Yes I have adjusted my model, and used the VIF to remove the variables that had a too high VIF.

But by doing a shapiro test on the residuals of my response variable which is a lifetime, I rejected the null hypothesis.

So I don't know if having residuals from a normal distribution is one of the application conditions or not?

Can you please tell me more?

PS: You can fit a Gaussian GLM via maximum likelihood and it should come out to the same coefficients as OLS I'm pretty sure.

beta_hat = (X'X)^-1*X'y

It will be the same formula (for both Max Like and OLS) even if there is one or several independent variabels.

GLM means generalized linear models. That includes distributions from the exponential family, like the normal distribution, the binomial distribution, the Poisson distribution, the exponential distribution, the gamma distribution and others.

Then the formula will be iteratively reweighted least squares.