What are we looking for when we want to determine which model is most efficient? In my course slide they often discuss which model is most efficient, but I don't really know what they are looking and how they determine efficiency.
Comparing the asymptotic efficiency of the OLS and the IV estimator makes sense only if the OLS estimator is also consistent. In that case, OLS is efficient by virtue of the Gauss-Markov Theorem, and IV is not efficient. This also makes sense intuitively as the IV estimator uses only correlation between the instrument and the endogenous (which is actually exogenous if OLS is consistent) variable to estimate its effect. OLS, on the other hand, uses the whole correlation, effectively instrumenting the independent variable with itself.
In finite samples, the mean squared error of the biased OLS estimator of an endogenous variable might, however, actually be smaller than the mean squared error of a correctly specified IV estimator. This is because of the efficiency loss (as described above) and because of the finite-sample bias of the IV estimator. Typical causes of this are many or weak instruments or both. However, it is generally hard to assess how well the OLS estimator is doing in comparison. One approach is to discard an IV estimator when the F-statistic from the first stage is less than 10.