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Are there ways to estimate the finite sample bias with instrumental variables? I guess this would be conditional on assuming some structure to the problem and also would involve simulation, but, at least in applied econometrics papers, I have never seen people do it.

PS: If possible, the answer could be illustrated with R code.

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Using the model with one regressor and many instruments(vector-matrix notation) $$y = x\beta +u$$ $$x = Z\gamma +v$$

Bound, Jaeger & Baker (1995) give the following approximation to the 2SLS finite sample bias:

$$B(\hat \beta_{IV}) = \frac {\operatorname{Cov}(u,v)}{\gamma'Z'Z\gamma}(K-2)$$

where $K$ is the number of instrument (the column dimension of matrix Z). Naturally, one uses the estimated quantities to calculate the above. This as a special case of results in A.L.Nagar (1959).
Other relevant literature is A.Buse (1992) and Staiger and Stock (1997).

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