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if the double arrows show that X and the error term are correlated, but that neither variable affects Y, is endogeneity a problem in this scenario? Why or why not?

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Assume the simplest linear regression model

$$y = bx + u$$

The OLS etimator for $b$ is

$$\hat b_{OLS} = \frac {\sum x_iy_i}{\sum x_i^2} = b + \frac {\sum x_ie_i}{\sum x_i^2}$$

Whatever the true $b$ is (it is zero in your case, as another answer pointed out) , the fact is that

$$\text{plim} \frac {n^{-1}\sum x_ie_i}{n^{-1}\sum x_i^2} \neq 0 $$

and so the OLS estimator will be inconsitent (and also, biased too).

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In a typical OLS model, $Y=\alpha+\beta X+\epsilon$, endogeneity exists when $E[\epsilon\,|\,X]\ne 0$, which results from $X$ and $\epsilon$ being correlated with one another.

In your case, $Y$ uncorrelated with $\epsilon$ implies only that $E[\epsilon]=0$, which is not the same as the exogeneity condition $E[\epsilon\,|\,X]= 0$. Moreover, $Y$ uncorrelated with $X$ just means that $\beta=0$. Again, this does not save you from the endogeneity problem. Actually, in addition to endogeneity, you now have another (bigger?) problem to deal with; that is, using a regressor that is a very poor predictor of the dependent variable.

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Watch out!!! This problem is very subtle.

The unobserved term $\epsilon$ cannot be the regression error term if $\epsilon$ is uncorrelated with $Y$. In the model $Y=\beta_0 + \beta_1 X + U$, the error term $U$ is a part of $Y$, and thus $U$ is 100% correlated with $Y$. In your case, $\epsilon$ is uncorrelated with $Y$, so it is not the error term unless $\epsilon = 0$.

[Edited from here]

The issue of endogeneity has nothing to do with whether $X$ and $Y$ are correlated. If $Y = \beta_0 + \beta_1 X + U$, $\beta_1 = 0$ and $X$ and $U$ are correlated, then $X$ is correlated with $Y$. On the other hand, if $\beta_1 \ne 0$ and $X$ and $U$ are "exactly" correlated, then it is possible that $X$ and $Y$ are uncorrelated.

The question only says that $X$ and $Y$ are uncorrelated. (It also says that $Y$ and $\epsilon$ are uncorrelated, but this information is irrelevant because it does not explain what $\epsilon$ means other than that it is unobserved.) So, endogeneity matters, not because $X$ and $\epsilon$ are correlated or whatever but because it says nothing about how $X$ and the error term are correlated.

If $\epsilon$ is the error term, then it is implied that $\epsilon = 0$ so we have a perfect fit and no regressor-error correlation.

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