Godness of fit
Summary
- In this section \(e_i\) are the OLS residuals and \({\hat{y}}_i\) are the OLS fitted values
- Residual sum of squares \(RSS\) :
\[RSS=\sum_{i=1}^{n}{ e_i^2 }\]
- Definition explained sum of squares , \(ESS\)
\[ESS=\sum_{i=1}^{n}{ {\left( {\hat{y}}_i-\bar{\hat{y}} \right)}^2 }=\sum_{i=1}^{n}{ {\left( {\hat{y}}_i-\bar{y} \right)}^2 }\]
- Definition of total sum of squares , \(TSS\) ,
\[TSS=\sum_{i=1}^{n}{ {\left( y_i-\bar{y} \right)}^2 }\]
- Result:
\[TSS=ESS+RSS\]
- This result will fail to hold if we do not include “1” in our regressor data.
- Definition of \(R^2\) :
\[R^2= \frac{ESS}{TSS}=1- \frac{RSS}{TSS}\]
- Result:
\[0≤R^2≤1\]
- This result will fail to hold if we do not include “1” in our regressor data.
- Definition of uncentered \(R^2\) :
\[R_U^2= \frac{\sum_{i=1}^{n}{ {\hat{y}}_i^2 }}{\sum_{i=1}^{n}{ y_i^2 }}=1- \frac{\sum_{i=1}^{n}{ e_i^2 }}{\sum_{i=1}^{n}{ y_i^2 }}\]
- \(0≤R_U^2≤1\) even if “1” is not in our regressor data.
- Definition of adjusted \(R^2\)
\[{\bar{R}}^2=1- \frac{1/(n-k)\sum_{i=1}^{n}{ e_i^2 }}{1/(n-1)\sum_{i=1}^{n}{ {\left( y_i-\bar{y} \right)}^2 }}\]
- Result:
\[{\bar{R}}^2=1-\left( 1-R^2 \right) \frac{n-1}{n-k}=R^2-\left( 1-R^2 \right) \frac{k-1}{n-k}\]