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I know this is a completely hypothetical scenario but I just want to understand how the effect of a variable could be held constant and how the coefficients of two independent variables are estimated in a multitple regression.

Let us think about this hypothetical story. Let us assume that there are two types of oils that increase hair length; olive and castor oil. Let's also assume that the two have totally different components and substances and function through totally different mechanisms, which means that they are not related at all, i. e there is no similar mechanism or substance in both that affect hair length. They are both totally unique and have a unique effect on the body, which increases hair length.

Let's assume that a scientist bought a mixture of both oils and observed hair growth. So he wanted to test how they affect hair growth. He did an experiment. When he used the mixture of both oils together he found out that for each 2 milligrams of the mixture (consisting of 1mg of each oil) , his hair grew by 1 cm. (Let's also for simplicity assume that there are no other factors affecting hair growth. Only the oils). He runs two simple linear regressions each with only 1 type of oil alone and the results of the regressions show him that for each milligram of oje type of oil; hair length increases by 1cm. However, he did not trust the results as he knows that this 1 cm increase is a result of both oils and not one. Therefore, he decided to do two seperate experiments to estimate the effect of each one alone and therefore keeping the other one constant.

When he used castor oil alone, he found that for each additional milligram; hair length increases by 0. 5 cm. He did the same experiment later using olive oil and also found the same effect, namely that an additional milligram of olive oil increases hair length by 0.5 cm. Thus the estimated effect of each one alone holding the other constant is 0.5. Now he knows that the 1 cm increase he witnessed before was partially because of olive and partially because of castor and was now successful to estimate each one's unique effect on hair length by running two different experiments.

The problem is trying to find the unique effect if it was a social science phenomenon. Let's assume these two oils are two oils that let the economy grow and not the hair anymore. Instead of castor oil and olive oil; they are human and physical capital (assuming that we know for sure that they increase the GDP which used to be the hair length in our model). Now the problem is that in a social science, we cannot manipulate an experiment like the natural scientist doing with both oils. An economic researcher cannot only increase human capital alone and see its effect on growth and do the same for physical capital while holding human capital constant. Instead, he deals with the data available and tries to control for variables. Therefore if he is in the natural scientist situation with exactly the same data but for the economic variables mentioned;he cannot do two seperate experiments. Instead, he only has the mixture of both increasing together exactly by the same amount. I know it sounds illogical to think that both independent variables are increasing together by the same amount but take the example of the oil mixture. Each time he adds it, it includes both oils, so they increase together. Since the social scientist can't seperate the oils and run two different experiments, he has to include both in a model to be able to hold one constant (as a control variable) and therefore see the effect of castor if oilve oil is held constant and see the effect of oilve oil if castor is held constant although in reality they were added together. Some softwares when running a multitple regression with both variables estimate the coefficient of one of the oils to be 1 and the other 0, while other softwares don't run the regression as it is a case of perfect multicollinearity as olive and castor oil increase together exactly by the same amount. The solution I know is dropping one of the variables, but the problem is that although they are mathematically correlated, in reality, each of them has a unique effect of 0.5 and if I drop one of them, then the effect of the other will be 1 although it is only 0.5.

So, is there anyway to solve this? How can social scientists estimate the effect of different variables on an dependent variable while holding other variables constant? especially when there is a high multicollinearity but just due to chance and not causation. It seems to me that in social sciences, we are following numbers and techniques that cannot estimate the truth.

I know I made many assumptions and it is too hypothetical. I know that in reality the effect could change when both are used together, but let's just assume this scenario for simplicty.

I would appreciate an answer that explains the concept of how to deal with this problem without going into too much math and algebra, which in this scenario failed to capture the true effects of both oils.

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  • $\begingroup$ @cure. Thank you so much. Will look into it $\endgroup$ – MMESAM Jun 11 at 20:07
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Mostly through a thing called natural experiment. Social scientists look for the situations in nature, that are not an experiment, but with use of (whole arsenal of) econometric tools, they may isolate the effect of one variable as it was an experiment.

Most popular methods are:

  • Instrumental variable.
  • Regression discontinuity design (also spatial version).
  • Matching methods (PSM and more).
  • Using changes in time (panel and DD methods).

A good thing to start would be Mostly Harmless Econometrics: An Empiricist's Companion. While most of this book is mathematics, i would suggest reading introductory parts to the chapters (pages 83-84 especially).

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