I'm taking an econometrics class, and my Professor included a quick proof sketch for why including independent variables increases $R^2$ (And decreases unconditional variance of CEF error). Here's their outline:
$$ \begin{align} &\text{ Let } z = \mathbb{E}[y|x_1, x_2], \text{ with Jensen's Inequality and LIE: } \\ & \quad \;\;\;\;\: (\mathbb{E}[z|x_1])^2 \leq \mathbb{E}[z^2|x_1] \\ &1) \Rightarrow \mathbb{E}[(\mathbb{E}[z|x_1])^2] \leq \mathbb{E}[\mathbb{E}[z^2|x_1]]\\ &2) \Rightarrow \mathbb{E}[(\mathbb{E}[z|x_1])^2] \leq \mathbb{E}[z^2] \\ &3) \Rightarrow \mathbb{E}[\mathbb{E}[y|x_1]^2] \leq \mathbb{E}[\mathbb{E}[y|x_1, x_2]^2]\\ &4) \Rightarrow Var(\mathbb{E}[y|x_1]) \leq Var(\mathbb{E}[y|x_1, x_2]) \end{align} $$
I'm fine with the whole sketch except for the left side from $2 \rightarrow 3$. How do we go from $\mathbb{E}[(\mathbb{E}[z|x_1])^2]$ to $\mathbb{E}[\mathbb{E}[y|x_1]^2]$? Am I right that this implies $\mathbb{E}[y|x_1] = \mathbb{E}[\mathbb{E}[y|x_1, x_2]|x_1]$? How does the conditional expectation work here? Thanks.
