\chapter{Methodology}
\label{ch:method}


\section{A Statistical Model: Logistic Regression}

I apply a statistical model to historical data on issuers' characteristics.  The particular form of statistical model is a discrete-time
event history model.  This model is designed to predict the risk of an event occuring, as a function of specified variables measured before the event occurs.  

The linear regression (a discrete time model) can be used to predict the risk of an event within a certain time period.  This is equalvent and estimated by applying a logistic regression to issuer-year of data.

The logistic regression takes the following form
\begin{equation}
\log \left(\frac{p}{1-p} \right) = \sum^K_{k=1} \beta_k\,x_k \label{eq:one}
\end{equation}
where $p$ is the probability of the even occuring, and $K$ independent variables, $x$, are each weighted by a coeffiecent, $\beta$.


From \ref{eq:one}, this different from the conventional linear regression model
\begin{equation}	\label{eq:two}
y = \sum^K_{k=1} \beta_k\,x_k 
\end{equation}

From \ref{eq:two} differs from \ref{eq:one}, because does not predict the value of the dependent variable, $y$, but rather the natural logarithm of the ratio of the probability of the event occurring to the probability of the event not occurring.  

This logit function preferable to the linear regression function because it limits $p$, the probability of the event, to be between 0 and 1 (or zero and $100\%$), where applying the conventional linear regression to dichotomous outcomes would instead allow nonsensical results like probabilities greater than 1 or less than zero.

A transformation of \ref{eq:one}, obtain the logistic model which gives the probability of the event as
\begin{equation}\label{eq:three}
p = \frac{e^{\sum{{\sum^K_{k=1} \beta_k\,x_k}}}}{1+e^{\sum^K_{k=1} \beta_k\,x_k}} 	
\end{equation}
\vspace{.25cm}

This difference in structure between this model and more familiar linear regression models gives rise to
differences in how the results of the model can be interpreted.  Changing one independent
variable by a fixed amount changes the level of the dependent variable by an amount
that is identical, no matter what the levels of the other independent variables.


If interpreting a linear regression to predict the risk of a bank failing, whether
its current risk was $10\%$, decreasing its ratio of net income to assets by a fixed quantity would increase its risk of failure by the same amount—let's say $1\%$, respectively $11\%$. 
\vspace{.25cm}

In contrast, in a logistic regression, a change in one factor changes the risk by an amount that is \textbf{proportional} to the level of the other factors. If the other factors produce a failure probability of $2\%$, decreasing the ratio of net income to assets might double the risk to $4\%$.


\section{Choosing a Significant Model}

Am estimating algorithm is used to find the coefficients, $\beta$'s that best satisfy the relationship expressed in the regression equation for the estimation data sample. The technique used to find those coefficients for logistic
regression, was using maximum likelihood estimation.   Basically, method tries coefficients until it finds the set that maximizes the value of a mathematical function that gives the joint probability of observing the given data.
\vspace{.25cm}

That function, $L$, the likelihood function, forms the basis of a statistical test of
how well the model fits the observed data:
\begin{equation}
L^2 = 2(\log L_T - \log L_B) 	\label{four}
\end{equation}
where $L_T$ is the likelihood function of the first model with smaller variables and the $L_B$ is the liklihood function of a baseline model.  


$L^2$ is a statistic that will be compared with the standard $\chi^2$ table to determine whether the tested model fits significantly better than a baseline model.  This procedure is completed to establish whether the added variables
significantly improve the fit to the data, or whether conversely the smaller subset is
equally sufficient. We use this procedure to omit from the model variables that do not
significantly improve our ability to predict insolvency.
