# Instrumental Variable: Bias & larger standard errors

Let's assume the instrument $$Z$$ satisfies instrumental exogeneity $$corr(Z, \epsilon)=0$$ and it's relevant $$corr(Z, X_{end})≠0$$. Where $$\epsilon$$ is error term and $$X_{end}$$ is the endogenous explanatory variable.

If my instrument is furthermore correlated with another exogenous covariate ($$X_{i \neq end}$$) which I specified in my model (it's not in the error term) would this result in bias ?

According to my intuition it should only induce larger standard error for the estimator of our endogenous variable and not induce bias. As our instrument is correlated with another exogenous covariate it will be unclear whether we can contribute (part of) the variation to our exogenous variable or endogenous variable, hence the standard error must be larger. Is my reasoning correct?

Thank you very much.

• Covariance of the instrument and an exogenous covariate won't add bias to the estimates. I'm not sure if you can necessarily says that it increases variance ( i.e. the standard error ) because it's a question of whether the respective diagonal element of $(X^{\prime}X)^{-1}$ becomes larger. I don't think one can assume that it does increase because the covariance is an off diagonal element of that matrix. There's a nice discussion of instrumental variables in a textbook that I was recently looking at. If I can find the title, I'll send it. Jan 16, 2023 at 5:08
• Well, my intuition tells me that it should have an effect... Because, doing two-stage regression, first I have to estimate $x_{end}$ with the use of $x_{exo}$ and $Z$. If they are strongly correlated, variance of estimates would be inflated, meaning the estimates might be less precise which then leads to higher probability of $Z$ not filtering out problematic information from $x_{end}$... Jan 19, 2023 at 19:45
• So I have simulated it right now... I do not think my previous idea was correct. I have gone from $corr(x_{exo}, Z) = 0$ to $corr(x_{exo}, Z) = 0.99$ and have seen no bias and no effect on standard errors. The simulation was performed on sample $n = 30000$ and replications $M = 1000$... Some more robust explanation might be needed here... Jan 19, 2023 at 20:21