Bernhard Kabelka - Mathematical Statistics: "Linear Regression

Multiple Linear Regression

In this case, we consider functions of the form

where we would like to estimate $a_1,\,...,\,a_k$ with the help of observations $(Y_i,\,x_{i1},\,...,\,x_{ik})$ ( $i=1,\,...,\,n$ ).

If we write

$Y & := & ( \begin{array}{ccc} Y_1 & *s & Y_k \en...... & \vdots \\x_{n1} & *s & x_{nk} \\\end{array} )$

we finally arrive at

$\widehat{a} & = & ( X^T X )^{-1} * X^T Y\\...... Y^T X * ( X^T X )^{-1} * X^T Y )$

which leads us (again for standard normal distributed errors ) to the confidence intervals

$[ \widehat{a}_i - t * \sqrt{\widehat{sigma}^2......at{sigma}^2 * ( X^T X )^{-1}_{ii} ]$

for and

$[ \widehatY(x) - t * \sqrt{\widehat{sigma}^2 ......sigma}^2 * x^T ( X^T X )^{-1} x} ]$

for , as well as to the prediction interval

$[ \widehatY(x) - t * \sqrt{\widehat{\......left( 1 + x^T ( X^T X )^{-1} x )} ]$

for , where stands for $t_{n-k;\frac{1+gamma}2$ .