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Investopedia says that:

Multicollinearity is a statistical concept where independent variables in a model are correlated.

Also Investopedia says that:

An endogenous variable is a variable in a statistical model that's changed or determined by its relationship with other variables within the model.

What's the difference between an endogenous variable and multicollinearity?

Does a pair of independent endogenous variables cause multicollinearity?

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    $\begingroup$ Suppose you have a regression model in which you use $x_1, \dots, x_k$ as leading indicators to try to predict $Y$ for the coming month. All of the variables $x_1, \dots, x_k$ might be called 'endogenous'. It is possible that some of the $x_i$ are correlated with others. Presumably, for an econometric model at least some will be significantly correlated because they all reflect aspects of the recent economy. But theoretically, the $x_i$ need not be significantly correlated with each other. If $x_i$ are highly correlated, you may want to drop some of them to get a less variable model. $\endgroup$
    – BruceET
    Commented Jun 22, 2020 at 18:22

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Both definitions are awful.

Normally when you specify a model you have the left hand side (LHS) of the equation depending on its right hand side (RHS). So the purpose of modeling is to find the coefficients that will help you determine what should be LHS for a given RHS. Endogeneity arises when RHS also depends on LHS. Without getting into details, this can mess up your coefficients.

The variables on RHS not only can be, but usually are correlated with each other. It's rarely the case when they are not. That's why the definition of multicollinearity that you cited is terrible. Multicollinearity is a condition when the correlations are so high that they cause issues. This situation is not uncommon though.

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  • $\begingroup$ From your first paragraph, is endogeneity and reverse causality bias the same thing? $\endgroup$ Commented Jun 22, 2020 at 18:42
  • $\begingroup$ How does "RHS also depends on LHS" can be reconciled with the definition provided by stackoverflow that "Endogeneity refers to a situation where an explanatory variable in a model is correlated with the error term"? $\endgroup$ Commented Jun 22, 2020 at 18:47
  • $\begingroup$ it reconciles, e.g. see simultaneity. I find it more difficult to explain correlation with errors than pointing to RHS that depends on LHS. it just tends to be easier to imagine $\endgroup$
    – Aksakal
    Commented Jun 22, 2020 at 19:12
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    $\begingroup$ There is probabilistic dependence and there is causal dependence. Your second paragraph does not specify which one you mean, but the correct one is the causal dependence. The first paragraph (first sentence) is quite correct, though :) $\endgroup$ Commented Jun 22, 2020 at 19:22
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Typically, when we talk about endogeneity, we are interested in interpreting the output from a model as causal. Of particular concern is that there are causal variables and relationships that exist in the universe, but are not included in our model. If so, that means that the model's estimates are biased and should not be interpreted causally (but may be fine as estimates of the marginal associations). It is also possible that there are variables in the model that have complex causal relationships that are not captured by the way the model is set up. For example, there may be a mediational relationship amongst the X-variables. Another possibility is that Y causes X, and the data are being modeled the wrong way around.

Multicollinearity is when predictor variables are highly correlated with each other. If it gets to the point where the correlation is strong enough, it becomes very difficult for the model to determine what the relationships really are. If you were to get another dataset and fit the same model, you might find that you get very different estimates. To account for this fact, the standard errors that the model estimates for those variables becomes very large. A way to assess this possibility is to compute the variance inflation factors (VIFs) for your variables. The VIF is a multiplicative factor that tells you how much larger the variance of the sampling distribution (${\rm SE}^2$) is than it would have been if the variables had been perfectly uncorrelated. By convention, it is said that you have a problem with multicollinearity if the VIF is >10. Considering the case where there are only 2 X-variables, that means their pairwise correlation is >.95.

  • Note that the problem with multicollinearity is just estimating the marginal association with reasonable precision. It isn't about whether it is appropriate to interpret that marginal association causally. Furthermore, variables can be correlated without either being a cause of the other.

  • If one variable is causally related to another, that correlation need not be very large at all to bias the interpretation of the estimate. On the other hand, there can be a causal relation between variables without leading to a problem with collinearity, because collinearity doesn't occur until the correlation is really strong.

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  • $\begingroup$ I diasagree with the model's estimates are biased and should not be interpreted causally. If we take them as estimates of causal effects, they are biased. If we take them as estimates of the best approximation (in case of a linear regression, a linear projection), they need not be unbiased. Thus they are either biased or noncausal, but not both (except when there are other sources of bias in addition to endogeneity). Also, are casually related and casual relation intentional or typos? $\endgroup$ Commented Jun 22, 2020 at 19:16
  • $\begingroup$ @RichardHardy, that's what I was trying to say. Let me see if I can edit it to make it clearer. $\endgroup$ Commented Jun 22, 2020 at 19:20

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