What does it mean if the controls in my IV model are correlated with my instrument? I am seeking to understand what it means for my 2SLS IV model if my controls are correlated with my instrument (such that when I add additional controls to my model that are positively correlated with Y, the coefficient on my instrument declines).

Is it correct to conclude that my instrument is &quot;invalid&quot;?
Or is my instrument still valid, and this is normal?

I am having difficulty locating an answer. Thank you!
This would not make instrument necessarily invalid. For some 2SLS instrument model of form:
$$y_i  = \beta_0 + \beta_1 \hat{x_i} + \beta_2 k_i +\epsilon_i$$
$$x_i = \pi_0 + \pi_1 z_i + \pi_2 k_i +e_i $$
where $y$ is dependent variable, $x_i$ is the endogenous regressor, $k$ some controls and $z$ instrument the main conditions for instrument validity are:

$z$ has to be correlated with the $x$ variable $cov(x,z)\neq 0$ and have causal effect on $x$

$z$ can affect the $y$ only through $x$ and $z$ itself has to be exogenous $cov(z,\epsilon)=0$


Furthermore, instrument should not be weak which would invalidate inference from the regression and so 1st stage should have $F$-statistics higher then $10$ as a rule of thumb (although some new results suggest that actually correct rule of thumb would be having $F&gt;105$ - see Lee, McCrary, Moreira, &amp; Porter 2020).
You can find more rigorous proof of these conditions in econometric hanbooks such as Verbeek (2008) guide to modern econometrics or Angrist &amp; Pischke (2009) Mostly Harmless Econometrics. The point is that none of the conditions directly say instrument cannot be correlated with other controls as long as it does not violates any other of the conditions. In fact omitting them could lead to omitted variable bias violating the condition that instrument should be endogenous if they are important. However, having highly correlated variables can inflate the standard errors of your model due to multicolinearity, this being said multicolinearity does not lead to bias it only reduces the precision of the estimates and reduces statistical power.