Exogeneity tests and weak-identification

Transcription

1 Exogeneity tests and weak-identification Firmin Doko Université de Montréal Jean-Marie Dufour McGill University First version: September 2007 Revised: October 2007 his version: February 2007 Compiled: February 5, 2008, :58am his work was supported by the William Dow Chair in Political Economy (McGill University), the Canada Research Chair Program (Chair in Econometrics, Université de Montréal), the Bank of Canada (Research Fellowship), a Guggenheim Fellowship, a Konrad-Adenauer Fellowship (Alexander-von-Humboldt Foundation, Germany), the Institut de finance mathématique de Montréal (IFM2), the Canadian Network of Centres of Excellence [program on Mathematics of Information echnology and Complex Systems (MIACS)], the Natural Sciences and Engineering Research Council of Canada, the Social Sciences and Humanities Research Council of Canada, the Fonds de recherche sur la société et la culture (Québec), and the Fonds de recherche sur la nature et les technologies (Québec), and a Killam Fellowship (Canada Council for the Arts). Université de Montréal, Département de Sciences Économiques, C.P. 628, succ. Centre-ville, Montréal, QC H3C 3J7. EL: () ext. 5283; Fax: () ; f.doko.tchatoka@umontreal.ca. William Dow Professor of Economics, McGill University, Centre interuniversitaire de recherche en analyse des organisations (CIRANO), and Centre interuniversitaire de recherche en économie quantitative (CIREQ). Mailing address: Department of Economics, McGill University, Leacock Building, Room 59, 855 Sherbrooke Street West, Montréal, Québec H3A 27, Canada. EL: () ; FAX: () ; jean-marie.dufour@mcgill.ca. Web page:

2 ABSRAC In a linear regression model, one often uses Hausman and Wu procedures to access whether some set of explanatory variables are uncorrelated with the disturbances. Instrumental variable methods are typically employed to make reliable these two procedures. However, the Hausman and Wu tests are based on the questionable assumption that the instrumental variables used are strong and valid and this raises the question: what happens to these tests when some of the instruments are weak? In this paper, we focus on structural models and analyze the effects of weak instruments (or weak identification) on the Hausman and Wu exogeneity tests. We show that if the model parameters are not identified or are close to not being identifiable, both procedures are valid but not consistent against the presence of endogenous explanatory variables. In particular, if the model parameters are not identified, the power of the tests converges to their nominal level α. In other words, if the parameters are unidentified or are weakly identified, both Ordinary Least Squares (OLS) and wo Stage Least Squares (2SLS) estimators are biased but the Hausman and Wu procedures are unable to detect the bias. Key words: structural model; exogeneity tests; weak-identification; robustness to weak instruments. i

3 Contents. Introduction 2. Framework 3. Statistics 3 4. Asymptotic theory with weak instruments Simplified model Extension to structural equations with exogenous variables Monte Carlo experiment 7 6. Conclusion 2 A. Appendix: Proofs 2 List of Assumptions, Propositions and heorems 4. heorem : Asymptotic distribution of the statistics under H heorem : Asymptotic distribution of the statistics under the alternative hypothesis heorem : Asymptotic distribution of the statistics under H 0 : extended model heorem : Asymptotic distribution of the statistics under the alternative hypothesis: extended model Proof of heorem Proof of heorem Proof of heorem Proof of heorem ii

4 . Introduction Since Durbin (954) and Wu (973), it has been showed growing interest on the inference about the vector of covariance between the stochastic explanatory variables and the disturbance term of a structural equation. Revankar and Hartley (973), Farebrother (976), Hausman (978), Revankar (978), Kariya and Hodoshima (980), Richard (980), Holly and Sagan (982) proposed tests for the hypothesis of independence between a full vector of stochastic explanatory variables and a disturbance term. Hwang (980) and Smith (984) studied likelihood ratio tests, Hausman and aylor (98), Spencer and Berk (98) and Wu (983) extended the tests previously studied by Wu (973) and Hausman (978), Engle (982) derived the LM tests while Dufour (987) proposed a generalized Wald procedures for the inference on the vector of covariance between the stochastic explanatory variable and the disturbance term in structural models. A drawback with all these procedures is that they assume strong identification of the model parameters, i.e. they assume that the instruments are strong. However, the last decade shows growing interest for weak instruments problems in the econometric literature, i.e. situations where instruments are poorly correlated with endogenous explanatory variables; see the reviews of Dufour (2003) and Stock, Wright and Yogo (2002). More generally, these can be viewed as situations where model parameters are not identified or close to not being identifiable, as meant in the econometric literature [see Dufour and Hsiao (2008)]. his raises the question: what happens to these procedures when some of the instruments are weak? In other words, how robust are these inference procedures to weak-identification? In the econometric literature, little attention is focused on Hausman (978) and Wu (973, 974) tests when the instruments used are weak. In this paper, we focus on structural models and analyze the effects of weak-identification (weak instruments) on these exogeneity tests. After formulating a general asymptotic framework which allows one to study these issues in a convenient way, we show that both test procedures studied are robust to weak-identification (in term of level control), but have no power. hese results means that both test procedures are valid in the presence of weakidentification, but are not able to detect whether or not the explanatory variables are endogenous in the model. When the model is not or weakly identified, both Ordinary Least Squared (OLS) and wo Least Squared (2SLS) estimators of the structural parameters are inconsistent but the Wu and Hausman type procedures are unable to detect this inconsistency. he paper is organized as follows. Section 2 formulates the model considered. Section 3 describes briefly the statistics. Section 4 studies the behavior of the statistics when some instruments are weak. Section 5 studies the finite-samples properties of the statistics with weak-identification. We conclude in section 6. Proofs are presented in the Appendix. 2. Framework We consider the following standard simultaneous equation framework: y = Y β + γ + u, (2.) Y = Π + 2 Π 2 +, (2.2)

5 where y is a vector of observations on the dependent variable, Y is a G matrix of observations on explanatory (possibly) endogenous variables (G ), is a k matrix of observations on the included exogenous variables, 2 is a k 2 (k 2 > G) matrix of observations on the excluded variables (instruments), = [, 2 ] = [ (),..., ( ) ] is a k (k = k + k 2 ) full-column-rank matrix of instruments, u = [u,..., u ] and = [,..., ] are respectively vector and G disturbance matrices with mean zero, β and γ are G and k vectors of unknown coefficients, Π and Π 2, are k G and k 2 G matrices of unknown coefficients. he usual necessary and sufficient condition for identification of this model is rank(π 2 ) = G. We also assume that u t = t a + ε t, t =,...,, (2.3) where ε t has mean zero and variances σ 2 ε and uncorrelated with t, a is a G vector of unknown coefficients. (2.3) can be rewritten in matrix form as: u = a + ε. (2.4) We make the following generic assumptions on the asymptotic behaviour of model variables [where A > 0 for a matrix A means that A is positive definite (p.d.), and refers to limits as ]: [ [ ] [ ] p Σ 0 ] ε ε 0 σ 2 > 0, (2.5) ε [ ε ] p 0, ε L S ε N [0, Σ ε ] (2.6) L S, vec(s ) N [0, Σ Σ ], (2.7) [ p Σ Σ Σ = 2 Σ 2 Σ 2 ] ] [ u ε [ L Su S ε ] > 0, (2.8) N [0, Σ S ], (2.9) S u N [ 0, σ 2 uσ ], Sε N [ 0, σ 2 εσ ], (2.0) where Σ is G G fixed matrix and σ 2 u is the variance of u. From the above assumptions, it is easy to see that: u p 0, [ u ] [ u ] p Σ = [ σ 2 u δ δ Σ ] > 0, (2.) where δ = Σ a, σ 2 u = a Σ a + σ 2 ε, S u = S a + S ε = S (Σ δ) + S ε (2.2) Σ Y = Σ Π, Σ Y = Π Σ Π + Σ. (2.3) 2

6 It is worthwhile to note that since Σ > 0, δ = 0 if and only if a = 0. Hence, testing the exogeneity of Y is equivalent to test whether a = 0 in this model. 3. Statistics We consider in this paper the problem of testing H 0 : δ = 0 (3.) and we analyze the behavior of the?and? statistics. he Wu and Hausman test-statistics for H 0 in equations (2.)-(2.2) are given by H = Q Q/, = Q /G Q /(k 2 G), 2 = Q /G Q 2 /( k 2G), (3.2) where 3 = Q /G Q 3 /( k G), 4 = Q /G Q/( k G), (3.3) Q = (ˆβ β) [(Y (M M)Y ) (Y M Y ) ] (ˆβ β), (3.4) Q = (y Y ˆβ) M (y Y ˆβ), Q = (y Y β) (M M)(y Y β), (3.5) Q 2 = Q Q, Q 3 = (y Y β) M (y Y β), (3.6) β = [Y (M M)Y ] Y (M M)y, ˆβ = [Y M Y ] Y M y, (3.7) M = I ( ), M = I ( ), and = [, 2 ]. (3.8) Note that ˆβ is the ordinary least squares estimator of β whereas β is the instrumental variables method estimator of β. Q/ is the ordinary least squares estimator of σ 2 u. From (2.)-(??), it easy to see that [ ] plim ˆβ = β + Π (Σ Σ Σ Σ )Π + Σ δ, (3.9) where Σ = [Σ, Σ 2 ]. If Π 0, i.e the instruments are strong, we have plim β = β. (3.0) However, if the model parameters are not identified or are close to being identifiable, like the Ordinary Least Squares, the wo Stage Least Squares estimator β of β may be inconsistent. In particular, if Π = 0, we have plim β = plim ˆβ = β + Σ δ, (3.) and when δ 0, both ˆβ and β are inconsistent estimators of β. If the model parameters are identified, with some regular conditions, the asymptotic distribution of H, 3, and 4 is a χ 2 (G) under H 0. Further, if u N[0, σ 2 I ], then F (G, k 2 G) and 2 F (G, k 2G) [see, 3

7 Hausman (978), Wu (973)]. However, if the parameters are not identified or are close to not being identifiable, the distributions of these statistics may be affected. Following Staiger and Stock (997), we refer to locally weak-instrument asymptotic setup by considering a limiting sequence of Π where Π is local-to-zero. In the following section, we study the asymptotic behavior of the statistics when some instruments may be weak. 4. Asymptotic theory with weak instruments We study in this section the large-sample properties of the statistics described above. In the first subsection, we consider the case where drops from the model (no excluded exogenous instruments) and in the second subsection, we extend the model to structural equations with exogenous variables. 4.. Simplified model We consider the following form of (2.)-(2.2) where is dropped from the model: y = Y β + u, (4.) Y = Π +, (4.2) where y and u are vectors, Y and are G matrices, is a k matrix, and β, Π are respectively a G vector and a k G matrix of unknown coefficients. wo setups are considered: () Π = Π 0 where Π 0 is a k G constant matrix, and (2) Π = Π 0 / ( ), where Π 0 is a k G constant matrix. heorem 4. below summarizes the asymptotic behavior of Hausman and Wu statistics for both setups described above. For a random variable W whose distribution depends on the sample size, the notation W L + means that P [W > x] as, for any x. heorem 4. ASYMPOIC DISRIBUION OF HE SAISICS UNDER H 0. Suppose that the assumptions (2.)-(2.0), and (4.)-(4.2) hold, and let δ = 0, then H L χ 2 (G), (4.3) L 2, 3, 4 G χ2 L (k G) (G) and F (G, k G). (4.4) G In the above theorem, no restriction is imposed on the rank of Π. In particular, the result holds even if Π is not a full-column rank matrix. When δ = 0, the limiting distributions of the statistics do not depend on nuisance parameters even if the model parameters are unidentified, i.e the statistics do not show size distortion even if the parameters are not identified or are close to not being identifiable. heorem 4.2 below summarizes the asymptotic behavior of the statistics when δ 0, i.e under the alternative hypothesis. 4

8 heorem 4.2 POHESIS. Suppose that the assumptions (2.)-(2.0), and (4.)-(4.2) hold, and let δ 0. (A) If Π = Π 0 where Π 0 is a k G constant matrix and if rank(π 0 ) = G, then ASYMPOIC DISRIBUION OF HE SAISICS UNDER HE ALERNAIE HY- H, i L +, i =, 2, 3, 4. (4.5) (B) If Π = Π 0 / where Π 0 is a k G constant matrix (Π 0 = 0 is allowed), then H L S δ = σ 2 ε (Π 0 Σ H S L χ 2 (G, µ ) where µ = σ 2 ε δ Σ S ε) Σ Λ (Λ Σ Λ ) Λ Σ (Π 0 Σ δ Σ S ε), δ Σ Π 0Σ Λ (Λ Σ Λ ) Λ Σ Π 0 Σ δ, (4.6) where 2, 3, 4 L G S δ and 2, 3, 4 S G χ2 (G, µ ), (4.7) L (k 2 G) G S δ L (k G) and S Λ δ G F (G, k G; ν, λ ), (4.8) λ = σ 2 ε Λ δ = (S Σ δ + S ε) [Σ Λ (Λ Σ Λ ) Λ ](S Σ δ + S ε), (4.9) δ Σ S [Σ σ 2 ε, Σ, Σ, S, and S ε are defined in (2.)-(2.2). Λ (Λ Σ Λ ) Λ ]S Σ δ, Λ = Π 0 + Σ S. (4.0) he results of heorem 4.2 show clearly that when the model parameters are identified, the Wu and Hausman procedures are consistent against the presence of endogenous explanatory variable in the regression model. However, when the model parameters are not identified or are close to not being identifiable (i.e when the instruments are weak), both procedures are not consistent against the presence of endogenous explanatory variables. hese results mean that when the instruments are weak, both Ordinary Least Squares and wo Stage Least Squares estimators of β are inconsistent but the above tests are unable to detect this inconsistency. It is worthwhile to note that if Π 0 = 0 in (B), we have ν = λ = 0, and H L χ 2 L (G), and i G χ2 (G), i =, 2, 3, 4. In this case, the power function of the tests H, i, i =, 2, 3, 4 converge to the nominal level α. Overall, our results underscore the importance of checking for the presence of weak instruments when applying exogeneity tests Extension to structural equations with exogenous variables In this subsection, we consider the more general model (2.)-(2.2). As in the previous subsection, two setups are examined: () Π = Π 0 where Π 0 is a k G constant matrix, and (2) Π = Π 0 / ( ), where Π 0 is a k G constant matrix, k = k + k 2. heorem 4.3 and heorem 4.4 below derive the asymptotic distributions of the statistics for both setups. 5

9 heorem 4.3 ASYMPOIC DISRIBUION OF HE SAISICS UNDER H 0 : EXENDED MODEL. Suppose that the assumptions (2.)-(2.0), and let δ = 0, then H L χ 2 (G), (4.) L 2, 3, 4 G χ2 L (k 2 G) (G) and F (G, k 2 G). (4.2) G heorem 4.4 POHESIS: EXENDED MODEL. Suppose that the assumptions (2.)-(2.0) hold, and let δ 0. (A) If Π = Π 0 where Π 0 is a k G constant matrix and if rank(π 0 ) = G, then ASYMPOIC DISRIBUION OF HE SAISICS UNDER HE ALERNAIE HY- H, i L +, i =, 2, 3, 4. (4.3) (B) If Π = Π 0 / where Π 0 is a k G constant matrix (Π 0 = 0 is allowed), then where H L S δ = σ 2 [Σ δ (Λ Σ Λ Λ Σ Λ ) (Λ S u Λ S u )] ε [Λ Σ Λ Λ Σ Λ ][Σ δ (Λ Σ Λ Λ Σ Λ ) (Λ S u Λ S u )], (4.4) 2, 3, 4 L G S δ, (4.5) L σ 2 ε (k 2 G) G S δ, (4.6) Λ δ Λ δ = S uσ S u S uσ S u + [Σ δ (Λ Σ Λ Λ Σ Λ ) (Λ S u Λ S u )] [Λ Σ Λ Λ Σ Λ ][Σ δ (Λ Σ Λ Λ Σ Λ ) (Λ S u Λ S u )], (4.7) σ 2 ε = σ 2 u δ Σ δ, Λ = Π 0 + Σ S, Λ = Σ Σ Π 0 + Σ S, (4.8) S = [S, S 2 ], S u = [S u, S 2u ], and S, S u,, Σ, Σ, Σ, Σ, σ 2 ε, σ 2 u are defined in (2.)-(2.2). he results of heorem 4.3 and heorem 4.4 are similar to those in heorem 4. and heorem 4.2 of the previous subsection. Clearly, when the model parameters are not identified or are close to not being identifiable (i.e when the instruments are weak), the Wu and Hausman procedures control the size but are not consistent against the presence of endogenous explanatory variables. 6

10 5. Monte Carlo experiment We study in this section the finite-samples properties of the statistics described above when some of the instruments used are weak. We consider the following data generating process: y = Y β + u, Y = Π +, (5.9) [ ] (u t, t ) i.i.d δ N(0, Σ), where Σ = (5.20) δ Σ and Σ = [σ ij ] i,j=,,g with σ ii = for all i =,..., G. is the k fixe matrix of instruments excluded in the structural equation such that is i.i.d N(0, I ). he k G matrix Π is such that Π = ηπ 0, where η takes the value 0 (design of non identification),.00 (design of weak identification) or (design of strong identification), and Π 0 is obtained from identity matrix by keeping the first k lines and the first G columns. he number of endogenous variable G varies from to 3 whereas the number of instruments k varies from 5 to 20. he parameters values are set at β = 5 for G =, (β = 5, β 2 = 2) for G = 2 and (β = 5, β 2 = 2, β 3 = 3 2 ) for G = 3. he sample size is = 00 and the number of replications is N = 0, 000. he covariance matrix Σ is chosen as: δ {.5,.25, 0,.25,.25,.5} and Σ = for G = ; Σ = δ δ 2 δ δ δ 2 δ 2 δ δ 2 where δ j {.5,.25, 0,.25,.25,.5} (j=,2) for G = 2; and finally Σ = δ δ 2 δ 3 δ δ δ 2 δ δ 3 δ 2 δ δ 2 δ 2 δ 3 δ 3 δ δ 3 δ 2 δ 3, (5.2), (5.22) where δ j {.5,.25, 0,.25,.25,.5} (j=,2,3) for G = 3. If δ i = 0 for all j, then H 0 is satisfied and the variables Y are exogenous in the model. Otherwise, they are endogenous. he nominal level of the tests is 5%. For each value of δ j (j=,2,3), we compute the empirical rejection probability of each statistic. Remark that for a given δ 0, the computed rejection probability is the power of the test at δ. For δ = 0, the empirical rejected frequency is the level of the tests. he results are presented in ABLE -ABLE 3 for the different value of G. In the first column of the tables, we report the values of η (which is the quality of the instruments) whereas in the second columns, we report the values of k (the number of instruments). Finally in the others columns, we report for each value of δ, the rejected frequencies at nominal level of 5 % for the statistics i (i=,2,3,4) and H. he main observation from these results is that excepted the statistic 3, the size of the others statistics are not affected by weak instruments. Which means that H,, 2 and 4 tests are valid even if the model parameters are not identified or close to not being identifiable. However, the power of the tests is dramatically affected by weak instruments (approximatively equal to 5 %). his 7

11 result means that when the model parameters are not identified or close to not being identifiable, the above exogeneity tests are inconsistent against the presence of endogenous explanatory variables in the model. 8

12 ABLE : Percent Rejected at nominal level of 5 % for G= δ =.50 δ =.25 δ = 0 k H H H η = η = η = δ =.25 δ =.25 δ =.50 k H H H η = η = η =

13 ABLE 2: Percent Rejected at nominal level of 5 % for G=2 δ =.50, δ2 =.50 δ =.25, δ2 =.25 δ = 0, δ2 = 0 k H H H η = η = η = δ =.25, δ2 =.25 δ =.25, δ2 =.25 δ =.50, δ2 =.50 k H H H η = η = η =

14 ABLE 3: Percent Rejected at nominal level of 5 % for G=3 δ =.50, δ2 =.50, δ3 =.50 δ =.25, δ2 =.25, δ3 =.25 δ = 0, δ2 = 0, δ3 = 0 k H H H η = η = η = δ =.25, δ2 =.25, δ3 =.25 δ =.25, δ2 =.25, δ3 =.25 δ =.50, δ2 =.50, δ3 =.50 k H H H η = η = η =

15 6. Conclusion In this paper, we have showed that in linear structural models, when the parameters are not identified or are close to not being identifiable, the usual exogeneity tests like Hausman (978) and Wu (973) type tests are valid but not consistent against the presence of endogenous explanatory variables. In particular, if the parameters are unidentified (i.e if the instruments are irrelevant), the power of these tests converges to their nominal level α. In other words, if the instruments are weak, both Ordinary Least Squares and wo Stage Least Squares estimators are biased, but the Hausman (978) and Wu (973) procedures are unable to detect the bias. A. Appendix: Proofs PROOF OF HEOREM 4. Note first that the statistics i, i =, 2, 3, 4; can be written as = (k 2 G) Q/ H, 2 = ( k 2G) G Q G H H, (A.) and 3 = Q/ G Q 3 /( k G) H, 4 = ( k 2G) H, G (A.2) where H is defined by (3.2). Suppose that δ = 0, From (4.)-(4.2), we have M = I and M M = P = ( ). (A) If Π = Π 0 where Π 0 is a k G constant matrix, then by the assumptions (2.)- (2.0), we have and Y P Y p Π 0Σ Π 0, Y Y Q = u u u Y ( Y Y p Π 0Σ Π 0 + Σ, ) Y u p σ 2 u, Y u = Π u 0 + (Σ δ) + ε = Π u 0 + ε, Y P u = where (A.5) holds because δ = 0. So, we have ( Y ) ( (A.3) (A.4) (A.5) ) ( ) u, (A.6) Y u L Π 0 S u + S ε, Y P u L Π 0S u. (A.7) 2

16 If rank(π 0 ) = G, we have Q L [(Π 0 Σ Π 0 + Σ ) (Π 0S u + S ε ) (Π 0Σ Π 0 ) Π 0S u ] [(Π 0Σ Π 0 ) (Π 0Σ Π 0 + Σ ) ] [(Π 0Σ Π 0 + Σ ) (Π 0S u + S ε ) (Π 0Σ Π 0 ) Π 0S u ], (A.8) where Q is defined by (3.4). hus, H L σ 2 [(Π 0Σ Π 0 + Σ ) (Π 0S u + S ε ) (Π 0Σ Π 0 ) Π 0S u ] u [(Π 0Σ Π 0 ) (Π 0Σ Π 0 + Σ ) ] [(Π 0Σ Π 0 + Σ ) (Π 0S u + S ε ) (Π 0Σ Π 0 ) Π 0S u ]. (A.9) Since /delta = 0 if and only if a = 0, from (2.2), we have σ 2 u = σ 2 ε and it is easy to see from assumptions (2.)- (2.0) that [ ] { [ Π 0 S u + S ε Π 0 S N 0, σ 2 Π 0 Σ Π 0 + Σ Π 0 Σ ]} Π 0 u u Π 0 Σ Π 0 Π 0 Σ Π 0 and (Π 0Σ Π 0 + Σ ) (Π 0S u + S ε ) (Π 0Σ Π 0 )Π 0S u N { 0, σ 2 u[(π 0Σ Π 0 ) (Σ + Π 0Σ Π 0 ) ] }. (A.0) hus Q L σ 2 uχ 2 (G) and H L χ 2 (G), 2, 4 L G χ2 (G). (A.) Furthermore, if δ = 0, we have plim ( Q 3 G ) = σ2 u, so Moreover, we can write Q as 3 L G χ2 (G). Q = u P M (P Y )P u = ũ MŶ ũ, (A.2) (A.3) where ũ = P u and Ŷ = P Y. So, we have Q = ũ MŶ ũ = u P u u ( ) Y (Y P Y ) Y ( ) u, (A.4) 3

17 and if rank(π 0 ) = G, from assumptions (2.)- (2.0), Q L S u Σ S u S uπ 0 (Π 0Σ Π 0 ) Π 0S u = S uσ /2 [I k P (P P ) P ]Σ /2 S u, (A.5) where P = Σ /2 Π 0 and the matrix I k P (P P ) P is idempotent with rank k G. Further, we have S u N[0, I k ], hence σ u Σ /2 Q L σ 2 u χ 2 (k G) and L (k G) G χ 2 (G) χ 2 (k G). (A.6) Further, since Q can be written as u A u where A = [Y (Y Y ) P Y (Y P Y ) ][(Y P Y ) (Y Y ) ] [(Y Y ) Y (Y P Y ) Y P ] (A.7) and from (A.3), we have Q = u P M (P Y )P u, it is clear that A (P M (P Y )P ) = 0. So, Q and Q are asymptotically uncorrelated, and because S u and S ε are normal variables, Q and Q are asymptotically independent, i.e the numerator and the denominator of are independent. hus L (k G) F (G, k G). (A.8) G (B) If Π = Π 0 / where Π 0 is a k G constant matrix (Π 0 = 0 is allowed), then if δ = 0, we have Y P Y L Ψ = (Σ Π 0 + S ) Σ (Σ Π 0 + S ), Y Y p Σ, (A.9) Y u p δ = 0, Y P u L (Σ Π 0 + S ) Σ S u, (A.20) Q = u u u Y ( Y Y ) Y u p σ 2 u δ Σ δ = σ2 u. (A.2) Because when δ = 0 S u and S are uncorrelated, we deduce from (2.)- (2.0) that S u is independent of S. Hence, H L σ 2 S uσ u (Σ Π 0 + S )Ψ Furthermore we have (Σ Π 0 + S ) Σ S u S N(0, σ 2 uψ ), so (Σ Π 0 + S ) Σ S u. (A.22) Ψ /2 (Σ Π 0 + S ) Σ σ S u S N(0, I G ), (A.23) u i.e, H S L χ 2 (G). Since the conditional distribution of H is independent of S, we have H L χ 2 (G) and 2, 3, 4 L G χ2 (G). (A.24) 4

18 By following the same steps as the proof in (A) and conditioning on S, we get L (k G) F (G, k G). (A.25) G PROOF OF HEOREM 4.2 Suppose that δ 0. As in the proof of heorem 4., we have from (4.)-(4.2) M = I and M M = P = ( ). (A) If Π = Π 0 where Π 0 is a k G constant matrix, then by the assumptions (2.)- (2.0), we have Y P Y p Π 0Σ Π 0, Y Y p Π 0Σ Π 0 + Σ, Y u p δ 0, (A.26) Y P u = ( Y ) ( he denominator of the Hausman statistic is such that ( Q = u u u Y Y Y ) Y u ) ( ) u and its numerator can be written as [ (Y Q ) Y Y ( u Y ) ] = P Y [ (Y Y P u P Y [ (Y ) Y Y ( u Y ) ] P Y Y P u. If rank(π 0 ) = G, we have [ (Y Y ) Y u ( Y P Y p 0. (A.27) p σ 2 u δ Σ δ, (A.28) ) ] [ (Y Y P u P Y [ (Y ) Y Y u ) ( Y ) ] Y ) ( Y Y ( Y P Y (A.29) ) ] ) ] Y P u p δ (Π 0Σ Π 0 + Σ ) [(Π 0Σ Π 0 ) (Π 0Σ Π 0 + Σ ) ] (Π 0Σ Π 0 + Σ ) δ > 0. (A.30) hus, Q L +, i.e H, i L +, i =, 2, 3, 4. (A.3) 5

19 (B) If Π = Π 0 / where Π 0 is a k G constant matrix (Π 0 = 0 is allowed), then Y Y In addition, p Σ, Y P Y L (Σ Π 0 + S ) Σ (Σ Π 0 + S ), Y u p δ 0 Y P u L (Σ Π 0 + S ) Σ S u. (A.32) plim ( Q ) = plim ( Q 3 ) = σ2 ε = σ 2 u δ Σ δ = σ2 u a Σ a > 0, (A.33) and we have H L S δ = σ 2 ε where Λ = Π 0 + Σ S δ = σ 2 ε [Λ Σ Λ Σ δ Λ S u ] [Λ Σ Λ ] [Λ Σ Λ Σ δ Λ S u )], (A.34) (Π 0 Σ S. Since S u = S a + S ε = S (Σ δ) + S ε, we have δ Σ S ε) Σ Λ (Λ Σ Λ ) Λ Σ (Π 0 Σ δ Σ S ε). (A.35) Moreover, from (2.)- (2.0), S ε N(0, σ 2 εσ ) and independent of S, so, we is easily to see that S δ S L χ 2 (G, µ ), i.e H S, converge to a non central non degenerate χ 2 (G, µ ), and 2 S, 3 S, and 4 S also converge to a G χ2 (G, µ ) distribution, where µ = σ 2 ε δ Σ By proceeding as the proof in heorem 4., we have Π 0Σ Λ (Λ Σ Λ ) Λ Σ Π 0 Σ δ Q = ũ M P Y ũ L Λ δ = (S Σ δ +S ε) Σ /2 M Σ /2 (S Σ δ +S ε) σ 2 εχ 2 (k G, λ ), (A.36) where M = I k P (P P ) P, P = Σ /2 Λ and λ = σ 2 δ Σ S Σ /2 M Σ /2 S Σ δ = ε σ 2 ε δ Σ S [Σ Λ (Λ Σ Λ ) Λ ]S Σ δ. So, we get S L (k G) G F (G, k G; µ, λ ) (A.37) PROOF OF HEOREM 4.3 Suppose that δ = 0. (A) If Π = Π 0, Π 0 is a k G constant matrix and if rank(π 0 ) = G, then the proof is the same as those in heorem 4.. 6

20 (B) If Π = Π 0 /, Π 0 is a k G constant matrix (Π 0 = 0 is allowed), then we have Y (M M)Y L Ξ = (Σ Π 0 +S ) Σ (Σ Π 0 +S ) (Σ Π 0 +S ) Σ (Σ Π 0 +S ), (A.38) where we partition S as S = [S, S 2 ], S is a k G random matrix and S 2 is a k 2 G random matrix. Y P Y p 0, Y M Y = Y Y Y P Y p Σ, Y M u p δ = 0, (A.39) Y (M M)u L (Σ Π 0 + S ) Σ ( Q = u u u M Y Y M Y S u (Σ Π 0 + S ) Σ S u, ) Y M u (A.40) p σ 2 u δ Σ δ = σ2 u, (A.4) where S u is defined as S u = [S u, S 2u ], S u is the first k components of S u and S 2u its last k 2 components. Hence, H L σ 2 (Λ S u Λ S u ) Ξ (Λ S u Λ S u ), (A.42) u where Λ = Π 0 + Σ S and Λ = Σ Σ Π 0 + Σ S. So, it easily see that Ξ = Λ Σ Λ Λ Σ Λ. Since δ = 0, from assumptions (2.)- (2.0), S u is independent of S. We can use this fact to show that Λ S u Λ S u S N(0, σ 2 uξ ), i.e σ u Ξ /2 (Λ S u Λ S u ) S N(0, I G ). Since S u and S u are normally distributed and S u is not dependent on S, we have [ ] (Λ S u Λ S u ) S = [Λ, Λ Su ] S N(0, σ 2 uφ ), (A.43) where Φ = Λ Σ Λ + Λ Σ Λ Λ Σ Λ Λ Σ Λ. We now show that Φ = Ξ. hat is, Λ Σ Λ = Λ Σ Λ. It is easy to see from the expressions of Λ and Λ that Φ = Ξ if and only if Σ Σ S = S. (A.44) Moreover, we have Let partition Σ S u [ Σ = [ Σ Σ Σ Σ 2 ], Σ = ] 2. (A.45) according to the partition of Σ, i.e Σ = [ Σ Σ 2 Σ 2 Σ 22 Σ 2 Σ 2 ], Σ Σ = I k. (A.46) 7

21 From (A.46), we have Σ Σ + Σ 2 Σ 2 = I k, Σ Σ 2 + Σ 2 Σ 22 = Σ 2 Σ 2 + Σ 2 Σ = 0, Σ 2 Σ 22 + Σ 2 Σ 2 = I k2, where I k and I k2 are identities matrices with order respectively k and k 2. Now, we have [ Σ Σ Σ S = [ Σ Σ Σ 2 ] 2 ] [ ] S Σ 2 Σ 22 S 2 = (Σ Σ + Σ 2 Σ 2 )S + (Σ Σ 2 + Σ 2 Σ 22 )S 2. (A.47) (A.48) So, it is clear from (A.47) that hus [Λ, Λ ] [ Su S u distribution of H is not dependent on S, we have Σ Σ S = S, i.e Ξ = Φ. (A.49) ] S N(0, σ 2 L uξ ). Hence, H S χ 2 (G). Since the conditional H L χ 2 (G), 2, 3, 4 L G χ2 (G). By following the same steps as in heorem 4., we get (A.50) and that Q and Q are asymptotically independent, thus Q L σ 2 u χ 2 (k 2 G), (A.5) L (k 2 G) F (G, k 2 G), G (A.52) PROOF OF HEOREM 4.4 Similar to the one in heorem 4.2 References Dufour, J.-M., 987. Linear Wald methods for inference on covariances and weak exogeneity tests in structural equations. In: I. B. MacNeill, G. J. Umphrey, eds, Advances in the Statistical Sciences: Festschrift in Honour of Professor.M. Joshi s 70th Birthday. olume III, ime Series and Econometric Modelling. D. Reidel, Dordrecht, he Netherlands, pp Dufour, J.-M., Identification, weak instruments and statistical inference in econometrics. Canadian Journal of Economics 36(4),

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