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<TITLE>GNU Octave - Optimization</TITLE>
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<H1><A NAME="SEC154" HREF="octave_toc.html#TOC154">Optimization</A></H1>
<P>
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<H2><A NAME="SEC155" HREF="octave_toc.html#TOC155">Quadratic Programming</A></H2>
<H2><A NAME="SEC156" HREF="octave_toc.html#TOC156">Nonlinear Programming</A></H2>
<H2><A NAME="SEC157" HREF="octave_toc.html#TOC157">Linear Least Squares</A></H2>
<P>
<DL>
<DT><U>Function File:</U> [<VAR>beta</VAR>, <VAR>v</VAR>, <VAR>r</VAR>] = <B>gls</B> <I>(<VAR>y</VAR>, <VAR>x</VAR>, <VAR>o</VAR>)</I>
<DD><A NAME="IDX762"></A>
Generalized least squares estimation for the multivariate model
<CODE><VAR>y</VAR> = <VAR>x</VAR> * <VAR>b</VAR> + <VAR>e</VAR></CODE> with <CODE>mean (<VAR>e</VAR>) =
0</CODE> and <CODE>cov (vec (<VAR>e</VAR>)) = (<VAR>s</VAR>^2)*<VAR>o</VAR></CODE>,
where
<VAR>Y</VAR> is a <VAR>T</VAR> by <VAR>p</VAR> matrix, <VAR>X</VAR> is a <VAR>T</VAR> by <VAR>k</VAR>
matrix, <VAR>B</VAR> is a <VAR>k</VAR> by <VAR>p</VAR> matrix, <VAR>E</VAR> is a <VAR>T</VAR> by
<VAR>p</VAR> matrix, and <VAR>O</VAR> is a <VAR>T</VAR><VAR>p</VAR> by <VAR>T</VAR><VAR>p</VAR>
matrix.
</P>
<P>
Each row of Y and X is an observation and each column a variable.
</P>
<P>
The return values <VAR>beta</VAR>, <VAR>v</VAR>, and <VAR>r</VAR> are defined as
follows.
</P>
<DL COMPACT>
<DT><VAR>beta</VAR>
<DD>
The GLS estimator for <VAR>b</VAR>.
<DT><VAR>v</VAR>
<DD>
The GLS estimator for <CODE><VAR>s</VAR>^2</CODE>.
<DT><VAR>r</VAR>
<DD>
The matrix of GLS residuals, <CODE><VAR>r</VAR> = <VAR>y</VAR> - <VAR>x</VAR> *
<VAR>beta</VAR></CODE>.
</DL>
</DL>
<P>
<DL>
<DT><U>Function File:</U> [<VAR>beta</VAR>, <VAR>sigma</VAR>, <VAR>r</VAR>] = <B>ols</B> <I>(<VAR>y</VAR>, <VAR>x</VAR>)</I>
<DD><A NAME="IDX763"></A>
Ordinary least squares estimation for the multivariate model
<CODE><VAR>y</VAR> = <VAR>x</VAR>*<VAR>b</VAR> + <VAR>e</VAR></CODE> with
<CODE>mean (<VAR>e</VAR>) = 0</CODE> and <CODE>cov (vec (<VAR>e</VAR>)) = kron (<VAR>s</VAR>,
<VAR>I</VAR>)</CODE>.
where
<VAR>y</VAR> is a <VAR>t</VAR> by <VAR>p</VAR> matrix, <VAR>X</VAR> is a <VAR>t</VAR> by <VAR>k</VAR>
matrix, <VAR>B</VAR> is a <VAR>k</VAR> by <VAR>p</VAR> matrix, and <VAR>e</VAR> is a <VAR>t</VAR>
by <VAR>p</VAR> matrix.
</P>
<P>
Each row of <VAR>y</VAR> and <VAR>x</VAR> is an observation and each column a
variable.
</P>
<P>
The return values <VAR>beta</VAR>, <VAR>sigma</VAR>, and <VAR>r</VAR> are defined as
follows.
</P>
<DL COMPACT>
<DT><VAR>beta</VAR>
<DD>
The OLS estimator for <VAR>b</VAR>, <CODE><VAR>beta</VAR> = pinv (<VAR>x</VAR>) *
<VAR>y</VAR></CODE>, where <CODE>pinv (<VAR>x</VAR>)</CODE> denotes the pseudoinverse of
<VAR>x</VAR>.
<DT><VAR>sigma</VAR>
<DD>
The OLS estimator for the matrix <VAR>s</VAR>,
<PRE>
<VAR>sigma</VAR> = (<VAR>y</VAR>-<VAR>x</VAR>*<VAR>beta</VAR>)' * (<VAR>y</VAR>-<VAR>x</VAR>*<VAR>beta</VAR>) / (<VAR>t</VAR>-rank(<VAR>x</VAR>))
</PRE>
<DT><VAR>r</VAR>
<DD>
The matrix of OLS residuals, <CODE><VAR>r</VAR> = <VAR>y</VAR> - <VAR>x</VAR> * <VAR>beta</VAR></CODE>.
</DL>
</DL>
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