The comprehensive approach
Another approach closely related to the Cox’s procedure is the comprehensive approach advocated by Atkinson (1970) whereby tests of nonnested models are based upon a third comprehensive model, artificially constructed so that each of the nonnested models can be obtained from it as special cases. Clearly, there are a large number of ways that such a comprehensive model can be constructed. A prominent example is the exponential mixture, HX, which in the case of the nonnested models (13.4) and (13.5) is defined by
/(yt| xt, Qt_a; 9)1-x g(yt| zt, Qt_a; у )x
S^f (yt xt, Qt_1; 9)1-x g(ytzt, Qt_1; y)x dyt’
where R, represents the domain of variations of yt, and the integral in the denominator ensures that the combined function, cX(yt xt, zt, Qt-1; 0, y), is in fact a proper density function integrating to unity over R,. The "mixing" parameter X varies in the range [0, 1] and represents the weight attached to model H/. A test of X = 0 (X = 1) against the alternative that X Ф 0 (X Ф 1) can now be carried out using standard techniques from the literature on nested hypothesis testing. (See Atkinson, 1970 and Pesaran, 1982a.) This approach is, however, subject to three important limitations. First, although the testing framework is nested, the test of X = 0 is still nonstandard due to the fact that under X = 0 the parameters of the alternative hypothesis, y, disappear. This is known as the Davies problem. (See Davies, 1977.) The same also applies if the interest is in testing X = 1. The second limitation is due to the fact that testing X = 0 against X Ф 0, is not equivalent to testing Hf against Hg, which is the problem of primary interest. This implicit change of the alternative hypothesis can have unfavorable consequences for the power of nonnested tests. Finally, the particular functional form used to combine the two models is arbitrary and does not allow identification of the mixing parameter, X, even if 0 and у are separately identified under H/ and Hg respectively. (See Pesaran, 1981.)
The application of the comprehensive approach to the linear regression models
and (13.20) yields:
where v-2 = (1 – X)a-2 + Xaf2. It is clear that the mixing parameter X is not identified.16 In fact setting к = Xv2/ю2 the above "combined" regression can also be written as
and a test of X = 0 in (13.31) can be carried by testing к = 0 in (13.32). Since the error variances о2 and ю2 are strictly positive X = 0 will be equivalent to testing к = 0. The Davies problem, of course, continues to apply and under Hf (к = 0) the coefficients of the rival model, p, disappear from the combined model. To resolve this problem Davies (1977) proposes a two-stage procedure. First, for a given value of P a statistic for testing к = 0 is chosen. In the present application this is given by the f-ratio of к in the regression of y on X and yp = Zp, namely
and where Mx is already defined by (13.26). In the second stage a test is constructed based on the entire random function of f^ZP) viewed as a function of p. One possibility would be to construct a test statistic based on
Рк = Max^Zp)}.
Alternatively, a test statistic could be based on the average value of f^ZP) obtained using a suitable prior distribution for p. Following the former classical route it is then easily seen that Рк becomes the standard Fz* statistic for testing b2 = 0, in the regression
y = Xb1 + Z*b2 + vf, (13.33)
where Z* is the set of regressors in Z but not in X, namely Z* = Z – X П Z.17 Similarly for testing Hg against Hf the comprehensive approach involves testing c1 = 0, in the combined regression
y = X*c1 + Zc2 + vg, (13.34)
where X* is the set of variables in X but not in Z. Denoting the F-statistic for testing c1 = 0 in this regression by Fx*, notice that there are still four possible outcomes to this procedure; in line with the ones detailed above for the Cox test. This is because we have two F-statistics, Fx* and Fz*, with the possibility of rejecting both hypotheses, rejecting neither, etc.
An altogether different approach to the resolution of the Davies problem would be to replace the regression coefficients, p, in (13.32) by an estimate, say U, and then proceed as if yp = ZU is data. This is in effect what is proposed by Davidson and MacKinnon (1981) and Fisher and McAleer (1981). Davidson and MacKinnon suggest using the estimate of P under Hg, namely ST = (Z’Z)-1Zy. This leads to the /-test which is the standard f-ratio of the estimate of к in the artificial regression18
H: y = Xa + кZpT + v^
For testing Hg against Hf, the /-test will be based on the OLS regression of y on Z and XaT, and the /-statistic is the f-ratio of the coefficient of Xa T (which is the vector of fitted values under Hf) in this regression.
The test proposed by Fisher and McAleer (known as the /А-test) replaces в by the estimate of its pseudo-true value under Hf, given by P*(aT)
S*(a T) = (Z’Z)-1Z’a T.
In short the /А-test of Hf against Hg is the f-ratio of the coefficient of yPa = Z(Z’Z)-1Z’aT in the OLS regression of y on X and yPa. Similarly, a /А-test of Hg against Hf can be computed.
Both the /- and the /А-test statistics, as well as their various variations proposed in the literature can also be derived as linear approximations to the Cox test statistic. See (13.28).
Various extensions of nonnested hypothesis testing have also appeared in the literature. These include tests of nonnested linear regression models with serially correlated errors (McAleer ef al, 1990); models estimated by instrumental variables (Ericsson, 1983; Godfrey, 1983); models estimated by the generalized method of moments (Smith, 1992); nonnested Euler equations (Ghysels and Hall, 1990); autoregressive versus moving average models (Walker, 1967; King, 1983); the generalized autoregressive conditional heteroskedastic (GARCH) model against the exponential – GARCH model (McAleer and Ling, 1998); linear versus loglinear models (Aneuryn-Evans and Deaton, 1980; Davidson and MacKinnon, 1985; Pesaran and Pesaran, 1995); logit and probit models (Pesaran and Pesaran, 1993; Weeks, 1996; Duncan and Weeks, 1998); nonnested threshold autoregressive models (Altissimo and Violante, 1998; Pesaran and Potter, 1997; Kapetanios and Weeks, 1999).