If the variables w k are strictly exogenous (with respect to the idiosyncratic error component), then any serial error correlation does not affect the validity of them (or any of their lags) as instruments. If there is serial error correlation due to the omission of relevant lags of w k as regressors, e.g. due to delayed direct effects of L2.(w k), then w k would not be strictly exogenous in the first place in a model with those omitted lags. Thus, saying that w k are strictly exogenous effectively is also a statement about the correct specification of the model dynamics.
In this regard, I wonder what your motivation is for including L.(w k) as regressors instead of w k. Sometimes, people do this to avert simultaneous feedback from the dependent variable. In that case, however, L.(w k) may not be endogenous any more, but they cannot be strictly exogenous either. At best, they would be predetermined (weakly exogenous). For predetermined variables, serial error correlation does matter for the validity of the instruments. Probably even more important, simply lagging the regressors for this argument typically creates model misspecification, which then puts the whole analysis in jeopardy.
In this regard, I wonder what your motivation is for including L.(w k) as regressors instead of w k. Sometimes, people do this to avert simultaneous feedback from the dependent variable. In that case, however, L.(w k) may not be endogenous any more, but they cannot be strictly exogenous either. At best, they would be predetermined (weakly exogenous). For predetermined variables, serial error correlation does matter for the validity of the instruments. Probably even more important, simply lagging the regressors for this argument typically creates model misspecification, which then puts the whole analysis in jeopardy.
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