# If x x is nonsingular multiplying both sides of the

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: and only if b∗ is a solution to the normal equations: X Xb = X y. If X X is nonsingular, multiplying both sides of the normal equations by (X X)−1 shows that the only solution to the normal equations is b∗ = (X X)−1 X y. If X X is singular, there are inﬁnitely many solutions that include (X X)− X y for all choices of generalized inverse of X X. X X[(X X)− X y] = X [X(X X)− X ]y = X PX y = X y ˆ Henceforth, we will use β to denote any solution to the normal equations. opyright c 2012 Dan Nettleton (Iowa State University) Statistics 511 26 / 27 Ordinary Least Squares Estimator of E(y) = Xβ ˆ ˆ ˆ We call Xβ = PX Xβ = X(X X)− X Xβ = X(X X)− X y = PX y = ˆ the y OLS estimator of E(y) = Xβ . ˆ It might be more appropriate to use Xβ rather than Xβ to denote our estimator because we are estimating Xβ rather than pre-multiplying an estimator of β by X. As we shall soon see, it does not make sense to estimate β when X X is singular. However, it does make sense to estimate E(y) = Xβ whether X X is singular or nonsingular. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics 511 27 / 27...
View Full Document

## This document was uploaded on 03/27/2014.

Ask a homework question - tutors are online