Unbiased IV estimation combining a endogenous variable and its endogenous lag

Here is a method of instrumenting a regression equation if there are no fully exogenous variables, but it is possible to make certain assumptions about the relation between the available endogenous variables. These assumptions are more likely to be met by lagged variables.

The unbiased estimation of the parameter B in the matrix equation Y = X.B + e presents difficulties if the available instruments W are correlated with the error term, since under IV estimation we have B(est) = B + E((W'X)^-1.(W'e)), and the expectation will be non-zero since E(W'e)<>0.

We may be able to find two instruments V and W such that

Var((V.X)^-1.V'e)=a^2.Var((W.X)^-1.W'e) + independent error

and

E((V.X)^-1.V'e)=a.E((W.X)^-1.W'e) + independent error.

The conditions say that the two instrumental variables are related in their behaviour relative to the error term, and may plausibly apply to W if it is the original variable X and V is one of its lags, when a would probably be expected to be less the unity. We can regress Var(V'e) on Var(W'e) to get an unbiased estimate of a. Then the first instrumental variable estimation using W on X gets E(B(est,W)) = B + delta where delta is the bias and E(B(est, V)) = B + a.delta, and we can calculate the bias as (B(est, V) - B(est, V))/(1-a(est)). This bias is asymptotically correct because of convergence in distribution of the numerator and in probability of the denominator. Thus, we can calculate the unbiased B as B(est, W) - bias(est).

Geometrically, the assumptions amount to allowing further projections of V on W beyond the usual IV ones. The assumptions no doubt could be weakened.