Journal of Econometrics

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1 Journa of Econometrics 70 (202) Contents ists avaiabe at SciVerse ScienceDirect Journa of Econometrics ourna homepage: wwweseviercom/ocate/econom Underidentification? Manue Areano a,, Lars Peter Hansen b, Enrique Sentana a a CEMFI, Casado de Aisa 5, E-2804 Madrid, Spain b Department of Economics, University of Chicago, 26 East 59th St Chicago, IL 60637, USA a r t i c e i n f o a b s t r a c t Artice history: Avaiabe onine 2 June 202 We deveop methods for testing that an econometric mode is underidentified and for estimating the nature of the faied identification We adopt a generaized-method-of moments perspective in a possiby non-inear econometric specification If, after attempting to repicate the structura reation, we find substantia evidence against the overidentifying restrictions of an augmented mode, this is evidence against underidentification of the origina mode o diagnose how identification might fai, we study the estimation of a one-dimensiona curve that gives the parameter configurations that provide the greatest chaenge to identification, and we iustrate this cacuation in an empirica exampe 202 Esevier BV A rights reserved Introduction It is natura to abandon without further computation the set of restrictions strongy reected by the (ikeihood ratio) test Simiary, it is natura to appy a test of identifiabiity before proceeding with the computation of the samping variance of estimates and to forego any use of the estimates, if the indication of nonidentifiabiity is strong Koopmans and Hood (953) (see p 84) It is common in econometric practice to encounter one of two different phenomena Either the data are sufficienty powerfu to reect the mode, or the sampe evidence is sufficienty weak so as to suspect that identification is tenuous he eary simutaneous equations iterature recognized that underidentification is testabe, but to date such tests are uncommon in econometric practice despite the fact that there are many situations of economic interest in which seemingy point identified modes may be ony set identified his is a substantiay revised and extended version of Areano et a (999) We thank Javier Avarez, Raque Carrasco, Rui Cui, Jesús Carro, Aessandro Gaesi, and especiay Eena Manresa, Javier Mencía and Francisco Peñaranda for abe research assistance at various stages of this proect Aso, we thank Andres Santos and Azeem Shaikh for hepfu conversations about our genera approach to this probem Finay, we thank Marine Carrasco and three referees for vauabe feedback on earier drafts of this paper Moreover, we benefited from vauabe comments from mutipe seminar audiences MA and ES gratefuy acknowedge financia support from the Spanish Ministry of Science and Innovation through grant ECO and LPH gratefuy acknowedges support from the Nationa Science Foundation through grant SES Corresponding author E-mai addresses: areano@cemfies (M Areano), hansen@uchicagoedu (LP Hansen), sentana@cemfies (E Sentana) We find it productive to pose this as an estimation probem where we seek to identify the ocation of the identification probem We adopt a generaized-method-of-moments (GMM) perspective and suppose that (under the nu hypothesis) we may identify a curve or a function of a scaar variabe that represents such a curve hus the target of estimation is this curve or the corresponding function For modes that are sufficienty inear, this curve represents a one-dimensiona subspace, but transated he resuting set can be easiy parameterized and estimated using a standard GMM approach In effect we buid an augmented structura mode in which the moment conditions are satisfied by a curve instead of a point his estimation is of direct interest because it isoates the dimension aong which identification of the origina mode is probematic he famiiar J test from the work of Sargan (958) or Hansen (982a) for overidentification of the augmented mode now becomes a test for underidentification of the origina mode If we can identify a curve or function representing that curve without statistica reection, then the origina mode is not we identified and we refer to this phenomenon as underidentification We refer to such a test as an I test In contrast, a statistica reection provides evidence that the parameter vector in the origina mode is indeed point identified, uness of course the famiiar J test continues to reect its over-identifying restrictions he idea of identifying a curve extends to estimation environments in which we may not be abe to represent the curve with a finite-dimensiona parameterization hus we aso suggest a more genera estimation approach, study the resuting statistica efficiency and discuss impementation Inferentia methods are necessariy atered, and we are ed to buid on the work of Carrasco and Forens (2000) in designing a GMM approach to this probem We consider in progression three different estimation environments: modes that are inear in parameters (Section 3), modes /$ see front matter 202 Esevier BV A rights reserved doi:006/econom

2 M Areano et a / Journa of Econometrics 70 (202) with noninear restrictions on the parameters (Section 4), and finay, modes with more fundamenta noninearities (Section 5) hroughout we deveop specific exampes to iustrate the nature and the appicabiity of our methods In Section 6 we show how to appy these methods to a consumption-based asset pricing mode fit to microeconomic data What foows is a more detaied overview of the paper 2 Overview As in Hansen (982a), suppose that {x t } is an observabe stationary and ergodic stochastic process and et P be a parameter space that we take to be a subset of R k Introduce a function f (x, ) : P R p for each x he function f is ointy Bore measurabe and it is continuousy differentiabe in its second argument for each vaue of x Finay suppose that E f (x t, β) < for each β P In ight of this assumption we define E[f (x t, β)] = f (β) for each β P GMM estimation uses the equation: f (β) = 0 () to identify a parameter vector β 0 When β 0 is identified, it is the unique soution to (), otherwise there wi be mutipe soutions o reate to standard anayses of identification and deveop tests for underidentification, we suppose that p k In discussing the ack of identification in non-inear modes in those circumstances, it is important to distinguish the different situations that may arise We say that β β 0 is observationay equivaent to β 0 if and ony if E[f (x t ; β )] = 0 he true vaue β 0 is ocay identifiabe if there is a neighborhood of β 0 such that in this neighborhood E[f (x t ; β)] = 0 ony if β = β 0 (Fisher, 966) he order condition p = dim (f ) dim (β) = k provides a first check of identification, but this is ony necessary A compement is provided by the rank condition: If E f (x, β)/ β is continuous at β 0, and rank{e f (x, β 0 )/ β } = k, then β 0 is ocay identified (Fisher, 966; Rothenberg, 97) In contrast to the order condition, this condition is ony sufficient But if rank{e f (x, β)/ β } is aso constant in a neighborhood of β 0, then the above rank condition becomes necessary too However, as argued in Sargan (983a,b), there are non-inear modes in which the rank condition fais, and yet β 0 is ocay identified (see Wright, 2003, for tests of the Jacobian rank condition in non-inear modes) In this paper we wi take a decidedy goba approach Goba identification requires that β 0 be the unique soution on P to the system of Eqs () o study underidentification, we foow Sargan (959) by imposing an expicit structure on the ack of identification his eads us to study an aternative estimation probem Specificay, we consider a parameterization of the form β = π(θ), where π is a continuous function with range P and θ R, which is some convenienty chosen domain For exampe, suppose that θ π(θ) =, (2) τ(θ) so that θ is the first component of the parameter vector We then expore a set of such functions that is restricted appropriatey 2 As an aternative identification condition, we require f [π(θ)] = 0 for a θ if, and ony if π = π 0 If we can successfuy identify a nonconstant function π 0 that reaizes aternative vaues As esewhere in the econometrics iterature, anaogous resuts can be obtained using other data generating processes For cross-sectiona and pane extensions of Hansen s (982a) formuation see the textbooks by Hayashi (2000) and Areano (2003), respectivey 2 See Section 5 for further detais in the parameter space, then we cannot uniquey identify a singe parameter vector β 0 from the moment conditions () hus the parameter vector β 0 is underidentified Our investigation of underidentification eads naturay to the question of how to estimate π 0 efficienty One approach woud be to use one of the standard GMM obective functions and try to construct an estimator of π 0 as the set of approximate minimizers of that obective In this paper we expore a rather different approach, one that depicts the identification faiure in the construction of π 0 It eads us naturay to ask what the efficiency gains are to estimating ointy aternative points aong a curve, say π 0 (θ) for aternative vaues of θ Initiay we iustrate these efficiency gains using a standard formuation of the GMM efficiency bounds In what foows we use parameterization (2) For each vaue of θ in a finite set n = {θ, θ 2,, θ n } we estimate the (k )- dimensiona parameter vector τ(θ) We can map this into a standard GMM probem where we simpy repicate the origina moment conditions Later we wi extend this discussion to the case in which the set of θ s that are of interest is an interva, but this finite set construction wi set the stage for a more genera treatment As posed, this is a standard GMM estimation probem, abeit one with a specia structure o anayze the gains to oint efficiency, we presume the foowing centra imit approximation: Assumption 2 t= f [x t, π 0 (θ)] converges to a Gaussian random vector {g(θ) : θ n } with mean zero for a θ n and covariance function: K(θ, ϑ) = E g(θ)g(ϑ) = im E f [x t, π 0 (θ)] f [x s, π 0 (ϑ)] t= In stating this assumption and in what foows we wi abuse the E notation by using it both for the origina probabiity space and for the probabiity space used in constructing the Gaussian process used in the centra imit approximation Construct f [xt, π 0 (θ)] 0k D(θ) = E, β I k where 0 k is a row vector of zeros and I k is an identity matrix of dimension k Assumption 22 D(θ) D(θ) is nonsinguar for each θ n With these ingredients, we appy directy the anaysis in Hansen (982a) and the earier anaysis in Sargan (958, 959), which invoves reducing the moment conditions by introducing a (k ) n by p n seection matrix A that picks among the possibe moment conditions: Ef [x t, π(θ )] = 0 for =, 2,, n he moment restrictions are thus broken into n bocks and the parameter vector π(θ ) ony appears in bock Our characterization of the GMM efficiency bound expoits this bock structure A partitioned seection matrix reduces the moments to be the same as the number of free parameters and has the estimation probem focus on: A A 2 A n Ef [x t, π(θ )] 0 A 2 A 22 A 2n Ef [x t, π(θ 2 )] 0 =, A n A n2 A nn Ef [x t, π(θ n )] 0 where A i has dimension (k ) p he choice of seection matrix A = [A i ] aters the resuting statistica efficiency of a s=

3 258 M Areano et a / Journa of Econometrics 70 (202) GMM estimation Some coections of seection matrices impy the same asymptotic efficiency, however For instance, forming a new seection matrix by premutipying a given seection matrix by a nonsinguar matrix in effect uses the same moment conditions for estimation and hence does not ater the asymptotic efficiency of the corresponding GMM estimator As a consequence, if we define D as a bock diagona matrix of dimension p n by (k ) n with D(θ ) in the th diagona position, seection matrices for our estimation probem can aways be restricted to satisfy (at east asymptoticay): AD = I (3) without atering the efficiency bound By imposing this restriction we simpify the formua for the asymptotic covariance matrix as we wi see Consider now the first row bock of (3): A D(θ ) = I, A D(θ ) = 0 = 2, 3,, n (4) It foows from Hansen (982a) that the asymptotic covariance matrix for the resuting GMM estimator of τ(θ ) is the covariance matrix of n A g(θ ) = he corresponding efficiency bound is soved by minimizing this covariance matrix by choice of A, A 2,, A n Whie covariance matrices are ony partiay ordered, this minimization probem turns out to have a we defined minimum he zero restrictions imposed in (4) contro for the fact that τ(θ ) for = 2,, n are estimated at the same time as τ(θ ) and hence imit the construction of the seection matrix Whie we have focused on the efficiency of τ(θ ), an anaogous argument appies for τ(θ ) for = 2,, n It wi be convenient for us to represent this minimization probem differenty We consider random variabes of the form n B g(θ ), = where D(θ ) B = 0 for =, 2,, n he zero restriction imits the basis random variabes B g(θ ) that we wi use in our construction of the bound Form n F n = y = B g(θ ) : D(θ ) B = 0, =, 2,, n = Next transform the random vectors {g(θ) : θ n } into: h(θ) = [D(θ) D(θ)] D(θ) g(θ) hen the efficiency bound for a given n is obtained by soving: γ min E h(θ ) 2 f f F n for any vector γ he soution is Pro γ h(θ ) F n, where Pro is the east squares proection operator and the minimized obective is the second moment of the east squares proection error 3 Whie the finite parameter GMM bound is we known, what foows gives a forma representation of that bound that wi have direct extension as we expand the number of moment conditions used in estimation 3 In what foows when we use the notation Pro appied to a random vector we mean the vector of proections obtained by proecting each coordinate on the reevant cosed inear space of random variabes that is being proected on Proposition 2 he GMM efficiency bound for estimating τ(θ ) is: h(θ E ) Pro[h(θ ) F n ] h(θ ) Pro[h(θ ) F n ] for =, 2,, n Proof Write h(θ ) Pro [h(θ ) F n ] = g(θ ) g(θ à à 2 2 ) à n, g(θ n ) where à D(θ ) = 0 for =, 2,, n Form A = à + [D(θ ) D(θ )] D(θ) à 2 à n Notice that A D = I 0 0 Let A be the first row bock of a seection matrix A that satisfies (3) Consider some other matrix A such that A D = I 0 0, which is the first row bock of a seection matrix A that satisfies (3) hen the entries of the random vector g(θ ) [A A ] g(θ 2 ) g(θ n ) are in F n Hence g(θ ) (A A )E g(θ 2 ) g(θ ) g(θ 2 ) g(θ n ) (A ) = 0 g(θ n ) since a vector of proection errors is orthogona to the space that is being proected onto hus g(θ ) g(θ A E 2 ) g(θ ) g(θ 2 ) g(θ n ) (A ) g(θ n ) g(θ ) A E g(θ 2 ) g(θ ) g(θ 2 ) g(θ n ) (A ) g(θ n ) where is the usua inequaity for comparing positive semidefinite matrices Remark 22 In this representation the covariance of h(θ ) is the asymptotic covariance matrix for a GMM estimator that uses ony moment conditions based on [D(θ ) D(θ )] D(θ ) Ef [x t, π(θ )] = 0, but ignores the possibe efficiency gains from oint estimation It even fais to sove the second-best probem of efficienty estimating τ(θ ) using inear combinations of the moment restrictions: Ef [x t, π(θ )] = 0 except in very specia circumstances

4 M Areano et a / Journa of Econometrics 70 (202) Remark 23 Our choice of using the seection matrix [D(θ ) D(θ )] D(θ ) in the construction of h(θ) is ony one possibiity Notice that if we had chosen an aternative seection matrix D such that D D(θ ) = I, then Pro[ D g(θ ) F n ] = Pro [D(θ ) D(θ )] D(θ ) g(θ ) F n D [D(θ ) D(θ )] D(θ ) g(θ ) hus D g(θ ) Pro[ D g(θ ) F n ] = h(θ ) Pro[h(θ ) F n ] resuting in the same east squares proection error As expected our choice of starting point is inconsequentia to our fina cacuation Focusing on a finite set n, whie pedagogicay reveaing, is too restrictive for some of our anaysis Given our regression-error characterization of the efficiency bound, there is a direct extension to estimating curves (Section 5) Nevertheess, for some important exampes that we consider in the next two sections, in which we consider modes that are inear in the parameters (Section 3) and modes in which the noninearity is concentrated in the parameters (Section 4), it suffices to focus on a finite number of θ s Some reated iterature Our work is reated to two different strands of the iterature that have gained prominence in recent years One is the weak instruments iterature (see eg Stock et a, 2002), which maintains the assumption that the rank condition is satisfied, but ony ust o reate to this ine of research, suppose that is an interva and consider an interior point θ Suppose that π is differentiabe at θ hen under appropriate reguarity conditions: f (β) β β dπ(θ) dθ θ = 0, where β = π(θ ) In other words, the matrix f (β) β β has reduced rank for any θ in the interior of In contrast the weak instruments iterature considers the reduced rank as the imit of a sequence of data generating modes indexed by the sampe size 4 In our anaysis such a sequence coud be interesting as a oca specification under the aternative hypothesis of identification We seek to infer the specific manner in which identification may fai whereas the weak instrument iterature focuses on deveoping reiabe standard errors and tests of hypotheses about a unique true vaue of β he other strand is the set estimation iterature (see eg Chernozhukov et a, 2007 or more recenty Yidiz, 202), which often aows for E[f (x; β)] = 0 for set vaues of β and whose obective is to make inferences about this set 5 In contrast, in this paper we expore the precise nature of the underidentification Given this focus, we are ed to add structure to the potentia underidentification that is considered By adding this structure to the possibe identification faiure, we are ed to ater the usua GMM obective in order to estimate efficienty the onedimensiona function π that parameterizes the potentia ack of identification 4 ypicay in this iterature the rank is not ust reduced but is zero in the imit 5 Some of this iterature aso considers moment inequaities as a source of underidentification Our anaysis does not cover this situation 3 Linearity in the parameters We first study the identification of an econometric mode that is inear in parameters, in which case we write () as: E(Ψ t ) = 0, (5) β where β is a k-dimensiona unknown parameter vector and Ψ t is an p by k + matrix of random variabes constructed from data 6 Suppose that there are two soutions to equation (5), say β [] and β [2] where the first entry of β [] is restricted to be one and the first entry of β [2] to zero 7 hen E(Ψ t ) π 0 (θ) = 0, for a θ R, where π 0 (θ) = θπ 0 (0) + ( θ)π 0 () (6) A feature of the inearity in parameters is that we can identify two distinct vaues of β that satisfy (5), in this case π 0 (0) = β [], 0 π 0 () = β [2] 0 hus to study underidentification, we focus on identifying β [] 0 and β [2] 0 that sove the dupicated moment conditions E(Ψ t ) β [] = 0 (7) E(Ψ t ) β [2] = 0 In this probem we envision β [] 0 and β [2] 0 as the target of a GMM estimation probem subect to the restrictions on the first entries that we mentioned previousy his eads us to estimating 2 (k ) free parameters Given estimates of β [] 0 and β [2] 0, we then infer a one-dimensiona curve (actuay a ine) using formua (6) In posing the above estimation probem, we imposed a normaization in the origina equation (5) In what foows we wi adopt a different and sighty more genera starting point by considering: E(Ψ t )α = 0, (8) where α is a (k+)-dimensiona unknown parameter vector in the nu space of the popuation matrix E(Ψ t ) If there is a soution α 0 to this equation, then any scae mutipe of α 0 wi aso be a soution hus from a statistica perspective, we consider the probem of identifying a direction o go from a direction to the parameters of interest requires an additiona scae normaization of the form q α =, where q is a k + vector that is specified a priori For instance, we coud choose q to be a member of the canonica basis, which woud restrict one of the components of α to be one as in: α = β, which we effectivey imposed in (5) Aternativey, we coud choose q = α so that α =, together with a sign restriction on one of the nonzero coefficients as in: α = + β 2 β, 6 herefore, we consider not ony modes which are inear in both variabes and parameters, but aso the non-inear in variabes but inear in parameters modes discussed in Chapter 5 of Fisher (966), which combine different non-inear transformations of the same variabes 7 We adopt these restrictions for convenience Normaizing a coefficient to unity is common practice, and normaizing the second one to have zero coefficient rues out the possibiity that the resuting coefficient vectors are proportiona Other normaizations are possibe

5 260 M Areano et a / Journa of Econometrics 70 (202) where β Neither of these approaches can be empoyed without oss of generaity, however he particuar appication dictates how to seect the parameters of interest from this direction 8 Suppose now that instead of a one-dimensiona subspace, we can actuay infer a two-dimensiona subspace of α s that satisfies (8) his eads us to efficienty estimate those α [] 0 and α [2] 0 for which E(Ψ t )α [] = 0 E(Ψ t )α [2] = 0 (9) Our parameterization π 0 given in (6) gives us one way to parameterize this two-dimensiona subspace It is the space spanned by the two vectors: π 0 (0) and by π0 () as required by (7) he dupicated moment conditions in (7) or (9) give us a direct ink to the rank condition famiiar in the econometrics iterature Suppose the order condition (p k) is satisfied, but not necessariy the rank condition hus the maxima possibe rank of the matrix E(Ψ t ) is min{p, k + } Mode (8) is said to be identified when E(Ψ t ) has rank k, in which case its nu space is precisey onedimensiona When p > k and the mode is identified, it is said to be overidentified because the rank of the matrix E(Ψ t ) now must not be fu Instead of having maxima rank k+, E(Ψ t ) has reduced rank k his impication is known to be testabe and statistica tests of overidentification are often conducted in practice In contrast, mode (8) is said to be underidentified when the rank of E(Ψ t ) is ess than k In this case the nu space of E(Ψ t ) wi have more than one dimension A singe normaization wi no onger seect a unique eement from the parameter space By focusing on (6), our approach puts an expicit structure on the ack of identification, as iustrated by (9) hus, we initiay make the foowing assumption (see Section 32 for other possibiities): Hypothesis 3 E(Ψ t ) has rank k Under this hypothesis the set of soutions to Eq (8) is twodimensiona o test for this ack of identification, we think of (9) as a new augmented mode We attempt to determine (α [], α [2] ) simutaneousy and ask whether they satisfy the combined overidentifying moment restrictions (9) If they do, then we may concude that the origina econometric reation is not identified or equivaenty is underidentified hus by buiding an augmented equation system, we may pose the nu hypothesis of underidentification as a hypothesis that the augmented equation system is overidentified Reections of the overidentifying restrictions for the augmented mode provide evidence that the origina mode is indeed identified Posed in this way, underidentification can be tested simpy by appying appropriatey an existing test for overidentification For instance, a standard J test for overidentification, such as those of Sargan (958) and Hansen (982a), is potentiay appicabe to the augmented mode his test wi be our I test he foowing exampe iustrates our formuation Exampe 3 Suppose that p = and k = Write E(Ψ t ) = a a 2 For there to be identification in the sense that we consider, at east one of the entries of this vector must be different from zero If we normaize the first entry of α = β to be one, then we obtain the more restrictive rank condition that a 2 0 he normaization rues out the case that E(Ψ t ) is of the form 0 a 2 and α = α 0 Our notion of identification incudes this possibiity o understand better impementation, in the remainder of this section we consider as exampes three specific situations: singe equation IV, mutipe equations with cross-equation restrictions, and sequentia moment conditions 3 Singe equation IV Exampe 32 Suppose the target of anaysis is a singe equation from a simutaneous system: y t α = u t, where the scaar disturbance term u t is orthogona to a p-dimensiona vector z t of instrumenta variabes: E (z t u t ) = 0 (0) Form: Ψ t = z t y t hen orthogonaity condition (0) is equivaent to α satisfying the moment reation (8) For this exampe we dupicate the moment conditions as in (9), and study the simutaneous overidentification of those 2p moment conditions o proceed with the construction of a test, we have to rue out the possibiity that α [] and α [2] are proportiona One strategy is to restrict α [2] to be orthogona to α [] wo orthogona directions can be parameterized with 2k parameters, k parameters for one direction and k for the orthogona direction However, there is not a unique choice of orthogona directions to represent a two-dimensiona space here is an additiona degree of fexibiity A new direction can be formed by taking inear combinations of the origina two directions and a corresponding orthogona second direction hus the number of required parameters is reduced to 2k 2, and the number of overidentifying restrictions for the I test of underidentification is 2p 2k + 2 In practice, we can impose the normaizing restrictions α [] = α [2] = by using spherica coordinates, forcing α [] α [2] = 0, and setting the first entry of α [2] to zero his works provided that a vectors in the nu space of E(z t y t ) do not have zeros in the first entry Aternativey, we coud restrict the top two rows (α [], α [2] ) to equa an identity matrix of order two his rues out the possibiity of a vector in the nu space that is identicay zero in its first two entries, but this may be of itte concern for some appications 9 When k =, both approaches boi down to setting (α [], α [2] ) = I 2 so that the 2p moment conditions: E z t y t = 0 can be represented without resort to parameter estimation As a resut, the identified set wi be the whoe of R 2 Exampe 3 coud emerge as a specia case of Exampe 32 with p = and k = Notice that our underidentification test in this case tests simutaneousy the restriction that a = 0 and a 2 = 0 More generay, when p 2 our test considers simutaneousy E(z t y,t ) = 0 and E(z t y 2,t ) = 0 he resuting I test is different from the test for the reevance of instruments 8 Sensitivity to the choice of normaization can be avoided in GMM by using the approach of Hiier (990) and Aonso-Borrego and Areano (999) or by using the continuousy-updated estimator of Hansen et a (996) As a consequence, our more genera rank formuation can be expored using such methods 9 Once again, it is desirabe to construct a test statistic of underidentification using a version of the test of overidentifying restrictions that is invariant to normaization

6 M Areano et a / Journa of Econometrics 70 (202) in a mode with a normaization restriction on one variabe to be estimated by, say, two-stage east squares Such a test woud examine ony E z t y 2,t = 0 In contrast, when k >, some parameters must be inferred as part of impementing the I test he estimated parameters can then be used for efficienty estimating the identified inear set by expoiting (6) o iustrate this point, consider a normaized reationship between three endogenous variabes with instrument vector z t : E z t y0,t β y,t β 2 y 2,t = 0 Now z t need not be uncorreated with a three endogenous variabes for there to be underidentification Lack of correation with two inear combinations of them is enough 0 For exampe, we may write the nu of underidentification as z t y0,t γ y 2,t H 0 : E z t y,t γ 2 y 2,t If H 0 hods, for any β = 0 E z t y0,t β y,t γ γ 2 β y2,t = 0, so that the observationay equivaent vaues β, β 2 are contained in the ine β 2 = γ γ 2 β A time series exampe is a forward-ooking Phiips curve as in Gaí et a (200), where the components of y denote current infation, future infation, and a measure of aggregate demand, whereas the components of z consist of ags of the previous variabes, and of other variabes such as the output gap and wage infation here are theoretica and empirica considerations to suggest that a nu ike H 0 is pausibe in this context For exampe, ack of higher-order dynamics in a new Keynesian macro mode has been shown to be a source of underidentification of a hybrid Phiips curve with agged infation (see Mavroeidis, 2005 and Nason and Smith, 2008) Reatedy, Cochrane (20) aso raises simiar concerns regarding the identification of ayor rues by Carida et a (2000) and others 3 Reated iterature ests of underidentification in a singe structura equation were first considered by Koopmans and Hood (953) and Sargan (958) When the mode is correcty specified and identified, the rank of E(z t y t ) is k Under the additiona assumptions that the error term u t is a conditionay homoskedastic martingae difference, an asymptotic chi-square test statistic of overidentifying restrictions with p k degrees of freedom is given by λ, where α Y Z Z Z Z Yα λ = min, α α Y () Yα and Z Y = t= z ty t, etc hus λ is the smaest characteristic root of Y Z Z Z Z Y in the metric of Y Y (See Anderson and Rubin, 949 and Sargan, 958) his a version of the J test for overidentification, and it does not require that we normaize α Koopmans and Hood (953) and Sargan (958) indicated that when the rank of E(z t y t ) is k instead, if λ 2 is the second smaest characteristic root, (λ + λ 2 ) has an asymptotic chisquare distribution with 2(p k) + 2 degrees of freedom hese authors suggested that this resut coud be used as a test of the hypothesis that the equation is underidentified and that any possibe equation has an iid error term he statistic (λ + λ 2 ) has a straightforward interpretation in terms of our approach Indeed, it can be regarded as a continuousy-updated GMM test of overidentifying restrictions of the augmented mode (9), subect to the additiona restrictions on the error terms mentioned previousy o see this, et A = α [] α [2] and consider the minimizer of α [] Y Z α [2] Y Z (A Y YA Z Z) Z Yα [] Z Yα [2] subect to A Y YA = I 2 he constraint restricts the sampe covariance matrix of the disturbance vector to be an identity matrix It uses the positive definite matrix Y Y to define orthogona directions when dupicating equations, which is convenient for this appication In ight of this normaization, the minimization probem may be written equivaenty as min α [] Y Z Z Z Z Yα [] + α [2] Y Z A Y YA=I 2 Z Z Z Yα [2], (2) and the minimized vaue coincides with λ + λ 2 (Rao, 973, p 63) A comparison of (2) with () makes it cear that the I test wi be numericay at east as arge as the J test, a resut that is a specia case of Coroary B2 in Appendix B his comparison aso shows that the estimate of α obtained from () coincides with the estimate of α [] obtained from (2), so that in this specia case the optima point estimate beongs to the optima inear set estimate More recenty, Cragg and Donad (993) considered singe equation tests of underidentification based on the reduced form For the singe equation mode, the rank of the matrix E (Ψ t ) is the same as that of L = E (Ψ t ) E(z t z t ) = E y t z t E(zt z t ) his is the matrix of coefficients of the reduced form system of popuation regressions of the entries of y t onto z t Suppose the second component of y t is the first component of z t Partition L as: Π Π L = 2 I 0 he nuity of L and hence E (Ψ t ) is the same as the nuity of Π 2 Cragg and Donad (993) construct a minimum chi-square test statistic that enforces the rank restriction in Π 2 heir statistic can aso be reated to our approach As we show in Appendix A, under the assumption that u t is a conditionay homoskedastic martingae difference, the Cragg Donad statistic minimizes α [] Y Z α [2] Y Z (A Y MYA Z Z) Z Yα [] Z Yα [2] subect to A Y MYA = I 2, where M = I Z Z Z Z Moreover, a Cragg Donad statistic that is robust to heteroskedasticity and/or seria correation can be reinterpreted as a continuousy updated GMM criterion of the augmented structura mode using MYA as errors in the weight matrix Since the difference between YA and MYA at the truth is of sma order, using one form of errors or the other is asymptoticay irreevant Whie the Cragg and Donad (993) approach is straightforward to impement in the singe-equation case, it is more difficut to impement in some modes with cross-equation restrictions his difficuty can emerge because we must simutaneousy impose the 0 Phiips (989) and Choi and Phiips (992) study the IV estimator of β and β 2 in the presence of identification faiure Cragg and Donad (993) aso considered an aternative nu of no identifiabiity in an equation with the coefficient of one of the endogenous variabes normaized to unity his is a rank restriction in the submatrix of Π 2 that excudes the row corresponding to the normaized entry

7 262 M Areano et a / Journa of Econometrics 70 (202) restrictions on the reduced form together with the rank deficiency In Exampe 32, this is easy to do, and it is aso feasibe in the appications to inear observabe factor pricing modes of asset returns carried out by Cragg and Donad (997) and Burnside (2007), but not in more genera modes as we wi iustrate in Sections 32 and Underidentification of a higher dimension Athough the nu Hypothesis 3 is the natura eading case in testing for underidentification, it is straightforward to extend the previous discussion to situations in which the underidentified set is of a higher dimension Suppose that the rank of E(Ψ t ) is k for some hen we can write a the admissibe equations as inear combinations of the ( + )p orthogonaity conditions E(Ψ t ) α [], α [2],, α [+] = 0 (3) If we impose ( + ) 2 normaizing restrictions on (α [], α [2],, α [+] ) to avoid indeterminacy, 2 the effective number of parameters is ( + )(k + ) ( + ) 2 = ( + )(k ) and the number of moment conditions is ( + )p under the assumption that there are no redundancies herefore, by testing the ( + )(p k + ) overidentifying restrictions in (3) we test the nu that α is underidentified of dimension against the aternative of underidentification of dimension ess than or identification Henceforth, we sha refer to those tests as I tests 32 Mutipe equations with cross-equation inear restrictions We next consider exampes with mutipe equations with common parameters 3 Exampe 33 Consider the foowing two-equation mode with cross-equation restrictions: α y,t = u y,t, 3,t α y2,t = u y 2,t, 3,t where y,t, y 2,t are scaars Let z t denote a p -dimensiona vector of common instrumenta variabes appropriate for both equations, so that E z t u,t = 0, E z t u 2,t = 0 Form: zt y Ψ t =,t z t y 3,t, z t y 2,t z t y 3,t so that p = 2p We transform this equation system to obtain an equivaent one by forming: Ψ zt (y t =,t y 2,t ) 0 (4) z t y,t z t y 3,t impying that E z t (y,t y 2,t ) = 0 (5) 2 For instance, we may make the top + rows of A [+] = (α [], α [2],, α [+] ) equa to the identity matrix of order + More generay, we can impose the (+) 2 normaizing restrictions A [+] A [+] = I (+) and a i = 0 for > i, where a i denotes the (i, )-th eement of A [+] 3 Interestingy, Kim and Ogaki (2009) suggest to use modes with cross-equation restrictions to try to break away from the potentia identifiabiity probems that affect singe-equation IV estimates In this exampe, dupicating (5) woud induce a degeneracy because Eq (5) does not depend on parameters Instead these p moment conditions shoud be incuded ust once he I test is impemented by again parameterizing a two-dimensiona subspace with 2k 2 free parameters here are 3p < 2p composite moment conditions to be used in estimating these free parameters hus the degrees of freedom of the I test are 3p 2k+ 2 his I test incudes (5) among the moment conditions to be tested even though these conditions do not depend on the unknown parameters If these moment conditions were excuded, then it woud matter if the second row bock of Ψ t in (4) is repaced by z t y 2,t z t y 3,t By incuding (5) among the moment conditions to be tested this change is inconsequentia An extended version of this exampe arises in og-inear modes of asset returns such as those studied by Hansen and Singeton (983) and others Such modes have a scaar y 3,t given by consumption growth expressed in ogarithms he variabes y,t and y 2,t are the ogarithms of gross returns In addition there are separate constant terms in each equation that capture subective discounting and ognorma adustments By differencing the equations we obtain a counterpart to (5) except that a constant term needs to be incuded Dupication continues to induce a degeneracy because this constant term is triviay identified Exampe 34 Consider a normaized four-input transog cost share equation system After imposing homogeneity of degree in prices and dropping one equation to take care of the adding-up condition in cost shares we have y,t = β, p,t + β,2 p 2,t + β,3 p 3,t + v,t ( =, 2, 3), where y,t denotes the cost share of input, and p,t is the og price of input reative to the omitted input 4 he underying cost function impies the foowing three cross-equation symmetry constraints β,k = β k, k Moreover, prices are endogenous (possiby due to data aggregation) and a p-dimensiona vector of instruments z t is avaiabe, so that: E(z t v,t ) = 0 ( =, 2, 3) (6) In the absence of the symmetry restrictions, the order condition is satisfied if p 3 It woud appear that the parameters may be ust identified with p = 2 when the symmetry restrictions are taken into account, for in that case the order condition is satisfied However, it turns out that such a system has reduced rank 5 by construction o test for underidentification, we dupicate the origina moment conditions, introduce suitabe normaizations, and drop redundant moments, obtaining E[z t (y,t γ,2 p 2,t γ,3 p 3,t )] = 0, ( =, 2, 3) (7) E[z t (p,t γ 0,2 p 2,t γ 0,3 p 3,t )] = 0 (8) Since there are 4p orthogonaity conditions and 8 parameters, with p = 2 the augmented set of moments does not introduce any overidentifying restrictions For arbitrary p, (7) (8) impy that (6) is satisfied for any β,, and for β,,2 β,3 ( =, 2, 3) such that β,2 = γ,2 β, γ 0,2 β,3 = γ,3 β, γ 0,3 (9) 4 See Berndt (99, p 472) For simpicity we abstract from intercepts and og output terms since they have no effect on our discussion

8 M Areano et a / Journa of Econometrics 70 (202) hus, if we do not impose symmetry, the identified set wi be of dimension three (β,, β, 2, β 3, ) and wi be characterized by the eight γ parameters in (7) (8) However, one restriction must be imposed on those parameters for the augmented mode to characterize observationay equivaent vaues of the origina β parameters satisfying the symmetry constraints o see this, note that, subect to the cross-restrictions, (7) (8) impy that (6) are satisfied as before for any β, (and for β,2 and β,3 as in (9)), but ony for β = 2, β,2 so that β 2, = γ,2 β, γ 0,2, and for β 2,2 and β 2,3 such that β = γ 2,2 2,2 (γ,2 β γ, 0,2)γ 0,2, β 2,3 = γ 2,3 (γ,2 β, γ 0,2)γ 0,3 Equay, they are satisfied ony for β = 3, β,3 so that β 3, = γ,3 β, γ 0,3, and for β 3,2 and β 3,3 such that β 3,2 = γ 3,2 (γ,3 β, γ 0,3)γ 0,2 β 3,3 = γ 3,3 (γ,3 β, γ 0,3)γ 0,3 Moreover, the restriction β = 3,2 β 2,3 impies that the admissibe vaues of the coefficients in the augmented mode must satisfy for any β, : γ 3,2 (γ,3 β, γ 0,3)γ 0,2 = γ 2,3 (γ,2 β, γ 0,2)γ 0,3, or γ 3,2 γ 2,3 = γ,3 γ 0,2 γ,2 γ 0,3 (20) hus, after enforcing symmetry the identified set is of dimension one (β, ) and depends on seven parameters ony he I test for this probem is a test of overidentifying restrictions based on the moments (7) (8) subect to (20) Enforcing (20) reduces the set of observationay equivaent parameters under the nu, but this is the right way to proceed since the existence of other β s that satisfy the instrumenta-variabe conditions but not the symmetry conditions shoud not be taken as evidence of underidentification of the mode 5 33 Sequentia moment conditions Consider next an exampe with an expicit time series structure he expectations are taken by averaging across individuas (over i) Exampe 35 Suppose that y i,t+2 = v i,t+2 v i,t+ v i,t for a scaar process {v i,t : t =, 2, } hus k = + 2 Form: α y i,t+2 = u i,t+2, where E z i,t u i,t+2 = 0 for t =, and α 0 hus E z i,t y i,t+2 α = 0 (2) he dimension of the vector z i,t varies with t his dependence is reevant in a pane data setting in which the number of time periods is sma reative to the number of individuas 6 Assume that there is no redundancy among the entries of z i,t hat is, E z i,t z i,t is nonsinguar Moreover, assume that the entries of zi,t are among the entries of z i,t 5 Note that when p = 2, the mode s parameters are not identified, but it is sti possibe to test the restriction (20) as a specification test of the mode 6 In a pure time series setting, there is ony one i, say i = but is arge For this mode to be underidentified, we must be abe to find an α α, both distinct from zero, such that α aso satisfies equation system (2) Since α and α are distinct and inear combinations of α and α must satisfy (2), it foows that E z i,t y i,t+ γ = 0 (22) for t =, 2,, where y i,t+ = v i,t+ v i,t v i,t and γ is not degenerate and has k entries Conversey, suppose that moment conditions (22) are in fact satisfied Notice that E z i,t y i,t+2 γ = 0 because E z i,t+ y i,t+2 γ = 0, where this atter equation is ust (22) shifted one time period forward As a consequence, both α = γ 0, α = 0 γ necessariy satisfy (2) hus the I test for underidentification naturay eads us to test an aternative set of moment conditions with one ess free parameter given by (22) Identification of the parameter vector α from (2) up to scae requires that we reect moment (22) up to scae In a pane data setting, the I test is buit from the moment conditions (22) for t =, 2,, and arge N his construction of the I does not simpy dupicate moment conditions, as this woud ead to a degeneracy or repetition of moment conditions Instead, the time series structure naturay eads to an aternative equation system to be studied Aso we coud construct a coection of reduced form equations by proecting y i,t+2 onto z i,t for each i and expore the restrictions imposed on coefficients he reducedform coefficients woud necessariy be time dependent, and they woud incude some impicit redundancies For this exampe, it is particuary convenient to work directy with the origina structura equation system A concrete exampe of this estimation comes from Areano and Bond (99) hey consider the estimation of a scaar autoregression with a fixed effect In this exampe there is an underying process {v i,t : t = 0,, } Form the scaar v i,t = v i,t v i,t and construct z i,t to incude v i,0, v i,,, v i,t By taking first differences the fixed effect is eiminated from the estimation equation When there is a unit root, this differencing reduces the order of the autoregression, but in genera the order is not reduced he I test checks whether in fact the order can be reduced We iustrate this using an AR(2) mode for pane data with an individua specific intercept η i : α (v i,t+2 η i ) = α 2 (v i,t+ η i ) α 3 (v i,t η i ) + u i,t+2 and E u i,t v i,,, v i,t ; η i = 0 (t = 3,, ), (23) aking the first differences of Eq (23) eiminates the fixed effect Foowing Areano and Bond (99), consider GMM estimation of α and α 2 based on a random sampe {v i,,, v i, : i =,, N} and the unconditiona moment restriction: E[z i,t (α v i,t+2 + α 2 v i,t+ + α 3 v i,t )] = 0 (t =,, 2) hus, we have a system of 3 equations with a set of admissibe instruments that increases with, but a common parameter

9 264 M Areano et a / Journa of Econometrics 70 (202) vector α With = 3 there is a singe equation in first differences with two instruments so that α is at best ust identified up to scae We may pin down the scae by etting the residua variance be zero or we coud normaize the first coefficient to be unity, in which case the remaining coefficients are the negatives of the famiiar autoregressive coefficients Returning to our origina specification (23), suppose that α + α 2 + α 3 = 0 hen α (v i,t+2 η i ) = α 2 (v i,t+ η i ) α 3 (v i,t η i ) + u i,t+2 (t = 3,, ) Under this parameter restriction the fixed effect is inconsequentia and can be dropped Imposing this zero restriction aows us to rewrite the equation as: α v i,t+2 = (α 2 + α )v i,t+ + v i,t+2 his first-order AR specification in first-differences is impicity the specification that is used in buiding the I test If this specification satisfies the orthogonaity restrictions, then the parameters of the origina mode cannot be identified using the approach of Areano and Bond (99) he hypothesis that underies the I test is thus equivaent to an AR(2) specification with a unit root Up unti now we have considered ony modes that are inear in the variabes We extend this discussion to incude modes with noninearities In this discussion, it is important to distinguish two cases In the first case there is a separation between variabes and parameters, and hence the noninearity is confined to the parameters In the second case, the noninearities between variabes and parameters interact in a more essentia way 4 Noninearity in the parameters We first extend our previous anaysis by repacing the parameter vector α by a noninear, continuousy differentiabe function φ : P R k+ where P is the cosure of an open set in R We study the noninear equation: Assumption 4 E (Ψ t ) φ(β) = 0 (24) for some β P he identification question is ony of interest when φ is a oneto-one (ie inective) function If there are two distinct parameter vaues β and β for which φ(β) = φ(β ) then we know a priori that we cannot te β apart from β on the basis of Assumption 4 We make the stronger restriction Assumption 42 For any two vaues of the parameter vector β β in P, φ(β) cφ(β ) for some rea number c We know that we can ony identify φ(β) up to a proportionaity factor In Assumption 42 we ask the noninear parameterization to eiminate scae mutipes from consideration We find it fruitfu to think of the function φ as imposing restrictions on a parameter vector α through the mapping φ(β) = α By thinking of α as the parameter to be estimated, we can use aspects of the approach described previousy Since φ is oneto-one, we can uncover a unique β for each α his eads us to construct the parameter space: Q = {α : α = φ(β) for some β P} Suppose now that two vaues β [] and β [2] satisfy Assumption 4 and are distinct hus both φ(β [] ) and φ(β [2] ) are in the nu space of the matrix E(Ψ t ) By Assumption 42, the vectors φ(β [] ) and φ(β [2] ) are not proportiona Any two inear combinations of φ(β [] ) and φ(β [2] ) must aso be in the nu space of E(Ψ i ) o study underidentification using our previous approach, we expand the parameter space as foows: Q = {α : α = c α + c 2 α 2, α Q, α 2 Q, c, c 2 R} (25) Notice that if E(Ψ t )α = 0 for two vaues of α in Q, then there is a set of soutions to this equation in Q his probem is not a specia case of our earier anaysis because Q may not be a inear space o iustrate how noninearity in parameters can ater the anaysis, we use an exampe that is cosey reated to the non-inear IV mode with seriay correated errors considered by Sargan (959) Nevertheess, it differs in an important way because in our case the vaid instrumenta variabes are predetermined but not necessariy stricty exogenous 7 Exampe 4 Consider a time series exampe: x t β = u t + γ w t, u t = β 2 u t + γ 2 w t, (26) where {w t } is a mutivariate martingae difference sequence Suppose aso that z t is a inear function of w t, w t 2, he process {u t } is unobservabe to the econometrician, but x t β β 2 (x t β ) = (γ + γ 2 ) w t β 2 γ w t Let Ψ t = z t 2 x t z t 2 x t, and consider identification of β based on: E(Ψ t )φ(β) = 0, where φ(β) = β β 2 β (27) o achieve identification requires that we impose an additiona normaization, say β = We may wish to restrict β 2 < Since we have not restricted γ w 2 t to be uncorreated with u t, the unobserved (to the econometrician) process {u t } can be stationary and sti satisfy Eq (26) hus when β 2 >, u t = (β 2 ) γ w 2 t+ = is a stationary process that satisfies (26) Notice, however, in this case u t + γ w t is orthogona to z t so there is an additiona moment restriction at our disposa As is we known the case of β 2 = requires specia treatment Consider two parameter choices (β, β 2 ) and (β, β 2 ) Without oss of generaity write β = cβ + dη (28) where c = β β, η = and η β, and impose that c 2 + d 2 = to guarantee that β = too In ine with the inear case assume that rank[e(ψ t )] = k so that it has a two-dimensiona nu space his means that if there are other observationay equivaent structures, they must satisfy cβ + dη E(Ψ t ) cβ β 2 + dβ η = 0 (29) 2 Given the party inear and party non-inear structure of the mode, underidentification emerges in three ways that we now consider 7 In his Presidentia address to the Econometric Society Sargan (983a) studied a static mode with the same mathematica structure, whie Sargan (983b) anayzed a dynamic mutivariate version

10 4 Ony β identified: here is one specia way in which identification can break down Suppose that E z t 2 x t β = 0, and hence E z t x t β = 0 (30) for some β his phenomenon can occur for one of two reasons First perhaps the choice z t 2 is unfortunate Aternativey, x t β may depend ony on current and possiby future vaues of the martingae difference sequence {w t } As we have seen, this may happen when β 2 > or in the degenerate case when u t is identicay zero (γ = 0) 8 For this same β, it is aso required that E z t 2 x t β = 0 ypicay, there wi be common entries in z t and z t 2 Let z t be a random vector formed after eiminating these redundancies in order that E z t z t is nonsinguar hen the I test for β 2 is based on: E z t x t β = 0 In other words, if the composite disturbance term u t + γ w t is orthogona to z t, then β 2 is not identified via the moment conditions his I test is impemented by estimating the econometric reationship without quasi-differencing, and then testing the resuting overidentifying restrictions Of course, if the nu hypothesis underying the I test is accepted, there are other moment conditions that coud be used to identify β 2 given β Notice in this case there is a continuum of vaues of the composite parameter vector β that satisfy the moment conditions under the nu hypothesis of the I test, but ony a singe vaue of β, which our procedure wi estimate efficienty hus the function π associated with this case is: β π(θ) = θ his test is cosey reated but not identica to the underidentification test proposed by Sargan (959) for the non-inear in parameters mode that he studied he augmented set of moment conditions that he considered were (30) and β E(Ψ t ) β = 0, 2 β where he impicity chose β 2 so that the sampe covariance matrix of x t β and (x t β 2 x t )β were 0 Apart from our emphasis on symmetric normaization and robustness to seria correation and heteroskedasticity, the main difference with his approach is that we impose the restriction β = β, which, in parae with a gain in estimation efficiency, eads to a reduction in the number of degrees of freedom and the resuting gain in power, and aso eiminates the need to choose two arbitrary vaues for β 2 As we mentioned previousy, we coud aow for the vaue of β 2 to have an absoute vaue greater than one In this case identification of β 2 wi fai uness we repace z t 2 by z t 42 Ony β 2 is identified: As another aternative suppose there is a vector β β such that 8 In the case in which β2 > we may identify β 2 from other moment conditions M Areano et a / Journa of Econometrics 70 (202) α = β β 2 β satisfies the moment conditions: E (Ψ t ) α = 0 Since any inear combination of α and α must satisfy moment conditions, we can choose c = 0 in (28) so that η β 2 η shoud aso satisfy the moment conditions (29) his gives rise to a second I test We parameterize two orthonorma directions η and β aong with a singe parameter β 2 When β has ony two components, we are free to set β and η equa to the two coordinate vectors and freey estimate ony the parameter β 2 In that case the moment conditions of the I test can be expressed as E[z t 2 (x i,t β 2 x i,t )] = 0, i =, 2 More generay, under the nu hypothesis associated with this I test there is a two-dimensiona pane of (non-normaized) vaues of the origina parameter vector β that satisfy the moment conditions, but ony one vaue of β 2 After normaization, the manifod of observationay equivaent structures wi be given by (28), and hence we may represent the observationay equivaent β s via: π(θ) = θβ + θ 2 η β 2 for θ where we introduce additiona restrictions that permit us to identify β and η used to represent π Note that if E[z t 2 (x i,t β 2 x i,t )] = 0 for some i, then a the β coefficients wi be identified except the one corresponding to x i,t Importanty, this test is different from a inear test of rank[e(ψ t )] = k derived aong the ines of Section 3, since such a test woud not impose that the observationay equivaent structures must satisfy (27) Once again, as a by-product of our procedure we wi obtain efficient GMM estimators of β 2, and the parameters β and η that characterize the identified set through (28) 43 Another possibiity In the two previous cases, we constructed functions π with reaized vaues that satisfied the moment conditions Another possibiity is that the rank[e(ψ t )] = k but that there are ony two distinct parameter vaues in P, say β [] and β [2] that satisfy: E(Ψ t )φ(β) = 0 In this case there is sti a two-dimensiona subspace of Q constructed in (25) With an additiona normaization, obtained, say, by restricting the magnitude of the vector β to have a norm equa to one, the curve is reduced to one dimension 9 5 Fundamenta noninearity In this section we expore the underidentification probem when there is a more fundamenta noninearity of the parameters in the moment conditions Reca that in the inear mode discussed 9 In the first-order underidentified case studied by Sargan (983a), there is ony one β 0 that satisfies the moment conditions (24) even though the rank of the matrix {E[Ψ t ] φ(β 0 )/ β } is ess than his case can be regarded as the imit of the isoated two-points case in which β [] and β [2] get coser and coser to each other in such a way that the dimension of the nuspace of E(Ψ t ) remains two