\documentclass[10pt]{article}

\usepackage{fullpage}
\usepackage{setspace}
\usepackage{parskip}
\usepackage{titlesec}
\usepackage[section]{placeins}
\usepackage{xcolor}
\usepackage{breakcites}
\usepackage{lineno}
\usepackage{hyphenat}





\PassOptionsToPackage{hyphens}{url}
\usepackage[colorlinks = true,
            linkcolor = blue,
            urlcolor  = blue,
            citecolor = blue,
            anchorcolor = blue]{hyperref}
\usepackage{etoolbox}
\makeatletter
\patchcmd\@combinedblfloats{\box\@outputbox}{\unvbox\@outputbox}{}{%
  \errmessage{\noexpand\@combinedblfloats could not be patched}%
}%
\makeatother


\usepackage[round]{natbib}
\let\cite\citep




\renewenvironment{abstract}
  {{\bfseries\noindent{\abstractname}\par\nobreak}\footnotesize}
  {\bigskip}

\titlespacing{\section}{0pt}{*3}{*1}
\titlespacing{\subsection}{0pt}{*2}{*0.5}
\titlespacing{\subsubsection}{0pt}{*1.5}{0pt}


\usepackage{authblk}


\usepackage{graphicx}
\usepackage[space]{grffile}
\usepackage{latexsym}
\usepackage{textcomp}
\usepackage{longtable}
\usepackage{tabulary}
\usepackage{booktabs,array,multirow}
\usepackage{amsfonts,amsmath,amssymb}
\providecommand\citet{\cite}
\providecommand\citep{\cite}
\providecommand\citealt{\cite}
% You can conditionalize code for latexml or normal latex using this.
\newif\iflatexml\latexmlfalse
\providecommand{\tightlist}{\setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}}%

\AtBeginDocument{\DeclareGraphicsExtensions{.pdf,.PDF,.eps,.EPS,.png,.PNG,.tif,.TIF,.jpg,.JPG,.jpeg,.JPEG}}


\usepackage[english]{babel}








%%% Your header encountered errors when converting with LaTeXML
%%% Please uncomment it incrementally and check for problems
%%%   If you need assistance, contact us at help@authorea.com
%%% 
%
%\iflatexml
%\documentclass{article}
%\fi
%
%% \usepackage{latexsym} % Not (yet) supported by LaTeXML.
%
%\usepackage{amsmath}
%
%\usepackage[active]{srcltx}
%
%\usepackage{epsfig}
%
%\usepackage{graphicx}
%
%% \usepackage{colortbl} % Not (yet) supported by LaTeXML.
%
%\usepackage{rotating}
%
%\usepackage{float}
%
%\usepackage{setspace}
%
%% \usepackage{lipsum} % Not (yet) supported by LaTeXML.
%
%% \usepackage{scrextend} % Not (yet) supported by LaTeXML.
%
%% \usepackage[T1]{fontenc} % Not (yet) supported by LaTeXML.
%
%\usepackage[latin9]{inputenc}
%
%\usepackage{geometry}
%
%\geometry{verbose,tmargin=2.5cm,bmargin=2.2cm,lmargin=2.5cm,rmargin=2.5cm}
%\usepackage{array}
%
%\usepackage{longtable}
%
%\usepackage{multirow}
%
%% \usepackage{standalone} % Not (yet) supported by LaTeXML.
%
%\usepackage{preview}
%
%\usepackage{xcolor}
%
%% \usepackage{mathtools} % Not (yet) supported by LaTeXML.
%
%\pagestyle{plain}
%\paperwidth = 8.5in
%\oddsidemargin = 0in
%\evensidemargin = 0in
%\textwidth = 6.9in
%\paperheight = 11in
%\topmargin = 0in
%\headheight = 0in
%\headsep = 0in
%\topskip = 0in
%\textheight = 9.5in
%\parindent = 0in
%\newcounter{qcounter}
%\newcounter{qcountera}


\begin{document}

\title{Unnamed Article}



\author[1]{Zung}%
\affil[1]{Affiliation not available}%


\vspace{-1em}



  
  \date{June 03, 2024}


\begingroup
\let\center\flushleft
\let\endcenter\endflushleft
\maketitle
\endgroup








\sloppy


1st paper: Conceptual framework let \(U_{1}\) and
\(U_{0}\) such that \(U_{1}-U_{0}>0\).

\begin{equation}
U_{1}=w^{{}^{\prime}}\beta_{1}+z^{{}^{\prime}}_{1}\gamma_{1}+\epsilon_{1}\>\>\>and\;\;\;U_{0}=w^{{}^{\prime}}\beta_{0}+z^{{}^{\prime}}_{0}\gamma_{0}+\epsilon_{0}\\
 \end{equation}

; \(z_{1}\) and \(z_{0}\) denote attributes of the two
choices; \(\epsilon_{1}\) and \(\epsilon_{0}\)

\begin{equation}
Prob[Y=1|w,z_{1},z_{0}]=Prob[U_{1}>U_{0}]\;\\
 \end{equation}

This is leading to:

\begin{equation}
Prob[Y=1|w,z_{1},z_{0}]=Prob[x^{{}^{\prime}}\beta+\epsilon>0|x]\\
 \end{equation}

where \(x^{{}^{\prime}}\beta\) and \(\epsilon\)

\selectlanguage{english}
\FloatBarrier
\end{document}

