Proof of Lemmas Lemma 1 Consider ξ nt = r

Transcript

1 Supplemetary Material to "GMM Estimatio of Spatial Pael Data Models with Commo Factors ad Geeral Space-Time Filter" (Not for publicatio) Wei Wag & Lug-fei Lee April 207 Proof of Lemmas Lemma Cosider = r l= C,lV l + D e, where C,l s are matrices ad D is a square matrix of dimesio. Let E[V l,i V k,i ] = σ lk, E[V l,i V k,i V s,i ] = µ 3,lks, E[V l,i V k,i V s,i V w,i ] = µ 4,lksw, for l, k, s, w =,..., r, ad E[e 3,i ] = µ 3e, E[e 4,i ] = µ 4e. For ay -dimesioal square matrix A, let vec D (A) = (a,,...a, ) deote the colum vector formed with the diagoal elemets of A, The, for ay two square matrices A ad B of dimesio, ad r r r E( A ) = C,l vec D (C,kA C,s )µ 3,lks + D vec D (D A D )µ 3e l= k= s= = E( A B ) r r r r [(µ 4,lksw σ lk σ sw σ ls σ kw σ lw σ ks )vec D(C,lA C,k )vec D (C,sB C,w ) l= k= s= w= +σ 2 e0 l= +σ ls σ kw tr(c,la C,k C,sB C,w ) + σ lw σ ks tr(c,la C,k C,wB C,s ) +σ lk σ sw tr(c,la C,k )tr(c,sb C,w )] r [tr(c,la C,l )tr(d B D ) + tr(c,lb C,l )tr(d A D ) +tr(c,l C,lA D D B s ) + tr(d D A C,l C,lB s )] +(µ 4e 3σ 4 e0)vec D(D A D )vec D (D B D ) +σ 4 e0[tr(d A D D B s D ) + tr(d A D )tr(d B D )]. Proof: Note that V l s are ucorrelated with oe aother, ad are all idepedet of e. The the result is straightforward from expasios of A ad A B ad Lemma A.2 i Lee (2007). Q.E.D. Lemma 2 Uder Assumptios F, F2,, 3 ad 4, for ay ostochastic UB matrix A, (i) E( A ) = O(); (ii) V ar( A ) = O(); (iii) A = O p () ad A E( A ) = o p (); (iv) for the IV matrix Q, plim Q A = 0. Proof: (i) E( A ) = tr(a Σ ) = r l= tr[a ((f l (α 0 )f l (α 0)) I )]+σ 2 e0tr[a (J K J )], where (f l (α 0 )f l (α 0)) I ad (J K J ) are UB uder Assumptios F, F2, 3 ad 4. Thus E( A ) = O() by Lemma A.3 i Lee (2007).

2 (ii) From Lemma, V ar( A ) = E( A A ) E( A )E( A ) r r r r = {(µ 4,lksw σ lk σ sw σ ls σ kw σ lw σ ks ) l= k= s= w= vec D[(f l (α 0 ) I )A (f k (α 0 ) I )]vec D [(f s(α 0 ) I )A (f w (α 0 ) I )] + +σ ls σ kw tr[a (f l (α 0 )f k0(α 0 ) I )A (f s (α 0 )f w(α 0 ) I )] +σ lw σ ks tr[a (f l (α 0 )f k0(α 0 ) I )A (f w (α 0 )f s(α 0 ) I )]} r σ 2 e0tr[(f l (α 0 )f l (α 0 ) I )A (J K J )As l= +(J K J )A ((f l (α 0 )f l (α 0 ) I )A s ] +(µ 4e 3σ 4 e0)vec D[P J A J P ]vec D [P J A J P ] +σ 4 e0tr[a (J K J )As (J K J )], As J P, (f l (α 0 )f l (α 0)) I, for l =,..., r, ad J K J are UB uder Assumptios F, F2, 3 ad 4, V ar( A ) = O() follows from Lemma A.3 i Lee (2007). (iii) These immediately follow from (i) ad (ii). (iv) As = r l= f l(α 0 ) V l + (J P )e, we have Q A = Q A r l= [f l(α 0 ) I ]V l + Q A (J P )e. Uder Assumptios F, 3 ad 4, elemets of Q A [f l (α 0 ) I ] are bouded ad J P is UB. Therefore, by Assumptio 4, a law of large umbers ca apply to each row of Q A [f l (α 0 ) I ]V l s ad Q A (J P )e, which i tur, implies that plim Q A = 0. Q.E.D. Lemma 3 (CLT for a geeralized liear-quadratic form) Suppose that, for l, k =,..., m, {A,lk } is a sequece of symmetric ostochastic UB matrices ad B,l = (b,l,...b,l, ) is a -dimesioal vector such that sup (/) i= b i,l 2+η < for some η > 0. Let σ 2 Q be the variace of Q where Q = m m l= k= V,l A,lkV,k + m l= B,l V,l, ad V,l s are ucorrelated radom vectors whose elemets v i,l s are i.i.d. with zero mea ad fiite variace σ 2 l, ad E( v i,l 4+δ0 ) ad E( v i,l v i,k 2+δ0 ) for some δ 0 > 0 exist ad are uiformly bouded for all i, l ad k. Assume that the variace σ 2 Q is bouded away from zero at the rate. The (Q E(Q ))/σ 2 D Q N(0, ). Mai steps of the proof: Defie σ-fields F i, =< v,,..., v i,, v,2,..., v i,2,..., v,m,...v i,m > geerated by v,,..., v i,, v,2,..., v i,2,..., v,m,...v i,m. Because (v i,,..., v i,m ) for all i of V,l s are mutually idepedet, it is easily see that E(Z i, F i, ) = 0. Thus {(Z i,, F i, ) i } forms a martigale differece double array. Therefore, σ 2 Q = i= E(Z2 i, ). Defie ormalized variables Zi, = Z i,/σ Q. The {(Zi,, F i,) i } is a martigale differece double array ad (Q E(Q ))/σ 2 Q = i= Z i,. I order for the martigale cetral limit theorem to be applicable, we ca establish the followig two steps: Step. There exists δ > 0 such that i= E( Z i, 2+δ ) teds to approach zero as goes to ifiity. Step 2. i= E(Z 2 i, F i,) p. The detailed proof is available upo request, which is a rather log ad mechaical extesio of Kelejia ad Prucha (200) ad Qu ad Lee (202). Q.E.D. Lemma 4 Suppose that A (α 0 ) is a matrix, where α 0 is a parameter vector of fiite dimesio. Let α be ay possible value of α 0, ad α i is the i-th elemet of α, assume that {A (α)} ad 2

3 { A (α)/ α i } are UB uiformly i α. If α is a cosistet estimator of α 0, the uder Assumptio 4, (i) [tr(a ( α)) tr(a (α 0 ))] = o p (), ad (ii) A ( α) A (α 0 ) = o p (). Proof: (i) By Taylor s expasio, [tr(a ( α)) tr(a (α 0 ))] = [ tr(a (α))/ α)]( α α 0 ), where α is a value betwee α ad α 0. As { A (α)/ α i } is UB uiformly i α, { A (α)/ α i } is UB. By Lemma A.3 i Lee (2007), tr( A (α)/ α i ) = O(). So tr(a (α))/ α i = tr( A (α)/ α i ) = O(). Because α α 0 = o p (), it follows that [ tr(a (α))/ α)]( α α 0 ) = o p (). (ii) Let e i be the idicator vector with the i-th elemet beig oe ad all other elemets beig zero. The A ( α) A (α 0 ) = max i j= e i [A ( α) A (α 0 )]e j. By the mea value theorem, e i [A ( α) A (α 0 )]e j = e i [ r k= max i ( ) A (α ij) α k A (α ij) α k ( α k α k0 )]e j, where α ij is a value betwee α ad α 0. Thus r A (α ij ) A ( α) A (α 0 ) = max i e i[ ( α k α k0 )]e j α j= k k= r ( ) A (α ij ) ( ) α k α k0 α k rc A (α ij ) α max i α k, j= k= where is the ij-th elemet of A (α ij) α ij k ad c α = max k { α k α k0 }. Because { A (α)/ α k } ( is UB uiformly i α, { A (α ij )/ α k } is UB so that max i A (α ij) j= α k is bouded. As α )ij ( α 0 = o p (), c α = o p (). Notig that r is fiite, rc α max i A (α ij) j= α k )ij = o p(). Therefore, A ( α) A (α 0 ) = o p (). Q.E.D. Lemma 5 Let θ ad θ be, respectively, the miimizers of F (θ) ad F (θ) i Θ. Suppose that (F (θ) F (θ)) coverges i probability to zero uiformly i θ Θ, where θ 0 is i the iterior of Θ, ad { F (θ)} satisfies the uiqueess idetificatio coditio at θ 0. If (F (θ) F (θ)) = o p () uiformly i θ Θ, the both θ ad θ coverge i probability to θ 0. I additio, suppose that 2 F (θ) ij coverges i probability to a well-defied limitig matrix, uiformly i θ Θ, which is osigular at θ 0, ad F (θ 0) = O p (). If F (θ 0) ( 2 F (θ) j= 2 F (θ) ) = o p () uiformly i θ Θ ad ( F (θ0) ) = o p (), the ( θ N θ 0 ) ad ( θ N θ 0 ) have the same limitig distributio. Proof: This is Lemma A.6 i Lee (2007), otig that T is fiite here. Q.E.D. Lemma 6 Suppose that z, ad z 2, are -dimesioal colum vectors of costats of which their elemets are uiformly bouded, the costat matrix A is uiformly bouded i colum sums i absolute value, ad the costat matrices B, ad B 2, are UB. Let θ be a -cosistet estimate of θ 0. Let C be either Σ, Σ (I T G j )Z π 0 or Σ (δ 0)/ δ l, where δ l is the l-th elemet of δ. Let Ĉ be the estimated couterparts of C. For these C matrices, C L represets its liear trasformed matrix that preserves UB property. The, uder Assumptios F, F2,, 3, 4 ad 6, (i) z, (Ĉ C )L z 2, = o p (); (ii) z, (Ĉ C )L A = o p (); (iii) B, (Ĉ C )L B 2, = o p (); (iv) (Ĉ C ) = o p (). Proof: Uder Assumptios F, F2,, 3, 4 ad 6, these results ca be obtaied from Lemmas 4 (ii) ad 8. Q.E.D. Lemma 7 Suppose that { W j } for j =,..., p ad { S }, where is a matrix orm, are bouded. The { S (λ) } is uiformly bouded i a eighborhood of λ0. 3 ij

4 Proof: This is Lemma C8 i Lee ad Liu (200). Q.E.D. Lemma 8 Uder Assumptios F, F2,, 3, 4 ad 6, {Σ (δ)}, { Σ (δ)/ δ l }, { Σ (δ)/ δ l}, ad { 2 Σ (δ)/ δ2 l } are UB uiformly i δ, where δ l is the l-th elemet of δ. Proof: The UB of {Σ (δ)} ad { Σ (δ)/ δ l } are directly from their expressios ad Assumptios F, F2,, 3, ad 4. As Σ (δ)/ δ l = Σ (δ) Σ (δ) δ l Σ (δ), from the expressio of Σ (δ)/ δ l ad Assumptios F, F2,, 3, 4 ad 6, its UB follows. Similarly, the UB of { 2 Σ (δ)/ δ2 l } ca be derived from its expressio ad the assumptios metioed above. Q.E.D. Proofs of Propositios 4 ad 5 Proof of Propositio 4: We shall show that F (θ) = ĝ (θ) Ω ĝ (θ) ad F (θ) = g (θ)ω will satisfy the coditios i Lemma 5. If so, the GMM estimator from the miimizatio of F (θ) will have the same limitig distributio as that of the miimizer of F (θ). The differece of F (θ) ad F (θ) ad its derivatives ivolve the differece of ĝ (θ) ad g (θ) ad their derivatives. Furthermore, oe has to cosider the differece of Ω ad Ω. First, cosider N (ĝ N (θ) g N (θ)). Let m = p + h + 2q + 2. Explicitly, (ĝ (θ) g (θ)) = { (ϑ)( Q Q ), [ (ϑ)( P, P, ) (ϑ) tr(( P, P, )Σ (δ))],..., [ (ϑ)( P m, P m, ) (ϑ) tr(( P m, P m, )Σ (δ))]}. g (θ) The (ϑ) is related to as (ϑ) = d (ϑ) + [I T S (λ)s ], where d (ϑ) = Z (π 0 π) + p j= (λ j0 λ j )(I T G j )Z π 0. To prove (ϑ)( Q Q ) = o p() uiformly i θ Θ, we eed to cosider each compoet i Q Q ad its product with (ϑ), otig that Q = Σ [Z, (I T G )Z π 0,..., (I T G p )Z π 0 ]. For j =,..., p, (ϑ)[ Σ (I T Ĝj)Z π Σ (I T G j )Z π 0 ] = (ϑ)[ Σ (I T Ĝj) Σ (I T G j )]Z π + (ϑ)σ (I T G j )Z ( π π 0 ) = [Z (π 0 π) + (λ j0 λ j )(I T G j )Z π 0 ] [ Σ (I T Ĝj) Σ (I T G j )]Z π j= + [I T S (λ)s ] [ Σ (I T Ĝj) Σ (I T G j )]Z π + [Z (π 0 π) + (λ j0 λ j )(I T G j )Z π 0 ] Σ (I T G j )Z ( π π 0 ) j= + [I T S (λ)s ] Σ (I T G j )Z ( π π 0 ). () The last three terms o the right had side of () are o p () uiformly i θ Θ by Assumptios 2-4 ad 4

5 Lemma 2 (iv) because θ p θ0. Note that S ( λ ) S = S ( λ )[S S ( λ )]S = ( λ ) ( λ j, λ j0 )W j S S ( λ ) S j= λ j, λ j0 Wj S = o p (), j= by Assumptio 3 ad Lemma 7 because λ p λ0, ad therefore, (I T Ĝj) (I T G j ) = I T (Ĝj G j ) = Ĝj G j (2) = W j S ( λ ) W j S S W j ( λ ) S = o p (). Moreover, by Lemmas 4 (ii) ad 8, Σ Σ = o p (). Thus, Σ (I T Ĝj) Σ (I T G j ) (3) = Σ [(I T Ĝj) (I T G j )] + ( Σ Σ )(I T G j ) Σ (I T Ĝj) (I T G j ) + Σ I T G j = o p (), which i tur implies that [Z (π 0 π) + Σ (λ j0 λ j )(I T G j )Z π 0 ] [ Σ (I T Ĝj) Σ (I T G j )]Z π = o p (). j= Therefore, (ϑ)[ Σ (I T Ĝj)Z π Σ (I T G j )Z π 0 ] = o p () for j =,..., p. Ad (ϑ)( Σ Z Σ Z ) = (ϑ)( Σ Σ )Z = o p () also follows from the above results. Thus, we coclude that (ϑ)( Q Q ) = o p() uiformly i θ Θ. Similarly, by expadig (ϑ) = d (ϑ) + [I T S (λ)s ], we have for l =,..., m, [ (ϑ)( P l, P l, ) (ϑ) tr(( P l, P l, )Σ (δ))] (4) = [ [I T S (λ)s ] ( P l, Pl, )[I T S (λ)s ] tr(( P l, Pl, )Σ (δ))] + 2 [ [I T S (λ)s ] ( P l, Pl, )[Z (π 0 π) + (λ j0 λ j )(I T G j )Z π 0 ] + [Z (π 0 π) + [Z (π 0 π) + j= (λ j0 λ j )(I T G j )Z π 0 ] ( P l, Pl, ) j= (λ j0 λ j )(I T G j )Z π 0 ]. j= The secod term o the right had side of (4) is o p () uiformly i θ Θ by Assumptios F, F2, ad -4 ad Lemma 2 (iv). Notig that Pl, Σ is δ l, Lemmas 4 (ii) ad 8 lead to the result that P l, P l, = 5

6 o p (). So the first ad last terms o the right had side of (4) are also o p () uiformly i θ Θ by Assumptios F, F2, ad -4. Therefore, [ (ϑ)( P l, P l, ) (ϑ) tr(( P l, P l, )Σ (δ))] = o p () uiformly i θ Θ for l =,..., m. We coclude that (ĝ (θ) g (θ)) = o p() uiformly i θ Θ. Cosider ext the derivatives of ĝ (θ) ad g (θ). We have ad, for j =,..., k, 2 g (θ) j = g (θ) = (ϑ)p s, Q (ϑ) (ϑ)pm s (ϑ), (ϑ) tr(p s, Σ (δ))/. tr(pm s, Σ (δ))/ Q 2 (ϑ) j (ϑ) j P, s (ϑ) + (ϑ)p,. 2 (ϑ) j (ϑ) j Pm s (ϑ), + (ϑ)pm 2 (ϑ), j where θ j is the j-th elemet of θ. The first order derivatives of (ϑ) are, 2 tr(p s, Σ (δ)) j 2 tr(p s m, Σ (δ)) j (ϑ) = [Z, (I T W )]Y,..., (I T W p )]Y, 0,..., 0], where Y = (I T S )Z π 0 + (I T S ). The secod order derivatives of (ϑ) are 2 (ϑ) λ π = 0, 2 (ϑ) δ l = 0, where δ l is the l-th elemet of δ. It follows from Lemmas 4 ad 6 that ( ĝ (θ) g (θ) ) = o p () ad ( 2 ĝ (θ) j 2 g (θ) j ) = o p () for j =,..., k, uiformly i θ Θ. with Cosider ( Ω Ω ), where m m = Ω N = ( Q Σ ) Q 0 0, m m tr(p, Σ P, s Σ N) tr(p, Σ Pm s, Σ )..... tr(p m, Σ P s, Σ ) tr(p m, Σ P s m, Σ ) Cosider the block matrix m m. That tr( P i, Σ s P Σ j, ) tr(p i, Σ Pj, s Σ ) = o p () for i, j =,..., m follows from Lemma 4. Next, cosider the block Q Σ Q. Lemma 6 implies that Therefore, ( Q Σ Q Q Σ Q ) = [ Q Σ ( Q Q ) + ( Q Q ) Σ Q ] = o p (). ( Q Σ Q Q Σ Q ) = ( Q Σ Q Q Σ Q ) + (Q Σ Q Q Σ Q ) = Q ( Σ Σ )Q + o p () = o p (),., 6

7 by Lemmas 4 ad 8. I coclusio, ( Ω Ω ) = o p(). As the limit of Ω exists ad is a osigular matrix, it follows that ( Ω ) ( Ω ) = o p () by the cotiuous mappig theorem. Furthermore, because (ĝ (θ) g (θ)) = o p(), ad [g (θ) E(g (θ)] = o p() uiformly i θ Θ, ad sup θ Θ E(g (θ) = O() (see the proof of Propositio 2), hece g (θ) ad ĝ (θ) are O p(), g uiformly i θ Θ. Similarly, (θ) ĝ, (θ), 2 g (θ) j ad 2 ĝ (θ) j for j =,..., k, are O p (), uiformly i θ Θ. With the uiform covergece i probability ad uiformly stochastic boudedess properties, the differece of F (θ) ad F (θ) ca be ivestigated. By expasio, (F (θ) F (θ)) = ĝ (θ) Ω (ĝ (θ) g (θ)) + g (θ)( Ω Ω )ĝ (θ) + g (θ)ω (ĝ (θ) g (θ)) = o p (), uiformly i θ Θ. Similarly, for each compoet θ j of θ, F ( 2 (θ) j 2 F (θ) j ) = 2 [ ĝ (θ) j Fially, because ( ĝ limit theorem i Lemma 3, (θ0) ( F (θ 0) (θ 0) = 2{ ĝ = 2 ĝ (θ 0) Ω g (θ0) ĝ Ω (θ) + ĝ ( g (θ) Ω g (θ) j + g = o p (), (θ) Ω (θ)ω 2 ĝ (θ) j 2 g (θ) j )] Ω ) = o p() as above, ad g (θ 0) = O p () by the cetral F (θ 0 ) ) Ω [ĝ (θ 0 ) g (θ 0 )] + [ ĝ (θ 0) ( Ω ) [ĝ (θ 0 ) g (θ 0 )] + o p (). Ω g (θ 0) Ω ] gn(θ 0 )} As [ĝ (θ 0) g (θ 0)] = o p () by Lemma 6, ( F (θ0) F (θ 0) ) = o p (). The desired result follows from Lemma 5. Q.E.D. Proof of Propositio 5: The ML estimator θ ml is characterized by the equatios (2)-(4) i the mai text, which are Z Σ (δ) (ϑ) = 0, (δ)(i T W j S (λ))z π] (ϑ)+ (ϑ)σ (δ)(i T W j S (λ)) (ϑ) tr[σ (δ)(i T W j S (λ))σ (δ)] = 0, [Σ ad (ϑ) Σ (δ) δ (ϑ) tr[ Σ (δ) Σ (δ)] = 0. l δ l 7

8 for j =,..., p ad l =,..., h + 2q + 2. Obviously, θ ml is the solutio of a ĝ ml, (θ) = 0, with a = I k x+t (k x+k z) I p I p 0, I h+2q+2 ad ĝ ml, (θ) = { (ϑ)σ ( δ ml )[Z, (I T W S ( λ ml ))Z π ml,..., (I T W p S ( λ ml ))Z π ml ], (ϑ)σ ( δ ml )(I T W S ( λ ml )) (ϑ) tr[σ ( δ ml )(I T W S ( λ ml ))Σ (δ)],..., (ϑ)σ ( δ ml )(I T W p S ( λ ml )) (ϑ) tr[σ ( δ ml )(I T W p S ( λ ml ))Σ (δ)], (ϑ) Σ ( δ ml ) α (ϑ) tr[ Σ ( δ ml ) Σ (δ)],..., (ϑ) Σ ( δ ml ) α α (ϑ) tr[ Σ ( δ ml ) Σ (δ)], h α h (ϑ) Σ ( δ ml ) γ (ϑ) tr[ Σ ( δ ml ) Σ (δ)],..., (ϑ) Σ ( δ ml ) γ γ (ϑ) tr[ Σ ( δ ml ) Σ (δ)], q γ q (ϑ) Σ ( δ ml ) η (ϑ) tr[ Σ ( δ ml ) Σ (δ)],..., (ϑ) Σ ( δ ml ) η η (ϑ) tr[ Σ ( δ ml ) Σ (δ)], q η q (ϑ) Σ ( δ ml ) ρ (ϑ) tr[ Σ ( δ ml ) Σ (δ)], (ϑ) Σ ( δ ml ) ρ (ϑ) tr[ Σ ( δ ml ) Σ (δ)]}. Ad it follows by similar argumets as i the proof of Propositio 4 that a ĝ ml, (θ) = 0 is asymptotically equivalet to a g ml, (θ) = 0 i the sese that their cosistet roots have the same limitig distributio, where g ml, (θ) = { (ϑ)σ [Z, (I T W S )Z π 0,..., (I T W p S )Z π 0 ], (ϑ)σ (I T W S ) (ϑ) tr[σ (I T W S )Σ (δ)],..., (ϑ)σ (I T W p S ) (ϑ) tr[σ (I T W p S )Σ (δ)], (ϑ) Σ α (ϑ) Σ γ (ϑ) Σ η (ϑ) Σ ρ (ϑ) tr[ Σ α (ϑ) tr[ Σ γ (ϑ) tr[ Σ η (ϑ) tr[ Σ ρ σ 2 e Σ (δ)],..., (ϑ) Σ Σ (δ)],..., (ϑ) Σ γ q Σ (δ)],..., (ϑ) Σ η q Σ (δ)], (ϑ) Σ σ 2 e σ 2 e α (ϑ) tr[ Σ Σ (δ)], h α h (ϑ) tr[ Σ Σ (δ)], γ q (ϑ) tr[ Σ Σ (δ)], η q (ϑ) tr[ Σ σ 2 Σ (δ)]}, e The vector of empirical momets g ml, (θ) cosists of liear ad quadratic fuctios of (ϑ), hece the correspodig optimal GMM estimator derived from mi g ml, (θ)ω g ml, (θ) is i the class of M. As the BGMM estimator is the most effi ciet estimator i M, the BGMM estimator is effi ciet relative to the ML estimator. O the other had, the ML estimator attais the lower boud of Fisher Iformatio uder ormality, it is the most asymptotic effi ciet estimator. Hece, we coclude that the BGMM estimator is asymptotically effi ciet as the ML estimator uder ormality. Q.E.D. 8