====== Standard Error of Regression Coefficient ======
see https://academicweb.nd.edu/~rwilliam/stats1/x91.pdf
$ \hat{Y} = a + b X $ 에서 b 에 대한 standard error 값을 말한다. 이 standard error는 샘플에서 구한 b값으로 모집단의 (population의) b값을 추정하는 데 쓰인다. 
\begin{eqnarray*}
\text{se}_{b} & = & \sqrt { \dfrac {\text{MSE}} {\text{ss(x)} } } \;\;, \text{ MSE } & = & \text{Mean Square Residuals (Errors)} \\
& = & \dfrac {s_{e}} {\sqrt{ss(x)} } \\
& = & \dfrac {s_{e}} {\Sigma{(x_{i}-\overline{x})^2}} \\
& = & \dfrac { \sqrt {\dfrac{ \Sigma{(y-\hat{y})^2} } {n-2} } } {\sqrt{\Sigma{(x_{i}-\overline{x})^2}} } \\
& = & \sqrt {\dfrac { \dfrac { \Sigma {(y-\hat {y})^2 } } {n-2} } { \Sigma{(x_{i}-\overline{x})^2} } } \\
\end{eqnarray*}

\begin{eqnarray*}
\text{se}_{b_{k}} & = & \sqrt { \dfrac {\text{MSE}} {\text{ss(x)} } } \;\;, \;\;\; \text{where} \\
\text{ MSE } & = & \text{Mean Square Residuals (Errors)} \\
\end{eqnarray*}

====== in Multiple Regression ======
In multiple regression, the SE of a coefficient \(\beta _{j}\) is calculated using matrix algebra as 
  * $SE(\hat{\beta}_j) = \displaystyle \sqrt{s^2 \cdot C_{jj}}$, 
    * where $(s^{2})$ is the residual variance and 
    * $(C_{jj})$ is the corresponding diagonal element of the variance-covariance matrix $((X^TX)^{-1})$.

how to calculate standard error of regression coefficient in multiple regression