Econometrics I. Professor William Greene Stern School of Business Department of Economics 21-1/67. Part 21: Generalized Method of Moments

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1 Ecoometrcs I Professor Wllam Greee Ster School of Busess Departmet of Ecoomcs 21-1/67

2 Ecoometrcs I Part 21 Geeralzed Method of Momets 21-2/67

3 I also have a questos about olear GMM - whch s more or less olear IV techque I suppose. I am rug a pael o-lear regresso (o-lear the parameters) ad I have L parameters ad K exogeous varables wth L>K. I partcular my model looks kd of lke ths: Y = b 1 *X^b 2 + e, ad so I am tryg to estmate the extra b 2 that do't usually appear a regresso. From what I am readg, to ru olear GMM I ca use the K exogeous varables to costruct the orthogoalty codtos but what should I use for the extra, b 2 coeffcets? Just some more possble IVs (lke lags) of the exogeous varables? I agree that by addg more IVs you wll get a more effcet estmato, but s't t oly the case whe you beleve the IVs are truly ucorrelated wth the error term? So by addg more "strumets" you are more or less mposg more ad more restrctve assumptos about the model (whch mght ot actually be true). I am askg because I have ot foud sources comparg olear GMM/IV to olear least squares. If there s o homoscadestcty/seral correlato what s more effcet/gve tghter estmates? 21-3/67

4 I m tryg to mmze a olear program wth the least square uder olear costrats. It s frst troduced by Aé & Gema (2000). It cossted o the mmzato of the sum of squared dfferece betwee the momet geeratg fucto ad the theoretcal momet geeratg fucto (The method was suggested by Quadt ad Ramsey the 1970s.) 21-4/67

5 Method of Momet Geeratg Fuctos For the ormal dstrbuto, the MGF s M(t, )=E[exp(tx)]=exp[t + 2 t ] Momet Equatos: exp( t ) exp[t + 1 jx j 2 t j ], j 1, 2. Choose two values of t ad solve the two momet equatos for ad. Mxture of Normals Problem: f(x,,,, ) N[, ] (1 ) N[, ] Use the method of momet geeratg fuctos wth 5 values of t M(t,,,, )=E[exp(tx)]= exp[t + t ] (1 )exp[t + t ] /67

6 Fdg the solutos to the momet equatos: Least squares 1 ˆM(t 1) exp( tx 1 1 ), ad lkewse for t 2,... Mmze(,,,, ) ˆM(t ) exp[t + ] (1 )exp[t + ] j 1 2t j 1 2 2t 2 2 Alteratve estmator: Maxmum Lkelhood L(,,,, ) log N[x, ] (1 ) N[x, ] /67

7 The Method of Momets Estmatg Parameters of Dstrbutos Usg Momet Equatos Populato Momet k E[x ] f (,,..., ) k k 1 2 K Sample Momet m x. m may also be h (x ), eed ot be powers 1 k 1 k 1 k 1 k Law of Large Numbers plm m f (,,..., ) k k k 1 2 K 'Momet Equato' (k = 1,...,K) m x f (,,..., ) 1 N k k N 1 k 1 2 K Method of Momets appled by vertg the momet equatos. ˆ g (m,...,m ), k = 1,...,K k k 1 K 21-7/67

8 Estmatg a Parameter Mea of Posso p(y)=exp(-λ) λ y / y! E[y]= λ. plm (1/)Σ y = λ. Ths s the estmator Mea of Expoetal p(y) = α exp(- α y) E[y] = 1/ α. plm (1/)Σ y = 1/ α 21-8/67

9 Mea ad Varace of a Normal Dstrbuto 2 1 (y ) p(y) exp Populato Momets E[y], E[y ] Momet Equatos y, y Method of Momets Estmators ˆ=y, ˆ y (y ) (y y) /67

10 Gamma Dstrbuto p(y) E[y] P exp( y)y (P) P P1 2 P(P 1) 3 P(P 1)(P 2) E[y ] E[y ] ad so o 2 3 E[1/ y] P 1 E[logy] (P) log, (P)=dl (P)/dP (Each par gves a dfferet aswer. Is there a 'best' par? Yes, the oes that are 'suffcet' statstcs. E[y] ad E[logy]. For a dfferet course...) 21-10/67

11 The Lear Regresso Model Populato y x 1 1 Populato Expectato E[ x ] 0, k = 1,...,K k Momet Equatos (y x x... x )x K K 1 (y x x... x )x K K 2 1 (y x1 1 x x K K)xK 0 Soluto : Lear system of K equatos K ukows. Least Squares 21-11/67

12 Istrumetal Varables Populato y x 1 1 Populato Expectato E[ z ] 0 for strumetal varables z... z. k 1 K Momet Equatos (y x x... x )z K K 1 (y x x... x )z K K 2 1 (y 1 x1 1 x x KK)zK 0 Soluto : Also a lear system of K equatos K ukows. -1 b = ( Z'X / ) ( Z'y / ) IV A exteso : What s the soluto f there are M > K IVs? 21-12/67

13 Maxmum Lkelhood 1 Log lkelhood fucto, logl = logf(y x,,..., ) Populato Expectatos logl E 0, k = 1,...,K k Sample Momets 1 1 logf(y x,,..., ) 1 K k K Soluto : K olear equatos K ukows. 1 logf(y x, ˆ,..., ˆ ) 1,MLE K,MLE 1 ˆ k,mle /67

14 Behavoral Applcato Lfe Cycle Cosumpto (text, pages ) 1 c t1 E t(1 r) 1t 0 1 ct dscout rate c t cosumpto formato at tme t t Let =1/(1+ ), R c / c, =- t t1 t t+1 t1 t E [ (1 r)r 1 ] 0 What s the formato set? Each pece of 'formato' provdes a momet equato for estmato of the two parameters. (1 r) 1 w 0, k=1,...,k T 1 ct 1 t1 tk 1 c t 21-14/67

15 Idetfcato Ca the parameters be estmated? Not a sample property Assume a fte sample Is there suffcet formato a sample to reveal cosstet estmators of the parameters Ca the momet equatos be solved for the populato parameters? 21-15/67

16 Idetfcato Exactly Idetfed Case: K populato momet equatos K ukow parameters. Our famlar cases, OLS, IV, ML, the MOM estmators Is the coutg rule suffcet? What else s eeded? Overdetfed Case Istrumetal Varables Uderdetfed Case Multcollearty Varace parameter a probt model Shape parameter a loglear model 21-16/67

17 Populato y x,,..., 1 K Populato Expectato 1 Overdetfcato E[ z ] 0 for strumetal varables z... z M > K. k 1 M There are M > K Momet Equatos - more tha ecessary (y x x... x )z K K 1 (y x x... x )z K K 2 1 (y x1 1 x x K K)zM 0 Soluto : A lear system of M equatos K ukows /67

18 Overdetfcato Two Equato Covarace Structures Model Coutry 1: Coutry 2: y y Two Populato Momet Codtos: E[(1/T) X '( y X )] E[(1/T) X '( y X )] X X (1) How do we combe the two sets of equatos? (2) Gve two OLS estmates, b ad b, how do we recocle them? 1 2 Note: There are eve more. E[(1/T) X '( y X )] /67

19 Uderdetfcato Model/Data Cosder the Mover - Stayer Model Bary choce for whether a dvdual 'moves' or 'stays' d 1( z u 0) Outcome equato for the dvdual, codtoal o the state: y (d 0) = y (d 1) x 0 0 = x (, ) ~ N[(0,0),(,, )] A dvdual ether moves or stays, but ot both (or ether). The parameter caot be estmated wth the observed data regardless of the sample sze. It s udetfed /67

20 Uderdetfcato - Normalzato Whe a parameter s udetfed, the log lkelhood s varat to chages t. Cosder the logt bary choce model exp( 0x) exp( 1x) Prob[y=0]= Prob[y=1]= exp( x) exp( x) exp( x) exp( x) Probabltes sum to 1, are mootoc, etc. But, cosder, for ay 0, exp[( 0 )x] exp( x) [exp( 0x)] Prob[y=0]= exp[( )x] exp[( )x] exp( x)[exp( x) exp( x)] exp[( 1 )x] exp( x) [exp( 1x)] Prob[y=1]= exp[( )x] exp [( )x] exp( x)[exp( x) exp( x)] exp( x) always cacels out. The parameters are udetfed. A ormalzato such as 0 s eeded /67

21 Uderdetfcato: Momets Nolear LS vs. MLE y ~ Gamma(P, ), exp( x ) f (y ) P E[y x] exp( y ) y ( P) P P1 We cosder olear least squares ad maxmum lkelhood estmato of the parameters. We use the Germa health care data, where y = come x = 1,age,educ,female,hhkds,marred 21-21/67

22 21-22/67 Nolear Least Squares --> NAMELIST ; x = oe,age,educ,female,hhkds,marred $ --> Calc ; k=col(x) $ --> NLSQ ; Lhs = hhc ; Fc = p / exp(b1'x) ; labels = k_b,p ; start = k_0,1 ; maxt = 20$ Momet matrx has become opostve defte. Swtchg to BFGS algorthm Normal ext: 16 teratos. Status=0. F= User Defed Optmzato... Nolear least squares regresso... LHS=HHNINC Mea = Stadard devato = Number of observs. = Model sze Parameters = 7 Degrees of freedom = Resduals Sum of squares = Stadard error of e = Varable Coeffcet Stadard Error b/st.er. P[ Z >z] B <====== B B *** B *** B *** B *** P <======= Nolear least squares dd ot work. That s the mplcato of the fte stadard errors for B1 (the costat) ad P.

23 Maxmum Lkelhood Gamma (Loglear) Regresso Model Depedet varable HHNINC Log lkelhood fucto Restrcted log lkelhood Ch squared [ 6 d.f.] Sgfcace level McFadde Pseudo R-squared (4 observatos wth come = 0 Estmato based o N = 27322, K = 7 were deleted so logl was computable.) Varable Coeffcet Stadard Error b/st.er. P[ Z >z] Mea of X Parameters codtoal mea fucto Costat *** AGE.00205*** EDUC *** FEMALE HHKIDS.06512*** MARRIED *** Scale parameter for gamma model P_scale *** MLE apparetly worked fe. Why dd oe method (ls) fal ad aother cosstet estmator work wthout dffculty? 21-23/67

24 E[ y x] P / exp( x ) Momet Equatos: NLS 2 y P x e e' e / exp( ) ee ' P ee ' e exp( x ) 0 2eP x 0 exp( x ) Cosder the term for the costat the model,. Notce that the frst order codto for the costat term s 1 2eP exp( x ) 0. Ths does't deped o P, sce we ca dvde both sdes of the equato by P. Ths meas that we caot fd solutos for both 1 ad P. It s easy to see why NLS caot dstgush P from. E[y x] = exp((logp- )...). There are a fte umber 1 1 of pars of (P, ) that produce the same costat term the model /67

25 1 Momet Equatos MLE The log lkelhood fucto ad lkelhood equatos are logl= P log log ( P) y ( P 1) log y log L d log ( P) log ( ) log 0, ( ) 1 P y P P dp log L P ; usg. 1 y x Recall, the expected values of the dervatves of the log lkelhood equal zero. So, a look at the frst equato reveals that the momet equato use for estmatg P s E[logy x ] ( P) log ad aother K momet P equatos, E y x 0 are also use. So, the MLE uses K+1 fuctoally depedet momet equatos for K+1 parameters, whle NLS was oly usg K depedet momet equatos for the same K+1 parameters /67

26 GMM Ageda The Method of Momets. Solvg the momet equatos Exactly detfed cases Overdetfed cases Cosstecy. How do we kow the method of momets s cosstet? Asymptotc covarace matrx. Cosstet vs. Effcet estmato A weghtg matrx The mmum dstace estmator What s the effcet weghtg matrx? Estmatg the weghtg matrx. The Geeralzed method of momets estmator - how t s computed. Computg the approprate asymptotc covarace matrx 21-26/67

27 The Method of Momets Momet Equato: Defes a sample statstc that mmcs a populato expectato: The populato expectato orthogoalty codto: E[ m () ] = 0. Subscrpt dcates t depeds o data vector dexed by '' (or 't' for a tme seres settg) 21-27/67

28 The Method of Momets - Example Gamma Dstrbuto Parameters p(y ) exp( y )y P P1 (P) Populato Momet Codtos P E[y ], E[logy ] (P) log Momet Equatos: E[m (,P)] = E[{(1/) y } P / ] 0 1 =1 E[m (,P)] = E[{(1/) logy } ((P) log )] 0 2 = /67

PSI Applcato Solvg the momet equatos Use least squares: Mmze {m E[m ]} {m E[m ]} 1 2 2 2 1 1 2 2 (m (P / ))

29 PSI Applcato Solvg the momet equatos Use least squares: Mmze {m E[m ]} {m E[m ]} (m (P / )) (m ( (P) log )) m m Plot of Ps(P) Fucto P 21-29/67

30 Method of Momets Soluto create ; y1=y ; y2=log(y)$ calc ; m1=xbr(y1) ; ms=xbr(y2)$ mmze; start = 2.0,.06 ; labels = p,l ; fc = (l*m1-p)^2 + (ms - ps(p)+log(l)) ^2 $ User Defed Optmzato Depedet varable Fucto Number of observatos 1 Iteratos completed 6 Log lkelhood fucto e Varable Coeffcet P L /67

31 Nolear Istrumetal Varables There are K parameters, y = f(x,) +. There exsts a set of K strumetal varables, z such that E[z ] = 0. The sample couterpart s the momet equato (1/) z = (1/) z [y - f(x,)] = (1/) m () = m() = 0. The method of momets estmator s the soluto to the momet equato(s). (How the soluto s obtaed s ot always obvous, ad vares from problem to problem.) 21-31/67

32 The MOM Soluto There are K equatos K ukows m( )= 0 If there s a soluto, there s a exact soluto At the soluto, m( )= 0, ad [ m( )]'[ m( )] = 0 Sce [ m( )]'[ m( )] 0, the soluto ca be foud by solvg the programmg problem Mmze wrt : [ m( )]'[ m( )] For ths problem, [ m( )]'[ m( )] = [(1/) ε'z] [(1/) Z'ε] The soluto s defed by [ m( )]'[ m( )] [(1/) ε'z ] [(1/) Z'ε] = 21-32/67

33 MOM Soluto [(1/) ε'z] [(1/) Z'ε] 2 (1/ ) G'Z [(1/) Z'ε] G = K matrx wth row equal to g f( x, ) For the classcal lear regresso model, f( x, ) x ' Z = X, G = X, ad the FOC are -2[(1/)( X'X)] [(1/) X' ε] = 0 ˆ -1 whch has uque soluto =( X'X) X'y 21-33/67

34 Varace of the Method of Momets Estmator The MOM estmator solves m( β)= 0 1 m β 1 ( )= m β Ω Ω ( ) so the varace s for some 1 Geerally, Ω = E[ m ( β) m ( β )'] Asy.Var[ βmom]=( G) Ω( G' ) where The asymptotc covarace matrx of the estmator s G= m( β) β' 21-34/67

35 Example 1: Gamma Dstrbuto m (y ) 1 P 1 1 m (log y (P) log ) Var(y ) Cov(y,logy ) Cov(y,logy ) Var(logy ) 1 P 1 N 2 G 1 1 '(P) 1 y y ˆ y log log 1 y y y log y log y 21-35/67

36 Example 2: Nolear IV Least Squares y f( x, ), z = the set of K strumetal varables Var[ ] m z Var[ m ] z z ' Var[ m( )]=(1/ ) z z ' ( / ) Z' Z 0 G (1/ ) z x ' 2 2 Wth depedet observatos, observatos are ucorrelated x 1 f( x, ). I the lear model, ths s just x. 0 0 G (1/ ) Z' X. 0 where x 1 ( G ) V( G )' [ (1/ ) Z' X ] [( / ) Z' Z][ (1/ ) X ' Z] = [ Z' X ] [ Z' Z][ X ' Z] s the vector of 'pseudo-regressors,' 21-36/67

37 Varace of the Momets How to estmate V = (1/) = Var[ m( )] Var[ m( )]=(1/)Var[ m( )] = (1/) Vˆ (1/ ) (1/ ) m ( ˆ) m ( ˆ)' Estmate Var[ m ( )] wth Est.Var[ m ( )] = (1/) m ( ) m ( )' The, =1 =1 For the lear regresso model, m x, Vˆ 2 (1/ ) (1/ ) x e e x ' (1/ ) (1/ ) e x x ' G (1/ ) X'X Est.Var[ b ] [(1/ ) X'X] [(1/ ) (1/ ) e x x '][(1/ ) X'X] = [ X'X] [ e x x '][ X'X] =1 = MOM = =1 (famlar?) 21-37/67

38 Cosstet? Propertes of the MOM Estmator The LLN mples that the momets are cosstet estmators of ther populato couterparts (zero) Use the Slutsky theorem to assert cosstecy of the fuctos of the momets Asymptotcally ormal? The momets are sample meas. Ivoke a cetral lmt theorem. Effcet? Not ecessarly Sometmes yes. (Gamma example) Perhaps ot. Depeds o the model ad the avalable formato (ad how much of t s used) /67

39 Geeralzg the Method of Momets Estmator More momets tha parameters the overdetfed case Example: Istrumetal varable case, M > K strumets 21-39/67

40 Two Stage Least Squares How to use a excess of strumetal varables (1) X s K varables. Some (at least oe) of the K varables X are correlated wth ε. (2) Z s M > K varables. Some of the varables Z are also X, some are ot. Noe of the varables Z are correlated wth ε. (3) Whch K varables to use to compute Z X ad Z y? 21-40/67

41 Choose K radomly? Choosg the Istrumets Choose the cluded Xs ad the remader radomly? Use all of them? How? A theorem: (Brudy ad Jorgeso, ca. 1972) There s a most effcet way to costruct the IV estmator from ths subset: (1) For each colum (varable) X, compute the predctos of that varable usg all the colums of Z. (2) Learly regress y o these K predctos. Ths s two stage least squares 21-41/67

42 2SLS Algebra Xˆ -1 Z(Z'Z) Z'X ˆ ˆ ˆ 1 b2sls ( X'X) X'y -1 But, Z(Z'Z) Z'X = ( I - M ) X ad ( I - M ) s dempotet. X'X ˆˆ = X' ( I - M )( I - M ) X = X' ( I - M ) X so b X'X ˆ X'y ˆ 2SLS 1 ( ) = a real IV estmator by the defto. Note, plm( X' ˆ /) = 0 sce colums of Z Z Z Z Z Xˆ are lear combatos of the colums of Z, all of whch are ucorrelated wth b X' I - M X X' I - M y -1 2SLS ( Z) ] ( Z) 21-42/67

43 Method of Momets Estmato Same Momet Equato m( β)= 0 Now, M momet equatos, K parameters. There s o uque soluto. There s also o exact soluto to m( β)= 0. We get as close as we ca. How to choose the estmator? Least squares s a obvous choce. Mmze wrt β : m( β )'m( β) E.g., Mmze wrt β : [(1/) ( β )'Z][(1/) Z' ( β)]=(1/ 2 ) ( β )'ZZ' ( β) 21-43/67

44 FOC for MOM Frst order codtos (1) Geeral m( β )'m( β)/ β = 2 G( β )'m( β) = 0 (2) The Istrumetal Varables Problem 2 2 (1/ ) ( β )'ZZ' ( β)/ β = - (2/ )( X ' Z)[ Z' ( y - Xβ)] = 0 Or, ( X ' Z)[ Z' ( y - Xβ)] = 0 (K M) (M )( 1) = 0 At the soluto, ( X ' Z)[ Z' ( y - Xβ)] = 0 But, [ Z' ( y - Xβ)] 0 as t was before /67

45 Computg the Estmator Programmg Program No all purpose soluto Nolear optmzato problem soluto vares from settg to settg /67

46 Asymptotc Covarace Matrx Geeral Result for Method of Momets whe M K Momet Equatos:E[ m( β)]= 0 Soluto - FOC: G( β )'m( β)= 0, G( β )' s K M Asymptotc Covarace Matrx ˆ -1-1 Asy.Var[ β] = [ G( β )' V G( β)], V = Asy.Var[ m β Specal Case - Exactly Idetfed: M = K ad -1 G( β) s osgular. The [ G( β)] exsts ad Asy.Var[ βˆ ] = [ G( β)] V [ G( β )'] -1-1 ( )] 21-46/67

47 MD More Effcet Estmato We have used least squares, Mmze wrt β : m( β )'m( β) to fd the estmator of β. Is ths the most effcet way to proceed? Geerally ot: We cosder a more geeral approach Mmum Dstace Estmato Let A be ay postve defte matrx: Let βˆ = the soluto to Mmze wrt β : q = m( β )' A m( β) Ths s a mmum dstace ( the metrc of A) estmator /67

48 MD Mmum Dstace Estmato Let A be ay postve defte matrx: Let βˆ = the soluto to Mmze wrt β : q = m( β )' A m( β) where E[ m( β)] = 0 (the usual momet codtos). Ths s a mmum dstace ( the metrc of A) estmator. βˆ s cosstet MD βˆ s asymptotcally ormally dstrbuted. MD Same argumets as for the GMM estmator. Effcecy of the estmator depeds o the choce of A /67

49 MDE Estmato: Applcato N uts, T observatos per ut, T > K y Xβ ε, E[ ε X ] 0 Cosder the followg estmato strategy: (1) OLS coutry by coutry, b produces N estmators of (2) How to combe the estmators? b1 β We hav b2 β e 'momet' equato: E 0... bn β How ca I combe the N estmators of β? β 21-49/67

50 Least Squares b1 β b1 β b2 β b2 β E 0. m( β)= bn β bn β To mmze m( β)' m( β) = ( b β )'( b β) b1 β m( β)' m( β) b2 β N 2[ I, I,..., I] 2 ( b β 0 ). 1 β... bn β N 1 N The soluto s ( b β) 0 or β = b b 1 1 N N /67

51 21-51/67 Geeralzed Least Squares The precedg used OLS - smple uweghted least squares. I A 0 I... 0 Also, t uses =. Suppose we use weghted, GLS I [ XX 1 ( 1 1) ] [ XX 2( 2 2 The, A = ) ] [ XX N( N N) ] The frst order codto for mmzg m( β)' Am( β) s N X X b β =1 {[ ( ) ] }( ) = N N XX =1 XX) ] } b =1 or β = {[ ( ) ] } {[ ( N = Wb = a matrx weghted average. =1

52 Mmum Dstace Estmato The mmum dstace estmator mmzes q = m( β )' A m( β) The estmator s (1) Cosstet (2) Asymptotcally ormally dstrbuted (3) Has asymptotc covarace matrx Asy.Var[ ˆ β ] [ G( β) 'AG( β)] [ G( β )'AVAG( β)][ G( β )'AG( β)] MD /67

53 Optmal Weghtg Matrx A s the Weghtg Matrx of the mmum dstace estmator. Are some A's better tha others? (Yes) Is there a best choce for A? Yes The varace of the MDE s mmzed whe A = {Asy. Var[ m( β)]} Ths defes the geeralzed method of momets estmator. A = V /67

54 GMM Estmato 1 m( β)= 1m (y, x,β) 11 Asy.Var[ m( β)] estmated wth W= 1m (y, x,β) m (y, x,β) The GMM estmator of β the mmzes q 1m (y, x,β ) 'W 1m (y, x,β) m( β) Est.Asy.Var[ˆ βgm M] [ G'W G], G = β 21-54/67

55 GMM Estmato Exactly detfed GMM problems 1 Whe m( β) = 1m (y, x,β) 0 s K equatos K ukow parameters (the exactly detfed case), the weghtg matrx q 1m (y, x, β ) 'W 1m (y, x,β) s rrelevat to the soluto, sce we ca set exactly m( β) 0 so q = 0. Ad, the asymptotc covarace matrx (estmator) s the product of 3 square matrces ad becomes [ G'W G] G WG' /67

56 A Practcal Problem Asy.Var[ m( β)] estmated wth 11 W= 1m (y, x, β) m (y, x,β) The GMM estmator of β the mmzes q 1m (y, x, β ) 'W 1m (y, x,β). I order to compute W, you eed to kow β, ad you are tryg to estmate β. How to proceed? Typcally two steps: (1) Use A = I. Smple least squares, to get a prelmary estmator of β. Ths s cosstet, though ot effcet. (2) Compute the weghtg matrx, the use GMM /67

57 Iferece Testg hypotheses about the parameters: Wald test A couterpart to the lkelhood rato test Testg the overdetfyg restrctos 21-57/67

58 Testg Hypotheses (1) Wald Tests the usual fasho. (2) A couterpart to lkelhood rato tests GMM crtero s q = m( β )'W m( β) whe restrctos are mposed o β q creases. d qrestrcted qurestrcted ch squared[j] (The weghtg matrx must be the same for both.) (3) Testg the overdetfyg restrctos: q would be 0 f exactly detfed. q - 0 > 0 results from the overdetfyg restrctos /67

59 Applcato: Iovato 21-59/67

60 Applcato: Iovato 21-60/67

61 Applcato: Multvarate Probt Model 5 - varate Probt Model y * x, y 1[y * 0] t t t t t x x x x x log [{(2 1), 1,...,5}, ] L y s t ds ds ds ds ds 5 t t Requres 5 dmesoal tegrato of the jot ormal desty. Very hard! But, E[y x ] ( x ). t t t Orthogoalty Codtos: E[{y - ( x )} x 0 t t t {y - ( x )} x {y - ( )} 1 x x Momet Equatos: {y - ( x )} x 1 {y - ( )} {y - ( x )} x x4 x equatos 8 parameters /67

62 Pooled Probt Igorg Correlato 21-62/67

63 Radom Effects: Σ=(1- ρ)i+ρ 21-63/67

64 Urestrcted Correlato Matrx 21-64/67

ECON 5360 Class Notes GMM

ECON 5360 Class Notes GMM ECON 560 Class Notes GMM Geeralzed Method of Momets (GMM) I beg by outlg the classcal method of momets techque (Fsher, 95) ad the proceed to geeralzed method of momets (Hase, 98).. radtoal Method of Momets