 A COMPANION TO Theoretical Econometrics

# Measures of Closeness and Vuong’s Approach

So far the concepts of nested and nonnested hypotheses have been loosely de­fined, but for a more integrated approach to nonnested hypothesis testing and model selection a more formal definition is required. This can be done by means of a variety of "closeness" criteria proposed in the literature for measuring the divergence of one distribution function with respect to another. A popular meas­ure employed in Pesaran (1987) for this purpose is the Kullback-Leibler (Kullback, 1959) information criterion (KLIC). This criterion has been used extensively in the development of both nonnested hypotheses tests and model selection procedures. Given hypotheses Hf and Hg, defined by (13.4) and (13.5), the KLIC measure of Hg with respect to Hf is written as

Ifg(0, у) = Ef{ln ft(0) - ln gt(y)}

ln j^jf». (13.39)

J Rf [ gt (Y) J

It is important to note that Ifg(0, у) is not a distance measure. For example, in general Ifg(0, y ) is not the same as Igf(y, 0), and KLIC does not satisfy the triangu­lar inequality, namely Ifg + Igh need not exceed Ifh as required if KLIC were a distance measure. Nevertheless, KLIC has a number attractive properties: Ifg(0, y) > 0, with the strict equality holding if and only if f( ) = g( ). Assuming that observations on yt are independently distributed then the KLIC measure is addi­tive over sample observations.

To provide a formal definition of nonnested or nested hypothesis we define two "closeness" measures: one measuring the closeness of Hg to Hf (viewed from the perspective of Hf), and another the closeness measure of Hf to Hg. These are respectively given by Cfg(0o) = Ifg(0o, y*(0o)), and Cgf(y0) = Igf(y0, 0* (y,)), where, as before, y*(0o) is the pseudo-true value of y under Hf, and 0*(yo) is pseudo-true value of 0 under Hg.

Definition 1. Hf is nested within Hg if and only if Cfg(0o) = o, for all values of 0o Є 0, and Cgf(yo) Ф o for some yo Є Г.

Definition 2. Hf and Hg are globally nonnested if and only if Cfg(0o) and Cgf (yo) are both non-zero for all values of 0o Є 0 and yo Є Г.

Definition 3. Hf and Hg are partially nonnested if Cfg(90) and Cgf(j0) are both non-zero for some values of 90 £ 0 and y0 £ Г.

Definition 4. Hf and Hg are observationally equivalent if and only if Cfg(90) = 0 and Cgf (y0) = 0 for all values of 90 £ 0 and y0 £ Г.

Using the above definitions it is easily seen, for example, that linear or nonlinear rival regression models can at most be partially nonnested, but exponential and lognormal distributions discussed in Section 3 are globally nonnested. For fur­ther details see Pesaran (1987).

We may also define a closeness measure of Hg to Hf from the perspective of the true model Hh and in doing so are able to motivate Vuong's approach to hypoth­esis testing and model selection. (See Vuong, 1989.) The primary focus of Vuong's analysis is to test the hypothesis that the models under consideration are "equally" close to the true model. As Vuong (1989) notes "If the distance between a speci­fied model and the true distribution is defined as the minimum of the KLIC over the distributions in the model, then it is natural to define the 'best' model among a collection of competing models to be the model that is closest to the true distribution". Thus, in contrast to the standard approach to model selection, a hypothesis testing framework is adopted and a probabilistic decision rule used to select a "best" model.

With our setup and notations the closeness of Hf to Hh viewed from the per­spective of the true model, Hh is defined by

Chf(9h*) = Ehjln ht(-) - ln ft(9h*)}.

Similarly, the closeness of Hg to Hh is defined by

Chg(Yh*) = Eh{ln ht(-) - ln gt(Yh*)}.

The null hypothesis underlying Vuong's approach is now given by

HV : Chf (9h*) Chg(yh*),

which can also be written as

Hv : Eh {ln f(9h*)} = Eh{ln gt(Yh*)}.

The quantity Eh{lnft(9h*) - ln gt(Yh*)} is unknown and depends on the unknown true distribution Hh, but can be consistently estimated by the average loglikelihood ratio statistic, T^1{Lf (0T) - Lg(yT)}. Vuong derives the asymptotic distribution of the average loglikelihood ratio under HV, and shows that it crucially depends on whether ft(9h*) = gt(Yh*), namely whether the distributions in Hf and Hg that are closest to the true model are observationally equivalent or not. In view of this a sequential approach to hypothesis testing is proposed. See Vuong (1989) for fur­ther details.

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