How to simulate a confounding variable?

Asked Jul 21 '21 at 19:31

Active Jul 21 '21 at 19:31

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Suppose we have $X_{true}$ and $Y_{true}$ with a given correlation coefficient $\rho$ between them.

We also have a third variable $W$. The three variables may have any distribution you like.

Now we want to find $\alpha_1, \alpha_2 \in R$ such that $X_{obs} = X_{true} + \alpha_1 W$ and $Y_{obs} = Y_{true} + \alpha_2W$ have another given correlation coefficient $\tilde \rho$ between them.

In other words, I am trying to add a confounding variable to my original variables in a way that I can control their observed correlation coefficient.

One of the reasons I want to do it is to simulate the instrumental variable estimation method (and other identification strategies as well). That is, I am going to simulate an instrument $Z$ that will be able to recover the true relationship between $X_{true}$ and $Y_{true}$ when only $X_{obs}$ and $Y_{obs}$ are observed. Please, feel free to say if I am going the wrong way to achieve this.

asked Jul 21 '21 at 19:31

Gustavo Rangel

simulate all three together that is $(X_{true},Y_{true},W)$ where $X_{true}$ and $Y_{true}$ are correlated and $Y_{true}$ and $W$ are correlated but $X_{true}$ and $W$ are independent – bdeonovic Jul 21 '21 at 19:43
@bdeonovic what about $X_{obs}$ and $Y_{obs}$? – Gustavo Rangel Jul 21 '21 at 19:45
You have written out what they are equal to... – bdeonovic Jul 21 '21 at 19:53
just plug into those equations (after also simulating $\alpha_1$ and $\alpha_2$) – bdeonovic Jul 21 '21 at 19:53
Sorry I think I misread something – bdeonovic Jul 21 '21 at 19:58
What is the reason for distinguishing observed and true variables? And what is precisely your question? Maybe adding what have you done and how does it not fit your needs will help... – cure Jul 21 '21 at 20:07
https://stats.stackexchange.com/a/313138/919 answers this question and provides working `R` code. – whuber Jul 21 '21 at 20:23

How to simulate a confounding variable?

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