Linear Model Theory :: SAS/STAT(R) 12.1 User's Guide

Linear Model Theory

Finding the Least Squares Estimators
Analysis of Variance
Estimating the Error Variance
Maximum Likelihood Estimation
Estimable Functions
Test of Hypotheses
Residual Analysis
Sweep Operator

This section presents some basic statistical concepts and results for the linear model with homoscedastic, uncorrelated errors in which the parameters are estimated by ordinary least squares. The model can be written as

$\bY = \bX \bbeta + \bepsilon \quad \quad \bepsilon \sim (0,\sigma ^2\mb {I})$

where $\bY$ is an $(n \times 1)$ vector and $\bX$ is an $(n \times k)$ matrix of known constants. The model equation implies the following expected values:

$\displaystyle \mr {E}[\bY ] =$	$\displaystyle \, \, \bX \bbeta$
$\displaystyle \mr {Var}[\bY ] =$	$\displaystyle \, \, \sigma ^2\bI \Leftrightarrow \mr {Cov}[Y_ i,Y_ j] = \left\{ \begin{array}{ll} \sigma ^2 & i = j \cr 0 & \mbox{otherwise} \end{array} \right.$