Page 76 - Compact Numerical Methods For Computers

Handling larger problems 65
products matrix are

(4.35)

Thus, the relevant information for the standard errors is obtained by quite simple
row sums over the V matrix from a singular-value decomposition. When the original
A matrix is rank deficient, and we decide (via the tolerance for zero used to select
‘non-zero’ singular values) that the rank is r, the summation above reduces to

(4.36)

However, the meaning of a standard error in the rank-deficient case requires careful
consideration, since the standard error will increase very sharply as small singular
values are included in the summation given in (4.36). I usually refer to the dispersion
measures computed via equations (4.33) through (4.36) for rank r < n cases as
‘standard errors under the condition that the rank is5 (or whatever value r currently
has)‘. More discussion of these issues is presented in Searle (1971) under the topic
‘estimable functions’, and in various sections of Belsley, Kuh and Welsch (1980).