[ ] 0. = (2) = a q dimensional vector of observable instrumental variables that are in the information set m constituents of u

Transcription

1 Genealized Mehods of Momens he genealized mehod momens (GMM) appoach of Hansen (98) can be hough of a geneal pocedue fo esing economics and financial models. he GMM is especially appopiae fo models ha yield implicaion of he following fom wih he veco of vaiables x obseved: ( θ ) Eh x, + n = () whee x + n= a k-dimensional veco of vaiables a dae + n, which ae saionay and egodic. θ = an -dimensional paamee veco o be esimaed k l m h : R R R E = he expecaions opeao condiional on infomaion a ime We can use he above model o geneae a family of ohogonaliy condiions. hese ohogonaliy condiions ae hen used o consuc a cieion funcion whose minimize is ou esimae of θ. he cieion funcion is consuced in manne ha guaanees ou paamee esimao is (i) consisen, (ii) asympoically nomal, and (iii) has an asympoic vaiance-covaiance maix ha can be esimaed consisenly.,, + n= + n hen Define u h( x θ ) [ ]. We now consuc a funcion f defined by he following E u + = () n (,, θ ) f x z = u z + n + n (3) whee f : k l q R R R R wih = mq z = a q dimensional veco of obsevable insumenal vaiables ha ae in he infomaion se m consiuens of u + nhave finie second momens q consiuens of z have finie second momens hus, an implicaion of () and (3) is ha E[ f( x +, z, θ )] =. (4) n

2 Applying he law of ieaed expecaions o (4) implies ha he uncondiional momen of f ( ) is zeo: E[ f( x, )]. + nzθ = (5) he above equaion (5) epesens a se of populaion ohogonaliy condiions fom which an esimao of θ can be consuced povided ha l. Le l g ( θ) = E[ f( x+ n, z, θ)] fo θ R and assume g ( θ ) does no depend on. Equaion (5) implies ha g ( θ ) = a θ = θ. hus, if he model undelying condiion () is ue hen g ( θ ), he mehod of momens esimao of g ( θ ), should be close o zeo when evaluaed a θ fo lage values of. he mehod of momens esimao is g θ = f( x, z, θ). (6) ( ) + n = hus, we wish o use g ( θ ) o ge an esimao θ of θ. he poblem is ha g ( θ ) is an ( ) veco and is usually geae han, l, he numbe of unknown paamees. Hence, hee is no way o find values of θ which se he ohogonaliy condiions o zeo, i.e. g ( θ ) =. I is possible, howeve, o se l linea combinaions of he ohogonaliy condiions equal o zeo. Specifically, we can selec θ such ha ou choice of θ ses. A ( ) g θ =. ( ) ( ) In his case, we will have l equaions in l unknowns and can find values of θ which se he above equaion o zeo.

3 Asympoic Resuls Now make he following assumpions and definiions.. Assume A, A whee A has ank l. u+ n u+ n. Define E z. θ hen assume D = z D. = θ 3. Define g ( θ) = g ( θ) g ( θ) g ( θ) ) (.... By applying an exac aylo Seies expansion abou following g ( θ) ( l) g θ we have he θ o ( ), ( θ) = ( θ) + ( θ θ) θ θ θ g g = θ ( θ) ( l ) g ( ) θ ( θ) = ( θ) + θ θ g g ( l) o, in maix fom, ( θ ) = θ + D ( θ )( θ θ ) ( l) g g ( ). (7) Using he equaion Ag ( θ ) =, we muliply boh sides of (7) by A o ge ( ) g ( ) Ag θ = A θ + AD( θ )( θ θ ) A g ( ) AD ( ) ( ) AD ( ) Ag ( ) ( ) θ = θ + θ ( θ θ ) θ θ = [ θ ] ( θ ) θ θ = [ AD θ ] A g ( θ ). (8) Equaion (8) hen implies, by Slusky s heoem, ( θ θ ) [ AD] A {asympoic disibuion of g ( θ ) ( ) [ AD θ ] exiss becauseθ can ( θ ) = D }. be made abiaily close o θ p lim[ D ] by assumpion () above, and p lim[ A ] = A by assumpion () above. hus, [ AD θ ] has ank l and is inveible. ( ) and

4 3 Wha is he asympoic disibuion of g ( θ )? Noe ha g f x z ( θ ) = (, θ, ) + n = = = f( x, θ, z ). By applying a cenal limi heoem o he above, we have he following esul + n = ( ) f( x, θ, z ) ~ N, S, + n + whee ( ) ( ) S = E[ f x, θ, z f x, θ, z ]. j= + n + n j he asympoic disibuion of ( θ θ ) can he be wien as ( θ θ )~ N( θ,[ AD] AS A[ AD] ). Wha is he bes choice of A? o answe his quesion, we need some meic by which o measue his choice. One naual meic is o choose he A which minimizes he asympoic vaiancecovaiance maix of he paamee esimaos. Lemma If S is nonsingula hen fo any A, [ AD] AS A[ AD] ( DS D). Poof: Fis, because S is posiive definie symmeic, you can faco i ino S = cc, whee c is nonsingula. Now define D= [ AD ] Ac [ DS D] Dc. Now noe ha Dc D = [ A D ] Acc D [ D S D ] D c c D = I I =

5 heefoe, [ AD] AS A[ AD] = [ AD] Acc[ A[ AD] = [ D+ [ DS D] Dc ][ D+ [ DS D] Dc ] = DD [ DS D]. posiive heefoe, DS D] is he lowe bound fo all A. Hence, if we choose A = ND S and le N = I, he ideniy maix, he above asympoic covaiance l l l maix can be wien as [ D S D] DS SS D[ DS D] = DS D DS D DS D = [ D S D] l l In class las ime, we discussed esing linea esicions on paamees, i.e. Rθ =. I uns ou ha nonlinea esicions can be esed also. In paicula, i is possible o show ha. Q Q [ Q( θ ) Q( θ) ] ~ N, [ DS D], θ θ whee Q is some funcion (linea o nonlinea) of he veco of paamees θ. Summay: Going back o ou oiginal fomulaion we choose θ o se Ag ( θ ) =, o seing A = DS we have ha, ( θ ) his, howeve, is equivalen o minimizing ( ) DS g =. (9) g θ S g ( θ) wih espec o θ. he way o inepe he GMM is ha fo a given se of momen condiions he GMM appoach finds he opimal linea combinaion of hese momens in ems of minimizing he vaiance-covaiance maix of he paamee esimaos. es fo Oveidenifying Resicions As menioned above, in many cases, he model is oveidenified (i.e. l > ). his allows a possible es of he model via a es fo oveideifying esicions.

6 5 Fom above, ecall ha we could wie Subsiuing in fo ( θ θ ) ( θ ) = θ + D ( θ )( θ θ ) g g ( ). we ge ha, ( ) g ( θ ) [ I D AD A g ( θ ) ) Now he opimal choice of A is DS, which gives us he mos efficien esimaos wihin he class of all linea combinaions of momen condiions. heefoe, ( ) ( ) θ ( θ ) g [ I D DS D DS ] g. Because S is posiive definie symmeic, we can faco can wie he above as S cc. = heefoe, we ( θ ) [ ( ] ( θ ) c g I c D D S D c c g A B I is possible o show ha A is an idempoen maix wih ank l, whee is he numbe of ohogonaliy condiions and l is he of paamees. B is a sandad nomal veco (noe he decomposiion of S ino cc ). Fom heoem discussed in class las ime, if z is a sandad nomal veco and A is idempoen wih ank m, hen z Az is χ m. heefoe, J g ( θ) S g ( θ) ~ χ l. When we se ou o do he esimaion, ecall ha we se l equaions o zeo fom equaions, i.e. ( θ ) g S g θ l = ( ). θ l Noe ha unde he null, all equaions ae close o zeo and ook l linea combinaions of hese o be zeo. heefoe, hee mus sill be l linealy independen ohe combinaions which ae close o zeo. he oveidenifying esicions saisic J ess whehe hese emaining combinaions saisfy he null hypohesis. One way o hink abou his saisic is in ems of a likelihood aio es. he J saisic epesens he value of he saisic unde he esiced model (i.e. by imposing

7 6 he null hypohesis diecly in esimaion). As an alenaive, howeve, we could ake he l emaining linea combinaions of he momen condiions and se hem equal o some veco of paamee δ. his will esul in a compleely idenified sysem, leading o he sum of squaes equal o zeo. his epesens he value of he saisic unde he unesiced model in which he null is no imposed (sinceδ does no have o equal zeo). he diffeence of hese wo saisics, i.e. he esiced minus he unesiced, is jus J and is analogous o a likelihood aio es. he lages poblem wih his oveidenifying es saisic comes fom he fac ha we have se a paicula linea combinaion of he momen condiions equal o zeo. he alenaive hypohesis, heefoe, is wih egads o he ohe linea combinaions no se equal o zeo. he difficuly is ha his alenaive may no be ineesing. In his conex hen, he es may no be consisen agains all alenaives. (See, fo example, Newey (986); Jounal of Economeics).

8 Examples 7 esing fo a Given Disibuion Suppose we wish o es whehe a seies of obsevaions x, =,... come fom a given disibuion F( X, θ ). Unde he null hypohesis ha his is ue, he momens x should coincide wih hose of F( X, θ ). of Specifically, E x y = =,... R () y ~ F X, θ. whee ( ) Assume ha x ae independenly disibued. Since he pocess x is saionay and egodic hen wih a long seies he sample momens of x should be close o is populaion momens, i.e. x E x,... R. = () = Combining equaions () and () yields x E y,... R. = () = Define f (, ) x θ as an R veco wih ypical elemen g = ( θ) f ( x θ) =,. x E y and le Hee g ( θ ) has ypical elemen given by (). he idea is o find he value of he paamees θ in he veco E[ y] R (,..., E y ) ha saisfy condiion (). If he numbe of paamees in θ (say M is less han R hen () places oveidenifing esicions on he seies x. In his case no values fo he paamees can se all he quesions in () equal o zeo. I is possible, howeve, o find he values of θ ha se M linea combinaions of he R veco g ( θ ) equal o zeo: Ag θ = (3) ( ). Unde he assumpion ha he Rh momen of x exiss, Hansen s (98) esuls deived above imply ha he opimal choice of he M R maix A is f( x, whee D θ = E θ S = E f x, θ f( x, θ). and () ( ) 8 DS

9 he esuling esimaes ˆ θ ae hen asympoically nomally nomal wih mean θ DS D A paiculaly aacive feaue of and vaiance-covaiance maix ( ). his pocedue is ha a es fo oveidenifying esicions is povided. Specifically, he saisic ( ˆ θ) ( ˆ θ) J = g S g is asympoically disibued as χr M unde he null hypohesis. We can ejec he esicions imposed by he disibuion if he saisic is highe han he chi squaed value a some easonable level of significance. In pacice, howeve, he vaiancecovaiance maix S is unknown because i depends upon he unobsevable paameesθ. All ha is necessay hough fo he asympoic esuls o be valid is consisen esimae fo which is eadily calculaed. S, he Nomal Disibuion One of he moe common assumpions in he social science lieaue concening a pocess disibuion is ha of nomaliy. Fo example, he eo em in mos linea models is assumed nomal. Addiionally, in finance he assumpion of nomaliy has pofound consequences fo asse picing models and ess of hese models. Fuhemoe, he assumpion ha sock euns ae nomally disibued helps fom measues of isk which may be inappopiae wih ejecion of he assumpions. ~, x µ ~ N, σ. Using he momen If x N ( µσ ) hen we can wie ( ) geneaing funcion of a nomal, he momens of x µ ae given by E ( x µ ) ( x µ ) n n n σ! = ( n) n n! ineges n. (4) Defining he sample momen veco of (4) yields ( x µ ) n ( ) n g (, ) n µσ = σ ( n)! ineges n. (5) = x u n n! Using he pocedue oulined above, we can es he nomal fo hee o moe momens in (5). Specifically, ecall ha we wish o choose he value of θ ha ses he linea combinaion given by equaion (3) equal o zeo. ha is, choose θ such ha DS g µσ, =. (6) ( ) Wihou loss of genealiy, conside esing fo nomaliy wih n = in equaion (5): 9

10 g ( µσ, ) ( x ) ( x µ ) ( x ) µ σ = 3 = x µ 4 4 µ ) 3σ Using equaion (4), he vaiance-covaiance maix of he momen condiions S and he deivaive maix D can be deived: S 4 σ 3σ 4 6 σ σ = 3σ 5σ 6 8 σ 96σ 4 6 D =. 6σ 3σ Subsiuing D and S ino equaion (6) yields he opimal GMM esimaos ˆµ and ˆ σ : ˆ µ = x, = σ µ ˆ = ( x ˆ ). = he coesponding asympoic vaiance-covaiance maix of hese esimaos, DS D is given by ( ), σ. 4 σ he GMM esimaos ˆµ and ˆ σ ae acually he maximum likelihood esimaos (MLE) fo nomaliy. In fac, noe ha he opimal weighs DS pick ou only he fis wo momen condiions he inuiion being ha hese momens descibe he enie nomal disibuion. heefoe, iespecive of he value of n chosen in equaion (4), he opimal weighs always choose only he fis wo momens in esimaions. o see his, ecall ha he GMM esimaion chooses he linea combinaion of he momens which minimize he nomal case, i is well known ha he vaiance-covaiance maix of he esimaos; in he nomal case, i is well known ha he Came-Rao lowe bound is achieved by ˆµ and ˆ σ. Hence, he opimal linea combinaion

11 mus be o weigh only fis wo momens independen of he ohe momen condiions included in esimaion. he J oveidenifying saisic akes on an especially ineesing fom. Because he fis wo momen condiions ae se exacly equal o zeo, J ess whehe he highe ode momens ae saisically equal o zeo. he exac fom of he saisic is a quadaic fom of he sample momens g ( ˆ, ˆ µσ ) weighed by he invese of he vaiance-covaiance maix of he momen condiions S, which is a measue of he pecision by which we esimae he momen esicions. As an illusaion, if he seies x followed a sandad nomal disibuion, hen he J saisic would place highe weighs on he low ode momens such as he hid. Fo example, he vaiance of he hid momen condiion is appoximaely 5% ha of he fouh momen and.% ha of he sixh momen. Esimaing he vaiance-covaiance Maix In he above fomulaion, i was agued ha i is easonable o esimae θ by l selecing he θ R ha makes g ( θ ) in ( 6 ) as close o zeo. ha is, we choose θ o minimize J θ g θ W g θ, (7) ( ) ( ) ( ) whee he weighing maix W defines he meic by which he sample ohogonaliy condiions g ( θ) ae made as close as possible o zeo. he poblem wih his minimizaion is ha W depends on θ. In pacice, we use a wo sep pocedue o esimae θ. hee ae examples by which we can pefom a one-sep minimizaion (as in he above example). Usually howeve, we need o obain a consisen esimao of θ. One way o do his is o fis choosew = I. Sage ( θ) I ( θ) min L= g g. θ We need o find he value of ˆ θ such ha ( ˆ) L θ =. θ Once we have he esimae of θ, we can hen esimae he vaiance-covaiance maix S, which poved above o be he opimal choice of W. All ha is equied fo he

12 asympoic opimaliy o hold is ha we obain esimae of S. Using ou esimae of ˆ, θ we can calculae he esimae of S in a saighfowad manne fo a finie numbe of n ems: R j = f x z ˆ f x z ˆ ( ) (,, θ) (,, θ) + n + n j j = j n ˆ = { () + ( ) + ( ) ]} j= W S R R j R j Some commens egading esimaion ofw ae ode. Fis, he infinie sum in W wih a finie se of daa does no necessaily imply ha he esimaion poblem is impacical. In many insances, he economeician need no calculae he finie sum - fo a numbe of poblems (see example below), he sum is deemined by he numbe of populaion auocovaiances, which depends on he ode of he moving aveage disubance em u. Second, hee is no guaanee ha when hese ems ae consisenly esimaed sepaaely he sum of hese ems will be posiive-semi definie. heefoe, unfounaely in finie samples, he vaiance-covaiance maix may ake nonsensical foms. hid, even when hee ae no a finie numbe of ems, he maix can be wien as a specal densiy maix (see Hansen (98) fo a discussion of fequency domain esimaion). In he lae wo cases, ecen wok by Newey and Wes (987; Economeica), Eichebaum, Hansen and Singleon (99;QJE) and Andews (99; Economeica) sugges alenaive esimaion pocedues fo he vaiance-covaiance maix. Essenially, he idea is o ape he weighs on he auocovaiances and unde ceain condiions consisency can be mainained, while managing o ensue posiive definieness. Sage We need o find he value of ( θ) min L= g( θ) S g. θ ˆ ˆ θ such ha ˆ ( ) L θ θ =, o equivalenly, g ( ˆ θ ) θ ( θ ) ˆ ˆ g =. S

13 he asympoic covaiance maix of ˆ θ can be wien, using he deivaions above, as [ DS D] and is esimaed by ˆ _ DS D, whee D h( xn, θ ) = θ = z. Similaly, he es fo oveidenifying esicions is given by ( ) g ( ) ( ) g ˆ ˆ ˆ ˆ θ θ θ ~ χ l. J S

14 3