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I am currently looking into the correlation between academic freedom (my independant variable) and university rankings (my dependant variable) using OLS.

I find a negative significant correlation, but since I only have one dependant variable, this results suffers from omitted variable bias. In order to take this into account, I want to introduce some control variables into my model (that are both correlated with my independant and dependant variable).

One of these controls is the presence of democracy/civil liberties (as measured by the Freedom House dataset) in the country. When I regress academic freedom on these variables, I find a very high correlation (the coefficient for both democracy and civil liberties is 0.95).

Because of this high correlation, I was wondering if it was a good idea to use democracy/civil liberties as a control variable to isolate the effect of academic freedom. Is there a risk of a biased coefficient because of this high correlation ? If there is a risk, would the use of proxies for qualities of institutions that are less correlated with academic freedom be a solution to still control for democracy in my model ?

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In OLS you have omitted-variable bias if the omitted predictors would have non-zero coefficients in the model and are correlated with the included predictors. Then the included predictors are correlated with the error term in the model. To minimize bias in your study, you want to include all predictors that are correlated with academic freedom and would still be associated with the outcome when academic freedom is taken into account.

The problem with including other predictors correlated with academic freedom isn't bias. Including a predictor whose omission would lead to omitted-variable bias would, if anything, reduce bias. The problem with including multiple correlated predictors is that their individual coefficient estimates will then have high variances, due to multicollinearity. Depending on the data, you might thus find that neither predictor individually has a "significant" association with outcome.

That's not necessarily a problem. You could, for example, evaluate the combination of the measures of academic freedom and democracy together instead of trying to isolate them individually. The negative covariances among the coefficient estimates can offset the high variances of the individual estimates when you consider them together.

An alternative might be to treat the measure of democracy as an instrumental variable, as explained in detail in a Wikipedia article and more heuristically on this Cross Validated page. In that case you want the instrumental variable to be highly correlated with the included predictor of primary interest but not with the model's error term and not to be associated with outcome except insofar as the instrument is associated with the included predictor. Whether this makes sense in your situation requires careful application of your understanding of the subject matter. If you want to take that approach you should consult with a statistician or econometrician experienced in such modeling, as there are many underlying assumptions and ways that you might be led astray.

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  • $\begingroup$ Thank you for your very clear answer ! When I regress ranking on both academic freedom and democracy, neither of them have a significant effet. Can I conclude from that that when democracy is held constant, there is no correlation between academic freedom and university rankings ? $\endgroup$ Commented May 14, 2023 at 17:51
  • $\begingroup$ @MullerI. no, that's the point of the second and third paragraphs. With highly correlated predictors you can't "hold one constant" very well while you change another. Thus you have very high error estimates for the predictors individually. If the model as a whole is significant then you can say that academic freedom and democracy together are associated with university rankings, just not which is individually important. If democracy is a valid instrument here, you could use instrumental-variable methods to evaluate the causal association of academic freedom with rankings. $\endgroup$
    – EdM
    Commented May 14, 2023 at 18:15
  • $\begingroup$ So the only scenarion where I can say something about democracy and academic freedom's correlation with the dependant variable is when both are significant, right ? If only one or neither are significant, there's not much I can deduce ? (and I don't think using democracy as an instrument would work, I thought about it but it wouldn't satisfy the exclusion restriction I think) $\endgroup$ Commented May 14, 2023 at 18:54
  • $\begingroup$ @MullerI. even if neither one is significant individually, it's possible for them together to be significant. For example, an anova() comparison of a regression model that contains both democracy and academic freedom as predictors against a corresponding model that contains neither of them as predictors could show that the combination of them is significantly associated with ranking. $\endgroup$
    – EdM
    Commented May 14, 2023 at 22:10

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