Computing condition number for OLS

Question

In https://www.statsmodels.org/0.8.0/examples/notebooks/generated/ols.html, if you scroll down to the section "Multicollinearity" (bolded), it shows that the condition number of $X$ is $4.86e+09$. Then right below that, it says

Condition number One way to assess multicollinearity is to compute the condition number. Values over 20 are worrisome (see Greene 4.9). The first step is to normalize the independent variables to have unit length

and then they compute another condition number using

norm_x = X.values
for i, name in enumerate(X):
    if name == "const":
        continue
    norm_x[:,i] = X[name]/np.linalg.norm(X[name])
norm_xtx = np.dot(norm_x.T,norm_x)

eigs = np.linalg.eigvals(norm_xtx)
condition_number = np.sqrt(eigs.max() / eigs.min())
print(condition_number)

and the result comes out to be 56240.8714071 .

I'm confused about several things here.

(1) Are we more interested in the condition number of $X$ or $X^TX$ when assessing the conditioning of the system for OLS? The linear system of equations is $X \beta = y$, but the normal equations are $X^TX\beta = X^Ty$, which is also a linear system of equations. I'm assuming this depends on the method since some methods for solving OLS don't even form the normal equations. I think statsmodels in Python uses SVD, which just performs the decomposition for $X$, so what's the point of computing the condition number for a normalized version of $X^TX$ here?

(2) Why are they normalizing the independent variables before computing the condition number?

Answer 1

The condition number for the design matrix X as reported by statsmodels is an indicator for numerical problems that can be caused by either multicollinearity or bad scaling.

The literature on multicollinearity usually uses standardized variables that remove scaling problems and focus on multicollinearity. The old low recommended thresholds are for the standardized version.

Statsmodels uses the non-standardized version of the design matrix to signal that there might be problems also in the case when the design matrix is badly scaled even without multicollinearity. There are several questions on crossvalidated and stack overflow related to scaling problems, for example Regression model constant causes multicollinearity warning, but not in standardized model .

There was also a debate about whether the constant should be included in the collinearity measures. Some authors use the correlation matrix, i.e. demeaned and standardized, while Belseley, Kuh and Welch argue for including constant to signal problems with the level of the variables.

The computation of the variance inflation factor of the current function in statsmodels also uses the user provided design matrix. This means that in the usual case the vif for the constant is computed.
Two cases of possible problems with the constant are variables with very little fluctuations that are almost constant, i.e. have small variance relative to (nonzero) mean, and secondly cases like dummy variable trap where users do not drop a reference category and a categorical variable is "collinear" with the constant. The correlation matrix of the slope variables would not detect those problems.