Least squares estimators in regression models

The least squares (LS) estimator of in the standard linear model y=X ß + ? , ? ~(0, ?2 I*) is given by ß=(X?X)-1X?y.According to Z'X ß = Z'y, ß=(Z'X)-1 Z'y, together with expectation value E(ß)=ß and variance Var(ß)=?2(Z'X)-1 Z'Z(X'Z)-1, is known as an instrumental variable estimator, where Z is a given by nxp matrix of ±1 and zeros. The sum of squared residuals ?'?=??i2 minimized by ß value, depend on the Householder transformations with ? = y - Xß.Generalized least squares estimator is ß=(X*'V-1X*)-1X*'V-1y* with X=L-1X*, y=L-1y* and LL'=V. Givens transformatios with generalized LS estimator is given by ß=U-1(t1-T12g2).In addition, Householder matrix transformations are applied in regression analysis for real dataset.