Nonnested Hypotheses

Nonnested Hypotheses M. Rodrigo Dupleich Ulloa and M. Hashem Pesaran University of Cambridge November 2006

1

Introduction

In economics, as in many other disciplines, there are competing explanations of the same phenomena, often characterized by alternative statistical models. Di¤erent models may represent, for example, di¤erent theoretical paradigms, or could be the result of alternative formulations from the same paradigm. Within the classical framework, the problem of model adequacy is approached through ‘general speci…cation tests’, the ‘diagnostic tests’, and the ‘non-nested tests’. All the three approaches can be used to test the same explanation or hypothesis of interest (the null or the maintained hypothesis), but they di¤er in their consideration of the alternative(s). General speci…cation tests intentionally consider a broad class of alternatives, whilst the alternatives considered under diagnostic and non-nested testing procedures are much more speci…c. In the case of nonnested tests the null hypothesis is contrasted to a speci…c alternative. Nonnested tests are appropriate when rival hypotheses are advanced for the explanation of the same economic phenomenon, and the aim is to devise a powerful test against a speci…c alternative. When the null hypothesis is nested within the alternative, standard classical procedures such as those based on the likelihood-ratio, Wald and Lagrange-multiplier (or Score) principles can be utilized. But if the null and the alternative hypotheses belong to ‘separate’families of distributions, classical testing procedures cannot be applied directly and need to be suitably modi…ed. This paper provides an overview of the concepts and some of the most widely used non-nested hypotheses tests and apply these procedures to the classical regression models. Our discussion of nonnested hypothesis testing will necessarily omit many topics. Survey articles on this subject include McAleer and Pesaran (1986), Gourieroux and Monfort (1994), and Pesaran and Weeks (2001).

2

Nonnested models

Suppose the object of interest is the process generating the random variable Y , observed over a sample of size n, y = (y1 ; y2 ; ::::; yn )0 . Assume that the true process generating y is characterized by a joint Prepared for the New Palgrave Dictionary, Second Edition.

1

probability density function, f0 (y), which is unknown, and two models (hypotheses) are advanced as possible explanations of Y , represented by the joint probability density functions: Hg = fg (y; ) ;

Hh = fh (y; ) ;

2

g;

(1)

2 g:

These functions are known but depend on a …nite number of unknown parameters denoted by 2 , respectively. The sets

and

2

and

represent the ‘admissible’parameter space for which the respective

densities g (y; ) and h (y; ) are well de…ned. The aim is to ascertain which of the two alternatives, Hg and Hh , if any, can be viewed as belonging to f0 (y). In this set up there is no natural null hypothesis, either of the two hypothesis under consideration can be taken as the null. In practice, the analysis of nonnested hypotheses is carried out with both alternatives taken in turn as the null hypothesis. Four outcomes are possible: (i) Hg rejected against Hh and not vice versa, (iii) Hh rejected against Hg and not vice versa, (iii) neither hypotheses is rejected against the other, and …nally (iv) both hypotheses are rejected against one another. The …rst two outcomes are familiar from the classical test results and are straightforward to interpret. The third outcome can arise when the two models are very close to f0 (y), and hence equivalent observationally. The fourth outcome suggests the existence of a third possible model which shares important features from both models under consideration.

2.1

Pseudo true values and closeness measures

Given the observations y, the maximum likelihood (ML) estimators of bn = arg maxLg ( ) ;

and

are given by,

b n = arg maxLh ( ) ; 2

2

where the corresponding log-likelihood functions are de…ned by Lg ( ) = log (g (y; )) and Lh ( ) = log (h (y; )). Throughout we shall assume that probability densities satisfy the usual regularity conditions as established, for example, in White (1982), such that bn and b have asymptotically normal limiting distributions under n

the ‘true’model, f0 (y). In the general case where neither of the models under consideration coincide with f0 (y) , bn and bn are known as quasi-ML estimators and their probability limits under f0 (y) are referred to as (asymptotic) pseudo-true values, such that, f

= arg maxEf fLg ( )g

f

2

= arg maxEf fLh ( )g

(2)

2

where Ef ( ) denotes expectations under the true density f0 (y). In what follows, we assume that the above asymptotic pseudo-true values exist; and

f

and

f

are the unique maxima to the respective optimization

problem given in (2), such that global identi…ability is ensured. For the case in which f0 (y) belongs to Hg , we have that

g

=

0

and

g

=

(

0 ),

where the ‘true’value of

symmetry of our setting, under Hh we have

h

=

(

0)

and

under Hg , is denoted by h

=

0,

where

0

0.

Given the

is the ‘true’value of

under Hh . The relationship between the parameters of the two models under consideration is given by the functions

g

=

(

0)

and

h

=

(

0 ),

known as the binding functions.

2

Using closeness measures and pseudo-true values, Pesaran (1987) provides a formalization of the concepts of nested and nonnested hypotheses. The closeness of Hg with respect to Hh is given by Cgh (

0)

= Igh ( =

0;

min Igh (

= Eg fLg ( where Igh (

0;

(

0 ))

0;

)

0)

(3)

Lh (

(

0 ))g

) is known as the Kullback-Leibler Information Criterion (KLIC) measure, introduced by

Kullback (1959). Similarly, the closeness of Hh to Hg , is de…ned by Chg ( Hg is nested within Hh if and only if Cgh ( 0

0)

= 0, for all values of

of

0

2

and

0

0

and

0

0)

and Chg (

0)

and Chg (

0)

2

0;

(

, and Chg (

0

0)

6= 0 for some

are both non-zero for all values

2 . and

0 )).

are both non-zero for some values of

Hg and Hh are observationally equivalent if and only if Cgh ( 2

0)

= Ihg (

2 .

Hg and Hh are partially nonnested if Cgh (

3

0)

2 :

Hg and Hh are globally nonnested if and only if Cgh (

0

(4)

0)

= 0 and Chg (

0)

0

2

= 0 for all values of

2 .

Tests of nonnested hypotheses

There are three general approaches to testing nonnested hypotheses. The …rst, due to the pioneering contributions of Cox (1961, 1962), involves centering the log-likelihood ratio statistic under the null hypothesis and then deriving its asymptotic null distribution. This is known as the Cox test. A second approach, also suggested by Cox (1962) and explored extensively by Atkinson (1970), is based on an arti…cially constructed general model. The basic idea is to introduce a third hypothesis in which both Hg and Hh are nested as special cases. A third approach, originally considered by Deaton (1982, 1983), and further developed by Mizon and Richard (1986) known as the encompassing procedure, focuses on the ability of one model in explaining particular features of an alternative model. In a related contribution, Gourieroux et. al. (1983) extend the Wald and Score type tests to nonnested models. Their statistics are based on the di¤erence between two estimators of the pseudo-true values. The Cox test statistic is derived by modifying the log-likelihood ratio statistic, Lg bn

Lh (b n ), so

that it is appropriately centered. Speci…cally, for testing Hg against Hh ; the numerator of the Cox statistic is given by

n b g Lg bn where E

Hh we have Cgh (

0)

n o n o b g Lg bn Sngh = Lg bn Lh (b n ) E Lh (b n ) ; (5) o Lh (b n ) is a consistent estimator of Cgh ( 0 ). In the case where Hg is nested within = 0 for all

0,

and Sngh reduces to the standard log-likelihood ratio statistic. An

application to linear regression models has been proposed by Pesaran (1974) and subsequently extended 3

to simultaneous nonlinear equations systems by Pesaran and Deaton (1978). As pointed out previously, since there is no natural null hypothesis in this testing framework, one also needs to consider the modi…ed log-likelihood statistic for testing Hh against Hg which is denoted by Snhg . Under a suitable normalization p (i.e. n), both statistics are asymptotically normally distributed under their respective nulls with a zero mean and a …nite asymptotic variance. When the null hypothesis of Hg is considered against Hh , we have

where Vgh is the asymptotic variance of

p gh nS Nngh = p n Vgh

p

nSngh and

a

a

N (0; 1)

denotes asymptotical equivalence in distribution (for

details see Pesaran and Deaton (1978)). Based on the results of the two statistics, Nngh and Nnhg , four outcomes are possible:

1. Reject Hg but not Hh if Nngh < c and Nnhg > c ; 2. Reject Hh but not Hg if Nngh > c and Nnhg < c ; 3. Reject both Hg and Hh if Nngh > c and Nnhg > c ; 4. Reject neither Hg and Hh if Nngh < c and Nnhg < c ; where the (1

) percent critical value of the standard normal distribution is denoted by c . In the

case of nonnested hypotheses, there is no way of ranking the models by the level of their generality. As a consequence, the test results may provide a consistent outcome such as the rejection of Hg (or Hh ) by both tests. But it is also not unusual, given the data, for both nonnested models to be simultaneously rejected or fail to be rejected. For the case of a simultaneous rejection of Hg and Hh , we need to …nd some other model that …ts the data better. If neither model is rejected, this may indicate lack of power. The second approach, named Atkinson’s comprehensive method (Atkinson (1970)), is based on an arti…cially nesting of the two models with a general model such as, 8 > > < (g (y; )) (h (y; ))1 fc (y; ; ; ) = Z ; > 1 > : (g (y; )) (h (y; )) dy

2 [0; 1] ;

2

;

2

9 > > = > > ;

(6)

Atkinson’s comprehensive approach considers families that are obtained by mixing the probability distributions of Hg and Hh . It requires the existence of the integral appearing in the denominator in equation (6). This component ensures that the combined function fc (y; ; ; ), is in fact a proper density function. In equation (6), the comprehensive model is based on an exponential combination (i.e. a geometric mean); alternatively the compound model can also be derived from an arithmetic mean (see for instance Quandt (1974)). In this set up, the hypothesis Hg is obtained by imposing by imposing

= 0. Thus, in principle, by testing

= 1 or

= 1, while the hypothesis Hh is obtained

= 0, we can test Hg or Hh , respectively. The

‘mixing’parameter, ; varies over the range [0; 1] and measures the relative weights attached to Hg and Hh . As a consequence, tests for the restriction of

= 1 ( = 0) against the alternative that 4

6= 1 ( 6= 0) can

be performed based on standard techniques from the literature of nested hypothesis testing (see Atkinson (1970) and Pesaran (1982b)). Atkinson’s approach is, however, subject to a number of drawbacks. The …rst one arises from the fact that under

= 1 (or

= 0), the unknown parameter vector

(or ) disappears from the combined model

written in (6). This is known as the Davies’s problem (Davies (1977)) which can be circumvented in various ways, as discussed, for example, in Pesaran (1982c). The second limitation is due to the fact that testing = 1 against

6= 1, is not equivalent to performing the test of Hg against Hh , which is the primary

object of the nonnested testing exercise. Finally, there is some degree of arbitrariness in the choice of the comprehensive model (see Pesaran (1981)). The encompassing approach generalizes Cox’s original idea and examines the extent to which Hg explains

one or more features of the rival model, Hh . When all the features of the model Hh can be explained by model Hg , then Hg is said to encompasses Hh . This condition is denoted by, Hg EHh :

f

=

(

f)

(7)

Likewise, Hh EHg implies that all features of model Hg can be explained by the model Hh , that is Hh encompasses Hg , such that,

Hh EHg : Recall that

f

and

f

f

are the pseudo-true values of

=

(8)

f

and , with respect to the true model Hf . Moreover,

( ) and

( ) are the binding functions linking the parameters of the models under Hg and Hh . The p bn (resp. encompassing hypothesis Hg EHh (resp. Hh EHg ) can be tested using the statistic n b n p b n n (b n ) ). Gourieroux and Monfort (1995) show that under the encompassing hypothesis, Hg EHh ; p bn and a set of regularity conditions, n b n is asymptotically normal with zero mean and a …nite covariance matrix. Based on this result, two testing procedures are proposed by Gourieroux and

Monfort (1995), the Wald encompassing test (WET) and the Score encompassing test (SET). In practice, the implementation of these tests tends to be di¢ cult. Firstly, the binding functions

( ) and

( ) are

not easy to derive and secondly, the variance-covariance matrices appearing in the test statistics tend to be di¢ cult to compute in practice. Chen and Kuan (2002) suggest the use of ‘pseudo-true score’as a way of avoiding the need to estimate pseudo-true values.

4

Voung’s model selection test

Vuong’s (1989) criterion is motivated by testing that Hg and Hh are observationally equivalent, using the Kullback-Leibler Information Criterion (KLIC) as a closeness measure.

The focus of this approach is to

test the hypothesis that the models under consideration are ‘equally’ close to the true unknown model, Hf : f0 (y). It provides a natural link between model selection and hypothesis testing approaches. Under model selection, a model is selected even if the ’‘best’ model happens to be very close to the second best model. Voung’s approach allows the statistical signi…cance of the di¤erences between models to be tested using classical testing procedures. It is based on the closeness measures of Hg and Hh with respect to the

5

true model, Hf , namely (for closeness of Hg to Hf ). Cf g (

f)

= Ef fLf ( )

Lg (

f )g ;

f

= Ef Lf ( )

Lh

f

and (for closeness of Hh to Hf ). Cf h

:

The null hypothesis of interest, ‘Hg and Hh are equivalent’, is then formally de…ned by HV : Cf g (

f)

= Cf h

which is equivalent to the unknown quantity HV : Ef Lg (

(9)

f

f)

Lh

= 0, that dependsnon f0 (y), the o unknown true distribution. However, the latter di¤erence can be consistently estimated by T 1 Lg bn Lh (b n ) , f

an average of the log-likelihood ratio statistic. Vuong derives an asymptotic standard normal distribution for the related test statistic under HV .

Rivers and Vuong (2002) provide a number of generalizations and show that the test can be applied to nonlinear dynamic models and other closeness measures.

5

Application to linear regression models

An important application of the nonnested tests in econometrics has been to linear regression models. Consider the following classical normal regression models: 2

Hg : y = X + ug ; ug s N 0;

In ; 0


> 0. The link between these strict inequality restrictions and the nesting

properties of the models in (10) will be made clear below. Suppose that neither Hg nor Hh belong to the true DGP, and the data is generated by, Hf : y = W + uf ; uf s N 0; v 2 In ; 0 < v 2 < 1: As before, assume that b ww = n their population values given by

for this case by maximizing Ef n f

=

1

(W0 W), b wx = n

ww , 1 Lg

f 2 f

wx

and

wz .

1

(W0 X) and b wz = n

v2 +

0

6

1

(W0 Z) can be replaced by

The pseudo-true values given in (2) can be obtained

( ) with respect to ! =

(11)

which yields,

1 xx

ww

xw wx

1 xx

xw

!

(12)

Similarly, for model Hh we have

f

!

f

=

! 2f

1 zz

=

0

v2 +

!

zw

ww

wz

1 zz

zw

(13)

Note that, for the case in which Hf belongs to the family of models given by Hg , the later result can be re-written as, =

g

!

g

! 2g

1 zz

=

2

zx 0

+

g

!

(14)

In terms of our previous discussion, these regression models are nonnested if it is not possible to write X as an exact linear function of Z and vice versa, or more formally if X " Z and Z " X. The model Hg is said to be nested in Hh if X Z

Z and Z " X. The two models are observationally equivalent if X

Z and

X. These conditions can be written in terms of the KLIC measure given by (3) as in McAleer and

Pesaran (1986), who derive the closeness measure of Hh with respect to Hg as, Cgh ( ) =

0

1 log 1 + 2

g 2

Similarly, the KLIC measure of closeness of Hg with respect to Hh is, Chg ( ) =

0

1 log 1 + 2

h

!2

It can be easily seen from this example, that a necessary and su¢ cient condition for Hg to be nested within Hh is

0

g

=

2

= 0 for all admissible values of

is implied by either

g

= 0 or

g

0

with

=! 2 6= 0 for some

h

= 0.

. Note that

0

g

=

2

=0

Given the linear set up and using results in Pesaran (1974), the adjusted log-likelihood ratio statistic for testing Hg against Hh is given by, Sngh

n = log 2

! b 2ng ! b 2n

!

(15)

b 2ng is the estimated pseudo-true value of the where ! b 2n is the estimate of ! 2 under the alternative Hh , and ! residual variance of Hh under Hg , such that, ! b 2n

! b 2ng

= n =

1

0

X b n ) (y

(y

b2n +

b0 b g b n n

X b n)

in which the estimates under the true model Hg are given by b2n = n (X0 X)

1

X0 y and b n = (Z0 Z)

1

1

y

Xbn

0

y

Xbn , bn =

Z0 y. As pointed out earlier, since we do not have a natural null hypothesis

in this framework, one also needs to evaluate Hg against Hh , for which the modi…ed log-likelihood statistic

is given by,

Snhg

n = log 2

7

b2nh b2n

!

(16)

For the statistic given by (15), the asymptotic variance of follows,

1

nSngh , denoted by Vgh , can be computed as

0 b2n b n X0 Mz Mx Mz X b n

Vgh = where Mx = In X0 (X0 X)

p

(17)

0

n b2n + b n b g b n

X0 and Mz = In Z0 (Z0 Z)

1

Z0 are orthogonal projection matrices. Combining p p (15) and (17), the associated standardized Cox statistic, Nngh = nSngh = Vgh , can now be calculated as

described in Pesaran (1974). Similar derivations lead to the analog statistic for the test of Hh against Hg , Nnhg . The application of the comprehensive approach to the above linear regression models, yields the following exponential combination as presented in (6), ) X + Z + u; u s N 0;

H : y = (1 where

=

2

=

2

and

2

= (1

positive, performing a test of

)

2

+ !

2

2

(18)

In 2

. Given that the error variances

= 0 is equivalent to testing

and ! 2 are strictly

= 0 when we consider the null hypothesis of

Hg against Hh . As pointed out earlier, the Davies problem arises when under the null hypothesis of Hg ( = 0), the unknown parameter vector

disappear from the mixture model. The presence of this nuisance

parameter results in a student-type of test statistic associated with such that, t ( )=

0

b2 (

that depends on the value of

Z0 Mx y

0 Z0 M Z x

,

(19)

1=2

)

where b2 denotes the usual estimator of the variance of the errors. One possible way to solve this identi…cation problem would be to construct a test statistic based on F = max t ( ).

A di¤erent approach to deal with the identi…cation problem was proposed by Davidson and MacKinnon

(1981), who propose a J test by replacing the nuisance parameter

by its estimator, b n ; under Hh . An

exact version of this test, proposed by Fisher and McAleer (1981) and known as the JA-test (indicating the Atkinson variation of this test), substitutes by the estimate of its pseudo-true value under Hh given in 1 b b b b (14), that is b zx n . By symmetry of our testing problem, the J and JA versions of the t-test n = zz

can also be calculated for Hh against Hg . Davidson and MacKinnon (1981) show that the t-ratio statistic, t ( b n ) ; has asymptotically a standard normal distribution under the null.

Based on the application of Roy’s union-intersection principle, McAleer and Pesaran (1986) show that

the test for

= 0 in (18) is equivalent to the standard F-statistic for the test of

model y = X

1 +Z 2

2

= 0 in the combined

+ u.

In order to frame the linear regression models into the encompassing type tests, we can focus on the discrepancy between the OLS estimator of the regression coe¢ cients, denoted by b n ; and the estimator of p p 1 the pseudo-true value in …nite samples, such that n b n b b n = n (Z0 Z) Z0 Mx y. Using this,

we can build an encompassing statistic for testing Hg EHh , as follows, p

n bn

b

b

n

=

p

n (Z0 Z)

1

8

p Z0 Mx W + n (Z0 Z)

1

Z0 Mx uf ;

if Hf is taken as the the true model given in (11). As a consequence, under some regularity conditions, p n b n b bn is asymptotically normally distributed with mean zero and the covariance matrix v2

1 xx

g

1 xx .

Using these results the WET statistic for testing Hg EHh , is given by, Egh =

y0 Mx Z1 (Z01 Mx Z1 ) vb2

1

Z01 Mx y

(20)

where Z1 are the components in Z that are orthogonal to X. Similarly, a variance encompassing test of Hg EHh can be constructed for the discrepancy between a consistent estimate of ! 2 and its pseudo-true value ! 2 , which takes the form of ! b2 ! b 2 bn . For the case in which Hg contains the true model, Hh ; the n

f

variance encompassing test is asymptotically equivalent to the Cox and the J tests. Voung’s test criterion for the comparison of Hg and Hh is computed as, Pn i=1 di Ggh = h i Pn 2 1=2 d d i i=1

where d = n

1

Pn

i=1

di ; di =

! 2n 1=2 b2n =b

1=2((b u2ig =b2n )

(21)

big and u bih are the estimated (b u2ih =b ! 2n )); and u

residuals of the underlying linear models given by (10). Under the null hypothesis HV , Hg and Hh are equivalent and Ggh is approximately distributed as a standard normal variate.

6

Extensions and empirical applications

Nonnested tests have also been derived for a number of other models, including tests of nonnested linear regression models with serially correlated errors (McAleer et. al. (1990)), regression models estimated by instrumental variables (Ericsson (1983)), models estimated by generalized method of moments (Smith (1992)), generalized empirical likelihood (Ramalho and Smith (2002)), conditional empirical likelihood (Kitamura (2003) and Otsu and Whang (2005)), nonnested Euler equations (Ghysels and Hall (1990)), autoregressive versus moving average models (Walker (1967)), autoregressive conditional heteroscedastic models (Bera and Higgins (1997) and McAleer and Ling (1998)), logit and probit models (Pesaran and Pesaran (1993)), nonnested threshold autoregressive models (Altissimo and Violante (2001) and Pesaran and Potter (1997)), and stochastic volatility models (Kim et. al. (1998)). Further theoretical contributions include a robust version of Cox-type statistics that controls for the e¤ect of contamination in the data (Victoria-Feser (1997)), conditional tests on su¢ cient statistics (Pace and Salvan (1990)), asymptotic improvements to Davidson and MacKinnon’s approach (Royston and Thompson (1995)), score-type statistics which are constructed from linear combinations of the likelihood functions (Santos Silva (2001)) and the enhancement of …nite-sample performance of nonnested tests by bootstrap methods (Godfrey (1998) and Davidson and MacKinnon (2002)). Various economic applications of nonnested hypothesis testing have appeared in the literature. Among them, savings and consumption functions (Deaton (1982)), Keynesian and new classical models of unemployment (Pesaran (1982a)), wage-employment bargaining models (Vannetelbosch (1996)), e¤ects of dividend taxes on corporate investment decisions (Poterba and Summers (1983)), money demand functions (McAleer

9

et. al (1982), and Elyasiani and Nasseh (1994)), autoregressive and moving-average schemes for unanticipated in‡ation series (Pagan et. al (1983)), exchange rates models (Backus (1984)), alternative crop response models (Ackello-Ogutu et. al. (1985) and Frank et. al (1990)), agricultural marketing margins (Lyon and Thompson (1993)), economic growth models (Ram (1986), Dowrick and Gemmell (1991) and Bleaney and Nishiyama (2002)), hedonic house prices (Dubin and Sung (1990), and Goodman and Dubin (1991)). In the literature of empirical industrial organization, nonnested hypotheses testing are applied to compare a Nash and collusive pricing in a industry with vertical product di¤erentiation (Bresnahan (1987)) Nonnested tests are also applied in game theoretic contexts by Gasmi et. al. (1992) and Sandler and Murdoch (1990), in sociological research by Halaby and Weakliem (1993), and in political science by Clarke (2001). Nonnested tests for rival linear regression models can be computed using various econometric packages. See, for example, Pesaran and Pesaran (1997).

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