Multiple-Equation GMM (II)

Transcription

1 Multiple-Equation GMM (II) Guochang Zhao RIEM, SWUFE Week 13, Fall 2015 December 10, / 34 Guochang Zhao RIEM, SWUFE Multiple-Equation GMM

2 Review Identification assumptions of the multiple-equation GMM Formula defined 2 / 34 Guochang Zhao RIEM, SWUFE Multiple-Equation GMM

3 Outline Large-sample properties and hypothesis tests Multiple-equation vs. single-equation estimation 3 / 34 Guochang Zhao RIEM, SWUFE Multiple-Equation GMM

4 Large-sample theory The large-sample theory for the multiple-equation GMM estimator is exactly the same as that for the single-equation GMM estimator. Cross-equation restriction: Take the wage equation in panel data as an example, we restrict β 1 = β 2 and π 1 = π 2. Test of overidentifying restrictions: the degrees of freedom for the J statistic are m K m m L m, and that for the C statistic is the total number of suspect instruments from different equations. 4 / 34 Guochang Zhao RIEM, SWUFE Multiple-Equation GMM

5 Large-sample theory (cont.) Consistent estimation of contemporaneous error cross-equation moments Let ˆδ m be a consistent estimator of δ m, and let ˆε im y Im z imˆδ m be the implied residual for m=1,2,...,m. Under assumptions of linearity, ergodic stationarity and the assumption that E(z) im z) ih exists and is finite for all m, h (=1,2,...,M), ˆσ mh p σ mh, where ˆσ mh = 1 n n ˆε imˆε ih and ˆσ mh = E[ε im ε ih ] i=1 5 / 34 Guochang Zhao RIEM, SWUFE Multiple-Equation GMM

6 his leaves Proposition 3.4 about consistently estimating S. The assumption ponding Large-sample to ssumption theory 3.6 (the (cont.) finite fourth-moment assumption) is ption Finite 4.6 (finite fourth fourth moments moments): E[(ximk z ~ ~ exists ~ ) and ~ is ] finite for (= 1,2,..., K,), j (= 1,2,..., Lh), m, h (= 1,2,..., M), where ximk is E[(x imk z ihj ) 2 ] exists and is finite for all k(= 1, 2,..., K m ), th element j(= 1, of 2, xi,..., and L h ), zih, m, is h(= the j 1, -th 2, element..., M), of zih. where x imk is the k-th element of x im and z ihj is the j-th element of z ih. ultiple-equation version of the formula (3.5.10) for consistently estimating The consistent estimate of S in the mulitple-equation model is me consistent for some estimate consistent iim of estimate ~ i,. ˆε im of ε im. 6 / 34 Guochang Zhao RIEM, SWUFE Multiple-Equation GMM

7 Large-sample theory (cont.) Table 4.1: Multiple-Equation GMM in the Single-Equation Format Sample nalogue of - - Orthogonality Conditions: gn(6) = s,, - Sxz6 = 0 GMM Estimator: i (G) = ( s ~ ~ ~ s ~ ~ ) - ~ s ~ Its Sampling Error: symptotic Variance of Optimal GMM: Its Estimator: J Statistic: i(g) - 6 = (s&gs,)-'s~,@~ ~var(i(s-l)) = (X;,S-~ zm)-l ~var(i(s-l)) = (S&S-'SXz)-' J(~(S-'), Spl) = n. gn (i(s-l))ls-lgn($(spl)) gi Single-Equation GMM applied to the Multiple-Equation GMM equation in question xi. ~i (4.1.4) 7 / 34 Guochang Zhao RIEM, SWUFE Multiple-Equation GMM

8 Its Estimator: Its Estimator: Large-sample theory (cont.) J Statistic: ~var(i(s-l)) = (S&S-'SXz)-' J(~(S-'), Spl) = n. gn (i(s-l))ls-lgn ($(Spl)) gi Single-Equation GMM applied to the Multiple-Equation GMM equation in question xi. ~i ~i (4.1.4) Size of W X xz xz KxK E(x~z:) Ern Km x E m (4.1.9) Km Estimator consistent under which 7 / 34 Guochang Zhao RIEM, SWUFE Multiple-Equation GMM

9 Its Estimator: ~var(i(s-l)) = (S&S-'SXz)-' Size of W KxK J Statistic: J(~(S-'), Spl) = n. gn (i(s-l))ls-lgn Ern Km x E ($(Spl)) m Km X xz E(x~z:) (4.1.9) Single-Equation GMM applied to the Multiple-Equation GMM Large-sample theory (cont.) equation in question Estimator gi consistent xi. ~i (4.1.4) under which assumptions? Estimator asymptotic normal under which X xz I Ern Km x E m S Size +p S of under W KxK Km 3.1, 3.2, 3.6, E(gig:) finite 4.1,4.2,4.6, E(gig:) finite which assumptions? I E(x~z:) (4.1.9) d.f. of J I K-L I E,(Km-Lm) Estimator consistent under which 7 / 34 Guochang Zhao RIEM, SWUFE Multiple-Equation GMM

10 Multiple-equation vs. single-equation estimation The difference between the multiple-equation GMM and the equation-by-equation GMM estimation of the stacked coefficient vector δ lies in the choice of the m K m m K m weighting matrix Ŵ. When are they equivalent? (a) If all equations are just identified, then the equation-by-equation GMM and the multiple-equation GMM are numerically the same and equal to the IV estimator. (b) If at least one equation is overidentified but the equation are unrelated in the sense of 4.4.3, then the efficient equation-by-equation GMM and the efficient multiple-equation GMM are asymptotically equivalent in that n times the difference converges to zero in probability. 8 / 34 Guochang Zhao RIEM, SWUFE Multiple-Equation GMM

11 Multiple-equation vs. single-equation estimation (cont.) Joint estimation can be hazardous Except for cases (a) and (b), joint estimation is asymptotically more efficient, but The small-sample properties of the coefficient estimates of the equation in question might be better without joint estimation; The asymptotic results presumes that the model is correctly specified, and chances of misspecification increases as you add equations to the system. 9 / 34 Guochang Zhao RIEM, SWUFE Multiple-Equation GMM

12 Special case of multiple-equation GMM ssumption 7: Conditional homoskedasticity uation GMM 275 E[ε im ε ih x im, x ih ] = σ im unconditional cross moment E(&irn&,h) equals amh by the Law of Total ons. You for should all m, have h no = trouble 1, 2,..., showing M. that The (m, h) block of E[g i g i lock of E(gig:) given in (4.1.11) ] E(E~~E~~x~~x:~) = E[ε im ε ih x im x ih] E[E(E~~ = E[E(ε im ε x im x E~~ xirnxih ( xim, xih)] (by the ih x Law im, of xtotal ih )](By Expectations) law of total expectation) E[E(E~~ cih = I xim E[E(ε, xih )xim xih ε ih ] x(by im, linearity x ih )x im of x conditional ih](by linearity expectations) of conditional expe.) E[arnhxirn xih = ] E[σ (by conditional im x im x homoskedasticity) ih](by conditional homoskedasticity) omh E(xirnxih) (by linearity of = σ im E[x im x expectations). (4.5.1) ih](by linearity of expecations) S in (4.1.11) can be written as Then the S can be written as 10 / 34 Guochang Zhao RIEM, SWUFE Multiple-Equation GMM

13 Full-information instrumental variables efficient (FIVE) ince by ssumption 4.5 S is finite, this decomposition implies that E(xirnxih) xists and is finite for all m, h (= 1, 2,..., M). ce by ssumption 4.5 S is finite, this decomposition The FIVE estimator of δ, denoted ˆδ implies that E(xirnxih) FIVE, is ists and is finite for all m, h (= 1, 2,..., M). ull-information Instrumental Variables Efficient (FIVE) n estimator of S exploiting the structure ˆδ FIVE of = fourth ˆδ(Ŝ 1 moments ) shown in (4.5.2) is ll-information Instrumental Variables Efficient (FIVE) estimator of S exploiting the structure of fourth moments shown in (4.5.2) is with Ŝ given by here, for some consistent estimator 6, of 6, and ere, for some consistent estimator 6, of 6, y Proposition for some 4.1, consistent Zmh estimator +p arnh provided ˆδ (in m addition of δ. to ssumptions 4.1 and 4.2) Proposition 4.1, Zmh +p arnh provided (in addition to ssumptions 4.1 and 4.2) at 11 E(zimzih) / 34 Guochang is finite. Zhao By ergodic RIEM, stationarity, SWUFE Multiple-Equation ximxih converges GMM in probabil- Ci

14 by (the multiple-equation adaptation of) Proposition 3.1. Because g is consistent for Large-sample S, the estimator properties is efficient by ofproposition FIVE 3.5. Thus, we have proved Suppose ssumptions hold. Suppose, furthermore, that Proposition E[z im z ih ] 4.4 exists (large-sample and is finite properties for all of m, FIVE): h(= 1, Suppose 2,..., M). ssumptions Let S and 4.7 hold. Suppose, furthermore, that E(zim zi, ) exists and is finite for all and Ŝ be as (4.5.2) and (4.5.3) respectively. Then rn, h (= l,2,..., M).3 Let S and IS be as in (4.5.2) and (4.5.3), respectively. Then (a) Ŝ p S; (a) s +, (b) ˆδ S; FIVE, defined as ˆδ(Ŝ 1 ), is consistent, asymptotically (b) inve, defined as i (g-'), is consistent, asymptotically normal, and efficient, normal and efficient, with var(ˆδ F IV E ) given by (4.3.3). with ~var(&~) given by (4.3.3); (c) The estimated asymptotic variance given in (4.3.4) is (c) The estimated asymptotic variance given in (4.3.4) is consistent forv~(6~~ ; consistent for var(ˆδ FIVE ). (d) (Sargan's statistic) (d) (Sargan s statistic) where where g,, (.) gis n given (.) isin given (4.2.1). in (4.2.1). Usually, the initial estimator im needed to calculate 12 / 34 Guochang Zhao RIEM, SWUFE Multiple-Equation GMM is the 2SLS estimator.

15 Stage Least Squares (3SLS) he Three-stage set of instruments least is the squares same across (3SLS) equations, the FIVE formula can lified somewhat. When thethe set of simplified instruments formula is the same is the across 3SLS equations, estimator, the denoted Multiple-Equation GMM 2 TO s.4 this end, FIVE let reduces to 3SLS estimator, denoted ˆδ 3SLS. Then the Error M x term: M matrix of cross moments of ~ i denoted, C, can be written as Multiple-Equation GMM 27 Then the M x M matrix of cross moments of ~ i denoted, C, can be written as The cross moments: To estimate C consistently, we need an initial consistent estimator of 6, for t purpose of calculating the residual iim. The term 3SLS comes from the fact th e stimator the 2SLS is due to estimator Brundy and of Jorgenson 6, is used (1971). as the initial estimator. Given the residuals th assumption s calculated, is needed a natural for estimator to be consistent. of C See is Proposition 4.1. stimator is due to Zellner and Theil(1962). To estimate and its C consistently, sample correspondence: we need an initial consistent estimator of 6, for th purpose of calculating the residual iim. The term 3SLS comes from the fact tha the 2SLS estimator of 6, is used as the initial estimator. Given the residuals thu calculated, a natural estimator of C is 13 / 34 Guochang Zhao RIEM, SWUFE Multiple-Equation GMM

16 which sion K, is then a matrix gi in whose (4.1.4), (m, S h) in element (4.5.2), is and the estimated in (4.5.3) cross can be moment written given compactl in (4.5.4). using the Kronecker product5 as If xi (= xil = xi2 =... = xim) is the common set of instruments with dimension K, then gi in (4.1.4), S in (4.5.2), gi and = Ei in 8 (4.5.3) Xi can be written compactly The g i, S and Ŝ can be written (MK XI) compactly using Kronecker using the Kronecker product5 as product as Three-stage least squares (3SLS) (cont.) gi = Ei 8 Xi (MK XI) Chap This implies that the x:=, xixi)-'. weighting The Kronecker matrix Ŵ product is such decomposition that (4.5.9) o S and the nonsingularity of S (by ssumption 4.5) imply. (; that kxix;)-l, Wmh (= (m, h) block of W) = a both C and E(xix (4.5 so are S-I nonsingular. = 2-I 8 (i x:=, xixi)-'. The Kronecker product decomposition i=l (4.5.9) of S and To the nonsingularity of S (by ssumption 4.5) imply that both C and E(xix:) where achieve ˆσ further rewriting of the 3SLS formulas, we need to go back t are (4.2.6), nonsingular. where which, Zmh mh is the (m, h) element of 1. is with the $? (m, = h) S-', element writes of 2-'. out in Substitute full the efficient this into (4.2.6) multiple-equation to obtain GMM To estimator. achieve further The formula rewriting (4.5.10) of the 3SLS implies formulas, that the we $? in need (4.2.6) to go is back such to that (4.2.6), which, with $? = S-', writes out in full the efficient multiple-equation so S-I = 2-I 8 (i 14 / 34 Guochang Zhao RIEM, SWUFE Multiple-Equation GMM

17 here Three-stage Zmh is the (m, h) least element Wmh squares of 2-'. Substitute (3SLS) this into (cont.) (4.2.6) to obtain The 3SLS estimator: (= (m, h) block of W) = a (4.5.11). (; kxix;)-l, where Zmh is the (m, h) element of 2-'. Substitute this into (4.2.6) to obtain i=l here where where imilarly, substituting into (4.3.3) the expression for C,, in (4.1.9) and the expresion for S in Similarly, (4.5.9) substituting and using into formula (4.3.3) (.6) the expression of the ppendix, for C,, in (4.1.9) we obtain and the the expres- xpressions sion for ~ar(c5~~~~): for S in (4.5.9) and using formula (.6) of the ppendix, we obtain the expressions for ~ar(c5~~~~): 15 / 34 Guochang Zhao RIEM, SWUFE Multiple-Equation GMM

18 Similarly, substituting into (4.3.3) the expression for C,, in (4.1.9) and the expre sion for Similarly, S in (4.5.9) substituting and using into formula (4.3.3) the (.6) expression of the for ppendix, C,, in (4.1.9) we obtain and theth expressions sion for for ~ar(c5~~~~): S in (4.5.9) and using formula (.6) of the ppendix, we ob The expressions asymptotic for ~ar(c5~~~~): variance: Three-stage least squares (3SLS) (cont.) where where mh = E(zincx;) [E(X~X;)]-~ E(X~Z:~), (4.5.1 mh = E(zincx;) [E(X~X;)]-~ E(X~Z:~), and amh is the (m, h) element of C-I. This is consistently estimated by nd and amh the estimate is the (m, of h) the element asymptotic of C-I. variance This is consistently estimated by (This sample (This version sample version can also can be also obtained be obtained directly directly by substituting by substituting (4.2.2) (4.2 an 16 / 34 Guochang Zhao RIEM, SWUFE Multiple-Equation GMM

19 (such as the n x K matrix X). Deriving such expressions is an analytical exercise to this Large-sample chapter. The properties preceding discussion of 3SLScan be summarized as Suppose ssumptions hold. nd suppose x im = x i Proposition (common4.5 set (large-sample of instruments). properties Suppose, of 3SLS): furthermore, Suppose ssumptions that exists and E[z 4.7 hold, im z and ih ] issuppose finite for xi, all = m, xi (common h(= 1, 2, set..., of M). instruments). Let Σ be Suppose, the furthermore, that E(zimzjh) exists and is finite for all m, h (= 1, 2,..., M). Let 5 M M matrix of estimated error cross moments calculated by be the M x M matrix of estimated error cross moments calculated by (4.5.7) using (4.5.7) using the 2SLS residuals. Then the 2SLS (a) ˆδ residuals. Then 3SLS, given by (4.5.12) is consistent, asymptotically (a) i3sls given by (4.5.12) is consistent, asymptotically normal, and efficient, with normal and efficient, with var(ˆδ 3SLS ) given by (4.5.12). ~var(i~~~~) given by (4.5.15). (b) The estimated asymptotic variance given in (4.5.17) is (b) The estimated asymptotic variance (4.5.17) is consistent for ~ var(i~~~~) consistent for var(ˆδ 3SLS ). (c) (Sargan's statistic) (d) (Sargan s statistic) where Ŝ = Σ ( (! xi 1 nxixi), i x K ix is i ), K is number of instruments, the number of common instruments, and and g n (.) is given in (4.2.1). where S = g,, (.) is given in (4.2.1). 17 / 34 Guochang Zhao RIEM, SWUFE Multiple-Equation GMM

20 e 18 give / 34 a proof Guochang of (4.5.16'). Zhao RIEM, Without SWUFEloss Multiple-Equation of generality, GMM suppose zim is the Seemingly unrelated regressions (SUR) Seemingly Unrelated Regressions (SUR) The ingly 3SLS Unrelated formula Regressions can further be (SUR) simplified if 3SLS formula The can 3SLSfurther can be be further simplified simplified if into SUR estimator if there is no endogeneous variables xi = union in of the (zil, sense.. of., Z~M). cross orthogonality: he difference between (4.5.20) and (4.5.21) is that the KWW equation is xi = union of (zil,..., Z~M). veridentified This is equivalent with EXPR to the as condition the additional that instrument. or is equivalent to the condition that cause SUR is a special E(zim case of. ~ih) 3SLS, = 0 formulas (m, h (4.5.12), = 1,2,.(4.5.15),.., M). and (4.5.17) (4.5.18' LS apply to isur. The implication of E(zim. ~ih) = 0 (m, the h SUR = 1,2, assumption..., M). (4.5.18) (4.5.18') That is, the The predetermined 3SLS is a special regressors case ofsatisfy FIVE, "cross" thus the orthogonalities: x i in the Âmh, is that, in - not ĉ ve expressions for mh, cmh, and mh, xi "disappears": mh only ar they predetermined and mh disappears. in each equation (i.e., E(zim - E ~ = ~ O), ) but also they ar is, the predetermined regressors satisfy "cross" orthogonalities: not only are predetermined in the other equations (i.e., E(zim. F ~ = ~ 0 ) for m # h). Th predetermined in each equation (i.e., E(zim - E ~ = ~ O), ) but also they are simplified formula is called the SUR estimator, to be denoted isur.6 etermined in the other equations (i.e., E(zim. F ~ = ~ 0 ) for m # h). The lified formula is called the SUR estimator, to be denoted isur.6

21 Review Question 7 of Section 3.8). So, for SUR, the initial estimator is the OLS estimator. The preceding discussion can be summarized as Large-sample properties of SUR Proposition Suppose ssumptions 4.6 (large-sample properties hold with of zsur): im = union Suppose of (z ssumptions i1,, z im ) Let and 4.7 Σ be hold the with M xi M = union matrix of of (zil, estimated..., z~m). error Let cross % be the moments M x M matrix of estimated calculated error by cross (4.5.7) moments usingcalculated the OLSby residuals. (4.5.7) using Thenthe OLS residuals. Then (a) (a) ˆδ SUR, given by (4.5.12) with Âmh and ĉ mh given by ( ) and isur given by (4.5.12) with imll and emh given by (4.5.13') and (4.5.14') for m, ( h = 1, ). for.., m, M h is = consistent, 1,..., M is asymptotically consistent, asymptotically normal, and efficient, normalwith ~var(&~) and efficient, given by with (4.5.15) var(ˆδ where 3SLS,/, ) given is given by (4.5.15) by (4.5.16'). where mh is given by ( ) (b) The estimated asymptotic variance (4.5.17) where xmh is given by (4.5.13') is consistent (b) The estimated for ~var(&~). asymptotic variance given in (4.5.17) is consistent for var(ˆδ 3SLS ). (c) (Sargan's statistic) (d) (Sargan s statistic) where where Ŝ = Σ ( 1 n xi xi), i x ix K i ), is Kthe isnumber numberof ofcommon instruments, instruments. and and gn (.) g is n given (.) is given in (4.2.1). in (4.2.1). = % D ( f xi 19 / 34 Guochang Zhao RIEM, SWUFE Multiple-Equation GMM

22 Since E(xix:) cannot be a zero matrix (it is assumed to be nonsingular), equations are "unrelated to each other if and only if crmh = 0. Therefore, SUR 20 / 34 Guochang Zhao RIEM, SWUFE Multiple-Equation GMM zim = SURxi) vs. into OLS(cont.) the expression for the point estimate (4.5.12). lso substituting (4.5.16') and (4.5.13') (again with zim = xi) into (4.5.15) and (4.5.17), we obtain, for multivariate regression, Since the regressors are predetermined, the system can also be estimated by the equation-by-equation OLS. The relation between SUR and equation-by-equation OLS is strictly analogous to the relation between the multiple-equation GMM and the equation-by-equation GMM. ) t least one Each equation equation is overidentified. is -Then multivariate SUR is regression. more efficient than equation-by-equation t leastols, one equation unless equations is overidentified: are "unrelated" The SUR to each is other in the sense of (4.4.3). more efficient In the present than equation-by-equation case of conditional homoskedasticity OLS, unless and the common set equations of instruments, are unrelated (4.4.3) becomes to each other in the the sense of crmhe(xix:) = O forallm # h.

23 SUR vs. OLS(cont.) Multiple-Equation GMM r1 efficient equation-by-equation GMM efficient multiple-equation GMM conditional homoskedasticity + u- II equation-by-equation 2SLS II SUR assumption (4.5.18), i.e., endogenous regressors + satisfy "cross" orthogonalities equation-by-equation OLS I E r l Figure 4.1 : OLS and GMM nother way to see the efficiency of SUR is to view the SUR model as a multivariate regression model with a priori exclusion restrictions. s an example, expand the two-equation system of Example 4.3 as 21 / 34 Guochang Zhao RIEM, SWUFE Multiple-Equation GMM

24 Relationships between multiple-equation estimators Table 4.2: Relationship between Multiple-Equation Multiple- equation GMM FIVE The model S (E var(g)) $ ssumptions (4.1.11) (4.3.2) ssumptions , ssumption 4.7 E(Z~,Z;~) finite (4.5.2) (4.5.3) (4.3.3) (4.3.3) (4.3.4) (4.3.4) ssumption 4.7 E(zirnzih) finite xi, xi, = xi for all m xi = u C 8 E(x~x:) C 8 (n-i C~X~X;) C C from 2SLS residuals C fr (4.5.12) with (4.5.13), (4.5.14) with (4.5.15) with (4.5.16) (4.5.17) with (4.5.13) 22 / 34 Guochang Zhao RIEM, SWUFE Multiple-Equation GMM

25 Relationships between multiple-equation estimators 2: Relationship between Multiple-Equation Table 4.2: Estimators Relationship between Multiple-Equation FIVE Multiple- equation GMM ptions , ssumption ssumptions 4.7 umption 4.7 The model E(zirnzih) finite ~,Z;~) finite xi, = xi for all m (4.5.2) S (E var(g)) C 8 E(x~x:) (4.1.11) (4.5.3) (4.3.3) (4.3.4) $ C 8 (n-i (4.3.2) C~X~X;) C from 2SLS residuals (4.5.12) with (4.5.13), (4.5.14) FIVE , ssumptions , ssumptions , ssumption 4.7 ssumption ssumption 4.7 xi, = xi for all m E(zirnzih) xi, = xi finite for all m xi, E(Z~,Z;~) finite xi = union of zil,..., Z ~M xi, Zim = xi = for xi all for m all m xi = u C (4.5.2) 8 E(x~x:) (4.5.15) (4.3.3) (4.3.3) (4.5.15) with (4.5.16) with (4.5.16') (4.5.17) (4.5.17) (4.3.4) (4.3.4) with (4.5.13) with (4.5.13') Multivariate regression C 8 E(x~x:) irrelevant C 8 (4.5.3) (n-'cixix:) C 8 (n-i C~X~X;) irrelevant C from OLS residuals C from 2SLS residuals (4.5.15) OLS formula with (4.5.16) (4.5.17) OLS formula with (4.5.13) C C fr (4.5.12) equation-by-equation (4.5.12) with (4.5.13'), (4.5.14') with (4.5.13), OLS (4.5.14) with 22 / 34 Guochang Zhao RIEM, SWUFE Multiple-Equation GMM

26 Common coefficients ssumption 1 : Linearity There are M linear equations: y im = z imδ + ε im (m = 1, 2,..., M; i = 1, 2,..., n) where n is the sample size, z im is the L m -dimensional vector of regressors, δ is the the conformable coefficient vector, and ε im is an unobservable error term in the m-th equation. Of the assumptions , only assumption 4.4 (identification) needs to be modified. 23 / 34 Guochang Zhao RIEM, SWUFE Multiple-Equation GMM

27 Of the other assumptions of the multiple-equation model, ssumptions , only ssumption 4.4 (identification) needs to be modified to reflect the common coefficient restriction. The multiple-equation version of g(wi; The multiple-equation version of g(w i ; δ) 8) is now is now Common coefficients (cont.) Multiple-Equation GMM so that E[g(wi; nd its 811 becomes expectation becomes where 24 / 34 Guochang Zhao RIEM, SWUFE Multiple-Equation GMM

28 Common coefficients (cont.) ssumption 4: identification with common coefficients The M m=1 K m L m matrix Σ xz defined in (4.6.4) is of full column rank. This is weaker than the assumption 4.4, since that requires that each equation of the system be identified. Indeed, a sufficient condition for identification is that E[x im z im ] be of full column rank for some m. 25 / 34 Guochang Zhao RIEM, SWUFE Multiple-Equation GMM

29 In order to relate this GMM estimator to popular estimators in the literature, it is necessary to write out the GMM formula in full. Substituting (4.6.5) into (4.2.3), The GMM estimator with common coefficients we obtain M M [ { ( 2 i m ) h ( 2 x i i h ) ] m=l h=l i=l i=l 26 / 34 Guochang Zhao RIEM, SWUFE Multiple-Equation GMM (4.6.6)

30 We summarize the result about the random-effects estimator as Large-sample properties of RE estimator position 4.7 (large-sample properties of the random-effects estimator): Suppose ssumptions 4.1, 4.2, 4.3, 4.4, 4.5 and 4.7 hold with pose ssumptions 4.1 ', 4.2, 4.3, 4.4 ', 4.5, and 4.7 hold with xi = union z im = union of (z i1,, z im ). Let Σ be the M M matrix l,..., zim). Let X be the M x M matrix whose whose (m,h) element is E[ε im ε ih ], and let Σ (m, h) element is E(cim cih be a consistent let 2 be a consistent estimate of X. Then estimate of Σ. Then i ~ given (a) ~ ˆδ RE by,, (4.6.8) given by is (4.6.8) consistent, consistent, asymptotically asymptotically normal, and normal efficient, wi var(ire) and given efficient, by (4.6.9). with var(ˆδ RE ) given by (4.6.9). The estimated (b) The estimated asymptotic asymptotic variance (4.6.10) variance is given consistent in (4.6.10) for var(ire). is consistent for var(ˆδ RE ). (Sargan's statistic) (d) (Sargan s statistic) where where S = 2 8 Ŝ (i = C Σ ( xixi), 1 n K i x ix is i ), K is number of instruments, the number of common instruments, an and g g,,(8) = s,, - s,,& n ( δ) = s with xy S s,, and xz δ with sxy and S S,, given by (4.6.5). xz given by (4.6.5). 27 / 34 Guochang Zhao RIEM, SWUFE Multiple-Equation GMM

31 so that Emh in (4.6.6) is given by (4.5.11), resulting in what should be called the Imposing conditional homoskedasticity ere C?mh is the 3SLS (m, estimator h) element with of common %-I. If, coefficients: in addition, the SUR condition 5.18) is assumed, Conditional then the "disappearance homoskedasticity of x" occurs + in common (4.6.7) and instruments: the efficient M estimator becomes what is called (for historical reasons) the random-effects imator: C?mh is the where (m, h) C?mh element is the of (m, %-I. h) element If, in addition, of %-I. the If, SUR in condition addition, the SUR condition ) is assumed, (4.5.18) then the is assumed, "disappearance then the of x" "disappearance occurs in (4.6.7) of and x" occurs the efficient in (4.6.7) and the efficient estimator asymptotic becomes GMM variance estimator what is is called becomes (for historical what is called reasons) (for the historical random-effects reasons) the random-effects tor: where Conditional C?mh is the homoskedasticity (m, h) element of + common %-I. If, instruments addition, the + cross estimator: SUR condition (4.5.18) orthogonality: is assumed, RE then estimator the "disappearance of x" occurs in (4.6.7) and the efficient GMM estimator becomes what is called (for historical reasons) the random-effects estimator: derive this, go back to the general formula (4.3.3) for the var of efficient mptotic M. Set variance X, as in is (4.6.4), S = X 3 E(xixi), use formula (.8) of ppendix Its with asymptotic variance is calculate the matrix product, and observe the "disappearance of x" that (4.5.16) omes (4.5.16') on page 280 under the SUR assumption (4.5.18).) It is consistly estimated by Its asymptotic variance is and rive this, go back to the general formula (4.3.3) for the var of efficient Set X, as (To in (4.6.4), derive S this, = X go 3 back E(xixi), to the use general formula formula (.8) of ppendix (4.3.3) for the var of efficient ulate 28 / the 34 matrix GMM. Guochang product, Set X, Zhao and as observe in RIEM, (4.6.4), the SWUFE "disappearance S = X 3 Multiple-Equation E(xixi), of x" use that formula GMM (4.5.16)(.8) of ppendix

32 For the SUR of the previous section, we obtained % from the residuals from t Pooled OLS write the formulas equation-by-equation (4.6.8)-(4.6.10) in more OLS. elegant The consistency forms. is all that is required for 2, so t same procedure for 5 works here just as well, but the finite For the SUR of the previous section, we obtained Σ sample distribution from RE might be improved if we exploit the a priori restriction that the coefficients the previous the section, residuals the same we obtained across fromequations % equation-by-equation from in the the residuals estimation OLS. finite sample properties of ˆδ from of 2. the So consider setting % to tion OLS. The consistency is all that is required RE for 2, might so the be improved if for 5 works we here exploit just as well, the but a prior the finite restriction sample distribution that the of coefficients be roved if we the exploit same the a across priori restriction equationsthat in the coefficients estimation be of Σ. quations in the estimation of 2. So consider setting % to By setting rather than the weighting matrix in the first step as Multiple-Equation GMM but not, then we have the pooled in the first-step OLS GMM estimator: estimation for the purpose of obtaining an initial consiste estimate of 6. The estimator is (4.6.8) with emh = 1 for m = h and 0 for m # which can be written as MM estimation for the purpose of obtaining an initial consistent e estimator is (4.6.8) with emh = 1 for m = h and 0 for m # h, tten which 29 / as 34 is simply Guochang the Zhao OLS estimator RIEM, SWUFE on the sample Multiple-Equation of size Mn GMM where observations

33 to the failure of the "cross" orthogonalities. But it is important to keep in mind that the standard errors printed out by the OLS package, which do not take into account the interequation cross moments (amh), are biased. For the pooled OLS estimator, which is a GMM estimator with a nonoptimal choice the correct formula for the asymptotic variance is (3.5.1) of Proposition 3.1. Setting W = IM 3 [E(xix:)]-', S = X 3 E(x;xi), X,, as in (4.6.4), and again observing the The disappearance asymptotic of x, we variance obtain of the pooled OLS estimator is Pooled OLS (cont.) which is consistently estimated by Since the pooled OLS estimator is consistent, the associated residual can be used to calculate emh (m, h = 1,..., M) in this expression. The correct standard errors of pooled OLS are the square roots of (lln times) the diagonal element of this matrix. 30 / 34 Guochang Zhao RIEM, SWUFE Multiple-Equation GMM

34 GMM estimation in Stata 31 / 34 Guochang Zhao RIEM, SWUFE Multiple-Equation GMM

35 GMM estimation in Stata 31 / 34 Guochang Zhao RIEM, SWUFE Multiple-Equation GMM

36 GMM estimation in Stata 31 / 34 Guochang Zhao RIEM, SWUFE Multiple-Equation GMM

37 GMM estimation in Stata 31 / 34 Guochang Zhao RIEM, SWUFE Multiple-Equation GMM

38 GMM estimation in Stata 31 / 34 Guochang Zhao RIEM, SWUFE Multiple-Equation GMM

39 Good book to recommend 32 / 34 Guochang Zhao RIEM, SWUFE Multiple-Equation GMM

40 Good book to recommend China has 56 ethnic groups, but the existing history books mainly talk about the Han ethnic group. Han and Huns; Tang and Turkic; Song and Khitan, Jurchen and Mongolia... What is the relationship between the ancient and contemporary minorities? 33 / 34 Guochang Zhao RIEM, SWUFE Multiple-Equation GMM

41 Good book to recommend Oxford Economic Papers, 67(2), 2015, doi: /oep/gpu032 dvance ccess Publication Date: 27 October 2014 Climate shocks, dynastic cycles and nomadic conquests: evidence from historical China By Qiang Chen School of Economics, Shandong University, 27 Shanda Nanlu, Jinan, Shandong Province, China, ; bstract Nomadic conquests have helped shape world history, yet we know little about why they occurred. Using a unique climate and dynastic data set from historical China dating from 221 BCE, this study finds that the likelihood of nomadic conquest increases with less rainfall proxied by drought disasters, which drove pastoral 34 / 34 Guochang Zhao RIEM, SWUFE Multiple-Equation GMM

42 School of Economics, Shandong University, 27 Shanda Nanlu, Jinan, Shandong Province, China, Good book to recommend ; bstract Nomadic conquests have helped shape world history, yet we know little about why they occurred. Using a unique climate and dynastic data set from historical China dating from 221 BCE, this study finds that the likelihood of nomadic conquest increases with less rainfall proxied by drought disasters, which drove pastoral nomads to attack agrarian Chinese for survival. Moreover, consistent with the dynastic cycle hypothesis, the likelihood of China being conquered increases with the number of years earlier that a Chinese dynasty had been established (and hence was weaker, on average) relative to a rival nomadic regime. These results survive a variety of robustness checks. JEL classifications: N45, O13 1. Introduction Nomadic conquests have helped shape world history. We may indulge ourselves for a moment to imagine the following counter-factuals: what if Western Europe did not fall to semi-nomadic Germanic tribes; Western Europe was conquered by the Huns, rabs, or Mongols; Kievan Rus did not succumb to Mongolian invaders; or Ming China did not give way to the Manchu Qing? Historians have provided vivid accounts of the rise of nomadic powers and their conquests (e.g., Grousset 1970), yet little is known about why nomadic 34 / 34conquests Guochang occurredzhao or did not RIEM, occur. SWUFE In this article, Multiple-Equation I use a unique dynastic GMM and climate data