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 he dea s to match the populato momets of a dstrbuto to the sample momets, usg as may momets as ecessary to estmate the ukow parameters. Let fx ; X ; :::; X g be a radom sample from the pdf f(x; ; :::; r ). Also, let m 0 k = X = xk be the k th sample momet ad 0 k = E(Xk ) the k th populato momet. he method of momets estmator for = ( ; :::; r ) 0 s therefore the soluto to the equatos m 0 = 0 () for = ; :::; r. Method of momets ca be mod ed to use cetered, as opposed to raw, momets. Whle cosstet, method of momets estmators are ot geerally e cet... Example Suppose fx ; X ; :::; X g s a radom sample from a gamma(; ) dstrbuto. he lkelhood fucto L() = ( () ) (x x x ) exp X x = = s d cult to evaluate wthout usg umercal methods. A method of momets estmator jotly solves X = X X = E(X) = ^ = X (X X) = E[(X ) ] = for ^ ad ^. hs gves ^ = X =^ ^ = ^ = X:
.. Varace of Method of Momets Estmator Let the sample momets be g k = X g k(x ) for k = ; :::; K ad g = (g ; :::; g K ) have asymptotc varace-covarace matrx V, wth elemets V jk = X (g j(x ) g j )(g k (X ) g k ) where j; k = ; :::; K. Now let G be the matrx G = 6 4 @g @g @ @ @g @ K @g @g @g @ @ @ K. @g K @ @g K @.... @g K @ K 7 5 KK : Sce the populato momets () are typcally a olear fucto, we wll learze usg a rst-order aylor approxmato to g k = k () aroud the true value g = () + G()(^ ) ) (^ ) = G ()(g ()): herefore, our estmate of the asymptotc varace s est:asy:var:(^) = ^G V ( ^G ) 0 :.. Gamma Example Cotued I the gamma dstrbuto example above, where g = X ad g = (X X), we have 6 ^ ^G = 4 ^ ^ 7 5 ^^ ad V = 6 4 dvar(g ) cov(g c ; g ) 7 5 : cov(g c ; g ) dvar(g )
. Geeralzed Method of Momets GMM exteds the classcal method of momets estmator to hadle cases where there are more momet codtos tha parameters to estmate (.e., the model s overdet ed)... Basc Framework Suppose there are K parameters to estmate = ( ; :::; K ) 0 ad L K momet codtos E[m l (y ; X ; Z ; )] = 0 () for l = ; :::; L. he sample aalog of () s m l (y ; X ; Z ; ) = X = m l(y ; X ; Z ; ) = 0 whch wll geerally have a uque soluto f L = K ad multple solutos f L > K. o recocle the multple solutos, cosder mmzato of q = m() 0 W m() where m() = ( m ; :::; m l ) 0 ad W s a postve de te weghtg matrx. If W = I, the mmzato of q s smply a least squares crtero. If W 6= I, the mmzato of q s smlar sprt to GLS, whch re-weghts the observatos accordg to the varace-covarace matrx of the errors. Aga, the sprt of GLS, Hase (98) shows that the optmal crtero (weghtg matrx) s to mmze q = m() 0 m() where = Asy:V ar:( p m): he resultg estmator, ^ GMM, wll have a asymptotc varace-covarace matrx equal to Est:Asy:V ar:(^ GMM ) = [ 0 ^ ] where s a matrx of partal dervatves smlar sprt to G above... Propertes of the GMM Estmator Assumg that the
. parameters are det able,. emprcal momets coverge probablty to ther populato couterparts (.e., m() p! 0), ad. the emprcal momets obey the cetral lmt theorem (.e., p m() d! N[0; ]), the ^GMM asy N[; ( 0 ) ]:.. Example #. Ordary Least Squares Exactly Idet ed Case Nearly all estmators we have covered ca be posed as method of momet estmators. Cosder GMM estmato of the bvarate lear regresso model y = + x + : wo momet codtos arsg from the Classcal assumptos are E[m (y ; x ; ; )] = E( ) = 0 E[m (y ; x ; ; )] = E( x ) = 0: he sample aalog of these populato momet codtos are X e = X (y ^ ^x ) = 0 X e x = X (y ^ ^x )x = 0; whch are, of course, the ormal equatos for OLS estmato of the classcal lear regresso model. I ths stace, the weghtg matrx W s rrelevat because both momet codtos ca be sats ed exactly. herefore, we have ^ GMM = ^ OLS = y bx P ^ GMM = ^ OLS = (y y)(x x) P (x x) :..4 Example #. Hall s Radom-Walk Cosumpto Hypothess I a famous 978 artcle the Joural of Poltcal Ecoomy, Robert Hall showed that uder certa codtos, cosumpto should be expected to follow a radom walk. Cosder a aget that chooses cosumpto 4
c t to maxmze dscouted, expected lfetme utlty E 0 ( + ) t 0:5[c c t ] ; where s the subjectve dscout rate, s a costat, ad c s the blss level of cosumpto, subject to A 0 = ( + r) t w t ) where A 0 s tal assets, r s the terest rate ad w t s the wage rate. Hall shows that f = r, the cosumpto follows a radom walk c t = c t + t where E t [ t ] = 0. Campbell ad Makw (989) test Hall s hypothess by posg a spec c alteratve agets smply cosume a gve fracto of ther curret come (.e., c t = w t ). he two hypotheses ca be ested accordg to c t c t = (w t w t ) + ( ) t c t = w t + t : I prcple, oe could just ru a regresso of the chage cosumpto o the chage come ad test whether the coe cet s d eret tha zero. he problem s that w t ad t are lkely to be correlated so that strumetal varables eed to be foud. Cosder usg the rst four lagged chages cosumpto: c t ; :::; c t 4. he momet codtos are therefore E[m ; w t ; c t ; )] = E[ t c t ] = 0 E[m ; w t ; c t ; )] = E[ t c t ] = 0 E[m ; w t ; c t ; )] = E[ t c t ] = 0 E[m 4 ; w t ; c t 4 ; )] = E[ t c t 4 ] = 0: he sample aalogs are m t(^) = m t(^) = m t(^) = m 4t(^) = ^w t )c t = 0 ^w t )c t = 0 ^w t )c t = 0 ^w t )c t 4 = 0: 5
he GMM estmate ^ GMM mmzes q = m() 0 W m() where W = s the asymptotc varace of p m(). See Gauss example # for OLS, SLS ad GMM estmates of...5 estg the Valdty of the Overdet cato Restrctos I a exactly det ed system, q = 0. I a overdet ed system, the momet restrctos mpled by theory wll ot all be sats ed exactly the data. herefore, q > 0. hs observato forms the bass for a test of overdetfyg restrctos. If q s substatally greater tha zero, the ths suggests that at least oe of the overdetfyg restrctos s lkely to be false. Smlar to the Wald test troduced earler chapters, we have q = [ p m(^)] 0 ^ [ p m(^)] asy [L K]: 6