% Some material is taken from 2015 slide set 15 (Powerpoint), and adapted based on text.
% For 2023, I decided to cut out most of the surrogate model stuff and leave it for later, which is how I organized it in the past. 2053f22 has the omitted material. 
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\mode<presentation>

\title{Structural Equation Models: The General Case\footnote{See last slide for copyright information.}}
\subtitle{STA431 Spring 2023}
\date{} % To suppress date

\begin{document}

\begin{frame}
  \titlepage
\end{frame}

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\begin{frame}
\frametitle{Features of Structural Equation Models}
%\framesubtitle{} 
  \begin{itemize}
    \item Multiple equations.
    \item All the variables are random.
    \item An explanatory variable in one equation can be the response variable in another equation.
    \item Models are represented by path diagrams.
    \item Identifiability is always an issue. 
    \item The statistical models are models of influence. They are \emph{causal models}.
  \end{itemize}
\end{frame}

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% This 2017 slide is commented out.
\begin{comment} 
\begin{frame}
\frametitle{Correlation versus Causation} \pause
%\framesubtitle{} 
\begin{itemize}
    \item The path diagrams deliberately imply influence. If $A \rightarrow B$, we are saying $A$ \emph{contributes} to $B$, or partly \emph{causes} it.  \pause
    \item Data are usually observational. The correlation-causation issue does not go away. \pause
    \item You may be able to argue on theoretical grounds that $A \rightarrow B$ is more believable than $B \rightarrow A$. \pause
    \item If you have a causal model, you may be able to test whether it's compatible with the data. 
\end{itemize}
\end{frame}
\end{comment}
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\begin{frame}
\frametitle{Modest changes in notation}
%\framesubtitle{} 
\begin{eqnarray*}
Y_{i,1} &=& \alpha_1 + \gamma_1 X_{i,1} + \gamma_2 X_{i,2} + \epsilon_{i,1} \\
Y_{i,2} &=& \alpha_2 + \beta Y_{i,1} + \epsilon_{i,2}
\end{eqnarray*}  \pause
\begin{itemize}
    \item Regression coefficients (links between exogenous variables and endogenous variables) are now called gamma instead of beta. 
    \item Betas are used for links between endogenous variables. 
    \item Intercepts will soon disappear. 
\end{itemize}
\end{frame}

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\begin{frame}
\frametitle{Example: A Path Model with Measurement Error} 
\begin{columns}  
\column{0.5\textwidth}
    \begin{center}
        \includegraphics[width=2.5in]{mepath} % Ugh! Powerpoint 2015. Sorry!
    \end{center}  
\column{0.5\textwidth}
%{\footnotesize
\begin{eqnarray*}
    Y_{i,1} &=& \alpha_1 + \gamma_1 X_i  + \epsilon_{i,1}  \\
    Y_{i,2} &=& \alpha_2 + \beta Y_{i,1} + \gamma_2 X_i  + \epsilon_{i,2}   \\
    W_i     &=& \nu_1 + \lambda_1X_i + e_{i,1}   \\
    V_{i,1} &=& \nu_2 + \lambda_2Y_{i,1} + e_{i,2}   \\
    V_{i,2} &=& \nu_3 + \lambda_3Y_{i,2} + e_{i,3}    
\end{eqnarray*}  
%} % End size
\end{columns} 
\end{frame}

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\begin{frame}
\frametitle{The General (original) Model: {\small Independently for $i=1, \ldots, n$, let}}
%\framesubtitle{Also called the ``original" model} 
\vspace{-8mm}
\begin{eqnarray*}\label{original2stage}
    \mathbf{y}_i &=& \boldsymbol{\alpha} + \boldsymbol{\beta} \mathbf{y}_i 
                     + \boldsymbol{\Gamma} \mathbf{x}_i +  \boldsymbol{\epsilon}_i \\
    \mathbf{F}_i &=& \left( \begin{array}{c}
                            \mathbf{x}_i  \\ \hline \mathbf{y}_i 
                            \end{array} \right) \nonumber \\
    \mathbf{d}_i &=& \boldsymbol{\nu} + \boldsymbol{\Lambda}\mathbf{F}_i + \mathbf{e}_i\pause, \mbox{ where}
\end{eqnarray*}

\vspace{-2mm}
{\scriptsize
\begin{itemize}
     \item $\mathbf{y}_i$ is a $q \times 1$ \emph{latent} random vector.
     \item $\boldsymbol{\alpha}$ is a $q \times 1$ vector of constants.
     \item $\boldsymbol{\beta}$ is a $q \times q$ matrix of constants \emph{with zeros on the main diagonal}.
     \item $\boldsymbol{\Gamma}$ is a $q \times p$ matrix of constants.
     \item $\mathbf{x}_i$ is a $p \times 1$  \emph{latent} random vector with expected value $\boldsymbol{\mu}_x$ and positive definite covariance matrix $\boldsymbol{\Phi}_x$. 
     \item $\boldsymbol{\epsilon}_i$ is a $q \times 1$ latent random vector with expected value zero and positive definite covariance matrix $\boldsymbol{\Psi}$. 
     \item $\mathbf{F}_i$ ($F$ for Factor) is a partitioned vector with $\mathbf{x}_i$ stacked on top of $\mathbf{y}_i$. It is a $(p+q) \times 1$ latent random vector whose expected value is denoted by $\boldsymbol{\mu}_F$, and whose variance-covariance matrix is denoted by $\boldsymbol{\Phi}$.
     \item $\mathbf{d}_i$ is a $k \times 1$ \emph{observable} random vector. The expected value of $\mathbf{d}_i$ will be denoted by $\boldsymbol{\mu}$, and the covariance matrix  of $\mathbf{d}_i$ will be denoted by $\boldsymbol{\Sigma}$.
     \item $\boldsymbol{\nu}$ is a $k \times 1$ vector of constants.
     \item $\boldsymbol{\Lambda}$ is a $k \times (p+q)$ matrix of constants.
     \item $\mathbf{e}_i$ is a $k \times 1$ latent random vector with expected value zero and covariance matrix $\boldsymbol{\Omega}$, which need not be positive definite.
     \item $\mathbf{x}_i$, $\boldsymbol{\epsilon}_i$ and $\mathbf{e}_i$ are independent.
\end{itemize}
} % End size
\end{frame}


\begin{frame}
\frametitle{Surrogate Models} 
\framesubtitle{Truth $\approx$ Original Model $\rightarrow$ Surrogate Model 1
$\rightarrow$ Surrogate Model 2 \ldots} 



\begin{itemize}
    \item We more or less accept the original model, but we can't identify the parameters. 
    \item So we re-parameterize, obtaining a surrogate model. Repeat. 
    \item We will carefully keep track of the \emph{meaning} of the new parameters in terms of the parameters of the original model.
\end{itemize}
\end{frame}


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\begin{frame}
\frametitle{The Original Model}
%\framesubtitle{} 

\begin{eqnarray*}\label{original2stage}
    \mathbf{y}_i &=& \boldsymbol{\alpha} + \boldsymbol{\beta} \mathbf{y}_i 
                     + \boldsymbol{\Gamma} \mathbf{x}_i +  \boldsymbol{\epsilon}_i \\
    \mathbf{F}_i &=& \left( \begin{array}{c}
                            \mathbf{x}_i  \\ \hline \mathbf{y}_i 
                            \end{array} \right) \nonumber \\
    \mathbf{d}_i &=& \boldsymbol{\nu} + \boldsymbol{\Lambda}\mathbf{F}_i + \mathbf{e}_i 
\end{eqnarray*}
where \ldots \pause

\begin{itemize}
    \item Carefully count the parameters that appear \emph{only} in $E(\mathbf{d}_i) = \boldsymbol{\mu}$ and not in $cov(\mathbf{d}_i)$. 
    \item There are more of these parameters than elements of $E(\mathbf{d}_i)$.
    \item Parameter count rule.
\end{itemize}
\end{frame}

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\begin{frame}
\frametitle{Center the model} 
%\framesubtitle{} 
\begin{itemize}
    \item There are too many expected values and intercepts to identify. 
    \item Center all the random variables in the model by adding and subtracting expected values. \pause
    \item Obtain a \emph{centered surrogate model} 
\begin{eqnarray*}
\stackrel{c}{\mathbf{y}}_i 
    &=& \boldsymbol{\beta} \! \stackrel{c}{\mathbf{y}}_i
         + \boldsymbol{\Gamma} \! \stackrel{c}{\mathbf{x}}_i +  \boldsymbol{\epsilon}_i \\
\stackrel{c}{\mathbf{F}}_i 
    &=& \left( \begin{array}{c}
        \stackrel{c}{\mathbf{x}}_i  \\ \hline \stackrel{c}{\mathbf{y}}_i 
         \end{array} \right) \\
\stackrel{c}{\mathbf{d}}_i 
    &=& \boldsymbol{\Lambda} \! \stackrel{c}{\mathbf{F}}_i + \mathbf{e}_i
\end{eqnarray*} \pause
    \item Same $\boldsymbol{\beta}$, $\boldsymbol{\Gamma}$ and $\boldsymbol{\Lambda}$, same variances and covariances.
\end{itemize}

\end{frame}


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\begin{frame}
\frametitle{Change of variables}
%\framesubtitle{} 
\begin{itemize}
    \item Centering is a change of variables. 
    \item Expected values and intercepts are gone, and the dimension of the parameter space is reduced.  
    \item Drop the little $c$ over the random vectors.
\end{itemize}
\end{frame}


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\begin{frame}
\frametitle{The General Centered Model}
% \framesubtitle{Stage 1 is the latent variable model and Stage 2 is the measurement model.}  
Independently for $i=1, \ldots, n$, 
\begin{eqnarray*}
    \mathbf{y}_i &=& \boldsymbol{\beta} \mathbf{y}_i + \boldsymbol{\Gamma} \mathbf{x}_i + \boldsymbol{\epsilon}_i \\ 
    \mathbf{F}_i &=& \left( \begin{array}{c}
                            \mathbf{x}_i  \\ \hline \mathbf{y}_i 
                            \end{array} \right)  \\
    \mathbf{d}_i &=& \boldsymbol{\Lambda}\mathbf{F}_i + \mathbf{e}_i  
\end{eqnarray*}  \pause % See, no intercepts
\begin{itemize}
     \item $\mathbf{d}_i$ (the data) are observable. All other variables are latent. 
     \item $\mathbf{y}_i = \boldsymbol{\beta} \mathbf{y}_i + \boldsymbol{\Gamma} \mathbf{x}_i + \boldsymbol{\epsilon}_i$ is called the \emph{Latent Variable Model}. 
     \item The latent vectors $\mathbf{x}_i$ and $\mathbf{y}_i$ are collected into a \emph{factor} $\mathbf{F}_i$. 
     \item $\mathbf{d}_i =  \boldsymbol{\Lambda}\mathbf{F}_i + \mathbf{e}_i$ is called the \emph{Measurement Model}.
\end{itemize} 
\end{frame}


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\begin{frame}
\frametitle{$\mathbf{y}_i= \boldsymbol{\beta} \mathbf{y}_i + \boldsymbol{\Gamma} \mathbf{x}_i + \boldsymbol{\epsilon}_i$ 
~~ $\mathbf{F}_i = \left( \begin{array}{c}
                            \mathbf{x}_i  \\ \hline \mathbf{y}_i 
                            \end{array} \right)$ 
~~ $\mathbf{d}_i = \boldsymbol{\Lambda}\mathbf{F}_i + \mathbf{e}_i$} \pause
% \framesubtitle{} 
\begin{itemize}
     \item $\mathbf{y}_i$ is a $q \times 1$ latent random vector. 
     \item $\boldsymbol{\beta}$ is a $q \times q$ matrix of constants with zeros on the main diagonal. 
     \item $\mathbf{x}_i$ is a $p \times 1$ latent random vector.
     \item $\boldsymbol{\Gamma}$ is a $q \times p$ matrix of constants.      
     \item $\boldsymbol{\epsilon}_i$ is a $q \times 1$ vector of error terms. 
     \item $\mathbf{F}_i$ ($F$ for Factor) is just $\mathbf{x}_i$ stacked on top of $\mathbf{y}_i$. It is a $(p+q) \times 1$ latent random vector. \pause
     \item $\mathbf{d}_i$ is a $k \times 1$ observable random vector. Sometimes, 
     $\mathbf{d}_i = \left( \begin{array}{c}
         \mathbf{w}_i  \\  \hline \mathbf{v}_i 
             \end{array} \right)$. \pause
     \item $\boldsymbol{\Lambda}$ is a $k \times (p+q)$ matrix of constants: ``factor loadings." \pause
     \item $\mathbf{e}_i$ is a $k \times 1$ vector of error terms. 
     \item $\mathbf{x}_i$, $\boldsymbol{\epsilon}_i$ and $\mathbf{e}_i$ are independent.
\end{itemize} 
\end{frame}

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\begin{frame}
\frametitle{Parameters}
\framesubtitle{More notation}  
{\small
\vspace{-6mm}
\begin{eqnarray*}
\mathbf{y}_i &=& \boldsymbol{\beta} \mathbf{y}_i + \boldsymbol{\Gamma} \mathbf{x}_i + \boldsymbol{\epsilon}_i \\
\mathbf{F}_i &=& \left( \begin{array}{c}
                            \mathbf{x}_i  \\ \hline \mathbf{y}_i 
                            \end{array} \right)  \\
\mathbf{d}_i &=& \boldsymbol{\Lambda}\mathbf{F}_i + \mathbf{e}_i  \\
&&\\ 
cov(\mathbf{x}_i) &=& \boldsymbol{\Phi}_x    \\
cov(\boldsymbol{\epsilon}_i) &=& \boldsymbol{\Psi}    \\
cov(\mathbf{F}_i) &=& \boldsymbol{\Phi} 
         =  \left( \begin{array}{c c}
            cov(\mathbf{x}_i)  & cov(\mathbf{x}_i,\mathbf{y}_i) \\
            cov(\mathbf{y}_i,\mathbf{x}_i)  & cov(\mathbf{y}_i)
            \end{array} \right)
         =  \left( \begin{array}{c c}
            \boldsymbol{\Phi}_{11}  & \boldsymbol{\Phi}_{12} \\
            \boldsymbol{\Phi}_{12}^\top  & \boldsymbol{\Phi}_{22} \\
            \end{array} \right)    \\
cov(\mathbf{e}_i) &=& \boldsymbol{\Omega}    \\
cov(\mathbf{d}_i) &=& \boldsymbol{\Sigma}    \\
\end{eqnarray*} 

\vspace{-8mm}
\begin{itemize}
    \item Collect the unique elements of $\boldsymbol{\beta}$, $\boldsymbol{\Gamma}$, $\boldsymbol{\Lambda}$, $\boldsymbol{\Phi}_x$, $\boldsymbol{\Psi}$ and $\boldsymbol{\Omega}$ into a parameter vector $\boldsymbol{\theta}$. 
    \item $\boldsymbol{\theta}$ is a \emph{function} of the original model parameters.
\end{itemize}
} % End size
\end{frame}

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\begin{frame} % A huge slide
\frametitle{Matrix Form}
%\framesubtitle{} 
%\begin{center}
\begin{tabular}{ccc}
\includegraphics[width=1in]{mepath} 
& \raisebox{.45in}{
\begin{minipage}{1.5in}
{\footnotesize
\begin{eqnarray*}
    Y_{i,1} &=& \gamma_1 X_i  + \epsilon_{i,1}  \\
    Y_{i,2} &=& \beta Y_{i,1} + \gamma_2 X_i  + \epsilon_{i,2}   \\
    W_i   &=& \lambda_1 X_i + e_{i,1}   \\
    V_{i,1} &=& \lambda_2 Y_{i,1} + e_{i,2}   \\
    V_{i,2} &=& \lambda_3 Y_{i,2} + e_{i,3}    
\end{eqnarray*} 
} % End size
\end{minipage}
} % End raisebox
& \raisebox{.45in}{
\begin{minipage}{1in}
{\footnotesize
\begin{eqnarray*}
\mathbf{y}_i &=& \boldsymbol{\beta} \mathbf{y}_i + \boldsymbol{\Gamma} \mathbf{x}_i + \boldsymbol{\epsilon}_i \\
\mathbf{F}_i &=& \left( \begin{array}{c}
         \mathbf{x}_i  \\ \hline \mathbf{y}_i 
             \end{array} \right)  \\
\mathbf{d}_i &=& \boldsymbol{\Lambda}\mathbf{F}_i + \mathbf{e}_i     
\end{eqnarray*} 
} % End size
\end{minipage}
} % End raisebox
\end{tabular} \pause
\vspace{-3mm}
{\small
\begin{displaymath}
\begin{array}{cccccccccc}
\mathbf{y}_i &=& \boldsymbol{\beta} & \mathbf{y}_i & + & \boldsymbol{\Gamma} & \mathbf{x}_i & + & \boldsymbol{\epsilon}_i \\ 
&&&&&&&&& \\
\left( \begin{array}{c}
Y_{i,1} \\ Y_{i,2} 
             \end{array} \right) & = &  
\left( \begin{array}{c c}
           0     & 0 \\
           \beta & 0 \\
             \end{array} \right) &
\left( \begin{array}{c}
Y_{i,1} \\ Y_{i,2}  
             \end{array} \right) & + & 
\left( \begin{array}{c}
\gamma_1 \\ \gamma_2 
             \end{array} \right) & X_i & + & 
\left( \begin{array}{c}
\epsilon_{i,1} \\ \epsilon_{i,2} 
             \end{array} \right)
\end{array} \pause
\end{displaymath}
\begin{displaymath}
\begin{array}{cccccc}
\mathbf{d}_i &=& \boldsymbol{\Lambda} & \mathbf{F}_i &+& \mathbf{e}_i \\
&&&&& \\
\left( \begin{array}{c}
W_i \\ V_{i,1} \\ V_{i,2} 
\end{array} \right)             & = &
 \left( \begin{array}{c c c}
           \lambda_1 & 0 & 0 \\
           0 & \lambda_2 & 0 \\
           0 & 0 & \lambda_3
          \end{array} \right)   &
\left( \begin{array}{c}
X_i \\ Y_{i,1} \\ Y_{i,2} 
\end{array} \right)             & + & 
\left( \begin{array}{c}
e_{i,1} \\ e_{i,2} \\ e_{i,3} 
\end{array} \right)
\end{array}
\end{displaymath} % \pause
} % End size
% The rest just will not fit. Make it HW.
% {\scriptsize
% \vspace{2mm}
% $cov(\mathbf{X}_i)=Var(X_i)= \phi$ \pause
% $cov(\boldsymbol{\epsilon}_i)=\boldsymbol{\Psi} = 
%      \left( \begin{array}{c c}
%            \psi_1 & 0 \\
%             0     & \psi_2 
%              \end{array} \right)$ \pause
% $cov(\mathbf{e}_i)=\boldsymbol{\Omega} = 
%       \left( \begin{array}{c c c}
%            \omega_1    & 0 & 0\\
%           0 & \omega_2 & 0 \\
%           0 & 0 & \omega_3
%          \end{array} \right)$
%} % End size
\end{frame}
% End of huge slide

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\begin{frame}
\frametitle{Observable variables in the ``latent" variable model $\mathbf{y}_i = \boldsymbol{\beta} \mathbf{y}_i + \boldsymbol{\Gamma} \mathbf{x}_i + \boldsymbol{\epsilon}_i$ }
\framesubtitle{Fairly common}  
\begin{itemize}
    \item These present no problem. 
    \item Let $P(e_j=0) = 1$, so $Var(e_j) = 0$. 
    \item And $Cov(e_i,e_j)=0$ \pause
    \item Because if $P(e_j=0) = 1$, 
\begin{eqnarray*}
    Cov(e_i,e_j) &=& E(e_ie_j) - E(e_i)E(e_j) \\ 
                 &=& E(e_i\cdot 0) - E(e_i)\cdot 0 \\ 
                 &=& 0-0=0   
\end{eqnarray*} \pause
    \item In $\boldsymbol{\Omega} = cov(\mathbf{e}_i)$, column $j$ (and row $j$) are all zeros. 
    \item $\boldsymbol{\Omega}$ singular, no problem.
\end{itemize}
\end{frame}

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\begin{frame}
\frametitle{What should you be able to do?} 
%\framesubtitle{} 
\begin{itemize}
    \item Given a path diagram, write the model equations and say which exogenous variables are correlated with each other. 
    \item Given the model equations and information about which exogenous variables are correlated with each other, draw the path diagram. 
    \item Given either piece of information, write the model in matrix form and say what all the matrices are. 
    \item Calculate model covariance matrices. 
    \item Check identifiability.
\end{itemize}
\end{frame}

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% In the STA2053f22 version, the next several slides are about surrogate models. This material is fairly brutal. Put it in the text and defer change of variables to the CFA unit. 

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\begin{frame}
\frametitle{Recall the notation}
\begin{eqnarray*}
\mathbf{y}_i &=& \boldsymbol{\beta} \mathbf{y}_i + \boldsymbol{\Gamma} \mathbf{x}_i + \boldsymbol{\epsilon}_i \\
\mathbf{F}_i &=& \left( \begin{array}{c}
         \mathbf{x}_i  \\ \hline \mathbf{y}_i 
             \end{array} \right)  \\
\mathbf{d}_i &=& \boldsymbol{\Lambda}\mathbf{F}_i + \mathbf{e}_i  \\
&&\\ 
cov(\mathbf{x}_i) &=& \boldsymbol{\Phi}_x    \\
cov(\boldsymbol{\epsilon}_i) &=& \boldsymbol{\Psi}    \\
cov(\mathbf{F}_i) &=& \boldsymbol{\Phi} 
         =  \left( \begin{array}{c c}
            cov(\mathbf{x}_i)  & cov(\mathbf{x}_i,\mathbf{y}_i) \\
            cov(\mathbf{y}_i,\mathbf{x}_i)  & cov(\mathbf{y}_i)
            \end{array} \right)
         =  \left( \begin{array}{c c}
            \boldsymbol{\Phi}_{11}  & \boldsymbol{\Phi}_{12} \\
            \boldsymbol{\Phi}_{12}^\top  & \boldsymbol{\Phi}_{22} \\
            \end{array} \right)    \\
cov(\mathbf{e}_i) &=& \boldsymbol{\Omega}    \\
cov(\mathbf{D}_i) &=& \boldsymbol{\Sigma}    \\
\end{eqnarray*} \pause
Calculate a general expression for $\boldsymbol{\Sigma}(\boldsymbol{\theta})$. 
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{For the latent variable model, calculate $\boldsymbol{\Phi} = cov(\mathbf{F}_i)$}
\framesubtitle{Have $cov(\mathbf{x}_i) = \boldsymbol{\Phi}_x$, need $cov(\mathbf{y}_i)$ and $cov(\mathbf{x}_i,\mathbf{y}_i)$}  
%{\footnotesize
\begin{eqnarray*}
    & & \mathbf{y}_i = \boldsymbol{\beta} \mathbf{y}_i + \boldsymbol{\Gamma} \mathbf{x}_i + \boldsymbol{\epsilon}_i \\ \pause
    &\Rightarrow& \mathbf{y}_i - \boldsymbol{\beta} \mathbf{y}_i = \boldsymbol{\Gamma} \mathbf{x}_i + \boldsymbol{\epsilon}_i \\ \pause
    &\Rightarrow& \mathbf{Iy}_i - \boldsymbol{\beta} \mathbf{y}_i = \boldsymbol{\Gamma} \mathbf{x}_i + \boldsymbol{\epsilon}_i \\ \pause
    &\Rightarrow& (\mathbf{I} - \boldsymbol{\beta} )\mathbf{y}_i = \boldsymbol{\Gamma} \mathbf{x}_i + \boldsymbol{\epsilon}_i \\ \pause
    & {\color{red} \Rightarrow} & (\mathbf{I} - \boldsymbol{\beta} )^{-1}(\mathbf{I} - \boldsymbol{\beta} )\mathbf{y}_i = (\mathbf{I} - \boldsymbol{\beta} )^{-1}(\boldsymbol{\Gamma} \mathbf{x}_i + \boldsymbol{\epsilon}_i) \\ \pause
    &\Rightarrow& \mathbf{y}_i = (\mathbf{I} - \boldsymbol{\beta} )^{-1}(\boldsymbol{\Gamma} \mathbf{x}_i + \boldsymbol{\epsilon}_i) 
\end{eqnarray*}  \pause
So, 
\begin{eqnarray*}
    cov(\mathbf{y}_i)  \pause
    &=& (\mathbf{I} - \boldsymbol{\beta} )^{-1}cov(\boldsymbol{\Gamma} \mathbf{x}_i + \boldsymbol{\epsilon}_i)(\mathbf{I} - \boldsymbol{\beta} )^{-1\top} \\ \pause
    &=& (\mathbf{I} - \boldsymbol{\beta} )^{-1}\left(cov(\boldsymbol{\Gamma} \mathbf{x}_i) + 
        cov(\boldsymbol{\epsilon}_i)\right) (\mathbf{I} - \boldsymbol{\beta}^\top )^{-1} \\ \pause
    &=& (\mathbf{I} - \boldsymbol{\beta} )^{-1}
        \left(\boldsymbol{\Gamma}\boldsymbol{\Phi}_x\boldsymbol{\Gamma}^\top + \boldsymbol{\Psi} \right)
        (\mathbf{I} - \boldsymbol{\beta}^\top )^{-1} 
\end{eqnarray*}
%} % End size
\end{frame}
% That's a lot cleaner than the way I was doing it before. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{Theorem: If the original model holds, $(\mathbf{I} - \boldsymbol{\beta} )^{-1}$ exists.} \pause
%\framesubtitle{} 
$\mathbf{y}_i = \boldsymbol{\alpha} + \boldsymbol{\beta} \mathbf{y}_i + \boldsymbol{\Gamma} \mathbf{x}_i + \boldsymbol{\epsilon}_i$ yields $ (\mathbf{I} - \boldsymbol{\beta} )\mathbf{y}_i = \boldsymbol{\alpha} +\boldsymbol{\Gamma} \mathbf{x}_i + \boldsymbol{\epsilon}_i$. \pause

Suppose $(\mathbf{I} - \boldsymbol{\beta} )^{-1}$ does not exist. \pause \vspace{2mm}

Then the rows of $\mathbf{I} - \boldsymbol{\beta}$ are linearly dependent, and there is a $q \times 1$ non-zero vector of constants $\mathbf{a}$ with $\mathbf{a}^\top (\mathbf{I} - \boldsymbol{\beta} ) = 0$. \pause So, 
\begin{eqnarray*}
 \mathbf{a}^\top(\mathbf{I} - \boldsymbol{\beta} )\mathbf{y}_i &=& 0 \pause = \mathbf{a}^\top\boldsymbol{\alpha} + \mathbf{a}^\top\boldsymbol{\Gamma} \mathbf{x}_i + \mathbf{a}^\top\boldsymbol{\epsilon}_i \\ \pause
\Rightarrow Var(0) &=& Var(\mathbf{a}^\top\boldsymbol{\Gamma} \mathbf{x}_i) + Var(\mathbf{a}^\top\boldsymbol{\epsilon}_i) \\ \pause
\Rightarrow 0 &=&   \mathbf{a}^\top \boldsymbol{\Gamma \Phi}_x \boldsymbol{\Gamma}^\top \mathbf{a} 
                  + \mathbf{a}^\top \boldsymbol{\Psi}\mathbf{a} \pause > 0. 
\end{eqnarray*} \pause

Contradicts $\mathbf{I} - \boldsymbol{\beta}$ singular.

%But the quantity on the right side is strictly positive, because while $\boldsymbol{\Gamma \Phi}_x \boldsymbol{\Gamma}^\top = cov(\boldsymbol{\Gamma} \mathbf{x}_i)$ is only guaranteed to be non-negative definite, $\boldsymbol{\Psi}$ is strictly positive definite according to the model. Thus, the assumption that $\mathbf{I} - \boldsymbol{\beta}$ is singular leads to a contradiction. This shows that $(\mathbf{I} - \boldsymbol{\beta} )^{-1}$ must exist if the model holds. ~$\blacksquare$
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{A hole in the parameter space}
%\framesubtitle{} 
$|\mathbf{I} - \boldsymbol{\beta}| \neq 0$ can create a hole in the parameter space. 
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{More calculations} 
%\framesubtitle{} 
\begin{itemize}
    \item  Have $ cov(\mathbf{y}_i) = (\mathbf{I} - \boldsymbol{\beta} )^{-1}
        \left(\boldsymbol{\Gamma}\boldsymbol{\Phi}_x\boldsymbol{\Gamma}^\top + 
        \boldsymbol{\Psi} \right)
        (\mathbf{I} - \boldsymbol{\beta}^\top )^{-1}$. \pause
    \item Know $cov(\mathbf{x}_i) = \boldsymbol{\Phi}_x$
    \item Easy to get $cov(\mathbf{x}_i,\mathbf{y}_i)$. \pause
    \item[]
\end{itemize}
\end{frame}

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\begin{frame}
\frametitle{For the measurement model, calculate $\boldsymbol{\Sigma} = cov(\mathbf{d}_i)$} 
%\framesubtitle{} 
%{\LARGE
\begin{eqnarray*}
    \mathbf{d}_i &=& \boldsymbol{\Lambda}\mathbf{F}_i + \mathbf{e}_i \\ \pause
\Rightarrow cov(\mathbf{d}_i) &=& cov(\boldsymbol{\Lambda}\mathbf{F}_i + \mathbf{e}_i) \\ \pause
 &=& cov(\boldsymbol{\Lambda}\mathbf{F}_i) + cov(\mathbf{e}_i) \\ \pause
 &=& \boldsymbol{\Lambda}cov(\mathbf{F}_i)\boldsymbol{\Lambda}^\top + cov(\mathbf{e}_i) \\ \pause
 &=& \boldsymbol{\Lambda}\boldsymbol{\Phi}\boldsymbol{\Lambda}^\top + \boldsymbol{\Omega} \\ \pause
 &=& \boldsymbol{\Sigma} 
\end{eqnarray*} 
%} % End size
\end{frame}

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\begin{frame}
\frametitle{Two-stage Proofs of Identifiability} 
\framesubtitle{Stage 1 is the latent variable model and Stage 2 is the measurement model.}  \pause
\begin{itemize}
    \item Show the parameters of the latent variable model $(\boldsymbol{\beta}, \boldsymbol{\Gamma}, \boldsymbol{\Phi}_x, \boldsymbol{\Psi})$ can be recovered from $\boldsymbol{\Phi} = cov(\mathbf{F}_i)$. \pause
    \item Solve $\left( \begin{array}{c c}
            cov(\mathbf{x}_i)  & cov(\mathbf{x}_i,\mathbf{y}_i) \\
            cov(\mathbf{y}_i,\mathbf{x}_i)  & cov(\mathbf{y}_i)
            \end{array} \right) = \boldsymbol{\Phi}
         =  \left( \begin{array}{c c}
            \boldsymbol{\Phi}_{11}  & \boldsymbol{\Phi}_{12} \\
            \boldsymbol{\Phi}_{12}^\top  & \boldsymbol{\Phi}_{22} \\
            \end{array} \right)$ for $(\boldsymbol{\beta}, \boldsymbol{\Gamma}, 
            \boldsymbol{\Phi}_x, \boldsymbol{\Psi})$? \pause
    \item Show the parameters of the measurement model $(\boldsymbol{\Lambda},\boldsymbol{\Phi},\boldsymbol{\Omega})$ can be recovered from $\boldsymbol{\Sigma} = cov(\mathbf{d}_i)$. \pause
    \item This means all the parameters can be recovered from $\boldsymbol{\Sigma}$. \pause
    \item Break a big problem into two smaller ones. 
    \item Develop \emph{rules} for checking identifiability at each stage.
    \item Just look at the path diagram.
\end{itemize}
\end{frame}

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\begin{frame}
\frametitle{Copyright Information}

This slide show was prepared by  \href{http://www.utstat.toronto.edu/~brunner}{Jerry Brunner},
Department of Statistical Sciences, University of Toronto. It is licensed under a 
\href{http://creativecommons.org/licenses/by-sa/3.0/deed.en_US}
     {Creative Commons Attribution - ShareAlike 3.0 Unported License}. Use any part of it as you like and share the result freely. The \LaTeX~source code is available from the course website:
\vspace{3mm}

\href{http://www.utstat.toronto.edu/brunner/oldclass/431s23} {\small\texttt{http://www.utstat.toronto.edu/brunner/oldclass/431s23}}

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\end{document}

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