Why can't the uncentered $R^2$ be negative?

Question

In vectorized OLS, let $\mathbf{y}$ and $\hat{\mathbf{y}}$ be the observations and predictions, and define the residuals as $\mathbf{e} = \mathbf{y} - \hat{\mathbf{y}}$. Then the uncentered $R^2$ is

$$ R^2_{uc} = 1 - \frac{\mathbf{e}^{\top} \mathbf{e}}{\mathbf{y}^{\top} \mathbf{y}}. $$

Now in his econometrics textbook, Hayashi writes:

Since both $\hat{\mathbf{y}}^{\top} \hat{\mathbf{y}}$ and $\mathbf{e}^{\top} \mathbf{e}$ are nonnegative, $0 \leq R^2_{uc} \leq 1$.

However, this restriction that $R^2_{uc}$ is nonnegative does not make senes to me. $R^2_{uc}$ should be negative when $\mathbf{e}^{\top}\mathbf{e} > \mathbf{y}^{\top}\mathbf{y}$. This is true when

$$ \begin{aligned} \hat{\mathbf{y}} &> 2\mathbf{y} \\ \hat{\mathbf{y}}^{\top} \hat{\mathbf{y}} &> 2\mathbf{y}^{\top} \hat{\mathbf{y}} \\ \mathbf{y}^{\top} \mathbf{y} + \hat{\mathbf{y}}^{\top} \hat{\mathbf{y}} - 2\mathbf{y}^{\top} \hat{\mathbf{y}} &> \mathbf{y}^{\top}\mathbf{y} \\ (\mathbf{y} - \hat{\mathbf{y}})^{\top} (\mathbf{y} - \hat{\mathbf{y}}) &> \mathbf{y}^{\top}\mathbf{y} \\ \mathbf{e}^{\top}\mathbf{e} &> \mathbf{y}^{\top}\mathbf{y} \end{aligned} $$

So why can't $R_{uc}^2$ be negative?

Answer 1

Vector (and scalar) inequalities do not work like that. Just a simple counterexample:

Take $\hat{\mathbf{y}} = (-2\, -2)^\top$ and ${\mathbf{y}} = (-2\,-2)^\top$. Then elementwise you can write that ${\mathbf{y}} = (-2\, -2)^\top > 2 {\mathbf{y}} = (-4 \, -4)^\top$, but $\hat{\mathbf{y}}^\top \hat{\mathbf{y}} = 8 < 2 {\mathbf{y}}^\top \hat{\mathbf{y}} = 16$.

The reason why the uncentered $R^2$ can not be negative is written just before this equation in the Hayashi book: ${\mathbf{y}}^\top {\mathbf{y}} = \hat{\mathbf{y}}^\top \hat{\mathbf{y}} + {\mathbf{e}}^\top {\mathbf{e}}$, hence, $$R^2_{uc} =\frac{\mathbf{y}^\top \mathbf{y}- {\mathbf{e}}^\top {\mathbf{e}}}{{\mathbf{y}}^\top{\mathbf{y}}}= \frac{\hat{\mathbf{y}}^\top \hat{\mathbf{y}}}{{\mathbf{y}}^\top{\mathbf{y}}},$$ where both the numerator and the denominator are non-negative.