pywafo/statsmodels/scikits/statsmodels/sandbox/mlogitmath.lyx

#LyX 1.6.2 created this file. For more info see http://www.lyx.org/
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\begin_layout Standard
Notes on mlogit.
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\begin_layout Standard
Assume that 
\begin_inset Formula $J=3$
\end_inset

, so that there are 
\begin_inset Formula $2$
\end_inset

 vectors of parameters for 
\begin_inset Formula $J-1$
\end_inset

.
 For now the parameters are passed around as 
\begin_inset Formula \[
\left[\beta_{1}^{\prime}\beta_{2}^{\prime}\right]\]

\end_inset


\end_layout

\begin_layout Standard
So if 
\begin_inset Formula $K=3$
\end_inset

 (including the constant), then the matrix of parameters is
\begin_inset Formula \[
\left[\begin{array}{cc}
b_{10} & b_{20}\\
b_{11} & b_{21}\\
b_{12} & b_{22}\end{array}\right]^{\prime}\]

\end_inset


\end_layout

\begin_layout Standard
(changed to rows and added prime above, so this all changes and the score
 is also just transposed and flattend along the zero axis.) This is flattened
 along the zero axis for the sake of the solvers.
 So that it is passed internally as 
\begin_inset Formula \[
\left[\begin{array}{cccccc}
b_{10} & b_{20} & b_{11} & b_{21} & b_{12} & b_{22}\end{array}\right]\]

\end_inset

Now the matrix of score vectors is
\begin_inset Formula \[
\left[\begin{array}{cc}
\frac{\partial\ln L}{\partial b_{10}} & \frac{\partial\ln L}{\partial b_{20}}\\
\frac{\partial\ln L}{\partial b_{11}} & \frac{\partial\ln L}{\partial b_{21}}\\
\frac{\partial\ln L}{\partial b_{12}} & \frac{\partial\ln L}{\partial b_{22}}\end{array}\right]\]

\end_inset


\end_layout

\begin_layout Standard
In Dhrymes notation, this would be column vectors 
\begin_inset Formula $\left(\partial\ln L/\partial\beta_{j}\right)^{\prime}\text{ for }j=1,2$
\end_inset

 in our example.
 So, our Jacobian is actually transposed vis-a-vis the more traditional
 notation.
 So that the solvers can handle this, though, it gets flattened but the
 score gets flattened along the first axis to make things easier, which
 is going to make things tricky.
 Now, in traditional notation, the Hessian would be 
\begin_inset Formula \[
\frac{\partial^{2}\ln L}{\partial\beta_{j}\partial\beta_{j}}=\frac{\partial}{\partial\beta_{j}}\vec{\left[\left(\frac{\partial\ln L}{\partial\beta_{j}}\right)\right]}\]

\end_inset


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\begin_layout Standard
where 
\begin_inset Formula $\vec{}$
\end_inset

 denotes a vectorized matrix, i.e., for a 
\begin_inset Formula $n\times m$
\end_inset

 matrix 
\begin_inset Formula $X$
\end_inset

, 
\begin_inset Formula $\vec{X}=\left(x_{\cdot1}^{\prime},x_{\cdot2}^{\prime},...,x_{\cdot m}^{\prime}\right)^{\prime}$
\end_inset

 such that 
\begin_inset Formula $x_{\cdot1}$
\end_inset

 is the first 
\begin_inset Formula $n$
\end_inset

 elements of column 1 of 
\begin_inset Formula $X$
\end_inset

.
 This matrix is 
\begin_inset Formula $mn\times n$
\end_inset

.
 In our case 
\begin_inset Formula $\ln L$
\end_inset

 is a scalar so 
\begin_inset Formula $m=1$
\end_inset

, so each second derivative is 
\begin_inset Formula $n\times n$
\end_inset

 and 
\begin_inset Formula $n=K=3$
\end_inset

 in our example.
 Given our score 
\begin_inset Quotes eld
\end_inset

matrix,
\begin_inset Quotes erd
\end_inset

 our Hessian will look like
\begin_inset Formula \[
H=\left[\begin{array}{cc}
\frac{\partial\ln L}{\partial\beta_{1}\partial\beta_{1}} & \frac{\partial\ln L}{\partial\beta_{1}\partial\beta_{2}}\\
\frac{\partial\ln L}{\partial\beta_{2}\partial\beta_{1}} & \frac{\partial\ln L}{\partial\beta_{2}\partial\beta_{2}}\end{array}\right]\]

\end_inset


\begin_inset Formula \[
H=\left[\begin{array}{cccccc}
\frac{\partial^{2}\ln L}{\partial b_{10}\partial b_{10}} & \frac{\partial^{2}\ln L}{\partial b_{10}\partial b_{11}} & \frac{\partial^{2}\ln L}{\partial b_{10}\partial b_{12}} & \frac{\partial^{2}\ln L}{\partial b_{10}\partial b_{20}} & \frac{\partial^{2}\ln L}{\partial b_{10}\partial b_{21}} & \frac{\partial^{2}\ln L}{\partial b_{10}\partial b_{22}}\\
\frac{\partial\ln L}{\partial b_{11}\partial b_{10}} & \frac{\partial\ln L}{\partial b_{11}\partial b_{11}} & \frac{\partial\ln L}{\partial b_{11}\partial b_{12}} & \frac{\partial\ln L}{\partial b_{11}\partial b_{20}} & \frac{\partial\ln L}{\partial b_{11}\partial b_{21}} & \frac{\partial\ln L}{\partial b_{11}\partial b_{22}}\\
\frac{\partial\ln L}{\partial b_{12}\partial b_{10}} & \frac{\partial\ln L}{\partial b_{12}\partial b_{11}} & \frac{\partial\ln L}{\partial b_{12}\partial b_{12}} & \frac{\partial\ln L}{\partial b_{12}\partial b_{20}} & \frac{\partial\ln L}{\partial b_{12}\partial b_{21}} & \frac{\partial\ln L}{\partial b_{12}\partial b_{22}}\\
\frac{\partial^{2}\ln L}{\partial b_{20}\partial b_{10}} & \frac{\partial^{2}\ln L}{\partial b_{20}\partial b_{11}} & \frac{\partial^{2}\ln L}{\partial b_{20}\partial b_{12}} & \frac{\partial^{2}\ln L}{\partial b_{20}\partial b_{20}} & \frac{\partial^{2}\ln L}{\partial b_{20}\partial b_{21}} & \frac{\partial^{2}\ln L}{\partial b_{20}\partial b_{22}}\\
\frac{\partial\ln L}{\partial b_{21}\partial b_{10}} & \frac{\partial\ln L}{\partial b_{21}\partial b_{11}} & \frac{\partial\ln L}{\partial b_{21}\partial b_{12}} & \frac{\partial\ln L}{\partial b_{21}\partial b_{20}} & \frac{\partial\ln L}{\partial b_{21}\partial b_{21}} & \frac{\partial\ln L}{\partial b_{21}\partial b_{22}}\\
\frac{\partial\ln L}{\partial b_{22}\partial b_{10}} & \frac{\partial\ln L}{\partial b_{22}\partial b_{11}} & \frac{\partial\ln L}{\partial b_{22}\partial b_{12}} & \frac{\partial\ln L}{\partial b_{22}\partial b_{20}} & \frac{\partial\ln L}{\partial b_{22}\partial b_{21}} & \frac{\partial\ln L}{\partial b_{22}\partial b_{22}}\end{array}\right]\]

\end_inset


\end_layout

\begin_layout Standard
But since our Jacobian is a row vector that alternate
\end_layout

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