If x x is nonsingular multiplying both sides of the

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

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