Optimal weighting matrix The optimal weighting matrix is the inverse of asymptotic variance of 8(0,Y It turn out to be S=limTE[(0 Y )1g(00: Y-1 =2I. t→) V=-00 Where r =E[h(o, y)lh(o,y-1) Ifh(0o, y,)Pe were serially uncorrelated, then the matrix S could be consistently estimated by =7∑Q,y)O2
Optimal weighting matrix • The optimal weighting matrix is the inverse of asymptotic variance of • It turn out to be • Where • If were serially uncorrelated, then the matrix S could be consistently estimated by ( , ) g θ 0 ΥT { } ∑ ∞ → ∞ =−∞ = Υ Υ = Γ v T T v t S TE [g ( ; )][ g ( ; )]' lim θ 0 θ 0 Γv = E {[ h (θ 0 , yt)][ h (θ 0 , yt− v )]' } { } ∞ t t=−∞ h ( , y ) θ 0 (){ } 1 / [ ( , )][ ( , )]' 0 0 1 * t t T t T S T ∑ h θ y h θ y = =
Optimal weighting matrix continued New-West (1987)estimate of S could be correlated, the If the vector process h(o, y)- is serially S=+∑/(q+1)kr Where =170mr Why? varu]=q9E(u)+(g-1)E(u1)+E(u1)+.+E(x-1)+E(a11 g∑B(n)
Optimal weighting matrix--- continued • If the vector process is serially correlated, the New-West (1987) estimate of S could be • Where • Why? • { } ∞ t t=−∞ h ( , y ) θ 0 { } ) ˆ ˆ 1 [ /( 1)] ( ˆ ˆ ' , , 1 0, v T v T q v T T S = Γ + ∑ − v q + Γ + Γ = ˆ 1 / {[ ( ˆ, )][ ( ˆ, )]' } 1 , t t v T t v v T T h y h y − = + Γ = ∑ θ θ ( ') var[ ] ( ) ( 1)[ ( ) ( )] ... [ ( ') ( ')] ' 1 ' 1 ' 1 t t k q v q t t t t t t t t q t q t q v v E u u q q v q u qE u u q E u u E u u E u u E u u − = − − − − − = ∑ ∑ − = = + − + + + +
Asymptotic distribution of the gmm estimates Ae the value that minimizes [g(0: Y)'S [8(8; Y)I With S- regarded fixed with respect 0 and S-PS The gmm estimates e is typically a solution to the following system of nonlinear equations ∫eg(G,Y) 6=0 xS7×g(n,Y)=0 In many situations( stationary of y, continuity of h(, and restriction on higher moments )it should be the case 7g(;Y)-→N0.S)
Asymptotic distribution of the GMM estimates • Let be the value that minimizes • With regarded fixed with respect θ and • The GMM estimates is typically a solution to the following system of nonlinear equations : • In many situations( stationary of y, continuity of h(), and restriction on higher moments) it should be the case θ T ˆ [ ( ; )] ˆ [ ( ; )]' 1 T T T g Υ S g Υ − θ θ 1 ˆ − S T S S p ˆ T ⎯⎯→ θ T ˆ [ ; ) ] 0 ˆ ( ˆ ' ' ( ; ) 1 ˆ × × Υ = ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ∂ ∂ Υ − = T T T T S g g T θ θ θ θ θ ( ; ) ( 0, ) 0 T g N S L θ ΥT ⎯⎯→
proposition With suitable conditions (2)7g{(;Yr)-→N0,S) ()For any sequence (e Pa satisfying 0, 0 · It is case that plim 08(Y, IXO ·Then √7(G1-)-2→N0,) where V=DS
proposition • With suitable conditions • (1) • (2) • (3) For any sequence satisfying • It is case that • Then • where 0 ˆ θ ⎯⎯→θ P T ( ; ) (0, ) 0 T g N S L θ ΥT ⎯⎯→ { }∞=1 *T T θ 0 * θ ⎯⎯→θ P T ' 0 * ' ( ; ) lim ' ( ; ) lim r a T T D g p g p T = = = × ⎭⎬⎫ ⎩⎨⎧ ∂ ∂ Υ = ⎭⎬⎫ ⎩⎨⎧ ∂ ∂ Υ θ θ θ θ θ θ θ θ ) (0, ) ˆ ( T 0 N V L θ T −θ ⎯⎯→ { } 1 1 ' − − V = DS D