Curly brackets below a word in an equation

You're almost there. Instead of

\underbrace{E_{t-2}v_t}_\textrm{=0}

type

\underbrace{E_{t-2}v_t}_{=0}

etc

---

I would also introduce some line breaks in the long equation.

\documentclass{article}
\usepackage{amsmath}
\DeclareMathOperator{\E}{E}
\begin{document}
\begin{multline*}
\E_{t-2}P_t=
\frac{1}{2}\Bigl(\E_{t-2}m_t
+\underbrace{\E_{t-2}v_t}_{=0}
-\underbrace{\E_{t-2}u_t}_{=0}\Bigr) \\
+\frac{(1-\zeta )}{2}\Bigl(
\underbrace{\E_{t-2}\E_{t-1}P_t}_{=\E_{t-2}}+
\underbrace{\E_{t-2}\E_{t-2}P_t}_{=\E_{t-2}}\Bigr) \\
+\frac{\zeta }{2}\Bigl(\E_{t-2}P_{t-1}+\E_{t-2}P_{t-2}\Bigr)
\end{multline*}
\end{document}

You can use Mathpix to generate LaTeX code for any mathematical equation by simply taking a screenshot of the equation. It will save you tons of time in future like it did for me after I discovered it!