How to estimate model where instrument is correlated with dependent variable I have the following problem: I would like to estimate the effect of price variation caused by uncertainty on an outcome variable. P is my price, X is the variable measuring uncertainty and Y is the outcome. Now, generally I would set it up as an instrumental variable approach. However, my instrument X might also be correlated with Y directly and I would like to measure this effect in the second stage by including X as well. In other words, there are two channels through which uncertainty might affect Y: 1) directly, 2) indirectly through P. And I want to measure both effects.
First stage:
$$
P = \beta_0 + \beta_1 X + e_1
$$
Second stage
$$
Y = \beta_3 + \beta_4 \hat P+ \beta_5 X + e_2
$$
Does this make sense at all? If not, I would appreciate any suggestions and advice. Alternatively, please feel free to point me towards a literature that might be helpful or specific papers that deal with similar problems.
 A: Just by denoting one variable $P$ and another $X$ you didn't make one of them the instrument. The way you described $X$ invalidates it as a choice of an instrument variable. It's not different from the price in terms of its suitability as an instrument.
A: From the first equation we get:
$$
\hat P = b_0 + b_1 X
$$
where $b_0, b_1$ are the estimated regression coefficients.
As a result, $corr(\hat P, X) = 1$ and I don't see how you can use both $\hat P$ and $X$ in the second equation.
A: X and phat will have a correlation of 1 and cannot be used together. Look at commonality analysis.
A: I do not believe that it makes a whole a lot of sense. @James already showed that your $\hat P$ would be totally explained by your other covariates. 
If you believe that your instrument is relevant for your second stage estimation, you will be problably solving your endogeneity problem by just including your "instrument" ($X$) in your structural equation as an independent variable and not run a first-stage regression. I will try to explain it by being a bit loose with the notation. By Angrist and Pischke (2008), we know that your $\beta_5$ in OLS estimation might also be represented as:
$$\beta_5=\frac {Cov(Y,\hat X)} {Var(\hat X)}$$
In which $\hat X$ is the estimated residual on a regression on all of your covariates. That is, if you have $P$ covariates, $\hat X$ would be:
$$X=\delta_0+\delta_1P+\hat X$$
In a multiple linear regression context, the effect of an independent variable is the "net" effect after taking out the effect of other variables included in the equation. The same logic applies to $P$ and so you would be having something similar to your first-stage regression, but you would be including the residuals instead of the projected values. So, your final estimation in my opinion should be just:
$$Y=\beta_3+\beta_4P+\beta_5X+\epsilon$$
Bibliography:
ANGRIST, J. D. PISCHKE, J. S. Mostly Harmless Econometrics. New Jersey:
Princeton University Press, 2008. 373 p.
