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\title{Supplemetary Material to\\Semi-parametric Bivariate Polychotomous Ordinal Regression}
%\title{A Common Framework for the Flexible Analysis of Categorical Response Variables in Recursive Systems of Equations}
\author{Francesco Donat\thanks{Corresponding Author. University College London, Gower Street, London, WC1E 6BT, UK. Tel.: +44 (0)2076791223. E-mail: \href{mailto:f.w.donat@gmail.com}{f.w.donat@gmail.com}}\mbox{ }}
\author{Giampiero Marra}
\affil{\normalsize{Department of Statistical Science\\University College London}}

\date{}

\begin{document}

\maketitle


% ----------------------------------------- %
% --- SUPPLEMENTARY MATERIAL --- %
% --------------------------------------- --%

%%%%%%%%%% Merge with supplemental materials %%%%%%%%%%
%\clearpage
%\begin{center}
%\textbf{\LARGE Supplementary Material to \\``Semi-parametric \textcolor{red}{Bivariate} Polychotomous Ordinal Regression"}
%\end{center}
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\section{Some Simulation Evidence}
The simulated scenario comprises a bivariate system of equations specified by the following Data Generating Process (DGP):
%
\begin{equation}
\begin{array}{ccr}
y_{1,i}^*&=&\textup x_{1,i}+2\textup x_{2,i}+\textup x_{3,i}+s_{1,1}(\textup v_{1,i})+s_{1,2}(\textup v_{2,i})+\epsilon_1\\
y_{2,i}^*&=&-0.3 y_{1,i}^* +\textup x_{1,i}-2\textup x_{2,i}+s_{2,1}(\textup v_{1,i})+\epsilon_2
\end{array}\mbox{ }\mbox{ }\mbox{ }\boldsymbol\epsilon_i\sim\mathcal N_2\left(\left(\begin{array}{c}0\\0\end{array}\right),\left[\begin{array}{cc}1 & \rho\\\rho & 1\end{array}\right]\right)\nonumber
\end{equation}
%
for various specifications of the correlation coefficient, $\rho=\{0.1,0.5,0.9\}$, indicating an increasing strength of the unmeasured confounding problem. The test functions are displayed in red in Figure \ref{fig:LatEnd.simSmooths} in the main text, and given by $s_{1,1}(\textup v_{1,i})=-0.7\{4\textup v_{1,i}+2.5\textup v_{1,i}^2+0.7\sin(5\textup v_{1,i})+\cos(7.5\textup v_{1,i})\}$, $s_{1,2}(\textup v_{2,i})=-0.4\{-0.3-1.6\textup v_{2,i}+\sin(5\textup v_{2,i})\}$ and $s_{2,1}(\textup v_{1,i})=0.6\{\exp\{\textup v_{1,i}\}+\sin(2,9\textup v_{1,i})\}$. Furthermore, the ordered values of $y_{j,i}$ have been computed following the observation rule:
%
\begin{equation}
y_{j,i}=\sum_{k_j\in\mathcal K_j}k_j\mathds 1\{c_{j,k_j-1}<y_{j,i}^*\le c_{j,k_j}\},\nonumber
\end{equation}
%
for every $j\in\{1,2\}$, and obtained by setting the threshold parameters at $\boldsymbol c_1:=(-2,-1, 0,2)^\top$ and $\boldsymbol c_2:=(-7,-2,-1,1,2)^\top$. Finally, sample sizes are specified at $500$, $3,000$, and $10,000$, and $N=100$ replications of each design are performed: in conclusion, this experiment encompassed 9 different simulated scenarios whose results are detailed below. In particular, we focus on the sampling properties of $\widehat{\psi}$ and $\widehat{\rho}$, namely the parameters that are neglected whenever a naive ordered regression model is fitted.

In general, the simulations show (Figure \ref{fig:LatEnd.SMboxplotsimulation2}) that the parameters of interest are unbiased starting from moderate sample sizes (i.e.\ 3,000) while, for $n=500$, our algorithm produced estimates for the correlation coefficient quite far from the corresponding simulated values: this tendency is nonetheless overcome once the number of observations is increased. We also stress that, as $n$ increases, the estimated parameter vector approaches its true value with a lower standard deviation and, interestingly, $\widehat\rho$ is on average closer to its simulated values at any given sample size and higher magnitude. This fact is not unexpected, and similar occurrences have been also reported by \Citet{ChibGreenberg_07} and \Citet{MarraRadice_11} in the context of a semi-parametric regression for dichotomous responses. In particular, they explained the situation by noticing that a higher correlation coefficient is a signal of a more severe role of the unobservables in determining the association between $Y_1$ and $Y_2$ which is more easily measured by a bivariate model. Stated differently, the joint estimation of the parameters in the model allows for the full use of all the information contained in the data.

\newpage

\begin{figure}[h!]\centering
\includegraphics[width=0.65\textwidth,angle=270]{Simulation2boxplot.eps}
\caption{Box plots corresponding to the estimates of $\psi$ and $\rho$ for different sample sizes and correlation coefficients (0.1, 0.5 and 0.9) where the true values are denoted by a red line in each of the panels. Results are obtained using 100 replications of the DGP detailed in this section.}
\label{fig:LatEnd.SMboxplotsimulation2}
%\clearpage
\end{figure}

\begin{figure}[h!]\centering
\includegraphics[width=0.32\textwidth,angle=270]{Simulation1smooths3000.eps}
\caption{Estimated smooth curves obtained from 50 replicates of a Monte Carlo experiment comprising 3,000 simulated observations (true curves in red). The DGP is given in this section, and $\psi$ and $\rho$ set to $-0.3$ and $0.2$, respectively. Refer to the caption of Figure 1 for more details.}
\label{fig:LatEnd.simSmooths3000}
\clearpage
\end{figure}

\newpage

\section{Further Details on the Empirical Illustration}

\begin{table}[h!]
\fontfamily{ptm}\selectfont\small
\centering
\begin{tabular}{m{2.5cm}m{3cm}m{5cm}m{3.5cm}}
\hline
Variable & BCS70 & Definition & Levels\\
\hline\hline
\texttt{edu} & HIACA00 & Respondent's highest education achieved & $=01$: $\le$ O-levels\\
\multicolumn{3}{c}{} & $=02$: A-levels\\
\multicolumn{3}{c}{} & $=03$: Higher Education\\
\texttt{drk} & drinking & Drinking frequency & $=01$: special occasions\\
\multicolumn{3}{c}{} & $=02$: 2-3 times/month\\
\multicolumn{3}{c}{} & $=03$: once a week\\
\multicolumn{3}{c}{} & $=04$: 2-3 days/week\\
\multicolumn{3}{c}{} & $=05$: most days\\
\texttt{drk5} & drinking, beer, spirits, wine, pops,sherry& Alcohol intake & $=01$: special occasions\\
\multicolumn{3}{c}{} & $=02$: 2-3 times/month\\
\multicolumn{3}{c}{} & $=03$: $<$ NHS limits\\
\multicolumn{3}{c}{} & $=04$: $\approx$ NHS limits\\
\multicolumn{3}{c}{} & $=05$: $>$ NHS limits\\
\hline
\texttt{BAS.tot} & i3504-i3644 & Total reported BAS (\texttt{edu}) & -- continuous\\
\texttt{region} & BD3REGN & Region of residence (\texttt{drk5/drk}) & $=01$: South East\\
\multicolumn{3}{c}{} & $=02$: Scotland\\
\multicolumn{3}{c}{} & $=03$: Wales\\
\multicolumn{3}{c}{} & $=04$: South West\\
\multicolumn{3}{c}{} & $=05$: missing/unknown\\
\multicolumn{3}{c}{} & $=06$: East Anglia\\
\multicolumn{3}{c}{} & $=07$: West Midlands\\
\multicolumn{3}{c}{} & $=08$: East Midlands\\
\multicolumn{3}{c}{} & $=09$: Yorks \& Humber.\\
\multicolumn{3}{c}{} & $=10$: North West\\
\multicolumn{3}{c}{} & $=11$: North\\
\hline
\end{tabular}
\caption{Description of the responses and the equation-specific covariates included in the study. The dependent variable \texttt{drk5} has been obtained by replacing levels 03-05 of \texttt{drk} by an equivalent (averaged) amount of alcohol units as based on the following conversion: 1 pint of beer: 2.8u; 1 glass of spirits/sherry: 1u; 1 glass of wine: 2.1u; 1 bottle of alcopop: 1.4u.}
\end{table}

\newpage

\begin{table}[t]
\fontfamily{ptm}\selectfont\small
\centering
\begin{tabular}{m{2.5cm}m{3cm}m{5cm}m{3.5cm}}
\hline
Variable & BCS70 & Definition & Levels\\
\hline\hline
\texttt{mum.not.pres.} & a5.1 & Relationship of mother figure & $=1$: other\\
\multicolumn{3}{c}{} & $=2$: missing\\
\multicolumn{3}{c}{} & $=3$: natural mother\\
\texttt{dad.not.pres.} & a6.1 & Relationship of father figure & $=1$: other\\
\multicolumn{3}{c}{} & $=2$: missing\\
\multicolumn{3}{c}{} & $=3$: natural father\\
\texttt{mum.edu} & c1.12-c1.20 & Mother's qualification & $=1$: $<$ O-levels\\
\multicolumn{3}{c}{} & $=2$: missing\\
\multicolumn{3}{c}{} & $=3$: O-levels\\
\multicolumn{3}{c}{} & $=4$: A-levels\\
\multicolumn{3}{c}{} & $=5$: professional\\
\multicolumn{3}{c}{} & $=6$: degree\\
\texttt{dad.edu} & c1.1-c1.11 & Father's qualification & $=1$: $<$ O-levels\\
\multicolumn{3}{c}{} & $=2$: missing\\
\multicolumn{3}{c}{} & $=3$: O-levels\\
\multicolumn{3}{c}{} & $=4$: A-levels\\
\multicolumn{3}{c}{} & $=5$: professional\\
\multicolumn{3}{c}{} & $=6$: degree\\
\texttt{s.class} & BD3PSOC & Social class from father's occupation & $=1$: professional\\
\multicolumn{3}{c}{} & $=2$: missing/no data\\
\multicolumn{3}{c}{} & $=3$: manual \& tech.\\
\multicolumn{3}{c}{} & $=4$: partially skilled\\
\multicolumn{3}{c}{} & $=5$: non-manual\\
\multicolumn{3}{c}{} & $=6$: manual\\
\multicolumn{3}{c}{} & $=7$: unskilled\\
\texttt{eth.child} & a12.1 & Ethnic group & $=1$: white European\\
\multicolumn{3}{c}{} & $=2$: missing\\
\multicolumn{3}{c}{} & $=3$: other\\
\texttt{mum.int.edu} & j097 & Mother's interest in child's education & $=1$: very/moderate\\
\multicolumn{3}{c}{} & $=2$: other\\
\multicolumn{3}{c}{} & $=3$: cannot say\\
\multicolumn{3}{c}{} & $=4$: no/very little\\
\texttt{dad.int.edu} & j098 & Father's interest in child's education& $=1$: very/moderate\\
\multicolumn{3}{c}{} & $=2$: other\\
\multicolumn{3}{c}{} & $=3$: cannot say\\
\multicolumn{3}{c}{} & $=4$: no/very little\\
\texttt{sex.b} & sex10 & Gender & $=0$: girl\\
\multicolumn{3}{c}{} & $=1$: boy\\
\texttt{home} & d2 & Home tenure & $=1$: owned outright\\
\multicolumn{3}{c}{} & $=2$: missing\\
\multicolumn{3}{c}{} & $=3$: rented/other\\
\multicolumn{3}{c}{} & $=4$: being bought\\
\texttt{mum.wrk.hr} & c5.1 & Mother's weekly working hours & -- continuous\\
\hline
\end{tabular}
\caption{Description of the covariates that are common to both equations.}
\clearpage
\end{table}

\begin{table}[t]
\fontfamily{ptm}\selectfont\small
\centering
\begin{tabular}{c|ccccc}
\hline
\multirow{2}{*}{Highest Education} & \multicolumn{5}{c}{Drinking Frequency}\\
\cline{2-6}
 & no/occasional & light & 1/week & 2-3/week & most days\\
\hline\hline
\multirow{2}{*}{Up to O-levels} & $0.2028$ & $0.1455$ & $0.2256$ & $0.3179$ & $0.1081$\\
& $(.1294;.2941)$ & $(.0535;.2203)$ & $(.2154;.2358)$ & $(.3090;.3263)$ & $(.1020;.1139)$\\
\multirow{2}{*}{A-levels} & $0.1621$ & $0.1328$ & $0.2225$ & $0.3480$ & $0.1345$\\
& $(.0990;.2441)$ & $(.0498;.1972)$ & $(.2124;.2324)$ & $(.3388;.3571)$ & $(.1279;.1413)$\\
\multirow{2}{*}{HE or equivalent} & $0.1426$ & $0.1237$ & $0.2159$ & $0.3615$ & $0.1563$\\
& $(.0853;.2188)$ & $(.0469;.1816)$ & $(.2063;.2256)$ & $(.3531;.3699)$ & $(.1516;.1609)$\\
\hline
\end{tabular}
\caption{Average predicted conditional probabilities when a measure of drinking frequency (\texttt{drk}) is used as response variable. More details are reported in the caption of Table 4.}
\clearpage
\end{table}

\begin{table}[t]
\fontfamily{ptm}\selectfont\small
\centering
\begin{tabular}{c|ccccc}
\hline
\multirow{3}{*}{Highest Education} & \multicolumn{5}{c}{Alcohol Consumption}\\
\cline{2-6}
&\multicolumn{2}{c}{} & \multicolumn{3}{c}{at least one drink per week}\\
\cline{4-6}
 & no/occasional & light & $<$ NHS limits & $\approx$ NHS limits & $>$ NHS limits\\
\hline\hline
\multirow{2}{*}{Up to O-levels} & $0.2320$ & $0.1424$ & $0.2550$ & $0.0926$ & $0.2780$\\
& $(.1525;.3285)$ & $(.0402;.2242)$ & $(.2447;.2648)$ & $(.0839;.1013)$ & $(.2699;.2856)$\\
\multirow{2}{*}{A-levels} & $0.2029$ & $0.1354$ & $0.2554$ & $0.0967$ & $0.3095$\\
& $(.1299;.2944)$ & $(.0388;.2105)$ & $(.2450;.2651)$ & $(.0877;.1058)$ & $(.3012;.3177)$\\
\multirow{2}{*}{HE or equivalent} & $0.1876$ & $0.1303$ & $0.2530$ & $0.0983$ & $0.3308$\\
& $(.1184;.2772)$ & $(.0378;.1998)$ & $(.2427;.2626)$ & $(.0889;.1075)$ & $(.3242;.3373)$\\
\hline
\end{tabular}
\caption{Average predicted conditional probabilities when the covariate \texttt{mum.wrk.hr} is dropped from the first equation. More details are reported in the caption of Table 4.}
\end{table}

\afterpage{
\begin{figure}[h]\centering
\includegraphics[width=0.32\textwidth,angle=270]{Shrinkage.eps}
\caption{Shrinkage method applied to the model specification of Section 4.1: \texttt{mum.wrk.hr} is not an influential predictor for children's education achievements.}
%\clearpage
\end{figure}
}

\newpage

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