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I understand that multicollinearity is a problem because the stronger the correlation, the more difficult it is to change one predictor without changing another and it becomes difficult for the model to estimate the relationship between each predictor and the response independently because the predictors tend to change in unison.

But why do the standard errors of the coefficients go up in the case of multicollinearity? Can someone give me some intuition to why this happens?

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Intuitively, the standard errors represent uncertainty in your coefficient estimates, so it's part of the exact problem you already laid out. Since the model has a harder time assigning the distinct effects of the collinear variables, there's more uncertainty in each of their estimates. That means the estimates are more imprecise, and you have larger standard errors.

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