x iu i E(x u) 0. In order to obtain a consistent estimator of β, we find the instrumental variable z which satisfies E(z u) = 0. z iu i E(z u) = 0.

Transcription

1 27 However, β MM is icosistet whe E(x u) 0, i.e., β MM = (X X) X y = β + (X X) X u = β + ( X X ) ( X u ) \ β. Note as follows: X u = x iu i E(x u) 0. I order to obtai a cosistet estimator of β, we fid the istrumetal variable z which satisfies E(z u) = 0. Let z i be the ith realizatio of z, where z i is a k vector. The, we have the followig: The β which satisfies Z u = z iu i E(z u) = 0. z iu i = 0 is deoted by β IV, i.e., z i(y i x i β IV ) = 0.

2 28 Thus, β IV is obtaied as: β IV = ( ) z ( i x i ) z iy i = (Z X) Z y. Note that Z X is a k k square matrix, where we assume that the iverse matrix of Z X exists. Assume that as goes to ifiity there exist the followig momet matrices: z i x i = Z X M zx, z iz i = Z Z M zz, z iu i = Z u 0.

3 29 As goes to ifiity, β IV is rewritte as: β IV = (Z X) Z y = (Z X) Z (Xβ + u) = β + (Z X) Z u = β + ( Z X) ( Z u) β + M zx 0 = β, Thus, β IV is a cosistet estimator of β. We cosider the asymptotic distributio of β IV. By the cetral limit theorem, Z u N(0, σ 2 M zz ) Note that V( Z u) = V(Z u) = E(Z uu Z) = E( E(Z uu Z Z) ) = E( Z E(uu Z)Z ) = E(σ2 Z Z) = E(σ 2 Z Z) E(σ 2 M zz ) = σ 2 M zz.

4 30 We obtai the followig asymmptotic distributio: (βiv β) = ( Z X) ( Z u) N(0, σ 2 M zx M zz M zx ) Practically, for large we use the followig distributio: where s 2 = k (y Xβ IV) (y Xβ IV ). β IV N ( β, s 2 (Z X) Z Z(Z X) ), I the case where z i is a r vector for r > k, Z X is a r k matrix, which is ot a square matrix. = Geeralized Method of Momets (GMM, )

5 3 5.2 Geeralized Method of Momets (GMM, ) Cosider the followig regressio model: Z y = Z Xβ + Z u, where Z, y, X, β ad u are r,, k, k ad matrices or vectors. Note that r k. y = Z y, X = Z X ad u = Z u deote r, r k ad r matrices or vectors, where r k. Rewrite as follows: y = X β + u, = r is take as sample size.

6 32 Mea ad variace of u are give by: E(u ) = 0 ad V(u ) = E(u u ) = σ 2 Z Z = σ 2 Ω. Usig GLS, GMM is obtaied as: β GMM = (X Ω X ) X Ω y = ( X Z(Z Z) Z X ) X Z(Z Z) Z y. β GMM is rewritte as: β GMM = ( X Z(Z Z) Z X ) X Z(Z Z) Z y = ( X Z(Z Z) Z X ) X Z(Z Z) Z (Xβ + u) = β + ( X Z(Z Z) Z X ) X Z(Z Z) Z u. Assume: X Z M xz, Z Z M zz, Z u 0.

7 33 The, β GMM is a cosistet estimator of β, which is show as follows: β GMM = β + ( ( X Z)( Z Z) ( Z X) ) ( X Z)( Z Z) ( Z u) β + (M xz M zz M xz) M xz M zz 0 = β. We derive the asymptotic distributio of β GMM. From the cetral limit theorem, Z u N(0, σ 2 M zz ). Accordigly, β GMM is distributed as: (βgmm β) = ( ( X Z)( Z Z) ( Z X) ) ( X Z)( Z Z) ( Z u) N ( 0, σ 2 (M x zm zz M xz) ).

8 34 Practically, for large we use the followig distributio: β GMM N ( β, s 2 (X Z(Z Z) Z X) ), where s 2 = k (y Xβ GMM) (y Xβ GMM ). The above GMM is equivalet to 2SLS. X: k, Z: r, r > k. Assume: X u = Z u = x iu i E(x u) 0, z iu i E(z u) = 0. Regress X o Z, i.e., X = ZΓ + V by OLS, where Γ is a r k ukow parameter matrix ad V is a error term,

9 35 Deote the predicted value of X by ˆX = Z ˆΓ = Z(Z Z) Z X, where ˆΓ = (Z Z) Z X. Note that 2SLS is equivalet to IV i the case of Z = ˆX, where this Z is differet from the above Z. This Z is a k matrix, while the above Z is a r matrix. Whe Z is a k istrumetal variable, the IV estimator is give by: β IV = (Z X) Z y, Z is replaced by ˆX. The, β 2S LS = ( ˆX X) ˆX y = ( X Z(Z Z) Z X ) X Z(Z Z) Z y = β GMM. GMM is iterpreted as the GLS applied to MM.

10 Geeralized Method of Momets (GMM, ) II Noliear Case Cosider the geeral case: E(h(θ; w)) = 0, which is the orthogoality coditio. A k vector θ deotes a parameter to be estimated. h(θ; w) is a r vector for r k. Let w i = (y i, x i ) be the ith observed data, i.e., the ith realizatio of w. Defie g(θ; W) as: g(θ; W) = where W = {w, w,, w }. g(θ; W) is a r vector for r k. h(θ; w i ),

11 37 Let ˆθ be the GMM estimator which miimizes: with respect to θ. g(θ; W) S g(θ; W), Solve the followig first-order coditio: with respect to θ. Computatioal Procedure: g(θ; W) S g(θ; W) = 0, θ There are r equatios ad k parameters. Liearizig the first-order coditio aroud θ = ˆθ, g(θ; W) 0 = S g(θ; W) θ g(ˆθ; W) θ θ = ˆD S g(ˆθ; W) + ˆD S ˆD(θ ˆθ), S g(ˆθ; W) + g(ˆθ; W) S g(ˆθ; W) (θ ˆθ) θ

12 38 where ˆD = g(ˆθ; W), which is a r k matrix. θ Note that i the secod term of the secod lie the secod derivative is igored ad omitted. Rewritig, we have the followig equatio: θ ˆθ = ( ˆD S ˆD) ˆD S g(ˆθ; W). Replacig θ ad ˆθ by ˆθ (i+) ad ˆθ (i), respectively, we obtai: ˆθ (i+) = ˆθ (i) ( ˆD (i) S ˆD (i) ) ˆD (i) S g(ˆθ (i) ; W), where ˆD (i) = g(ˆθ (i) ; W) θ. Give S, repeat the iterative procedure for i =, 2, 3,, util ˆθ (i+) is equal to ˆθ (i). How do we derive the weight matrix S?

13 39 I the case where h(θ; w i ), i =, 2,,, are mutually idepedet, S is: S = V ( g(θ; W) ) = E ( g(θ; W)g(θ; W) ) = E (( = which is a r r matrix. Note that h(θ; w i ) )( j= E ( h(θ; w i )h(θ; w i ) ), h(θ; w j ) ) ) = (i) E ( h(θ; w i ) ) = 0 for all i ad accordigly E ( g(θ; W) ) = 0, (ii) g(θ; W) = h(θ; w i ) = h(θ; w j ), j= (iii) E ( h(θ; w i )h(θ; w j ) ) = 0 for i j. E ( h(θ; w i )h(θ; w j ) ) j= The estimator of S, deoted by Ŝ is give by: Ŝ = h(ˆθ; w i )h(ˆθ; w i ) S.

14 40 Takig ito accout serial correlatio of h(θ; w i ), i =, 2,,, S is give by: S = V ( g(θ; W) ) = E ( g(θ; W)g(θ; W) ) = E (( h(θ; w i ) )( j= h(θ; w j ) ) ) = E ( h(θ; w i )h(θ; w j ) ). j= Note that E ( h(θ; w i ) ) = 0. Defie Γ τ = E ( h(θ; w i )h(θ; w i τ ) ) <, i.e., h(θ; w i ) is statioary. Statioarity: (i) E ( h(θ; w i ) ) does ot deped o i, (ii) E ( h(θ; w i )h(θ; w i τ ) ) depeds o time differece τ. = E ( h(θ; w i )h(θ; w i τ ) ) = Γ τ

15 4 S = E ( h(θ; w i )h(θ; w j ) ) j= = ( ( E h(θ; w )h(θ; w ) ) + E ( h(θ; w )h(θ; w 2 ) ) + + E ( h(θ; w )h(θ; w ) ) E ( h(θ; w 2 )h(θ; w ) ) + E ( h(θ; w 2 )h(θ; w 2 ) ) + + E ( h(θ; w 2 )h(θ; w ) ). E ( h(θ; w )h(θ; w ) ) + E ( h(θ; w )h(θ; w 2 ) ) + + E ( h(θ; w )h(θ; w ) )) = (Γ 0 + Γ + Γ Γ Γ + Γ 0 + Γ + + Γ 2. Γ + Γ 2 + Γ Γ 0 )

16 42 = = Γ 0 + = Γ 0 + ( Γ0 + ( )(Γ + Γ ) + ( 2)(Γ 2 + Γ 2 ) + (Γ + Γ )) i (Γ i + Γ i) = Γ 0 + q ( i ) (Γi + Γ q + i). ( i ) (Γi + Γ i) Note that Γ τ = E ( h(θ; w i τ )h(θ; w i ) ) = Γ( τ), because Γ τ = E ( h(θ; w i )h(θ; w i τ ) ). I the last lie, is replaced by q +, where q <. We eed to estimate Γ τ as: ˆΓ τ = h(ˆθ; w i )h(ˆθ; w i τ ). i=τ+ As τ is large, ˆΓ τ is ustable. Therefore, we choose the q which is less tha.

17 43 S is estimatated as: Ŝ = ˆΓ 0 + = the Newey-West Estimator q ( i ) (ˆΓ i + ˆΓ q + i), Note that Ŝ S, because ˆΓ τ Γ τ as. Asymptotic Properties of GMM: GMM is cosistet ad asymptotic ormal as follows: (ˆθ θ) N ( 0, (D S D) ), where D is a r k matrix, ad ˆD is a estimator of D, defied as: D = g(θ; W) θ, ˆD = g(ˆθ; W) θ.

18 44 Proof of Asymptotic Normality: Assumptio : ˆθ θ Assumptio 2: g(θ; W) N(0, S ), i.e., S = lim V ( g(θ; W) ). The first-order coditio of GMM is: g(θ; W) S g(θ; W) = 0. θ The GMM estimator, deote by ˆθ, satisfies the above equatio. Therefore, we have the followig: g(ˆθ; W) Ŝ g(ˆθ; W) = 0. θ

19 45 Liearize g(ˆθ; W) aroud ˆθ = θ as follows: g(ˆθ; W) = g(θ; W) + g(θ; W) where D =, ad θ is betwee ˆθad θ. θ = Theorem of Mea Value ( ) g(θ; W) (ˆθ θ) = g(θ; W) + D(ˆθ θ), θ Substitutig the liear approximatio at ˆθ = θ, we obtai: which ca be rewritte as: 0 = ˆD Ŝ g(ˆθ; W) = ˆD Ŝ ( g(θ; W) + D(ˆθ θ) ) = ˆD Ŝ g(θ; W) + ˆD Ŝ D(ˆθ θ), ˆθ θ = ( ˆD Ŝ D) ˆD Ŝ g(θ; W).

20 46 Note that D = g(θ; W) θ, where θ is betwee ˆθ ad θ. From Assumptio, ˆθ θ implies θ θ Therefore, (ˆθ θ) = ( ˆD Ŝ D) ˆD S g(θ; W). Accordigly, the GMM estimator ˆθ has the followig asymptotic distributio: (ˆθ θ) N ( 0, (D S D) ). Note that ˆD D, D D, Ŝ S ad Assumptio 2 are utilized.

21 47 Computatioal Procedure: q ( () Compute Ŝ (i) i = ˆΓ 0 + (ˆΓ i + ˆΓ q + ) i), where ˆΓ τ = q is set by a researcher. (2) Use the followig iterative procedure: h(ˆθ; w i )h(ˆθ; w i τ ). i=τ+ ˆθ (i+) = ˆθ (i) ( ˆD (i) Ŝ (i) ˆD (i) ) ˆD (i) Ŝ (i) g(ˆθ (i) ; W). (3) Repeat () ad (2) util ˆθ (i+) is equal to ˆθ (i). I (2), remember that whe S is give we take the followig iterative procedure: ˆθ (i+) = ˆθ (i) ( ˆD (i) S ˆD (i) ) ˆD (i) S g(ˆθ (i) ; W), where ˆD (i) = g(ˆθ (i) ; W) θ. S is replaced by Ŝ (i).

22 48 If the assumptio E ( h(θ; w) ) = 0 is violated, the GMM estimator ˆθ is o loger cosistet. Therefore, we eed to check if E ( h(θ; w) ) = 0. From Assumptio 2, ote as follows: J = ( g(ˆθ; W) ) Ŝ ( g(ˆθ; W) ) χ 2 (r k), which is called Hase s J test. Because of r equatios ad k parameters, the degree of freedom is give by r k. If J is small eough, we ca judge that the specified model is correct.

23 49 Testig Hypothesis: Remember that the GMM estimator ˆθ has the followig asymptotic distributio: (ˆθ θ) N ( 0, (D S D) ). Cosider testig the followig ull ad alterative hypotheses: The ull hypothesis: H 0 : R(θ) = 0, The alterative hypothesis: H : R(θ) 0, where R(θ) is a p k vector fuctio for p k. p deotes the umber of restrictios. R(θ) is liearized as: R(ˆθ) = R(θ) + R θ (ˆθ θ), where R θ = R(θ), which is a p k θ matrix.

24 50 Note that θ is bewtee ˆθ ad θ. If ˆθ θ, the θ θ ad R θ R θ. Uder the ull hypothesis R(θ) = 0, we have R(ˆθ) = R θ (ˆθ θ), which implies that the distributio of R(ˆθ) is equivalet to that of R θ (ˆθ θ). The distributio of R(ˆθ) is give by: R(ˆθ) = R θ (ˆθ θ) N ( ) 0, R θ (D S D) R θ. Therefore, uder the ull hypothesis, we have the followig distributio: R(ˆθ) ( R θ (D S D) R θ) R(ˆθ) χ 2 (p). Practically, replacig θ by ˆθ i R θ, D ad S, we use the followig test statistic: R(ˆθ) ( Rˆθ( ˆD Ŝ ˆD) R ˆθ) R(ˆθ) χ 2 (p). = Wald type test

25 5 Examples of h(θ; w):. OLS: Regressio Model: y i = x i β + ɛ i, E(x i ɛ i) = 0 h(θ; w i ) is take as: h(θ; w i ) = x i(y i x i β). 2. IV (Istrumetal Variable, ): Regressio Model: y i = x i β + ɛ i, E(x i ɛ i) 0, E(z i ɛ i) = 0 h(θ; w i ) is take as: h(θ; w i ) = z i(y i x i β), where z i is a vector of istrumetal variables.

26 52 Whe z i is a k vector, the GMM of β is equivalet to the istrumetal variable (IV) estimator. Whe z i is a r vector for r > k, the GMM of β is equivalet to the two-stage least squares (2SLS) estimator. 3. NLS (Noliear Least Squares, ): Regressio Model: f (y i, x i, β) = ɛ i, E(x i ɛ i) 0, E(z i ɛ i) = 0 h(θ; w i ) is take as: h(θ; w i ) = z i f (y i, x i, β) where z i is a vector of istrumetal variables.