1

Suppose I have this model:

$$y = \beta_1x + \beta_2x^2 + \epsilon$$

I would like to fit it using OLS. In my data the correlation between $x$ and $x^2$ is $0.91$. After I rescale $x$ to zero mean and unit variance, the correlation is still $0.88$.

I'm worried that the model will be unstable, in the sense that a small shift in data will cause large changes in the coefficients. Is there anything I can do?

badmax
  • 1,659
  • 7
  • 19
  • 1
    Yes: Use orthogonal polynomials which are uncorrelated by design. You'll find a number of posts on this site on them. – COOLSerdash Dec 13 '21 at 21:10
  • 1
    You have no problem here: correlations of $0.88$ and $0.91$ are not going to create issues. The reason comes down to a natural hierarchy of the variables $x$ and $x^2:$ you will either consider both of them together or just $x$ alone, *but not $x^2$ alone.* See https://stats.stackexchange.com/questions/304831 and https://stats.stackexchange.com/questions/28730 for extended explanations and discussion. – whuber Dec 13 '21 at 22:34

0 Answers0