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--- standard_error_of_regression_coefficient [2026/05/29 07:52] – hkimscil
+++ standard_error_of_regression_coefficient [2026/07/01 06:02] (current) – hkimscil
@@ Line 1: / Line 1: @@
 ====== 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{se}_{b} & = & \sqrt { \dfrac {\text{MSE}} {\text{ss(x)} } } \;\;, \text{ MSE } & = & \text{Mean Square Residuals (Errors)} \\
-\text{where} \\
+& = & \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