\chapter{Results}
\label{ch:results}

\section{Saturated Logistic Regression Model 1}

Preliminary analysis began by creating a "saturated" model including all predictors (ratio) described in the previous section.  There were concerns of that several ratios are correlated.  The correlation matrix of the variables is provided below.  A decision was made to to include only parameters with insignificant correlation in the model.  It should be noted, however, that in the aggregate the results would have been essentially identical with either variable, given that the two ratios are empirically virtually indistinguishable.

A correlation matrix is provided.  In preliminary analysis, the correlations between \textbf{EBITA/TA} and \textbf{EBIT/TA} was high.  Running the model with each variable, it was determined to use only \textbf{EBITA/TA}.  Also, \textbf{NI/TA}, also showed higher correlations across several other parameters.  The variable was dropped from the model possibilities, and furthermore, the variable was found to be insignificant, which will be illustrated in the outputs.

\begin{flushleft}
\begin{figure}
\includegraphics[scale=0.4]{correlations.png} 
\caption{Correlation Matrix}
\end{figure}
\end{flushleft}

\newpage

\subsection{Saturated Logistic Regression Model 1: Output and Diagnostics}

Using SAS and Excel, I obtained the following outputs.  SAS is listed below and Excel is provided later.  Both programs confirmed the results.  In the saturated model it was found that \textbf{EQY SH OUT} (Equity Share Values), \textbf{WC/TA} (Working Capital/Total Assets), \textbf{RE/TA} (Retained Earnings/Total Assets). and \textbf{EBIT/TA} to be insignificant to the model.  These parameters were taken out and the likelihood ratio will be calculated with the second model.  Although the p-values were insignificant on \textbf{EBITA/TA} and \textbf{PX LAST} (Last trading price), each was left in the analysis to determine a reduced model.  
\begin{flushleft}
\begin{figure}
\includegraphics[scale=0.4]{logmodelSAS1.png} 
\caption{SAS Output: Saturated Logistic Model}
\end{figure}
\end{flushleft}
\subsection{Saturated Logistic Regression Model 1 Interpretation}
\begin{itemize}
\item Then if \textbf{ME/TL} changes increases by 1, the odds that the default takes the value 1 increase $e^{-0.2396}=76.69\%$
\end{itemize}

\newpage
\section{Logistic Regression Reduced Model 2}

After rerunning the model with parameters removed In the saturated model it was found that all remaining parameters to be significant to the model. However, analyzing the p-value from the likelihood ratio test, it was determined that we can not reject the null hypothesis and there is no different between either models.    The likelihood ratio test for the hypothesis for LR- Test for b(EQY SH OUT)=b(WC/TA)=b(RE/TA)=b(EBIT/TA)=0 in model 1 is based on the log likelihoods f the two models.  It has a $\chi^2$=distribution with 5 degrees degrees of freedom because imposed 4 restrictions

Because of the variables and what each represents, the tendency may toward model 2.  Further validation techniques will be discussed in a later chapter to compare all three models with accuracy ratios and a bootstrapping method will be used to create confidence intervals from the predictions.  From the Excel output, the LR=4.81 and the SAS output LR=4.82.  

From the Excel output, the p-value can be interpreted as if were to add the eliminated variables to model 2, there is a probability of $30.76\%$ that we do not add explanatory power.  
\subsection{Logistic Regression Model 2: Output and Diagnostics}
\begin{flushleft}
\begin{figure}
\includegraphics[scale=0.4]{logmodelSAS2.png} 
\caption{SAS Output: Reduced Logistic Model 2}
\end{figure}
\end{flushleft}
\subsection{Logistic Regression Model 2: Interpretation}
\begin{itemize}
\item Then if \textbf{ME/TL} changes increases by 1, the odds that the default takes the value 1 increase $e^{-0.200}=81.87\%$
\item Then if \textbf{SA/TA} changes increases by 1, the odds that the default takes the value 1 increase $e^{0.0638}$
\end{itemize}
\newpage
\section{Logistic Regression Reduced Model 3}

To analyze the market variables in comparison to a model that is solely on financial balance sheet information, model 3 is compared to model 2.  The incorporation of market was included in the model building process to accommodate for large market movements, such as the the financial distress in recent times of 2008.  The variable, \textbf{PX LAST}, is the last trade price of the issuer on the market.

Further validation techniques will be discussed in a later chapter to compare all three models with accuracy ratios from the predictions.  From the Excel output, the LR=10.04 and the SAS output LR=7.17 with a degree of freedom of 1.  In this model, with the p-value < $1\%$, model 3 can be interpeted as a better model.  

From the Excel output, the p-value can be interpreted as if were to add the eliminated variable to model 3 \textbf{PX LAST}, there is a probability of $0.15\%$ that we do not add explanatory power. This is a smaller contribution, so model 3 will be considered, however, further validation will be compeleted as well as predicting and comparing results to other statistical and structural model used in the financial market.  
\subsection{Logistic Regression Model 3: Output and Diagnostics}
\begin{flushleft}
\begin{figure}
\includegraphics[scale=0.4]{logmodelSAS3.png} 
\caption{SAS Output: Reduced Logistic Model 3}
\end{figure}
\end{flushleft}

\subsection{Logistic Regression Model 3: Interpretation}
\begin{itemize}
\item Then if \textbf{ME/TL} changes increases by 1, the odds that the default takes the value 1 increase $e^{-0.1887}=81.87\%$
\item Then if \textbf{SA/TA} changes increases by 1, the odds that the default takes the value 1 increase $e^{0.0641}$
\item Then if \textbf{EBITA/TA} changes increases by 1, the odds that the default takes the value 1 increase $e^{50.3147}$
\end{itemize}

\subsection{Logistic Regression Model: Excel Output}

\begin{flushleft}
\begin{figure}
\includegraphics[scale=0.2]{logmodelexcel.png} 
\caption{Excel Output: Logistic Models}
\end{figure}
\end{flushleft}