# Tests of Hypotheses

To test a hypothesis on a single parameter, we can perform a standard normal test using the asymptotic normality of either MLE or the MIN /2 estimator. A linear hypothesis can be tested using general methods discussed in Section 4.5.1. The problem of choosing a model among several alternatives can be solved either by the Akaike Information Criterion (Section 4.5.2) or by Cox’s test of nonnested hypotheses (Section 4.5.3). For other criteria for choosing models, see the article by Amemiya (1981).

Here we shall discuss only a chi-square test based on Berkson’s MIN /2 estimator as this is not a special case of the tests discussed in Section 4.5. The test statistic is the weighted sum of squared residuals (WSSR) from Eq.

(9.2.30) defined by

WSSR = 2 aJ2[F-Pt) – х’\$]г. (9.2.42)

i=i

In the normal heteroscedastic regression model у ~ N(Xfi, D) with known D, (y — X/}ayD~у — хДз) is distributed as Xt-k-From this factwecan deduce that WSSR defined in (9.2.42) is asymptotically distributed as Xt-k – We can use this fact to choose between the unconstrained model P, = F(x#0) and the constrained model P, = F(x’ufil0), where xlt and fil0 are the first К — q elements of x, and fi0, respectively. Let WSSRU and WSSRC be the values of (9.2.42) derived from the unconstrained and the constrained models, respectively. Then we should choose the unconstrained model if and only if

WSSRC – WSSRU > x2.a, (9.2.43)

where x2 a denotes the a% critical value of/^. Li (1977) has given examples of the use of this test with real data.

To choose among nonnested models, we can use the following variation of

the Akaike Information Criterion:

AIC = { WSSR + K. (9.2.44)

This may be justified on the grounds that in the normal heteroscedastic regression model mentioned earlier, (y — Xfia)’D~l(y — XfiG) is equal to —2 log L aside from a constant term.

Instead of (9.2.42) we can also use

2 «ЛАП – АГЧА – (9.2.45)

i-i

because this is asymptotically equivalent to (9.2.42).