Introduction to Statistical Modeling with SAS/STAT Software

Linear Model Theory

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

bold upper Y equals bold upper X bold-italic beta plus bold-italic epsilon bold-italic epsilon tilde left-parenthesis 0 comma sigma squared bold upper I right-parenthesis

where bold upper Y is an left-parenthesis n times 1 right-parenthesis vector and bold upper X is an left-parenthesis n times k right-parenthesis matrix of known constants. The model equation implies the following expected values:

StartLayout 1st Row 1st Column normal upper E left-bracket bold upper Y right-bracket equals 2nd Column bold upper X bold-italic beta 2nd Row 1st Column normal upper V normal a normal r left-bracket bold upper Y right-bracket equals 2nd Column sigma squared bold upper I left right double arrow normal upper C normal o normal v left-bracket upper Y Subscript i Baseline comma upper Y Subscript j Baseline right-bracket equals StartLayout Enlarged left-brace 1st Row 1st Column sigma squared 2nd Column i equals j 2nd Row 1st Column 0 2nd Column otherwise EndLayout EndLayout
Last updated: March 08, 2022