\begin{align*}
y’y &= (X\hat\beta+\hat u)'(X\hat\beta + \hat u)\\&=\hat\beta’ X’ X \hat \beta + \hat u’ \hat u \end{align*}
Intuition: Variation in $y =$ Explained variation + Unexplained variation
\begin{align*}y’y – \hat u’ \hat u &=\hat\beta’ X’ X \hat \beta
\end{align*}
Define uncentered $R^2$. Then,
\begin{align*}
1 – \dfrac{\hat u’ \hat u}{y’y} &=\dfrac{\hat\beta’ X’ X \hat \beta}{y’y}
\end{align*}
Center $y$ around its sample mean $ \bar y = \frac{1}{n} \sum_{t=1}^n Y_t $
\begin{align*}
y’y – n\bar y^2 &=\hat\beta’ X’ X \hat \beta -n\bar y^2 + \hat u’ \hat u \\
1 – \dfrac{\hat u’ \hat u}{y’y – n\bar y^2 } &=\dfrac{\hat\beta’ X’ X \hat \beta}{y’y – n\bar y^2 } = R^2
\end{align*}
$R^2$ is the coefficient of determination, a measure of how well the data fits the model. Ideal: $R^2 = 1$ means the model fits the data perfectly and all data is on the regression line.
The fit of different models may be compared using the Akaike, Bayesian or Hannan-Quinn information criteria (AIC, BIC and HQC.)