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During a multiple linear regression analysis, I found correlation between intercept (beta-0) and two of the other regression coefficients. Is there any problem or concern in this case? If no, please explain me why.

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Such correlations are guaranteed if you have not standardized your predictors to take the value 0 at their means. However, this correlation is not really a mathematical/statistical problem, per se, but it may be easier to interpret the coefficients if you first standardize the variables.

Therefore, the short answer is no, such a correlation is not a problem, its just the interpretation. See this link as well for a good discussion on this issue. The reason the correlation is not a problem, statistically, is that standardizing is a linear transformation (add/multiply by a constant), which should not affect how well the line fits. As a concrete example...a linear regression of, say, Reaction Rate vs Temperature, (for some chemical reaction) should not depend on the choice of temperature units (Kelvin, Celsius, Farenheit). Essentially, the units you express a relationship in should not affet the accuracy or validity of your linear regression, since different units of some property (e.g., length, temp, time) are related to each other via a linear transformation, and it would make no sense for a particular, arbitrary set of units to yield more accuracy than another set of units.

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Well, I guess, you are saying - there is a big data set, and you divided the whole data set into multiple small parts. Then you ran linear regressions on each of these small subsets of data and calculated the correlation. But I am trying to guess the purpose. At least, I can think of one use of it you may thought of. I guess if the correlation values are too small or highly negative, then the subsets you selected are all very different in nature/distribution or the linear model does not fit good to all these subsets. But these can be better concluded from analyzing the whole data and comparing the goodness of fit measures viz. adjusted R square etc. And difference in distributions of the subsets can be directly obtained from analyzing the correlations of the variables or simply scatter plots of the subsets. So, for me, the approach solves no extra purpose.

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