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How to deal with multicollinearity in polynomial regression? Suppose I have $x$, $x^2$ and $x^3$ as independent variables in my regression equation. How can I calculate and remove multicollinearity from my model?

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    $\begingroup$ If interpretation of the coefficients is not a huge concern, demeaning $x$ has helped me in practice. doing so essentially removes all correlation between $x$ and $x^2$. If you further standardize $x$ (by dividing by standard deviation) than your resulting coefficients can be interpreted as change per standard deviation, squared standard deviation, and cubed standard deviation from the mean of $x$ respectively. If your looking for more flexible curve fitting, splines and kernel regressions have there advantages too... $\endgroup$ Apr 20 '15 at 16:45
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Don't use polynomial transformations "as such", because they will be collinear, as you note. Instead, transform them into orthogonal polynomials. In R, use the poly() command.

Even better, don't use higher order polynomials at all, since they will become unstable at the boundaries of your data space. Instead, use splines. In R, look at the splines package. For more information, look at Frank Harrell's Regression Modeling Strategies.

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    $\begingroup$ From the standpoint of pure prediction, extreme algebraic co-linearities do not hurt. $\endgroup$ Apr 20 '15 at 12:21

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