How does the covariance matrix gets the confidence intervals of regression coefficients?

[TomHsiung]

Hi, everyone

I don't quite understand the math behind the way the covariance matrix determines the CIs of regression coefficients. It is too hard for me because I know little about the matrix math. I guess there are some theories similar to the Delta method to get the results of CIs. However, I don't know where I should start. I intend to understand the proof and I would like to hear your suggestions. Thanks.

[sbxkoenk]

To obtain confidence intervals (CI)
, you first need to know about standard errors on (ML) parameter estimates.

Standard errors for maximum likelihood estimation
By Rick Wicklin on The DO Loop November 6, 2023
https://blogs.sas.com/content/iml/2023/11/06/stderr-mle.html

Koen

[JacobSimonsen]

I assume that with "covariance matrix" you mean the covariance between the parameter estimates.

Then you find the standard errors by taking squareroot to the diagonal elements in the covariance matrix. 

Then you find CL by calculating the estimates +/- 1.96 x std errors.

[TomHsiung]

Hi, Jacob and all responders

Thank you for answering my question. Yep, so far I now this matrix method could be used to estimation the interval estimation of regression coefficients. However, I would like to understand the proof of this method because it is so fundamental and important for regression technology.

Tom