Hello good people,

I am dealing with the following Linear Feedback Model:


y_{it} = a*y_{it-1} + exp[ X_{it}*B + h_i ] + u_{it} (1)

which leads to

y_{it} = a*y_{it-1} + exp[ X_{it}*B ]*V_i + u_{it} (2)


Where lambda = exp[ X_{it}*B ], and the fixed effects V_i = exp(h_i) may be seen as multiplicative and the error term u_{it} is additive, and B is a column vector of coefficients.


I am interested to estimate the above model with Chamberlain (1992) defined moment conditions, which first transforms the error term u_{it} to expunge the fixed effects:

s_{it} = [ u_{it} - a*u_{it-1} ] * exp[ ( X_{it-1} - X _{it} ) * B ] - [ u_{it-1} - a*u_{it-2} ]


and the subsequent moment conditions (assuming X is predetermined, that is the error term u_{it} is uncorrelated with current and past values of X):

E( s_{it} | y_{it-2}, x_{it-1} ) = 0


However if instead I assume that X is endogenous , that is the error term u_{it} is correlated with current and future values of X, then can I not use the following moment conditions ?

E( s_{it} | y_{it-2}, x_{it-2} ) = 0


In the literature Frank Windmeijer states that the above Chamberlain moment conditions only ever hold with predetermined X, something I have difficulty grasping.


This is important as Jeff Woolridge also suggested another such transformation of the error term: q_{it} = u_{it} / lambda_{it} - u_{it-1} / lambda_{it-1}; However the subsequent GMM estimation with this transformed q_{it} has computational problems.

I would highly appreciate any feedback / help as to why the above defined Chamberlain moment conditions is said to be invalid with endogenous X as described above.