A COMPANION TO Theoretical Econometrics

Examples of Nonnested Models

We start with examples of unconditional nonnested models. One such example, originally discussed by Cox (1961) is that of testing a lognormal versus an expo­nential distribution.

Hf : f(y I 0) = f(0) = y-1(2n02)-1/2 exp j-(ln^ 9l)2 J, ^ > 02 > 0, yt > 0.

Hg : g(y11Y) = gt(Y) = Y-1 exp(-y,/y), Y > 0, yt > 0.

These hypotheses (models) are globally nonnested, in the sense that neither can be obtained from the other either by means of suitable parametric restrictions or by a limiting process.6 Under Hf the pseudo-true value of y, denoted by yf* is obtained by solving the following maximization problem

Yf* = arg max Ef{ln g t(Y)}.

But7

Ef {ln g t(Y)} = - ln Y - Ef (y t)/Y = - ln y - exp(01 + 0.502)/y,

which yields

Yf* = Y *(00) = exp(010 + 0.5020). Similarly, under Hg we have8

0?(V) = ln Y0 - 0.5772, 0*^) = 1.6449.

Other examples of nonnested unconditional models include lognormal versus Weibull and Pereira (1984) and lognormal versus gamma distribution, Pesaran (1987).

The most prominent example of conditional nonnested models is linear normal regression models with "rival" sets of conditioning variables. As an example consider the following regression models:

Hf: yt = a'xt + Uf, utf ~ N(0, a2), ^ > a2 > 0, (13.10)

Hg: yt = P'zt + utg, utg ~ N(0, ю2), > ю2 > 0. (13.11)

The conditional probability density associated with these regression models are given by

Hf:f(yt|xf; 0) = (2na2)-1/2 exp j^(yt - a'xf)2j, (13.12)

Hg: g(yt |zt; 0) = (2пю2)-1/2 exp I2—-(yt - e'zt)2 і (13.13)

where 0 = (a', a2)', and у = (в', ю2)'. These regression models are nonnested if it is not possible to write xt as an exact linear function of zt and vice versa, or more formally if xt ^ zt and zt ^ xt. Model Hf is said to be nested in Hg if xt C zt and zt ^ xt. The two models are observationally equivalent if xt C zt and zt C xt. Suppose now that neither of these regression models is true and the DGP is given by

Hh: yt = 5'wt + и^, Uh ~ N(0, u2), ^ > u2 > 0.

It is then easily seen that conditional on {x t, z t, wt, t = 1, 2,..., T}

where

Maximizing Eh{T 2Lf(0)} with respect to 0 now yields the conditional pseudo-true values:

Similarly,

(13.16)

t =1

When the regressors are stationary, the unconditional counterparts of the above pseudo-true values can be obtained by replacing Xww, Xxx, Xwx, etc. by their popu­lation values, namely Xww = E(wtw't), Xxx = E(xtx'), Xwx = E(wtx'), etc.

Other examples of nonnested regression models include models with endog­enous regressors estimated by instrumental variables (see, for example, Ericsson, 1983; and Godfrey, 1983), nonnested nonlinear regression models and regression models where the left-hand side variables of the rival regressions are known transformations of a dependent variable of interest. One important instance of this last example is the problem of testing linear versus loglinear regression models and vice versa.9 More generally we may have

Hf : f( yt) = a + utf, Uf ~ N(0, a2), > a2 > 0,

Hg : g( yt) = P'zt + Utg, Utg ~ N(0, ю2), > ю2 > 0,

where f( yt) and g( yt) are known one-to-one functions of yt. Within this more general regression framework testing a linear versus a loglinear model is charac­terized by f(yt) = yt and g(yt) = ln(yt); a ratio model versus a loglinear model by f(yt) = У/qt and g(yt) = ln(yt), where qt is an observed regressor, and a ratio versus a linear model by f(yt) = y/qt and g(yt) = yt. For example, in analysis of aggre­gate consumption a choice needs to be made between a linear and a loglinear specification of the aggregate consumption on the one hand, and between a loglinear and a saving rate formulation on the other hand. The testing problem is further complicated here due to the linear transformations of the dependent variable, and additional restrictions are required if the existence of pseudo-true values in the case of these models are to be ensured. For example, suitable trun­cation restrictions need to be imposed on the errors of the linear model when it is tested against a loglinear alternative.

Other examples where specification of an appropriate error structure plays an important role in empirical analysis include discrete choice and duration models used in microeconometric research. Although the analyst may utilize both prior knowledge and theory to select an appropriate set of regressors, there is gener­ally little guidance in terms of the most appropriate probability distribution. Nonnested hypothesis testing is particularly relevant to microeconometric research where the same set of regressors are often used to explain individual decisions but based on different functional distributions, such as multinomial probit and logit specifications in the analysis of discrete choice, exponential and Weibull distributions in the analysis of duration data. In the simple case of a probit (Hf) versus a logit model (Hg) we have

where yt, t = 1, 2,..., T are independently distributed binary random variables taking the value of 1 or 0. In practice the two sets of regressors xt used in the probit and logit specifications are likely to be identical, and it is only the form of the distribution functions that separate the two models. Other functional forms can also be entertained. Suppose, for example, that the true DGP for this simple discrete choice problem is given by the probability distribution function H(5'xt), then pseudo-true values for 0 and у can be obtained as functions of 5, but only in an implicit form. We first note that the loglikelihood function under Hf, for example, is given by

and hence under the assumed DGP we have

Eh {T (9)} = T -1 X H(54 )log^(94)]

t=1

+ T -1 X [1 - H(5' xt )]log[1 - Ф(9' xt)].

t =1

Therefore, the pseudo-true value of 0, namely 0*(5) or simply 0*, satisfies the following equation

where ф(0*xt) = (2n)-1/2exp[-y-( 9*x t)2]. Using results in Amemiya (1985, pp. 271-2) it is easily established that the solution of 0* in terms of 5 is in fact unique, and 0* = 5 if and only if Ф( ) = H( ). Similar results also obtain for the logistic specification.