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?


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