Theil’s Corrected Я2

Theil (1961, p. 213) proposed a correction of R2 aimed at eliminating the aforementioned weakness of R2. Theil’s corrected R2, denoted by R2, is de­fined by

Подпись: (2.1.6)l-R2^j^(l-R2).

image129 Подпись: (2.1.7)

Because we have from (1.2.9)

where M = I — X(X’X) ^X’ and L = I — T ЧГ as before, choosing the equation with the largest R 2 is equivalent to choosing the equation with the
smallest a2 = (Г — AT)_1y, My. Coming back to the choice between Eqs.

(2.1.1) and (2.1.2), Theil’s strategy based on his R 2 amounts to choosing Eq.

(2.1.1) if

a<a, (2.1.8)

where a = (T— £,)-1y’M, y,

м^і-хда. г1^,

о:і = (Г-А2)-,у’М2у, and M2 = I — Х2(Х^Х2)-‘Х^.

The inequality (2.1.8) can be regarded as a constraint on у and hence defines a subset in the space of y. Call it SQ that is, S0 — {y|(?i < a). This choice of S can be evaluated in terms of the Bayesian minimand (2.1.5). Suppose Eq.

(2.1.2)

image131 Подпись: У'М2у T-K2 Подпись: (2.1.9)

is the true model. Then we have

ft’2X’2MlX2fi2 + 2u2M, X2y?2 + u2M, u2 u2M2u2 T-Kt T-K2

Therefore

E{5 – <t2|2) = ***** > 0 (2110)

Therefore, in view of the fact that nothing a priori is known about whether o — a is positively or negatively skewed, it seems reasonable to expect that

Р(уЄ5,0|2)<і (2.1.11)

For a similar reason it also seems reasonable to expect that

Дує Soil) <*. (2.1.12)

These inequalities indicate that S0 does offer an intuitive appeal (though a rather mild one) to the classical statistician who, by principle, is reluctant to specify the subjective quantities Ln, Lj,, P( 1), and P(2) in the posterior risk (2.1.5).

As we have seen, Theil’s corrected R2 has a certain intuitive appeal and has been widely used by econometricians as a measure of the goodness of fit. However, its theoretical justification is not strong, and the experiences of some researchers have led them to believe that Theil’s measure does not correct sufficiently for the degrees of freedom; that is, it still tends to favor the
equation with more regressors, although not as much as the uncorrected R2 (see, for example, Mayer, 1975). In Section 2.1.5 we shall propose a measure that corrects more for the degrees of freedom than Theil’s R2 does, and in Section 2.1.6 we shall try to justify the proposed measure from a different angle.

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