# Singularity issue, caused by polynomial regression, in the wald test of significance

I am trying to explain the housing sales prices and need for it to include the squared terms of some variables, for instance distance to the nearest forest, in order to capture the real effect. This lead my GVIF (generalized variance inflation factor) to exceed 10. I have read in the litterature that multicolinearity is not an issue when it concerns a variable and its squared term, so I moved on.

However, when I've tried to run a wald test of significance of my model, that contains heteroskedasticity robust standard error, I've got an error message telling that there's a singularity problem. I know that it is induced by the inclusion of some squared term, since when they are removed the problem disappear. I've tried "poly(x, 2, raw = TRUE)" and "poly(x, 2, raw = FALSE)" but the problem still remains.

So my questions are :

• Is it indispensable to conduct a Wald test at this stage if the F-test of my initial model was good ?
• How can I run a significance test of the model in the presence of heteroskedasticity and multicollinearity between x and x square ?

Thank you !

• How do you know it is induced by the inclusion of a squared term? – jbowman Jan 8 at 18:00
• @jbowman because when I exclude them it works. – Barry Jan 8 at 21:09
• What are the ranges of the values being squared? If they aren't large, the collinearity will be high, and there will be little extra information in the square term. That blanket statement about multicollinearity not being an issue is only true if the range of the term being squared is not small relative to its typical value, e.g., squaring numbers between 10 and 11 will give you a very high correlation (> 0.9999 if the numbers are uniformly distributed.) – jbowman Jan 8 at 21:35
• Thank you for your answer. It seems that the ranges are high enough between : 9-8282 and 18-11379. (Note that they are measuring distance in meters.) – Barry Jan 8 at 22:12
• One alternative would be to use a spline of distance instead of the linear and quadratic terms. – Peter Flom Jan 9 at 12:33