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I had some superficial understanding of multicollinearity, that two highly correlated variables in the regression model are not what we want, as the estimated coefficient would be biased.

Control variables are just like different names of the independent variables in the regression model that we are keen to keep constant to determine the marginal effect of the interest predictor.

Until I saw this video about the control variable Control variable , the main captured is attached.

enter image description here

Then isn't the multicollinearity and control variables is a dilemma

I also have seen this Does multicollinearity among control variables matter?

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    $\begingroup$ "two highly correlated variables in the regression model are not what we want, as the estimated coefficient would be biased." I would say that is this wrong. The coefficients will not be biased. If you control for a variable, that's a different effect. The effect of temperature on ice cream eating, and the effect of temperature on ice cream eating, controlling for shorts wearing, are different things. If you're talking about determining causation, that's different, but OLS does not "Mistakenly assign an effect" because OLS is not about causal estimates. $\endgroup$
    – Jeremy Miles
    Commented Feb 6 at 19:04
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    $\begingroup$ IMHO: Don't watch Youtube videos by random people Read books and papers which will have been at least superficially reviewed by someone vaguely knowledgeable. $\endgroup$
    – Jeremy Miles
    Commented Feb 6 at 19:05
  • $\begingroup$ @JeremyMiles thanks for your reply. Would you please enlighten me more about "The effect of temperature on ice cream eating, and the effect of temperature on ice cream eating, controlling for shorts wearing, are different things. If you're talking about determining causation, that's different, but OLS does not "Mistakenly assign an effect" because OLS is not about causal estimates." How are they different with/without"controlling for shorts wearing"? and regard your last sentence correct me if I am wrong. Isn't causal effect also obtained by OLS but with a subtle experiment -manipulation? $\endgroup$
    – LJNG
    Commented Feb 6 at 19:41

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There is so much wrong with what you quoted!

First, collinearity is not the same as correlation and can involve more than two variables.

Second, there is nothing wrong with any of the equations, and regression doesn't assign "effect" only association. What's wrong is using regression from an observational study to infer causation, but the regression didn't do that. "The fault, dear researcher, lies not in our statistics programs but in ourselves."

Third, the "true $\beta_1$" is probably not 0. Even at a given temperature, there is probably a relationship between shorts wearing and ice cream eating. People who feel warm when it is 75 degrees F are going to be more likely to wear shorts and eat ice cream at that temperature.

Fourth, this is not a multivariate regression, but a multiple one. Multivariate regression is when you have more than one dependent variable.

Now, to your question: You can easily have control variables with no problematic collinearity. The independent variables do not have to be perfectly orthogonal to each other. You can use collinearity diagnostics to see if you do have problematic collinearity (I prefer condition indexes to VIF, but they are both OK) and there are various ways to deal with it if you do have it (e.g. ridge regression, partial least squares, principal component regression, etc.).

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    $\begingroup$ I have you come across the Quantitude podcast? Their solution to collinearity is to "stop being a whiny-ass baby." $\endgroup$
    – Jeremy Miles
    Commented Feb 6 at 22:56

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