GMM Method (Single-equation) Pongsa Pornchaiwiseskul Faculty of Economics Chulalongkorn University
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1 GMM Method (Sngle-equaton Pongsa Pornchawsesul Faculty of Economcs Chulalongorn Unversty Stochastc ( Gven that, for some, s random COV(, ε E(( µ ε E( ε µ E( ε E( ε (c Pongsa Pornchawsesul, Faculty of Economcs, Chulalongorn Unversty (c Pongsa Pornchawsesul, Faculty of Economcs, Chulalongorn Unversty 3 Covered opcs Volaton of C(,ε Instrument Varable ( Method Generalzed Method of Moments (GMM Stochastc ( If some or all s are random, addtonal assumptons about are needed. One of them s E( ε, > ˆ s consstent or C(,ε (c Pongsa Pornchawsesul, Faculty of Economcs, Chulalongorn Unversty (c Pongsa Pornchawsesul, Faculty of Economcs, Chulalongorn Unversty 4
2 Volaton of C(,ε OLS estmator s nconsstent (nvald. hy? ˆ OLS ( ( ( + ε + ( ε (c Pongsa Pornchawsesul, Faculty of Economcs, Chulalongorn Unversty 5 Volaton of C(,ε ae condtonal varance on both sdes. ˆ V ( OLS ( V ( ε ( σ ( V ( n ˆ OLS σ n Uncondtonal varance?? (c Pongsa Pornchawsesul, Faculty of Economcs, Chulalongorn Unversty 7 Volaton of C(,ε ae expectaton on both sdes. ˆ E( OLS + E( ( ε ˆ s based n general OLS Defnton Proxy of s proxy of + ξ, ε > E( f and and E( E( ξ, ξ (c Pongsa Pornchawsesul, Faculty of Economcs, Chulalongorn Unversty 6 (c Pongsa Pornchawsesul, Faculty of Economcs, Chulalongorn Unversty 8
3 Regresson Model w/ volaton + + L+, ε > OLS s nconsstent (c Pongsa Pornchawsesul, Faculty of Economcs, Chulalongorn Unversty 9 + ε Regresson Model w/o volaton +, ε ε ε + + L+ ξ > OLS s consstent + ε + ξ How to get Proxy of ( Note that t s equvalent to splttng the varable nto parts, one part (γ wth no correlaton wth the orgnal error term(ε and the other part (ξ wth the correlaton. (c Pongsa Pornchawsesul, Faculty of Economcs, Chulalongorn Unversty How to get Proxy of ( ssumng that the set of varables are uncorrelated wth ε, we can defne the proxy as follows: γ + ξ γ s a proxy of How to get Proxy of (3 Note that γ wll be a good proxy f t can explan most varaton of. Need hgh R. 3 s called the set (vector of nstrument varables. (c Pongsa Pornchawsesul, Faculty of Economcs, Chulalongorn Unversty (c Pongsa Pornchawsesul, Faculty of Economcs, Chulalongorn Unversty
4 Instrument Varables ( Canddates all s whch s non-random or random but uncorrelated wth ε. any varable outsde the regresson model whch s uncorrelated to ε. (c Pongsa Pornchawsesul, Faculty of Economcs, Chulalongorn Unversty 3 Estmaton ( Steps run OLS regresson for γ + ξ get the ftted values for ˆ γˆ γˆ ( (c Pongsa Pornchawsesul, Faculty of Economcs, Chulalongorn Unversty 5 Instrument Varables ( Choces of s avalablty of data underlyng theores pure assumpton Estmaton ( 3 use them as the proxy for + ˆ + ε + + L 4 run OLS on the above regresson model to get the estmate for (c Pongsa Pornchawsesul, Faculty of Economcs, Chulalongorn Unversty 4 (c Pongsa Pornchawsesul, Faculty of Economcs, Chulalongorn Unversty 6
5 Estmaton (3 Matrx Notaton Cov + ε Uα + Vδ + ε α [ U V] and δ ( U, ε but V, ε (c Pongsa Pornchawsesul, Faculty of Economcs, Chulalongorn Unversty 7 Estmaton (5 Estmator ˆ ( ( ( ( ( ( ( + ε Note that + ( ε (c Pongsa Pornchawsesul, Faculty of Economcs, Chulalongorn Unversty 9 symptootcally zero Proxy of V Estmaton (4 Vˆ γˆ ( Uα + Vδ + ε Note that subset of ( U Uˆ U + ε Uα + Vδ + ε V because U s a (c Pongsa Pornchawsesul, Faculty of Economcs, Chulalongorn Unversty 8 Estmaton (6 ae varance on both sdes V(ˆ Vˆ (ˆ σ ( σ ( ˆ σ ( ˆ σ n ( ˆ ( (c Pongsa Pornchawsesul, Faculty of Economcs, Chulalongorn Unversty ( ˆ
6 Estmaton (7 Equvalence OLS ˆ EQOLS ( + ε In EVews, method s the same as wo-sage Least Square (SLS s not n Instrument lst have non-zero correlaton wth error terms ˆ (c Pongsa Pornchawsesul, Faculty of Economcs, Chulalongorn Unversty Estmaton (9 he model wth proxy or transformed wth matrx may not have one. does not need normalty of the error term. 3 a constant term s also an (c Pongsa Pornchawsesul, Faculty of Economcs, Chulalongorn Unversty 3 Estmaton (8 Note that SLS estmaton may yeld negatve R-squared even f there s a constant term or equvalent n the orgnal regresson equaton. hy? Estmaton ( 4 If all plus other s from outsde, method s exactly the same as OLS. th ˆ ˆ ˆ ˆ ( ˆ ( ˆ OLS (c Pongsa Pornchawsesul, Faculty of Economcs, Chulalongorn Unversty (c Pongsa Pornchawsesul, Faculty of Economcs, Chulalongorn Unversty 4
7 eghted SLS ( Unweghted Model and + υ Uα + Vγ + υ υ ε w V, υ or V( υ σ w ˆ Vˆ(ˆ ˆ σ eghted SLS (3 ( ( ˆ σ ( n ( ˆ ( ˆ (c Pongsa Pornchawsesul, Faculty of Economcs, Chulalongorn Unversty 5 (c Pongsa Pornchawsesul, Faculty of Economcs, Chulalongorn Unversty 7 eghted SLS ( eghted Model Uα + Vγ + υ w M w O L L O O w n (c Pongsa Pornchawsesul, Faculty of Economcs, Chulalongorn Unversty 6 M ε GMM ( s an mprovement over estmaton method does not fully utlze the assumpton or nowledge of zero correlaton between and the error term (ε. (c Pongsa Pornchawsesul, Faculty of Economcs, Chulalongorn Unversty 8
8 GMM ( method employs only one proxy from the whole set of Instrument Varables. dfferent subset of s can yeld a dfferent proxy. e don t have to use the whole set of s. e can gve dfferent weght to each. (c Pongsa Pornchawsesul, Faculty of Economcs, Chulalongorn Unversty 9 GMM (4 For each m, t s expected that E( m ε for all m. hat s, E( (- E( (- : E( (- M #of s [,,..., M ] } Moment Condtons (c Pongsa Pornchawsesul, Faculty of Economcs, Chulalongorn Unversty 3 GMM (3 Concept Gvng dfferent weghts to each s equvalent to mxng dfferent subset of s For each, there wll be assocated matrx. Need at least Instrument Varables (c Pongsa Pornchawsesul, Faculty of Economcs, Chulalongorn Unversty 3 GMM (5 Sample nalogy ˆ GMM arg mn ( w ( + ( w ( M + ( w M M ( w m weght for m (c Pongsa Pornchawsesul, Faculty of Economcs, Chulalongorn Unversty 3
9 GMM (6 More general weghtng ˆ GMM arg mn ( ( MxM symmetrc PD weght matrx for Note that general allows not only own weghts but also cross weghts (c Pongsa Pornchawsesul, Faculty of Economcs, Chulalongorn Unversty 33 GMM (8 Estmaton Step I (MxM dentty matrx Step Mnmze ( Step 3 Estmate for next teraton Ωˆ n Ω ˆ ( n ( Bac to Step untl convergence occurs ˆ (c Pongsa Pornchawsesul, Faculty of Economcs, Chulalongorn Unversty 35 GMM (7 Ideal eght matrx Ω Ω Var( [ ] hy? Smlar reason to GLS. (c Pongsa Pornchawsesul, Faculty of Economcs, Chulalongorn Unversty 34
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