The encompassing approach

This approach generalizes Cox’s original idea and asks whether model Hf can explain one or more features of the rival model Hg. When all the features of model Hg can be explained by model Hf it is said that model Hf encompasses model Hg; likewise model Hg is said to encompass model Hf if all the features of model Hf can be explained by model Hg. A formal definition of encompassing can be given in terms of the pseudo-true parameters and the binding functions defined in Section 2.

Model Hg is said to encompass model Hf, respectively defined by (13.5) and (13.4), if and only if

HgEHf : Єя* = 0*(yh*). (13.36)

Similarly, Hf is said to encompass Hg (or Hg is encompassed by Hf) if and only if

HfEHg : Yh* = Y*(0h*).

Recall that 0h* and Yh* are the pseudo-true values of 0, and y with respect to the true model Hh, and 0*() are y*() are the binding functions linking the parameters
of the models Hf and Hg. For example, in the case of the linear rival regression models (13.10) and (13.11), and assuming that the true model is given by (13.14) then it is easily seen that the functions that bind the parameters of model Hg to that of Hf are

0*(Yh*) =

£ xx £ xzPh*

h* + Ph*(£zz £zx £ xx £xz )eh*,

0*(Yh*) =

image348

Using (13.16) to substitute for the pseudo-true values в h* and ®h* we have

Therefore, conditional on the observation matrices X, Z, and W, model Hf encom­passes model Hg if and only if £ xZx £ xw 5

image349 Подпись: - £ £-1£ )5 + 5'£ £-1(£ - £ У1£ )У £ wz zz zw wz zz zz zx xx xz zz z Подпись: 5j

£xx £ xz£ zz £zw 5

These conditions are simplified to

Подпись: (13.37)£xw 5 = £xz £zz £zw5,

image353 Подпись: (13.38)

and

But it is easily verified that (13.37) implies (13.38), namely encompassing with respect to the regression coefficients imply encompassing with respect to the error variances. Therefore, Hf is encompassed by Hg if and only if (X’MzW)5 = 0. This condition is clearly satisfied if either Hf is nested within Hg, (X’Mz = 0), or if Hg contains the true model, (MzW = 0). The remaining possibility, namely when (X’MzW) = 0, but the true value of 5, say 5 0, is such that (X’ MzW)50 = 0, is a rather a low probability event.

The encompassing hypothesis, HgEHf, (or HfEHg) can now be tested using the encompassing statistics, VT [0T – 0*(fT)j, (or л/T [yT – Y*(0T)j). Gourieroux and Monfort (1995) show that under the encompassing hypothesis, 0h* = 0*(Yh*), and assuming certain requilarity conditions are met, xJt [0t – 0*(yT)j is asymptotic­ally normally distributed with zero means and a variance covariance matrix that in general depends in a complicated way on the probability density functions of the rival models under consideration. Complications arise since Hg need not belong to Hh. Two testing procedures are proposed, the Wald encompassing test
(WET) and the score encompassing test (SET), both being difficult to implement. First, the binding functions 0*() and y*() are not always easy to derive. (But this problem also afflicts the implementation of the Cox procedure, see below.) Sec­ond, and more importantly, the variance-covariance matrices of л/Т [0t – 0*(yT)j, (or Vt [yT – Y*(0t)]), are, in general, non-invertible and the construction of WET and SET statistics involve the use of generalized inverse and this in turn requires estimation of the rank of these covariance matrices. Alternative ways of dealing with these difficulties are considered in Gourieroux and Monfort (1995) and Smith (1993).

In the case of linear regression models full parameter encompassing (namely an encompassing exercise involving both regression coefficients and error vari­ances) is unnecessary.19 Focusing on regression coefficients the encompassing statistics for testing HgEHf are given by

VT [«■ – a*(St)] = VT (X’X)-1X’Mzy.

Under Hh, defined by (13.14),

VT [«t – a*(St)] = ЛІТ (X’X)-1(X’MZW)5 + VT (X’X)-1X’Mzuh,

image355

where uh ~ N(0, u2IT).20 Hence, under the encompassing hypothesis, (X’MzW)5 = 0, the encompassing statistic 4T [aT – a*(ST)j is asymptotically normally distrib­uted with mean zero and the covariance matrix v2I,-l(I<xx – Х^Д^Х^Х-1. There­fore, the construction of a standardized encompassing test statistic requires a consistent estimate of u2, the error variance of the true regression model, and this does not seem possible without further assumptions about the nature of the true model. In the literature it is often (implicitly) assumed that the true model is contained in the union intersection of the rival models under consideration (namely W = X U Z) and u2 is then consistently estimated from a regression of y on X U Z. Under this additional assumption, the WET statistic for testing HgEHf, is given by

where о2 is the estimate of the error variance of the regression of y on X U Z, and (X’MzX) is a generalized inverse of X’MzX. This matrix is rank deficient whenever X and Z have variables in common, namely if X П Z = Q Ф 0. Let X = (X1, Q) and Z = (Z1, Q), then

Подпись: X'M zX"X1MzX1 0"

v 0 0y.

But it is easily seen that E gf is invariant to the choice of the g-inverse used and is given by

e y’MzX1(X1MzX1)-1X1Mzy

Egf = 02

and is identical to the standard Wald statistic for testing the statistical signifi­cance of Xa in the OLS regression of y on Z and Xa. This is perhaps not surprising, considering the (implicit) assumption concerning the true model being a union intersection of the rival regression models Hf and Hg.

Other encompassing tests can also be developed depending on the parameters of interest or their functions. For example, a variance encompassing test of HgEHf compares a consistent estimate of о2 with that of its pseudo-true value о h*, namely 6T – о*(уT) = 6T – [raT + T^1SrZ, MIZ0T].21 Under the encompassing hypothesis this statistic tends to zero, but its asymptotic distribution in general depends on Hh. In the case where Hg contains the true model the variance encompassing test will become asymptotically equivalent to the Cox and the /-tests discussed above.

The encompassing approach can also be applied to the loglikelihood functions. For example, to test HgEHf one could use the encompassing loglikelihood ratio statistic T^1{Lf(0T) – Lf(0*(yT))}. This test can also be motivated using Cox’s idea of a centered loglikelihood ratio statistic, with the difference that the centering is now carried out under Hh rather than under Hg (or Hf). See Gourieroux and Monfort (1995) and Smith (1993) for details and difficulties involved in their implementation. Other relevant literature include Dastoor (1983), Gourieroux et al. (1983) and Mizon and Richard (1986).

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