Linear Models: Least Squares and Alternatives by C. Radhakrishna Rao & Helge Toutenburg

Author:C. Radhakrishna Rao & Helge Toutenburg [Rao, C. Radhakrishna & Toutenburg, Helge]
Language: eng
Format: epub
Published: 2011-02-19T05:00:00+00:00

=

σ2pii , (7.5)

V(ˆ)

=

V (I − P )y = σ2(I − P ) , (7.6)

var(î)

=

σ2(1 − pii) (7.7)

and for i = j

cov(î, ˆj) = −σ2pij .

(7.8)

The correlation coefficient between î and ˆj then becomes

−p

ρ

ij

ij = corr(î, ˆj ) = √

.

(7.9)

1 − pii

1 − pjj

Thus the covariance matrices of the predictor Xb0 and the estimator of error âre entirely determined by P . Although the disturbances i of the model are i.i.d., the estimated residuals î are not identically distributed and, moreover, they are correlated. Observe that T

ˆ

yi =

pijyi = piiyi +

pijyj

(i = 1, . . . , T ) ,

(7.10)

j=1

j=i

implying that

∂ ˆ

yi

∂ ˆ

y

= p

i

ii

and

= pij .

(7.11)

∂yi

∂yj

Therefore, pii can be interpreted as the amount of leverage each value yi has in determining ˆ

yi regardless of the realized value yi. The second relation of (7.11) may be interpreted, analogously, as the influence of yj in determining ˆ

yi.

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