Discussion
Which of the following is correctly defined as coefficient of determination ?
Answer: A
The coefficient of determination is \(1 - \frac{\Sigma \hat{u}_{i}^{2}}{\Sigma y_{i}^{2}}\).
Here \(y_{i}\) are deviations of the dependent variable from its mean, so \(\Sigma y_{i}^{2}\) is the total sum of squares (TSS), and \(\Sigma \hat{u}_{i}^{2}\) is the residual sum of squares (RSS). Since TSS = ESS + RSS,
\[ R^{2} = \frac{ESS}{TSS} = 1 - \frac{RSS}{TSS} \]
\(R^{2}\) is the share of the variation in \(Y\) explained by the regression. The ratio \(\frac{\Sigma \hat{u}_{i}^{2}}{\Sigma y_{i}^{2}}\) alone is the unexplained share, \(1 - R^{2}\), and the options with \(\Sigma x_{i}^{2}\) or \(\Sigma \hat{x}_{i}^{2}\) use the wrong sums of squares.
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