Roman numerals for sections and subsections

You need to append the section number to the subsection number using:

\documentclass{article}
\renewcommand{\thesection}{\Roman{section}} 
\renewcommand{\thesubsection}{\thesection.\Roman{subsection}}
\begin{document}
\section{First section}
\subsection{First subsection}
\subsection{Second subsection}
\section{Second section}
\subsection{First subsection}
\subsection{Second subsection}
\end{document}

\renewcommand{\thesection}{\Roman{section}}
\renewcommand{\thesubsection}{\thesection.\Roman{subsection}}

I was looking for the 3rd tier (subsubsection) but couldn't find it anywhere. I managed to figure it out though. So, for anyone who's in the same spot as i was, here is the subsubsection version:

\renewcommand{\thesubsubsection}{\thesection.\thesubsection.\Roman{subsubsection}}