Introduction Model secication tests are a central theme in the econometric literature. The majority of the aroaches fall into two categories. In the r

Transcription

1 Reversed Score and Likelihood Ratio Tests Geert Dhaene Universiteit Gent and ORE Olivier Scaillet Universite atholique de Louvain January 2 Abstract Two extensions of a model in the resence of an alternative model are roosed. The extensions are based on the score function of the alternative model. It is shown that the encomassing hyothesis is equivalent to standard conditions on the score of each of the extended models. The condition on the rst extension gives rise to the standard score encomassing test, while the condition on the second extension induces a so-called reversed score encomassing test. A similar logic is alied to the likelihood ratio, thus generating a likelihood ratio and a reversed likelihood ratio encomassing test. The ensued test statistics can be based on simulations if certain calculations are to dicult to carry out analytically. We study the rst-order asymtotic roerties of the roosed test statistics under general conditions. JEL classication 5, 3. Key-words score test, likelihood ratio test, encomassing, simulation-based inference. We are grateful to hristian Gourieroux for helful comments. Address corresondence to Geert Dhaene, Universiteit Gent, Faculteit Economie en edrijfskunde, oveniersberg 24, -9 Gent, elgium. Tel. (+32) Fax (+32) geert.dhaenerug.ac.be.

2 Introduction Model secication tests are a central theme in the econometric literature. The majority of the aroaches fall into two categories. In the rst aroach the model under test is confronted with another, often non-nested, model (see Gourieroux and Monfort [994] for a review), and therefore the tests are oriented towards this articular alternative model. The constraint underlying most of these tests is in fact the encomassing condition (see e.g. Mizon and Richard [986], endry and Richard [99], Gourieroux and Monfort [995], Dhaene [997], Dhaene, Gourieroux and Scaillet [998]), but not always (see Vuong [989]). The second aroach exloits orthogonality conditions imlied by the model under test without having a secic alternative model in mind (see ierens [994] for a review), and is known as conditional moment testing (Newey [985], Tauchen [985], ierens [99]). The aroach taken in this aer falls into the rst category, where an arbitrary conditional arametric model is tested against another arbitrary, ossibly non-nested, conditional arametric model. We exand on results reorted in Gourieroux and Monfort [995] and Dhaene [997], where score and likelihood ratio encomassing tests were roosed. These tests, and the new tests we roose, are generated by exonentially tilting the model under test in two alternative directions, each one involving the score function of the alternative model. Intuitively, the new tests we roose are obtained from reversing the roles of the true distribution generating the data and the seudo-true distribution of the model under test. This leads to what we call reversed score and likelihood ratio tests. The tests rely on simulations in order to avoid the need for analytic calculations of certain exectations in any articular alication. The framework is briey resented in Section 2. Section 3 introduces two extensions of the model under test, obtained by exonential tilting. It also restates the encomassing condition in terms of these extensions and gives the intuition underlying the reversed score and likelihood ratio tests. The basic test statistics are resented in Section 4. Their rst-order asymtotic roerties are studied in Section 5, in descending order of generality. Section 6 concludes. 2 Framework We consider an arbitrary air of conditional, ossibly non-nested, ossibly missecied, arametric models for indeendent and identically distributed data. Let X and Y be random vectors taking values x and y in IR k and IR l, resectively, and let P X be the true marginal distribution of X and P Y jx the true conditional distribution of Y, given X. Assume that the available data are T indeendent drawings (x t y t ), t = T from P X and P Y jx. Let G = ff G () j 2 IR m g and = ff () j 2 IR n g be arametric models of P Y jx. It is assumed that the

3 distributions F G (), F () and P Y jx admit conditional density functions f G (yjx ), f (yjx ) and (yjx), resectively, relative to some measure not deending on x, and. It is also assumed that the exectations of the log density functions exist whenever they are taken. Accounting for the ossibility thatg is missecied, i.e. P Y jx 62 G, and likewise for, it is also of interest to dene the seudo-true values of and with resect to P X and P Y jx (see e.g. Sawa [978]) = arg max E X E log f G (Y jx ) 2 = arg max E X E log f (Y jx ) 2 where the mathematical exectations E X and E are taken with resect to P X and P Y jx, resectively. We assume that and exist, are unique and interior to and, resectively. We shall be interested in testing G against. Therefore, we also dene the seudotrue value of with resect to P X and F G (), = arg max 2 E X E log f (Y jx ) where the mathematical exectation E is taken with resect to F G (). We assume that exists, is unique and interior to and is continuously dierentiable with resect to. y denition, G encomasses, written GE,if =. It is well known that the imlicit null hyothesis of many tests of G against is characterized by the condition that GE. See e.g. Mizon and Richard [986], Gourieroux and Monfort [995], and Dhaene [997]. Note that the underlying distributions P X and P Y jx are crucial in determining whether or not GE. The score functions of G and are dened as and s G (yjx ) = log f G(yjx ) s (yjx ) = log f (yjx ) resectively. It is assumed that the score functions are continuously dierentiable in the arameters, that their exectations exist whenever they are taken, that E X E s G (Y jx ) = only if = E X E s (Y jx ) = only if = E X E s (Y jx ) = only if = and that the matrices E X E [s G (Y jx )s G (Y jx )], E X E [s (Y jx )s (Y jx ]) and E X E [s (Y jx )s (Y jx ]) exist and are ositive denite. Then, dening the score quantity s = E X E s (Y jx ) 2

4 and the likelihood ratio (LR) quantity l = E X E [log f (Y jx ) ; log f (Y jx )] it is obvious that GEis equivalent tos = and also to l =. This roerty has led to the develoment of score encomassing tests, based on estimates of s (Gourieroux and Monfort [995]), and LR encomassing tests, based on estimates of l (Smith [994] and Dhaene [997]). The urose of this aer is to introduce tests that are based on quantities similar to s and l, in articular the quantities obtained from s and l by reversing the roles of P Y jx and F G ( ). A heuristic argument for doing so is resented in the next section. 3 Model extensions onsider the following extension of G G = ff G( ) j ( ) 2 IR n g where the distribution F G( ) has the following density function relative to fg(yjx )= f G(yjx ) ex( s (yjx )) E ex( s (yjx )) The density fg(yjx ) is obtained from f G (yjx ) by exonential tilting (arndorff-nielsen and ox [989]). Observe that G G and that the arameter vector ( ) need not be identied. Instead of utting = in the random vector s (Y jx ), one may alternatively ut =, leading to another extension of G G 2 = ff 2 G( 2 ) j ( 2 ) 2 IR n g where the distribution F 2 G( 2 ) has the following density function relative to fg(yjx 2 2 )= f G(yjx ) ex( s 2 (yjx )) E ex( 2s (yjx )) The density fg(yjx 2 2 ) is also obtained from f G (yjx ) by exonential tilting, but in a dierent direction. As before, G G 2 and ( 2 ) need not be identied. The motivation for considering the extended models G and G 2 comes from the following roosition. Proosition The following equivalences hold GE () E X E log f G(Y jx ) has a local maximum at ( )=( ) () E X E s (Y jx )= GE () E X E log f 2 G(Y jx 2 ) has a local maximum at ( 2 )=( ) () E X E s (Y jx )= 3

5 Proof. The score functions associated with G and G 2 are and s G(yjx )= s 2 G(yjx 2 )= s G (yjx ) ; E [s G (Y jx ) ex( s (Y jx ))] E ex( s (Y jx )) s (yjx ) ; E [s (Y jx )ex( s (Y jx ))] E ex( s (Y jx )) s G (yjx ) ; E [s G (Y jx )ex( 2 s (Y jx ))] E ex( 2s (Y jx )) s (yjx ) ; E [s (Y jx )ex( 2 s (Y jx ))] E ex( 2s (Y jx )) resectively. Putting ( )=( 2 )=( ) and taking exectations yields E X E s G(Y jx ) = = E X E s G (Y jx ) ; E X E s G (Y jx ) E X E s (Y jx ) ; E X E s (Y jx ) E X E s (Y jx )! A A! and E X E s 2 G(Y jx ) = = E X E s G (Y jx ) ; E X E s G (Y jx ) E X E s (Y jx ) ; E X E s (Y jx ) E X E s (Y jx )!! Given the assumtions made earlier, it follows that GE if and only if the functions E X E log fg(y jx ) and E X E log fg(y 2 jx 2 ) have a stationary oint at ( ) = ( ) and ( 2 ) = ( ), resectively. Now we need to show that, if =, the stationary oint ( ) is indeed a local maximum of the functions involved. First, xing =, E X E log fg(y jx ) attains a global maximum at =,by denition. Secondly, xing =, we nd, if =, " 2 E X E log fg(y jx ) # = = ;E X E [s (Y jx )s (Y jx )] The latter matrix is negative denite by assumtion, hence E X E log f G(Y jx ) attains a local maximum at =. The roof is comlete by noting that the functions f G and f 2 G are identical when =. (Q.E.D.) The roosition shows that GEif and only if the extensions of G using the score function of do not alter the seudo-true value associated with G, at least not locally. 4

6 In a sense, the extensions are thus ineective in bringing G closer to P Y jx, according to the Kullback-Leibler (95) Information riterion. Further, the condition GEis restated in terms of roerties of the score function s in relation to the distributions P Y jx and F G ( ). Interestingly, the two roerties mirror each other in the sense that each one, comared to the other, reverses the roles of P Y jx and F G ( ). After all, this should not come as a surrise since, for given, the distributions P Y jx and F G ( ) lay a symmetric role in the denition of encomassing. Thus, we are led to dene the reversed score quantity s 2 = E X E s (Y jx ) and, alying the same logic, the reversed LR quantity l 2 = E X E [log f (Y jx ) ; log f (Y jx )] The quantities s 2 and l 2 share the roerty with s and l that GE is equivalent to s 2 = and also to l 2 =. This roerty enables us to develo reversed score encomassing tests, based on estimates of s 2, and reversed LR encomassing tests, based on estimates of l 2. One may wonder whether the same reasoning of reversing the roles of P Y jx and F G ( ) can also be alied to the Wald encomassing test to yield something interesting. The Wald encomassing test (Gourieroux and Monfort [995]) is based on estimates of the Wald quantity, dened as w = ;. The reversed Wald quantity would then be w 2 = ; = ;w, which obviously does not lead to an interesting new test. The reason for this nding is that P Y jx and F G ( ) lay similar roles in w, aart from the sign. ence, reversing their roles doesn't lead to anything new. Looking back now at s and l, we clearly see that P Y jx and F G ( ) lay essentially dierent roles. This is why reversing them haens to be fruitful. 4 Test statistics Given the samle (x t y t ), t = T, of indeendent observations from P X and P Y jx, we seek to develo tests of the hyothesis that GE. It follows from the roerties derived in the revious section that estimates of the quantities s, l, s 2 and l 2 and of their covariance matrices naturally lead to tests of GE. Note that this hyothesis is weaker than the hyothesis that G is correctly secied, i.e. P Y jx 2G. ence estimates of the same quantities are also suited for testing the hyothesis that G is correctly secied. A distinguishing feature between tests of GEand tests of P Y jx 2Gis that, for the latter tests the distribution theory is usually based on the assumtion that G is correctly secied, whereas for the former tests the distribution theory can at most be based on the assumtion that GE. The distribution theory resented in this aer considers the most general case, i.e. where G ossibly does not encomass. 5

7 and The seudo-maximum likelihood estimators ^ and ^ solve max 2 T max 2 T TX t= TX t= log f G (y t jx t ) log f (y t jx t ) resectively. Under regularity conditions such as given in White [982], ^ as! and ^ as!. For any 2, let yt h (), t = T and h =, be indeendent drawings from F G (), given x t. For any h =,thesimulated seudo-maximum likelihood estimator ^ h is dened to solve TX max log f (y h t ()jx t ) 2 T t= Under similar regularity conditions, ^ h as as! and!. ere and in the sequel, stochastic limits are taken as T!, with xed, ossibly at. Then, dene the simulated score and reversed score statistics as ^s = T ^s 2 = T X TX h= t= X TX h= t= ^ h^ s (y t jx t ^ h^) s (y h t (^)jx t ^) resectively, and the simulated LR and reversed LR statistics as ^l = T ^l 2 = T X h= t= X h= t= TX h log f (y t jx t ^ h^) ; log f (y t jx t ^) i TX h log f (y h t (^)jx t ^) ; log f (y h t (^)jx t ^ h^) i as as resectively. We have ^s! s, ^s 2! s 2, ^l as! l and ^l as 2! l 2. The rst-order limit distributions of ^s, ^s 2, ^l and ^l 2 are investigated in the next section. 5 Limit distributions We need to introduce some additional notation. Let l G () = T TX t= log f G (y t jx t ) 6

8 and l () = T TX t= log f (y t jx t ) be the normalized log likelihood functions of G and based on the observed data (x t y t ), t = T, and let l h ( ) = T TX t= log f (y h t ()jx t ) be the normalized log likelihood function of based on the simulated data (x t y h t ()), t = T. orresondingly, dene the normalized score functions and s G () = l G() s () = l () s h ( ) = lh ( ) 5. Limit distributions under general conditions For suciently large T, ^ satises the rst-order condition s G (^) =. Exanding s G (^) in a Taylor series around s G ( ), taking the robability limit of [s G ()= ] = and rearranging yields the well known result (White [982]) where T (^ ; )= TK ; G s G( )+o () " # K G = ;E X E s G() = Similarly, T ( ^ ; )= TK ; s ( )+o () where and where K = ;E X E " s () # = T ( ^h ; )= T ~ K ; sh ( )+o () ~K = ;E X E " sh ( ) 7 # = =

9 Further, exanding ^ h^ around ^ h yields T ( ^h ^ ; ) = T ( ^ h a ; )+ T(^ ; )+o () where (see Dhaene [997]) with = T K ~ ; sh ( )+K ; s G( )+o () = " # = K ~ ; = ~ J G " # ~J G = E X E log f (Y jx ) log f G(Y jx ) Now, exanding s ( ^ h^) around s ( ) gives where T (^s ; s ) = T (s ( ) ; s ) ; T X h= G = = K ( ^ h^ ; )+o () = T (s ( ) ; s ) ; TK K ; G s G( ) ; TK ~ K ; X h= " # K = ;E X E s () s h ( )+o () = Exanding s h ( ^ ^) around s h ( ) gives T X T (^s2 ; s 2 ) = (s h ( ) ; s 2 ) ; T K ~ ( ^ ; ) h= T JG ~ (^ ; )+o () where = T X h= (s h ( ) ; s 2 ) ; T K ~ K ; s ( ) + T J ~ G K ; G s G( )+o () ~J G = E X E " log f (Y jx ) ~K = ;E X E " sh ( ) # # log f G(Y jx ) = = = = = ~ J G 8

10 This comletes the asymtotic exansions for ^s and ^s 2. Turning to ^l, exanding l ( ^ h^) around l ( ) gives T (^l ; l ) = T (l ( ) ; l ( ) ; l )+ T ; Ts ( ) ( ^ ; )+o () X h= s ( ) ( ^ h^ ; ) = T (l ( ) ; l ( ) ; l )+ Ts K; G s G( ) + Ts ~ K ; X h= where it was used that s ( ) as!. Finally, for ^l 2, s h ( )+o () T (^l2 ; l 2 ) = = T T ; T X h= (l h ( ) ; l h ( ) ; l 2 )+ X h= X h= T X h= s h ( ) ( ^ ; ) s h ( ) ( ^ h^ ; )+ T (~! G ; ~! G ) (^ ; )+o () (l( h ) ; l( h ) ; l 2 )+ Ts 2 K; s ( ) + T (~! G ; ~! G ) K ; G s G( )+o () using s h ( ) as!, with ~! G = E X E " log f G(Y jx )logf (Y jx ) ~! G = E X E " log f G(Y jx )logf (Y jx ) To summarize the exansions, let ^d = ^s ^s 2 ^l ^l2 A d = s s 2 l l 2 A # # = = 9

11 and Then, w t = s G (y t jx t ) s (y t jx t ) s (y t jx t ) ; s P h= s (y h t ( )jx t ) ; s 2 P s h= (yt h ( )jx t ) log f (y t jx t ) ; log f (y t jx t ) ; l P h= log f (y h t ( )jx t ) ; P h= log f (y h t ( )jx t ) ; l 2 A = ;KK ; G I ;K ~ K ; ~J G K ; G ; K ~ K ; I s K; G s ~ K ; (~! G ; ~! G ) K ; G s 2 K; T ( ^d ; d) = T T X t= Aw t + o () Observe that E X E w t =. Assuming the existence of V = E X E (w t w t), T ( ^d ; d) d! N( AVA ) A A by the central limit theorem. Note that all the submatrices in A can be consistently estimated, and hence A itself, by relacing E X E by T P T t=, E X by T P T t=, E by P E ^ or by h= and using yt h (^) in lace of y t, by ^, by ^, P by h= ^ h^, and (^s ^s 2 ^l ^l 2 )by (s s 2 l l 2 ), successively. Similar relacements in w t yield ^w t and P ^V = T T t= ^w t ^w t as a consistent estimator of V. A consistent estimator of AV A follows. Insection of Aw t reveals that no general asymtotic equivalences hold between subvectors of ^d. More recisely, there does not exist in general a xed non-zero matrix such that T( ^d;d) =o (), because V is not of reduced rank in general and A has not reduced row rank in general. This imlies, in articular, that no general asymtotic equivalences exist between ^s, ^s 2, ^l and ^l 2. This nding, and the full characterization of the joint rst-order limit distribution of ^s,^s 2, ^l and ^l 2 oens ersectives for jointly exloiting the evidence contained in these statistics against any ofthehyotheses GE and P Y jx 2 G, thereby gaining in ower comared to the standard score or LR test. The unresolved roblem for doing this is to control the (asymtotic) size of the joint

12 test. A fully joint test would tyically take a quadratic form in T ^d, weighted by a consistent estimate of (AV A ) +, and refer to the 2 distribution with aroriate degrees of freedom. As we show below, asymtotic equivalences do aear when GE (a fortiori when P Y jx 2G), making AV A a singular matrix. In many cases of interest, consistent estimates of AV A have an asymtotic rank that exceeds the rank of AV A, which makes consistent estimation of (AV A ) + a dicult task (see also Andrews [989]). In other words, the main diculty for building a test on the full vector ^d is that the rank of his covariance matrix deends on whether or not GE, which is recisely the hyothesis being tested. 5.2 Limit distributions under the condition GE The rst-order limit distribution of ^d when GEis easily obtained using the results of the revious subsection. We then have d =and w t = s G (y t jx t ) s (y t jx t ) s (y t jx t ) P s h= (yt h ( )jx t ) P s h= (yt h ( )jx t ) Further, K = K, ~ K = ~ K, ~! G = ~! G, = ~ K ; A = A ~ J G and ;K K ~ ; ~ J G K ; G I ;K K ~ ; ~J G K ; G ; K ~ K ; I from which we obtain T ^l = o () = T ^l 2 and the asymtotic equivalence T ^s = ;K ~ K ; T ^s2 + o () We can be more recise about the limiting behaviour of ^l and ^l 2 by considering the exansions Tl ( ^ h^) = Tl ( ^) ; T 2 ( ^ h^ ; ^) K ( ^ h^ ; ^)+o () Tl h ( ^ ^) = Tl h ( ^ h^ ^) ; T 2 ( ^ h^ ; ^) ~ K ( ^ h^ ; ^)+o () A

13 wherefrom ;2T ^l = T ;2T ^l 2 = T X h= X h= ( ^ h^ ; ^) K ( ^ h^ ; ^)+o () ( ^ h^ ; ^) ~ K ( ^ h^ ; ^)+o () Uon gathering revious results, T ( ^h ^ ; ^) = T K ~ ; sh ( )+ T K ~ ; ~ J G K ; G s G( ) ; TK ; s ( ) = ; TK ; ^s + o () = ; T K ~ ; ^s 2 + o () yielding the asymtotic equivalences ;2T ^l = T ^s K; ^s + o () = T ^s ~ 2 K ; K ~ K ; ^s 2 + o () ;2T ^l 2 = T ^s ~ 2 K ; ^s 2 + o () = T ^s ~ K; K K ; ^s + o () Note that ;2T ^l and ;2T ^l 2 are not in general asymtotically equivalent. The limit distributions can be summarized as follows. Let and v t = s G (y t jx t ) s (y t jx t ) P s h= (yt h ( )jx t ) D = JG ~ K ; G ; K ~ K ; Now Ev t =,and letting = E(v t v t)we have I A T d ^s! N( K K ~ ; ~ DD K ; K ) T d ^s2! N( DD ) ;2T ^l d! M(( ~ K ; K ~ K ; DD )) ;2T ^l 2 d! M(( ~ K ; DD )) where M((W )) is the distribution of a weighted sum of indeendent 2 variates with weights equal to the eigenvalues of W. The matrices D and and the necessary eigenvalues can be consistently estimated by the rocedure outlined in the revious 2

14 subsection. If we can determine the rank of the asymtotic covariance matrices of T ^s and T ^s 2, asymtotic score and reversed score encomassing tests follow readily. Asymtotic LR and reversed LR encomassing tests follow also from the limit distribution given above. They require the calculation of critical values of weighted sumofchi-squares distributions, which can easily be obtained by simulation. Note that LR and reversed LR tests do not require the determination of the rank of a matrix. 5.3 Limit distributions under the condition P Y jx 2 G Further simlications occur when P Y jx 2 G. We have F G ( ) = P Y jx, wherefrom ~K = K, yielding T ^s = ; T ~ JG K ; G s G( )+ T s ( ) ; T ^s2 = T ~ J G K ; G s G( ) ; T s ( ) ; and the asymtotic equivalences T ^s = ; T ^s 2 + o () X h= X h=! s h ( ) + o () s h ( )! + o () and ;2T ^l = T ^s K; ^s + o () = T ^s 2 K; ^s 2 + o () = ;2T ^l 2 + o () Note also that s ( ) and s h ( ), h =, are conditionally indeendent and identically distributed, given x t, t = T. Asymtotic score and reversed score tests and asymtotic LR and reversed LR tests of P Y jx 2 G can be constructed along the same lines as given in the revious subsection, taking advantage of the simlications just mentioned. 6 onclusion We have outlined two alternative rocedures to the standard score and LR encomassing tests, resectively. They follow from restating the encomassing condition in terms of a roerty regarding exonentially tilted models. Intuitively, the alternative rocedures are obtained from reversing the roles of the true distribution generating the data and the seudo-true distribution of the model under test. Alication requires the models 3

15 to be estimable by the method of maximum likelihood. No analytic calculations are needed beyond the analytic rst and second derivatives of the log likelihood functions. The need calculate mathematical exectations analytically is avoided by the use of any nite number of simulations from the model under test. References [] Andrews, D.W.K. (987), \Asymtotic Results for Generalized Wald Tests", E- conometric Theory, 3, 348{358. [2] arndor-nielsen, O.E. and D.R. ox (989), Asymtotic Techniques for Use in Statistics, London, haman and all. [3] ierens,. (994), Estimation, Testing, and Secication of ross-section and Time Series Models, ambridge University Press, ambridge. [4] ierens,. (99), \A onsistent onditional Moment Test of Functional Form", Econometrica, 58, 443{458. [5] Dhaene, G. (997), Encomassing Formulation, Proerties and Testing, Lecture Notes in Economic and Mathematical Systems, n o 446, Sringer Verlag, erlin. [6] Dhaene, G.,. Gourieroux and O. Scaillet (998), \Indirect Encomassing and Instrumental Models", Econometrica, 66, 673{689. [7] Gourieroux,. and A. Monfort (994), \Testing Non-Nested yotheses", in andbook of Econometrics, IV, ed. by R. Engle and D. McFadden, Elsevier Science, North-olland, Amsterdam, 2583{2637. [8] Gourieroux,. and A. Monfort (995), \Testing, Encomassing and Simulating Dynamic Econometric Models", Econometric Theory,, 95{228. [9] endry, D.F. and J.-F. Richard (99), \Recent Develoments in the Theory of Encomassing", in ontributions to Oerations Research and Econometrics, The Twentieth Anniversary of ORE, ed. by. ornet and. Tulkens, MIT Press, ambridge. [] Mizon, G.E. and J.-F. Richard (986), \The Encomassing Princile and its Alications to Testing Non-nested yotheses", Econometrica, 54, 657{678. [] Newey, W. (985), \Maximum Likelihood Secication Testing and onditional Moment Tests", Econometrica, 53, 47{7. 4

16 [2] Kullback, S. and R.A. Leibler (95), \On Information and Suciency," Annals of Mathematical Statistics, 22, 79{86. [3] Sawa, T. (978), \Information riteria for Discriminating among Alternative Regression Models", Econometrica, 46, 273{292. [4] Smith, R. (994), \onsistent Tests of the Encomassing yothesis", REST-DP 943. [5] Tauchen, G. (985), \Diagnostic Testing and Evaluation of Maximum Likelihood Models", Journal of Econometrics, 3, 45{443. [6] Vuong, Q. (989), \Lihelihood Ratio Tests for Model Selection and Non Nested yotheses", Econometrica, 57, 33{333. [7] White,. (982), \Maximum Likelihood Estimation of Missecied Models", E- conometrica, 5, {26. 5