Why including extra instrument variables in 2SLS changes my regression coefficients? For example, when I include $A_i$ as my first instrument variable, I got the causal parameter estimate $=0.0592$ and if I included one more IV, $B_i$, my causal parameter estimate changes to $0.0505$. Why is this happening? Shouldn't they be the same (at least theoretically)?
My thinking is that, since in the first stage, we need to regress on our regressor using the IVs and we will get the different model when using different numbers of IVs such that when we use the fitted values for stage 2, we get different estimates.
Is my thinking correct here? Thanks!
 A: Mechanically, recall that TSLS can be thought as a two step estimation: in the first step, we use the instruments to 'predict' treatment, and in the second step, loosely speaking we replace actual treatment with predicted treatment, and the causal effect is the coefficient of predicted treatment when we regress outcome on predicted treatment. As such, it's very natural that if our prediction of treatment changes, then we are running the second regression on a different variable (its predicted treatment based on two instruments rather than one), and so there's no reason to expect the coefficient to be the same.
As for theoretical underpinnings, without getting into too much detail, the interpretation of the estimate as a "causal effect" only follows under suitable assumptions. Furthermore, what is a "causal effect"? It's easy to think there is one causal effect, but the effect may differ for different populations! In the simple binary instrument case, Imbens and Angrist (1994) showed that TSLS (with a monotonicity assumption) identifies a causal effect of treatment for those induced by the instrument (so called Local Average Treatment Effects, or LATEs). For example, if your instrument was, say, whether you are randomly asked to participate in treatment, then TSLS gets a causal effect for those who would participate in treatment if asked, and would not if not asked. In particular, this causal effect is not for those who do not participate in treatment if asked, and it's perfectly reasonable to expect treatment for those individuals to be very different.
Extending this intuition to your setting (which is not technically true, as non-binary instruments and multiple instruments somewhat complicate what you identify), what we have is that under the first model with one instrument, we got the causal effect for those induced by the first instrument, and in the second one with two instruments, we got the causal effect for those induced by some combination of the two instruments. Depending on what those instruments are, it's perfectly natural to expect the effects to be different across these two groups!!
Now, in practice, it's common to assume constant treatment effects, so that the effect for any subpopulation is the same. In this case, you'd have to chalk up this difference to noise, or issues with the instruments, or something else. But that constant treatment effect assumption is an extremely strong assumption.
