%                       Large sample tools for Applied Stat I
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% Comment this out for handout
\title{Omitted Variables\footnote{See last slide for copyright information.}}
\subtitle{STA302 Fall 2013}
\date{} % To suppress date

\begin{document}

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


\begin{frame}
\frametitle{The fixed $x$ regression model}

$Y_i = \beta_0 + \beta_1 x_{i,1} + \cdots + \beta_k x_{i,p-1} + \epsilon_i, \mbox{ with }
    \epsilon_i \sim N(0,\sigma^2)$

\vspace{15mm}

Think of the model as \emph{conditional} given $\mathbf{X}_i = \mathbf{x}_i$.
\end{frame}

\begin{frame}
\frametitle{Independence of $\epsilon_i$ and $\mathbf{X}_i$}

  \begin{itemize}
    \item The statement $\epsilon_i \sim N(0,\sigma^2)$ is a statement about the \emph{conditional} distribution of $\epsilon_i$ given $\mathbf{X}_i = \mathbf{x}_i$.
    \item It says the density of $\epsilon_i$ given $\mathbf{X}_i = \mathbf{x}_i$ does not depend on $\mathbf{x}_i$.
    \item For convenience, assume $\mathbf{X}_i$ has a density.
  \end{itemize}

% \vspace{5mm}

\begin{eqnarray*}
     & & f_{\epsilon|\mathbf{x}}(\epsilon|\mathbf{x}) = f_{\epsilon}(\epsilon) \\
     & \Rightarrow & 
     \frac{f_{\epsilon,\mathbf{x}}(\epsilon,\mathbf{x})}{f_{\mathbf{x}}(\mathbf{x})} 
     =  f_{\epsilon}(\epsilon) \\
     & \Rightarrow & 
    f_{\epsilon,\mathbf{x}}(\epsilon,\mathbf{x}) 
    = f_{\mathbf{x}}(\mathbf{x}) f_{\epsilon}(\epsilon)
\end{eqnarray*} 
Independence!
\end{frame}


\begin{frame}
\frametitle{The fixed $x$ regression model}
\framesubtitle{$Y_i = \beta_0 + \beta_1 x_{i,1} + \cdots + \beta_k x_{i,p-1} + \epsilon_i, \mbox{ with }\epsilon_i \sim N(0,\sigma^2)$}

\vspace{5mm}

  \begin{itemize}
    \item If viewed as conditional on $\mathbf{X}_i = \mathbf{x}_i$, this model implies independence of $\epsilon_i$ and $\mathbf{X}_i$.
    \item What is $\epsilon_i$? \emph{Everything else} that affects $Y_i$. 
    \item So the usual model says that if the independent varables are random, they have \emph{zero covariance} with all other variables that are related to $Y_i$, but do not appear in the model.
    \item For observational data, this assumption is almost always violated.
    \item Does it matter?
  \end{itemize}
\end{frame}


\begin{frame}
\frametitle{Example}
Suppose that the variables $X_2$ and $X_3$ have an impact on $Y$ and are correlated with $X_1$, but they are not part of the data set. The values of the dependent variable are generated as follows:
\begin{displaymath}
    Y_i = \beta_0 + \beta_1 X_{i,1} + \beta_2 X_{i,2} + \beta_2 X_{i,3} + \epsilon_i,
\end{displaymath}
independently for $i= 1, \ldots, n$, where $\epsilon_i \sim N(0,\sigma^2)$. The independent variables are random, with expected value and variance-covariance matrix
\begin{displaymath}
    E\left[ \begin{array}{c} X_{i,1} \\ X_{i,2} \\ X_{i,3} \end{array} \right] =
     \left[ \begin{array}{c} \mu_1 \\ \mu_2 \\ \mu_3 \end{array} \right]
\mbox{ ~and~ }
V\left[ \begin{array}{c} X_{i,1} \\ X_{i,2} \\ X_{i,3} \end{array} \right] =
 \left[ \begin{array}{rrr}
\phi_{11} & \phi_{12} & \phi_{13} \\ 
          & \phi_{22} & \phi_{23} \\
          &           & \phi_{33}
\end{array} \right], 
\end{displaymath}
where $\epsilon_i$ is independent of $X_{i,1}$, $X_{i,2}$ and $X_{i,3}$.
\end{frame}

\begin{frame}
\frametitle{Absorb $X_2$ and $X_3$}
\begin{columns}   % Use Beamer's columns to make narrower margins!
\column{1.1\textwidth}
Since $X_2$ and $X_3$ are not observed, they are absorbed by the intercept and error term.
{\small
\begin{eqnarray*}
    Y_i &=& \beta_0 + \beta_1 X_{i,1} + \beta_2 X_{i,2} + \beta_2 X_{i,3} 
    + \epsilon_i \\
        &=& (\beta_0 + \beta_2\mu_2 + \beta_3\mu_3) + \beta_1 X_{i,1} + (\beta_2 X_{i,2} + \beta_3 X_{i,3} - \beta_2\mu_2 - \beta_3\mu_3 + \epsilon_i) \\
        &=& \beta^\prime_0 + \beta_1 X_{i,1} + \epsilon^\prime_i.
\end{eqnarray*}
} % End size

And,
\begin{displaymath}
    Cov(X_{i,1},\epsilon^\prime_i) = \beta_2\phi_{12} + \beta_3\phi_{13} \neq 0
\end{displaymath}
\end{columns}
\end{frame}

\begin{frame}
\frametitle{The ``True" Model}
\framesubtitle{Almost always true for observational data} 
{\LARGE
\begin{displaymath}
     Y_i = \beta_0 + \beta_1 X_i + \epsilon_i,
\end{displaymath}
 } % End Size
 
\vspace{5mm}

where $E(X_i)=\mu_x$, $Var(X_i)=\sigma^2_x$,  $E(\epsilon_i)=0$, $Var(\epsilon_i)=\sigma^2_\epsilon$, and $Cov(X_i,\epsilon_i)=c$.

\vspace{5mm}

Under this model,
\begin{displaymath}
     \sigma_{xy} = Cov(X_i,Y_i) = Cov(X_i,\beta_0 + \beta_1 X_i + \epsilon_i) 
             = \beta_1 \sigma^2_x + c
\end{displaymath}
\end{frame}

\begin{frame}
\frametitle{Estimate $\beta_1$ as usual}

\begin{eqnarray*}
\widehat{\beta}_1 &=& \frac{\sum_{i=1}^n(X_i-\overline{X})(Y_i-\overline{Y})}
                           {\sum_{i=1}^n(X_i-\overline{X})^2}  \\ 
                  &=& \frac{\frac{1}{n}\sum_{i=1}^n(X_i-\overline{X})(Y_i-\overline{Y})}
                           {\frac{1}{n}\sum_{i=1}^n(X_i-\overline{X})^2} \\
                  &=& \frac{\widehat{\sigma}_{xy}}{\widehat{\sigma}^2_x} \\
                  &\approx& \frac{\sigma_{xy}}{\sigma^2_x} 
                  \mbox{ for large } n\\
                  &=& \frac{\beta_1 \sigma^2_x + c}{\sigma^2_x} \\
                  &=& \beta_1 + \frac{c}{\sigma^2_x}
\end{eqnarray*}
\end{frame}


\begin{frame}
\frametitle{$\widehat{\beta}_1 \approx \beta_1 + \frac{c}{\sigma^2_x}$}
\begin{itemize}
    \item $\widehat{\beta}_1$ is biased, even as $n\rightarrow\infty$.
    \item It's inconsistent.
    \item It could be almost anything, depending on the value of $c$, the covariance between $X_i$ and $\epsilon_i$.
    \item The only time $\widehat{\beta}_1$ behaves properly is when $c=0$.
    \item Probability of Type I error goes almost surely to one.
    \item What if $\beta_1 < 0$ but $\beta_1 + \frac{c}{\sigma^2_x} > 0$?
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{All this applies to multiple regression}
\framesubtitle{Of course}

\emph{When a regression model fails to include all the independent variables that contribute to the dependent variable, and those omitted independent variables have non-zero covariance with variables that are in the model, the regression coefficients are biased and inconsistent}.
\end{frame}


\begin{frame}
\frametitle{Correlation-Causation}
\begin{itemize}
    \item The problem of omitted variables is the technical version of the correlation-causation issue.
    \item The omitted variables are ``confounding" variables.
    \item With random assignment and good procedure, $x$ and $\epsilon$ have zero covariance.
    \item But random assignment is not always possible. 
\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 Statistics, 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:
\href{http://www.utstat.toronto.edu/~brunner/oldclass/302f13} {\small\texttt{http://www.utstat.toronto.edu/$^\sim$brunner/oldclass/302f13}}

\end{frame}


\end{document}


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\begin{frame}
\frametitle{Example}

\end{frame}

\begin{frame}
\frametitle{}
% \framesubtitle{}
\begin{itemize}
    \item 
    \item 
    \item 
\end{itemize}
\end{frame}

{\LARGE
\begin{displaymath}
    
\end{displaymath}
 } % End Size
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