Hat Matrix H Is Given By
H X XTX 1XT and determines the fitted or predicted values since The diagonal elements of H hii are called leverages and satisfy.
Hat matrix h is given by. Y X XT1 y Hy where H XXTX1 XT is the n nHat Matrix and the vector of residuals is given by. Using the fact that Tr. Since the cross product is a linear transformation it can be represented as a matrixThe hat operator takes a vector and transforms it into its equivalent matrix.
Properties and Interpretation Week 5 Lecture 1 1 Hat Matrix 11 From Observed to Fitted Values The OLS estimator was found to be given by the p 1 vector b XT X 1XT y. The vector of tted values of the dependent variable is given by. View Answer Push-through identity.
Then1 vector of ordinary predicted values of the response variable is yHyˆwhere thennprediction or Hat matrixH is given by 14 HXXX1X. Check that Yb iY j H ij. Hence if you multiply H by y you project your observed values in y onto the space that is spanned by the variables in X.
But I have no idea how to calculate h Im betting the first equality works out because H is a symmetric idempotent matrix. A B Tr. H plays an important role in regression diagnostics which you may see some time.
I also have no clue what Tr. H X inv X X lambda I X. If np and the X matrix is invertible show that the hat matrix H is given by the pxp identity matrix.
Write H on board. So H X X X 1 X is actually a projection matrix. Some simple properties of the hat matrix are important in interpreting least squares.
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