Add parentheses under equations?

This is one possibility:

\documentclass[12pt,a4paper]{article}
\usepackage{amsmath}
\begin{document}

\begin{alignat*}{4}
&Y_t = {}&&\hat{\beta}_0 + {}&&\hat{\beta}_1 X_1 + \dots + {}&&\hat{\beta}_k X_k, \\
&(t)     &&()                && ()                           &&()  
\end{alignat*}

\end{document}

Another option using a new command,

\newcommand{\stacktwo}[2]{\begin{array}[t]{@{}c@{}}#1\\#2\end{array}}

can be like this:

\begin{equation}
\stacktwo{Y_t}{(t)} = \stacktwo{\hat{\beta}_0}{()} + \stacktwo{\hat{\beta}_1}{()} X_1 + \dots + \stacktwo{\hat{\beta}_k}{()} X_k,
\end{equation}

Also, you can load the stackengine package and use the command \stackMath\stackunder{Y_t}{(t)}, etc. from this package.

\documentclass[12pt,a4paper]{article}
\usepackage{amsmath,stackengine}
\stackMath
\renewcommand\useanchorwidth{T}
\renewcommand\stacktype{L}
\def\xbeta_#1{\stackunder{\beta_#1}{()}}
\begin{document}
\[
\stackunder{Y_t}{(t)} = \xbeta_0 + \xbeta_1 X_1 + \dots + \xbeta_k X_k, 
\]
\end{document}

Here's a solution that uses the \underset macro of the amsmath package. In the following screenshot, the parenthetic material is centered below the corresponding \beta particles (pun intended) in the preceding row. Simply adjust the closing curly braces if the parenthetic material should be centered on \beta_i X_.

\documentclass{article}
\usepackage{mathtools}
\newcommand\under[2]{\underset{\mathclap{\textstyle(#1)}}{#2}}
\begin{document}
\[
\under{t}{Y_t} =  \under{u}{\hat{\beta}_0} 
                + \under{v}{\hat{\beta}_1} X_1  
                + \dots 
                + \under{wxyz}{\hat{\beta}_k} X_k
\]
\end{document}