# MLE of regression coefficients depends only on correlation matrix and single-predictor z-scores

Assume a standard linear model $$\boldsymbol{y}=\boldsymbol{X}\boldsymbol{\lambda}+\boldsymbol{\varepsilon},$$ where $$\boldsymbol{y}$$ and the columns of $$\boldsymbol{X}$$ are standardized. In some texts, I've seen the MLE of regression coefficients written as $$\boldsymbol{\hat{\lambda}}=\boldsymbol{(X^T X)^{-1}X^T y}=n^{-1/2}\sigma \boldsymbol{R^{-1}\hat{z}},$$ where $$R=n^{-1}X^TX$$ is the predictor correlation matrix and $$\hat{z}=(n\sigma^2)^{-1/2}X^Ty$$ is the "vector of single-predictor z-scores". I have trouble understanding the latter term. Can someone explain what the interpretation of $$\hat{z}$$ is?