects estimator is consistent for b.

[Lucas Bordure]

Your explanation is really clear. Thank you very much Joro Kolev.