Motivation for nonnested statistics

From a statistical view point the main difference between the nested and nonnested hypothesis testing lies in the fact that the usual loglikelihood ratio or Wald stat­istics used in the conventional hypothesis testing are automatically centered at zero under the null when the hypotheses under consideration are nested while this is not true in the case of nonnested hypotheses. However, once the conven­tional test statistics are appropriately centered (at least asymptotically) the same classical techniques can be applied to testing of nonnested hypotheses. Using the two nonnested linear regression models in (13.19) and (13.20) we first demon­strate the problems with standard test statistics by focusing on a simple compari­son of sums of squared errors.

Consider the following test statistic:

^ = Z g – б 2, (13.21)

where

6 f = ef ef/(T – kf)

6 g = egeg/(T – kg),

and ef is the OLS residual vector under Hf such that = Mfy. Note that (13.21) represents a natural starting point being the difference between the mean sum of squared errors for the two models.

Подпись: Now under Hf, Подпись: (T - kf)z 2 = uf Mf uf. eg = My = Mg(Xa + f Подпись: (13.22)

In general the exact distribution of ^T will depend on the unknown parameters. To see this, first note that under Hf, ef = Mf (uf + Xa) therefore, (since Mf X = 0), we have

or

eg = Mg Xa + Mguf

and

(T – kg)Z g = egeg

= (uf + a’ X’ )Mg(Xa + uf)

= uf Mguf + 2a’X’Mguf + a’X’MgXa. (13.23)

Using (13.22) and (13.23) in (13.21) and taking expectations (under Hf) we have which we denote by pT = (a’ X’ Mg Xa)/(T – kg). Since ^T does not have mean zero under the null hypothesis Hf, then £,T cannot provide us with a suitable test – statistic. Notice, however that when Hf is nested within Hg, then MgX = 0 and £,T will have mean zero (exactly) under Hg. In this case, if we also assume that Uf is normally distributed, it can be easily shown that

(T-kf )£t. F

r. g k

where FrJ_kg is distributed as a (central) F with r and T – kg degrees of freedom; r here stands for the number of restrictions that we need to impose on Hg in order to obtain Hf.

A fundamental tenet of classical hypothesis testing is that the distribution of the test statistic is known under a well specified null hypothesis. Thus, in this context if Hf is nested within Hg then under the null of Hf the normalized differ­ence between the sum of squared errors have a zero expectation. When Hf is not nested within Hg we may adopt a number of alternate approaches. First, a suit­able test statistic that has zero mean asymptotically will be

ZT — {T

where {T is a consistent estimator of pT under Hf. More specifically where 7 — (X’X)-1X’y. Equation (13.25) represents an example of centering of a test statistic such that the distribution of zT is known (asymptotically). Cox (1961, 1962) utilized this approach to center the loglikelihood ratio statistic for two nonnested models. When the models are nested the loglikelihood ratio statistic is properly centered (at least asymptotically). For example, if we let Lf (9) and Lg(y) denote, respectively the loglikelihood functions for Hf and Hg, and we if assume that Hf is nested within Hg, then under Hf the loglikelihood ratio stat­istic, 2[Lg(y T) – Lf (0T)], does not require any centering and the test defined by the critical region

2[Lg(TT) Lf(PT)] ^ XfL-a)(r),

where r is the number of parameter restrictions required to obtain Hf from Hg, asymptotically has the size a and is consistent. In the case of nonnested models the likelihood ratio statistic is not distributed as a chi-squared random variable. The reason for this is simple. The degrees of freedom of the chi-square statistic for the LR test is equal to the reduction in the size of the parameter space after imposing the necessary set of zero restrictions. Thus, if neither Hf nor Hg nests the other model, the attendant parameter spaces and hence the likelihoods are unre­lated. In Section 5.2 we examine the application of centering (or mean adjust­ment) of the likelihood ratio statistic to obtain a test statistic that has a known asymptotic distribution. Given that in most instances the form of mean adjust­ment involves analytically intractable expectations in Section 7.1 we examine the use of simulation methods as a method of circumventing this problem.

Following seminal work by Efron (1979), an alternative approach conducts infer­ence utilizing the empirical distribution function of the test statistic. In this instance there is, in general, no need to center £,T using {T. Instead we take £,T as the observed test statistic, and given a null hypothesis, we simulate a large number, say R, of the og, z2 pairs. The empirical distribution function for ^T is then constructed based on 6gr and zfr, r — 1, 2,…, R. In Section 7.2 we examine the use of bootstrap procedures for conducting nonnested hypothesis tests. We also consider the case for combining the type of mean adjustment in (13.25) with bootstrap procedures.

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