# Asymtotically normal estimators

The assumption of normality can be relaxed for the situations that allow asymptotically normal estimators for p. This is because the inference theory developed for problems (25.3) or (25.8) is asymptotically valid for much broader classes of models and hypotheses. In fact, when consistent estimators of Q are available and, in (25.18), v has the stated distribution asymptotically, an asymptotically exact test of size a is based on the same x2-distribution given above. To obtain this result one needs to replace Q in the optimization problems with its corresponding consistent estimator. This is routinely possible when Q is a continuous function of a finite set of parameters other than p.

The three tests are not identical in this situation, of course, but have the same asymptotic distribution. Furthermore, the usual inequality, viz. > £LR > £LM is still valid, see GHM (1982). Often the test which can avoid the QP problem is preferred, which means the LM test for the null of equality of the restrictions, and the Wald test when the null is one of inequality and the alternative is unrestricted. But much recent evidence, as well as invariance arguments, suggest that the LR test be used.

In the general linear regression models with linear and/or nonlinear inequality restrictions, other approaches are available. Kodde and Palm (1986, 1987), Dufour (1989), Dufour and Khalaf (1995), and Stewart (1997) are examples of theoretical and empirical applications in economics and finance. Dufour (1989) is an alternative "conservative" bounds test for the following type situation:

H0 : Rp Є Г0 against H1 : P Є Г1,

where Г0 and Г are non-empty subsets, respectively of Rp and RK. This also allows a consideration of such cases as h(RP) = 0, or h(RP) > 0. Dufour (1989) suggests a generalization of the well known, two-sided F-test in this situation as follows:

where there are T observations from which ББ{, i = 0, 1, are calculated as residual sums of squares under the null and the alternative, respectively. Thus the traditional p-values will be upper-bounds for the true values and offer a conservative bounds testing strategy. Dufour (1989), Dufour and Khalaf (1993) and Stewart

(1997) , inter alia, consider "liberal bounds", and extensions to multivariate/simul – taneous equations models and nonlinear inequality restrictions. Applications to demand functions and negativity constraints on the substitution matrix, as well as tests of nonlinear nulls in the CAPM models show size and power improvements over the traditional asymptotic tests. The latter are known for their tendency to overreject in any case. Stewart (1997) considers the performance of the standard LR, the Kodde and Palm (1986) bounds for the X2-distribution, and the Dufour-type bound test of negativity of the substitution matrix for the demand data for Germany and Holland. Stewart looks at, among other things, the hypothesis of negativity against an unrestricted alternative, and the null of negativity when symmetry and homogeneity are maintained. It appears that, while the Dufour test did well in most cases, certainly reversing the conclusions of the traditional LR test (which rejects everything!), the Kodde and Palm bounds test does consistently well when the conservative bounds test was not informative (with a = 1). Both the lower and upper bounds for the x 2-squared distribution are available, while the "liberal"/lower bounds for the Dufour adjustment are not in this case.

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