11 THE GMM ESTIMATION

Transcription

1 Cotets THE GMM ESTIMATION 2. Cosistecy ad Asymptotic Normality Regularity Coditios ad Idetificatio The GMM Iterpretatio of the OLS Estimatio The GMM Iterpretatio of the MLE The GMM Estimatio i the Over-Idetificatio Case The GMM Iterpretatio of the Istrumetal Variable Estimatio The Restricted GMM Estimatio Comparig Restricted ad Urestricted GMM Estimators Hypothesis Testig Wald Test: The Miimum χ 2 Test: The Lagrage Multiplier Test: The GMM Iterpretatio of the Restricted OLS Estimatio

2 Chapter THE GENERALIZED METHOD OF MOMENTS ESTIMATION Give that a radom sample x, x 2,..., x are draw from a populatio which is characterized by the parameter θ whose true value is θ. If we ca idetify a vector of fuctios g(x; θ) of the radom variable x ad the parameter θ such that the true parameter value θ uiquely solves the followig populatio momet coditio E g(x; θ) = 0, (.) while the estimator ˆθ is the uique solutio to the sample momet coditio g(x i ; θ) = 0, (.2) the, uder some regularity coditios, we ca show that ˆθ is cosistet ad asymptotically ormal A ˆθ N (θ, G(θ ) (θ ) G(θ ) ), (.3) where θ deotes the true value of the parameter θ, g(x; θ) G(θ) = E, (.4) ad (θ) = E g(x; θ)g(x; θ). (.5) Ay estimator defied i such a setup is referred to as a Geeralized Method of Momet (GMM) estimator. The approach of first idetifyig some momet coditio ad the derivig the correspodig GMM estimator from its sample couterpart has become a very popular way of geeratig ew estimators i ecoometrics. A Simple Example Give the radom sample x, x 2,..., x draw from a uspecified populatio with a populatio mea µ ad variace σ 2, we have derived the asymptotic properties of the sample mea x as a estimator of µ i such a case by directly applyig law of large umbers ad the cetral limit theorem. We ow show that the asymptotic aalysis of the sample mea ca fit ito the GMM framework. 2

3 CHAPTER. THE GMM ESTIMATION 3 Let s cosider the fuctio g(x i ; µ) x i µ which gives the followig populatio momet coditio: Eg(x i ; µ) = E(x i ) µ = 0. It is obvious that the oly solutio of µ is the true value of the populatio mea µ to which E(x i ) is equal. The sample couterpart of the populatio momet coditio is g(x i ; µ) = x i µ = 0, ad the solutio of µ is othig but the sample mea x. So the sample mea x is actually a GMM estimator of µ. Cosequetly, we ca apply the geeral results for the GMM estimator to establish the cosistecy ad asymptotic ormality of the GMM estimator x. It is also easy to prove that the asymptotic variace of the GMM estimator x is σ 2. Although the GMM argumet here appears tedious, the idea is importat ad has wide applicability.. Cosistecy ad Asymptotic Normality The followig argumet for provig the cosistecy of the GMM estimator helps illustrate the key idea of the GMM approach. We first ote, if the secod momet of g(x; θ) exists, the law of large umbers implies g(x i ; θ) p E g(x; θ). (.6) It meas that the populatio momet coditio (.) ca be approximated by the sample momet coditio (.2). If the estimator ˆθ solves the sample momet coditio (.2) irrespective of the sample size, the its probability limit, say, θ must also solve the probability limit of (.2) which is the populatio momet coditio (.). But by defiitio the true parameter value θ uiquely solves the populatio momet coditio (.), so the probability limit θ must be equal to the true parameter value θ. That is, ˆθ is a cosistet estimator of θ. I other words, if we kow the true parameter value is the solutio to certai populatio momet coditio, the the solutio to its sample couterpart will be a cosistet estimator. The proof of asymptotic ormality is based o Taylor expasio ad the cetral limit theorem. Give that ˆθ coverges i probability to θ ad that g is differetiable with respect to θ, the for sufficietly large the first-order Taylor expasio of (.2) aroud the true value θ gives the followig approximatio or 0 = g(x i ; ˆθ) ( ˆθ θ ) g(x i ; θ ) + g(x i ; θ ) g(x i ; θ ) ( ˆθ θ ) (.7) g(x i ; θ ). (.8)

4 CHAPTER. THE GMM ESTIMATION 4 Provided that the secod momet of g(x i ; θ )/ exists, law of large umbers agai implies ad the cetral limit theorem implies { g(x i ; θ ) g(x i ; θ ) E g(x; θ ) } p G(θ ), (.9) d u N (0, (θ )), (.0) where E g(x; θ ) = 0. Cosequetly, ( ˆθ θ ) which implies (.3). d G(θ ) u N (0, G(θ ) (θ ) G(θ ) ), (.) The Estimator of the Asymptotic Variace-Covariace Matrix Based o law of large umbers ad (.9), it is readily see that the followig statistic is a cosistet estimator of the asymptotic variace-covariace matrix G(θ ) (θ ) G(θ ) of the GMM estimator ˆθ: g(x i ; ˆθ) g(xi ; ˆθ)g(x i ; ˆθ).2 Regularity Coditios ad Idetificatio g(x i ; ˆθ). (.2) I provig cosistecy ad asymptotic ormality of the GMM estimator, we have used law of large umbers ad the cetral limit theorem. Obviously, certai assumptios are required before we ca apply these theorems. The assumptios that esure the validity of the GMM estimatio are called regularity coditios ad they ca be divided ito four categories:. Coditios that esure the differetiability of g(x; θ) with respect to θ. For example, g(x; θ) is usually assumed to be twice cotiuously differetiable with respect to θ. 2. Coditios that restrict the momets of g(x; θ) ad its derivatives with respect to θ. For example, the secod momets of g(x; θ) ad its first derivative are usually assumed to be fiite. 3. Coditios that restrict the rage of the possible values which the parameter θ ca take. For example, θ is ot allowed to have ifiite value ad the true value θ may ot be at the boudary of the permissible rage of θ (if θ is o the boudary of the permissible rage of θ, the covergece to θ caot take place freely from all directios). 4. The solutio to the populatio momet coditio E g(x; θ) = 0 must be uique ad the uique solutio must be the true value θ of the parameter.

5 CHAPTER. THE GMM ESTIMATION 5 The first three categories of regularity coditios are somewhat techical ad are routiely assumed. However, we do eed to make special efforts to check the validity of the last oe i each applicatio. This last coditio is referred to as the idetificatio coditio because it allows us to idetify the true parameter value θ for estimatio. A obvious ecessary coditio for idetificatio is that the row umber, say, m of the vector g(x; θ) is o less tha the row umber, say, k of the parameter vector θ. That is, the umber of idividual populatio momet coditios caot be smaller tha the umber of parameters to be estimated. If m < k, the the populatio momet coditio will have multiple solutios of which all but oe ca be the true value so that the resultig GMM estimator does ot ecessarily coverge to the true parameter value. This is the so-called uder-idetificatio problem. The idetificatio coditio is implicitly assumed i the previous aalysis of the GMM estimatio. I fact, we have made a stroger assumptio that m = k so that the derivative G(x; θ) of g(x; θ) with respect to θ is a square matrix ad ivertible. This is the so-called justidetificatio case. I sectio.4 we will examie the over-idetificatio case with m > k..3 The GMM Iterpretatio of the OLS Estimatio For the liear regressio model y i = x i β + ε i, (.3) let s assume that the sample {y i, x i }, i =,...,, are i.i.d., ad that E(ε i) = 0. I the preset framework, the explaatory variables x i are stochastic ad, followig the argumets i Chapter 0, we have to assume the followig populatio momet coditio: E(x i ε i ) = Ex i (y i x i β) = 0. (.4) The dimesios of x i ad the zero vector o the right had side are both k. So we have i fact k populatio momet coditios which are just eough for us to estimate the k parameters i β, i.e., we have a just-idetificatio case. 2 The correspodig sample momet coditio is which ca be writte as x i (y i x i β) = 0, (.5) X (y Xβ) = 0 or X Xβ = X y. (.6) We treat β ot oly as the otatio for the regressio coefficiets but also as their true values. Such otatioal ambiguity has bee existig throughout the earlier chapters. Better otatios for the true values of the regressio coefficiets may be β. 2 If the x i cotais the costat term, the oe of the momet coditios is E(y i x i β) = 0 or E(y i ) = E(x i ) β.

6 CHAPTER. THE GMM ESTIMATION 6 But this is equivalet to the first-order coditio for the OLS estimatio. Hece, the OLS estimator, which are solved from the above sample momet coditios, ca be cosidered as a GMM estimator. I order to apply the asymptotic theory for the GMM estimatio, we eed to first evaluate 3 ad (β) Ex i ε i ε i x i = E E(ε 2 i x i)x i x i = E(σ 2 x i x i ) = σ 2 E(x i x i ) (.7) xi (y i x i G(β) E β) β = E(x i x i ). (.8) It is importat to ote that oe of the assumptios (Assumptio 5) we made for a multiple liear regressio model is lim X X = lim x i x i = Q, with Q beig a fiite ad p.d. matrix, we ca equate E(x i x i ) to Q. That is, we have (β) = σ 2 Q ad G(β) = Q. Now, followig the geeral asymptotic theory for the GMM estimatio, we the have that the OLS estimator b is cosistet: p b β. (.9) ad (b β) d N (0, σ 2 Q ). (.20).4 The GMM Iterpretatio of the MLE Suppose the sample {x i }, i =,..., are i.i.d. with the desity fuctio f (x θ ), where θ is a ukow k-dimesioal parameter to be estimated, the we have show i Chapter 9 that l f (xi θ ) E = 0, (.2) which ca be viewed as k populatio momet coditios that are just eough for us to estimate the k-dimesioal parameter θ. The correspodig sample couterpart is l f (x i θ) = 0, (.22) ad the solutio, deoted by ˆθ, is certaily the MLE of θ. I other words, the MLE ca be viewed as a GMM estimator. 3 Here, we have further assumed that E(ε 2 i x i) = σ 2 (i.e., ε i is homoscedastic with respect to x i ), which will be true if x i is assumed to be ostochastic.

7 CHAPTER. THE GMM ESTIMATION 7 ad I order to apply the asymptotic theory for the GMM estimatio, let s first defie l f (xi θ) l f (xi θ) (θ) = E (.23) 2 l f (x i θ) G(θ) = E. (.24) It has also bee show i Chapter 9 that (θ ) = G(θ ). Now followig the geeral asymptotic theory for the GMM estimatio, we the have the well-kow results that the MLE ˆθ is cosistet ad 4 ( ˆθ θ ) d N (0, (θ ) ). (.25).5 The GMM Estimatio i the Over-Idetificatio Case If i the populatio momet coditio E g(x; θ ) = 0 (.26) the row umber of g is strictly greater tha the row umber of the parameter vector θ, the it is ot possible to solve its sample couterpart g(x i ; θ) = 0, (.27) because the umber of equatios is greater tha the umber of parameters to be solved. What we could do i such a case is to fid a value of θ that makes the sample momet coditio as close to zero as possible based o the followig quadratic form: mi θ g(x i ; θ) W g(x i ; θ) where W is some positive defiite weightig matrix of costats. (.28) 4 I Chapter 9 we did ot assume the sample to be idetically distributed; i.e., the desity fuctios f i (x i θ ) have the subscript i, idicatig they are all differet. I such a case, the variace-covariace matrix of the asymptotical distributio is the iverse of lim E 2 l f i (x i θ ) It is readily see that such a matrix reduces to (θ ) = G(θ ) i the preset i.i.d. case..

8 CHAPTER. THE GMM ESTIMATION 8 Give the assumptio that G(θ) has full colum rak ad some additioal regularity coditios, ˆθ is cosistet. To see this we ote that the first-order coditio for the miimizatio problem (.28) is g(x i ; θ) W which ca be viewed as the sample couterpart of the momet coditios g(x i ; θ) = 0, (.29) G(θ) W E g(x; θ) = 0. (.30) Give that G(θ) has full colum rak ad that W is osigular, the oly the ture parameter value θ ca satisfy these momet coditios, which i turs implies that the GMM estimator ˆθ is cosistet. We ca further show that ˆθ is asymptotically ormal: 5 ˆθ A N (θ, G(θ ) W G(θ ) G(θ ) W (θ ) W G(θ ) G(θ ) W G(θ ) ). (.3) Obviously, differet weightig matrix W will give differet estimators with differet asymptotic variace-covariace matrices. That is, the efficiecy of the resultig GMM estimators 5 Give that ˆθ coverges i probability to θ ad that g is twice differetiable with respect to θ, the for sufficietly large the Taylor expasio of (.29) aroud the true value θ gives the followig approximatio 0 = or ( ˆθ θ ) g(x i ; ˆθ) W g(x i ; θ ) W { g(x i ; ˆθ) g(x i ; θ ) + g(x i ; θ ) W where S is a k k matrix i which the jth colum is { g(x i ; θ ) W } g(x i ; θ ) + S 2 g(x i ; θ ) W j g(x i ; θ ). } g(x i ; θ ) + S ( ˆθ θ ) g(x i ; θ ) W We ote that (.6) implies that S coverges i probability to zero. Thus, by (.9) ad (.0), we have g(x i ; θ ), ( ˆθ θ ) d G(θ ) W G(θ ) G(θ ) W u N (0, G(θ ) W G(θ ) G(θ ) W (θ ) W G(θ ) G(θ ) W G(θ ) ).

9 CHAPTER. THE GMM ESTIMATION 9 depeds o the weightig matrix W. It ca be show that 6 G(θ ) W G(θ ) G(θ ) W (θ ) W G(θ ) G(θ ) W G(θ ) G(θ ) (θ ) G(θ ) (.32) for ay positive defiite W. This fidig implies that the most efficiet GMM estimator ˆθ ca be obtaied by settig W = for ay cosistet estimator of (θ ) = E g(x; θ )g(x; θ ) ad the solvig the followig miimizatio problem: mi θ g(x i ; θ) g(x i ; θ). (.33) The resulitg GMM estimator is deoted agai as ˆθ which from ow o will represet such a efficiet GMM estimator. It is readily see that ˆθ is cosistet ad asymptotically ormal ˆθ A N (θ, G(θ ) (θ ) G(θ ) ). (.34) The derivatio of the GMM estimator ˆθ with over-idetified momet coditio essetially requires a two-stage procedure because a prelimiary estimator is eeded for calculatig the weightig matrix. A particularly simple choice of is g(x i ; θ)g(x i ; θ) (.35) where θ is a prelimiary estimator of θ which ca be ay cosistet estimator of θ. A commo oe ca be derived by solvig the followig simpler miimizatio problem mi g(x i ; θ) g(x i ; θ). (.36) θ That is, the prelimiary cosistet estimator θ itself is a GMM estimator based o a especially simple weightig matrix W = I. The asymptotic variace-covariace matrix ca be cosistetly estimated by g(x i ; ˆθ) g(x i ; ˆθ)g(x i ; ˆθ) g(x i ; ˆθ) which ca be compared to the oe for the just-idetified case i (.2).. (.37) 6 Let G G(θ ) ad (θ ), the (G WG) G W WG(G WG) (G G) = (G WG) G W WG G WG(G G) G WG (G WG) = (G WG) G W G(G G) G WG(G WG) which ca be expressed i the form of A A with A = (G WG) G W G(G G) G. Sice A A is ecessarily a p.d. matrix, we therefore have (G WG) GW WG(G WG) (G G).

10 CHAPTER. THE GMM ESTIMATION 0.6 The GMM Iterpretatio of the Istrumetal Variable Estimatio For the liear regressio model (.3), suppose the stochastic explaatory variables x i are edogeous; i.e., E(x i ε i ) = Ex i (y i x i β) = 0. (.38) The aalysis i Chapter 0 idicates that the OLS estimatio will ot be cosistet. To estimate the regressio coefficiet β, we eed to employ certai istrumetal variables z i such that Cov(z i, x i ) = O ad Cov(z i, ε i ) = E(z i ε i ) = Ez i (y i x i β) = 0. (.39) Here, let s assume the dimesios m of z i is greater tha or equal to k, the umber of explaatory variables i x i. The coditio (.39) ca ow be viewed as the (over-idetified or just-idetified) momet coditios we eed for coductig the GMM estimatio for the liear regressio model (.3). I order to implemet the GMM estimatio, we eed to first evaluate 7 (θ ) E(z i ε i ε i z i ) = E(ε2 i z iz i ) = E E(εi 2 z i)z i z i = E(σ 2 z i z i ) = σ 2 E(z i z i ). (.40) The GMM estimatio for β is the based o mi β ( z i (y i x i β) σ 2 ) z i z i z i (y i x i β). (.4) which ca be writte as 8 mi β (y Xβ) Z(Z Z) Z (y Xβ), (.42) where Z = z z 2... z. It is readily see that the solutio to this miimizatio problem is ˆβ = X Z(Z Z) Z X X Z(Z Z) Z y, (.43) which is also referred to as the istrumetal variable (IV) estimator of β. 9 7 Here, we have further assumed that E(ε 2 i z i) = σ 2 ; i.e., ε i is homoscedastic with respect to z i. β. 8 We drop the scalar σ 2 from the expressio. Doig so will ot affect the derivatio of the GMM estimator of 9 I Chapter 0 we suggested a two-stage estimatio for usig the istrumetal variables. It is readily see that the resultig two-stage estimator is idetical to (.43).

11 CHAPTER. THE GMM ESTIMATION I order to apply the asymptotic theory for the GMM estimatio, we eed zi (y i x i G(β) E β) = E(z i x i ). (.44) β The geeral asymptotic theory for the GMM estimatio implies that the IV estimator ˆβ is cosistet ad d ( ˆβ β) N (0, σ 2{ } E(x i z i ) E(z i z i ) ). E(zi x i ) (.45) Note that the asymptotic variace-covariace matrix for the IV estimator β ca be estimated by where s 2 is some cosistet estimator of σ 2. s 2 X Z(Z Z) Z X,.7 The Restricted GMM Estimatio Suppose other tha the over-idetified momet coditio, we have aother set of coditios that we believe the true parameter θ should satisfy. Let s also assume such extraeous coditios ca be expressed as a J-vector of fuctios of the parameter θ: h(θ ) = 0. (.46) We ote these coditios do ot ivolve the radom variable x i so that they are fudametally differet from the momet coditio. If these coditios are true, the we certaily wat the GMM estimator to satisfy them. The way to impose these coditios to the GMM estimator is to cosider the restricted miimizatio with the coditio h(θ ) = 0 imposed as a set of restrictios: or mi θ g(x i ; θ) mi θ g(x i ; θ) g(x i ; θ) subject to h(θ) = 0, (.47) g(x i ; θ) + h(θ) λ, (.48) where λ is a J-vector of Lagrage multipliers. The solutio to such a problem, deoted as ˆθ, is called the restricted GMM estimator as opposed to the urestricted GMM estimator ˆθ. Whe we derive the GMM estimator based o the over-idetified momet coditio E g(x; θ ) = 0, the momet coditio is ever exactly satisfied by either the restricted or the urestricted GMM estimator. But it should be poited out the restrictio h(θ ) = 0, i cotrast, is exactly satisfied by the restricted GMM estimator. So the momet coditio ad restrictio are ot treated symmetrically although both are coditios o the parameter value.

12 CHAPTER. THE GMM ESTIMATION 2 It ca be proved that, just like the urestricted GMM estimator, the restricted GMM estimator is cosistet ad has a asymptotic ormal distributio. However, while both estimators are cosistet, their asymptotic ormal distributios are ot the same. I particular, the asymptotic variace-covariace matrix of the restricted GMM estimator is always smaller tha or equal to that of the urestricted GMM estimator. This result simply reflects the fact that the restricted GMM estimator, by icorporatig more iformatio from the restrictio h(θ ) = 0, is more (asymptotically) efficiet. The preset discussio is very similar to the oe we had o the the relatioship betwee the urestricted MLE ad the restricted MLE. As a matter of fact, the derivatio of the asymptotic distributio of the restricted GMM estimator is parallel to that of the restricted MLE..7. Comparig Restricted ad Urestricted GMM Estimators To simplify our expositio here, let s deote (half of) the objective fuctio for miimizatio i defiig the GMM estimator with over-idetified momet coditio by q(θ) 2 g(x i ; θ) ad the first order derivative of q(θ) by s(θ) g(x i ; θ) g(x i ; θ). (.49) g(x i ; θ). (.50) Because of the differece betwee the restricted GMM estimator ˆθ ad the urestricted GMM estimator ˆθ, we observe the followig iequalities h( ˆθ ) = 0 = h( ˆθ), q( ˆθ ) q( ˆθ), s( ˆθ ) = 0 = s( ˆθ). (.5) The secod iequality is due to the fact that the restrictio h(θ) = 0 restricts the possible values of θ for miimizatio. The third iequality results from the fact that the first order coditio for the restricted miimizatio is s( ˆθ ) + H( ˆθ ) ˆλ = 0, where H(θ) = h(θ). (.52) while the first order coditio for the urestricted miimizatio is s( ˆθ) = 0. (.53) These three sets of iequalities i h, q, ad s hold for ay radom sample of a fiite sample size. Let s first deote the probability limit of the restricted GMM estimator ˆθ by θ, the, like ˆθ for every sample size, θ must also satisfy the restrictio: h(θ ) = 0. It is obvious

13 CHAPTER. THE GMM ESTIMATION 3 that whether θ is equal to θ, so that the restricted GMM estimator is cosistet, depeds o whether h(θ ) = 0 is correct or ot. The theory for the restricted GMM estimatio metioed i the previous subsectio is based o the implicit assumptio that the restrictio h(θ ) = 0 is correctly specified. We should also ote that the urestricted GMM estimator is always cosistet irrespective of whether the restrictio h(θ ) = 0 is correct or ot. We ca ow coclude that if the restrictio h(θ ) = 0 is correct, the h(θ ) = 0 = h(θ ), q(θ ) = q(θ ), s(θ ) = 0 = s(θ ). (.54) But if the restrictio h(θ ) = 0 is icorrect, the h(θ ) = 0 = h(θ ), q(θ ) > q(θ ), s(θ ) = 0 = s(θ ). (.55) The direct implicatio of the above iequalities is that, depedig o whether h(θ ) is equal to 0, or whether q(θ ) is greater tha q(θ ), or whether s(θ ) is equal to 0, we ca judge whether the restrictio h(θ ) = 0 is correct or ot. Therefore, eve though for a fiite sample size we have h( ˆθ) = 0, q( ˆθ ) q( ˆθ), ad s( ˆθ ) = 0, the differeces are expected to become small as the sample size becomes large if, ad oly if, the restrictio h(θ ) = 0 is correct. This coclusio is importat because it helps us formulate three formal tests for the hypothesis about the truthfuless of the restrictio h(θ ) = 0, as will be explaied ext..8 Hypothesis Testig Give the GMM estimator ˆθ that is based o over-idetified momet coditio, there are three asymptotically equivalet tests for testig H 0 : h(θ ) = 0 agaist H : h(θ ) = 0, where h is a J-vector of fuctios of the parameter θ. To explai the motivatio of the tests, we eed to thik the ull hypothesis h(θ ) = 0 as a set of restrictios o the true parameter value θ..8. Wald Test: Wald test is based o the idea of usig the differece betwee h( ˆθ) ad 0 to decide whether the ull hypothesis is true. To determie whether h( ˆθ) is sigificatly close to 0 or ot, we eed the followig result which ca be proved easily: A h( ˆθ) N (h(θ ), H(θ ) G(θ ) (θ ) G(θ ) H(θ ) ), where H(θ) = h(θ). (.56) Whe the ull hypothesis is true so that h(θ ) = 0, the we have the followig distributio result for the quadratic form W : { W h ( ˆθ) H( ˆθ) G( ˆθ) ( ˆθ) G( ˆθ) H( ˆθ) } h( ˆθ) A χ 2 (J), (.57)

14 CHAPTER. THE GMM ESTIMATION 4 where J is the umber of restrictios or the umber of rows i the vector h. This result forms the basis for the Wald test. Give the size of the test α ad the correspodig critical value c α from the χ 2 (J) distributio, the ull hypothesis is rejected if h( ˆθ) is sigificatly differet from 0 or, equivaletly, the value of W is greater tha the critical value c α..8.2 The Miimum χ 2 Test: The miimum χ 2 test is based o the idea of usig the differece betwee q( ˆθ ) ad q( ˆθ) to decide whether the ull hypothesis is true. Specifically, we have the followig asymptotic result: if the ull hypothesis h(θ ) = 0 is true, the MC 2 q( ˆθ ) q( ˆθ) A χ 2 (J). (.58) Hece, the ull hypothesis is rejected if the value of MC is greater tha the critical value c α..8.3 The Lagrage Multiplier Test: Lagrage multiplier test is based o the idea of usig the differece betwee s( ˆθ ) ad 0 to decide whether the ull hypothesis is true. It ca be show that the quadratic form L M s( ˆθ ) G( ˆθ ) ( ˆθ ) G( ˆθ ) s( ˆθ ) A χ 2 (J), (.59) if the ull hypothesis is true. Hece, we reject the ull hypothesis if the value of L M is greater tha the critical value c α. 0 The three test statistics W, MC, ad L M are asymptotically equivalet ad have the same asymptotic distributio χ 2 (J) whe the ull hypothesis is true. But i fiite sample applicatios, these three tests may give coflictig results ad there is o cosesus about how to resolve such coflicts whe they occur. Fially, sice the three tests are asymptotically equivalet, there is o eed to compute all three test statistics all the time. We ote the Wald test statistic W oly requires the urestricted GMM estimator ˆθ, the Lagrage multiplier test statistic L M oly requires the restricted GMM estimator ˆθ, while the miimum χ 2 test statistic MC requires both restricted ad urestricted GMM estimators. 0 The reaso for the ame Lagrage-Multiplier test is because the first-order coditio for the restricted GMM estimator implies s( ˆθ ) = H( ˆθ ) ˆλ so that L M ˆλ H( ˆθ ) G( ˆθ ) ( ˆθ ) G( ˆθ ) H( ˆθ ) ˆλ, which is a test statistic based o the Lagrage multiplier λ.

15 CHAPTER. THE GMM ESTIMATION 5.9 The GMM Iterpretatio of the Restricted OLS Estimatio As metioed earlier i Subsectio.3, the OLS estimatio of the multiple liear regressio model is based o the just-idetified populatio momet coditio (.4) ad its sample couterpart (.5). The immediate cosequece of imposig the liear restrictio Rβ = q (.60) to the GMM estimatio is that the restricted GMM estimator caot be exactly solved from the just-idetified momet coditio aloe sice the liear restrictio also eeds to be satisfied. The restricted GMM estimatio has chaged from the just-idetified case to a over-idetified case. To costruct the objective fuctio for derivig the GMM estimator i the over-idetified case requires the sample couterpart of (β) i (.7) which is s 2 X X, (.6) where s 2 is ay cosistet estimator of σ 2. The objective fuctio for the over-idetified GMM estimatio is a quadratic fuctio i the sample momets with the iverse of the above term as the weightig matrix: q(β) = 2s 2 (y Xβ) X(X X) X (y Xβ), (.62) Recall that the objective fuctio for the OLS estimatio is S(β) = (y Xβ) (y Xβ). It is easy to show 2s 2 q(β) = S(β) y My, (.63) where M = X(X X) X. Because of the equivalece betwee q(β) ad S(β) (both 2s 2 ad y My do ot ivolve β), we coclude that the OLS estimator ad the GMM estimator are idetical. We should uderstad that i the preset framework the OLS estimator is derived as the GMM estimator from the over-idetified momet coditios. The approach is differet from the oe i Subsectio.3 where the OLS estimator is derived as the GMM estimator from the just-idetified momet coditios. It is iterestig to ote that if we plug the OLS estimator b ito the quadratic objective fuctio (.62), we get q(b) = (y My y My)/2s 2 = 0, which is the smallest possible value of that quadratic fuctio. This special result reflects that the objective fuctio (.62) actually is built from just-idetified, istead of over-idetified, momet coditios. We ow tur to the restricted GMM estimator subject to the liear restrictio (.60) which is to be solved from mi q(β) s.t. Rβ = q. (.64) β Because q(β) ad S(β) are equivalet, the first-order coditio for the restricted GMM estimatio is also equivalet to the oe for the restricted OLS estimatio so that, similar to the case of the urestricted estimatio, the restricted GMM estimator is the same as the restricted OLS estimator b.

16 CHAPTER. THE GMM ESTIMATION 6 Three Asymptotic Tests Give that the urestricted ad restricted OLS estimators both have the GMM iterpretatios, they ca be used to costruct the three asymptotically equivalet tests for testig H 0 : Rβ = q agaist H : Rβ = q.. Wald Test: based o the asymptotic result we ca immediately get the Wald test statistic which is Rb A N (q, σ 2 R(X X) R ), (.65) W = (Rb q) R(X X) R (Rb q) s 2 A χ 2 (m), (.66) uder the ull hypothesis H 0, where s 2 is ay cosistet estimator of σ The Miimum χ 2 Test: give the objective fuctio (.62) for the GMM estimatio, the miimum χ 2 test statistic is MC = 2 q(b ) q(b) = 2 q(b ) = (Rb q) R(X X) R (Rb q) s 2 A χ 2 (m), (.67) uder the ull hypothesis H 0, where s 2 is ay cosistet estimator of σ 2. Note that q(b) is idetically equal to The Lagrage Multiplier Test: give the score fuctio the Lagrage multiplier test statistic is s(β) = s 2 X (y Xβ), (.68) L M = (y Xb ) X(X X) X (y Xb ) s 2 = (Rb q) R(X X) R (Rb q) s 2 A χ 2 (m), (.69) uder the ull hypothesis H 0, where s 2 is ay cosistet estimator of σ 2. It is iterestig to see that these three asymptotic tests are idetically equal ad W = MC = L M = m F, (.70) where F is the F test statistic discussed i Chapter 6 ad m is the umber of restrictios. It should also be poited out that, i cotrast to the F test, the three asymptotic tests do ot hige o the ormality assumptio ad they are valid oly whe the sample size is sufficietly large. The Lagrage multiplier test statistic ca also be derived from the fact that the asymptotic distributio of the Lagrage multiplier estimator c which is c = R(X X) R (Rb q) the ull hypothesis H 0. See (6.20) i Chapter 6. A N (0, σ 2 R(X X) R ), uder