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Econometrics II. Lecture Notes 1 ESTIMATING SYSTEMS OF EQUATIONS BY OLS, GLS and GMM 1. Introduction: SUR and Linear Panel Data models. 2. System OLS estimation of Multivariate Linear Systems. 3. GLS and FGLS estimation of Multivariate Linear Systems. 4. Examples (a) The SUR model. (b) Panel Data Model 5. A General Linear System of Equations with Endogenous Regressors 6. Generalized Method of Moments Estimation (a) The System 2SLS Estimator (b) Optimal Estimates (c) The 3SLS (FIV) Estimator 7. Testing using GMM 8. Optimal instruments 1 Econometrics II-1. Systems of Equations. 2009/10 UC3M. Master in Economic Analysis 1.1 Introduction We consider diﬀerent methods of estimation of systems of linear equations: system OLS, GLS and FGLS. GLS is more eﬃcient at the cost of more stringent assumptions, while system OLS may have interpretations in terms of single equation OLS. EX. 1 (SUR: Seemingly Unrelated Regressions) Population model of G linear equa- tions y1 = x1 β 1 + u1 y2 = x2 β 2 + u2 xg β g g = 1, . . . , G (1.1) Kg ×1 Kg ×1 ··· yG = xG β G + uG xg might be the same for each equation, but could have diﬀerent dimensions. The regres- sions are seemingly unrelated because the parameter vectors β g are diﬀerent. However there could be correlation across the errors ug . Random draws: yig = xig β g + uig , i = 1, . . . , n. Inferences are done as n tends to inﬁnity. For the study of the properties of diﬀerent estimates of β g we need assumptions on the relationship of the explanatory variables (x1 , x2 , . . . , xG ) and the unobservables ug . If the system is structural (without omitted variables, errors-in-variables or simultaneity), then we can assume that E [ug |x1 , x2 , . . . , xG ] = 0, g = 1, . . . , G. (1.2) (1.2) implies that ug is uncorrelated with the explanatory variables in all equations. • If the regressors are the same for all equations then the assumption is only E [ug |x] = E [ug |xg ] = 0. •If the xg are not the same, then the variables excluded for equation g, have no eﬀect on yg once xg has been taken into account: E [yg |x1 , x2 , . . . , xG ] = E [yg |xg ] = xg β g , g = 1, . . . , G. 2 Econometrics II-1. Systems of Equations. 2009/10 UC3M. Master in Economic Analysis EX. 2 (Panel Data Models) For each cross section unit we observed data on all the set of variables for T periods: yt = xt β + ut t = 1, . . . , T xt β (1.3) K×1 K×1 The model is static if all the variables in xt are contemporaneous (no lagged variables). • In (1.1) each equation explains a diﬀerent dependent variable for the same cross sec- tion unit. • In (1.3) there is only a single dependent variable, but observed -together with the ex- planatory variables- in several periods. However the statistical properties of estimates can be studied under the same set-up. Diﬀerent types of exogeneity: • Contemporaneous exogeneity: E [ut |xt ] = 0, t = 1, . . . , T. (1.4) • Strict exogeneity: E [ut |x1 , x2 , . . . , xT ] = 0, which is stronger than contemporaneous exogeneity, and together with the model (1.3) implies E [yt |x1 , x2 , . . . , xT ] = E [yt |xt ] = xt β. Strict exogeneity may fail if the regressors contain lagged endogenous variables, xt = (1, yt−1 ) , or in the presence of a ﬁnite distributed lag model. Which condition is assumed, determines the consistency of the estimation procedures of β, and the validity of inference rules. 3 Econometrics II-1. Systems of Equations. 2009/10 UC3M. Master in Economic Analysis 1.2 System OLS Estimation of Multivariate Linear Systems We have IID cross section observations Xi , yi : i = 1, 2, . . . , n K×G G×1 of the model y = X β + u, with β (1.5) K×1 The idea is to use the form of the covariance matrix of u to obtain more eﬃcient estimates than single equation methods. EX. 1 (SUR) y = (y1 , y2 , . . . , yG ) , u = (u1 , u2 , . . . , u) , x1 0 0 · · · 0 β1 0 x2 0 0 β2 ... . . , β = X = 0 0 . K×G . . K×1 . . . . 0 βG 0 0 0 · · · xG where K = K1 + K2 + · · · + KG . EX. 2 (Panel Data Models) Here X = (x1 , x2 , . . . , xT ) , K×T so all equations have the restriction of having the same parameter vector. 4 Econometrics II-1. Systems of Equations. 2009/10 UC3M. Master in Economic Analysis ASS. 1 (Orthogonality) E [Xu] = 0. This Assumption is similar to the orthogonality condition for OLS estimation of single equations, though it has diﬀerent meanings for each application in terms of the compo- sition of X. In most applications some elements of X are equal to 1, so ASS. 1 implies that E [u] = 0. EX. 1 (SUR) Here Xu = (x1 u1 , x2 u2 , . . . , xG uG ) , so ASS. 1 holds iﬀ E [xg ug ] = 0, g = 1, 2, . . . , G, but does not require xg and uh to be orthogonal for h = g. EX. 2 (Panel Data Models) Here T Xu = xt ut , t = 1, 2, . . . , T, t=1 so a suﬃcient natural condition for ASS. 1 to hold is E [xt ut ] = 0, t = 1, 2, . . . , T. Like (1.4), this allows xt and us to be correlated when s = t, but is weaker than strict exogeneity. ASS. 1 is the weakest for get consistent consistent estimates of β in a regression frame- work. Much stronger is the assumption that E [u|X] = 0 (1.6) which implies that every element of u and every element of X are uncorrelated. 5 Econometrics II-1. Systems of Equations. 2009/10 UC3M. Master in Economic Analysis Under ASS. 1 the vector β satisﬁes E [X (y − X β)] = 0, (1.7) or E [XX ] β = E [Xy] . Since XX is (random) symmetric, positive semideﬁnite (psd), then E [XX ] is also K × K symmetric, psd matrix. To be able to estimate β we need that it is the only vector that satisﬁes (1.7). ASS. 2 (Rank Condition) E [XX ] is nonsingular (has rank K). Then, under ASS. 1 and ASS. 2 we can write −1 β = E [XX ] E [Xy] , so these assumptions identify the vector β. By the analog principle we can estimate β by the sample analogue ˆ −1 β n = En [XX ] En [Xy] , which is the System Ordinary Least Squares (SOLS) Estimator, given En [XX ] is positive deﬁnite. Consistency of the SOLSE follows by taking probability limits and the WLLN: THM. 1 (Consistency of SOLS) Under ASS. 1 and 2 ˆ β n →p β as n → ∞. Depending on the structure of β and X the solved OLS problem will be diﬀerent. 6 Econometrics II-1. Systems of Equations. 2009/10 UC3M. Master in Economic Analysis EX. 1 (SUR) Here En [x1 x1 ] 0 0 ··· 0 En [x1 y1 ] 0 En [xG xG ] 0 0 En [x2 y2 ] .. . . En [XX ] = 0 0 . . , En [Xy] = . . . . . . . 0 En [xG yG ] 0 0 0 · · · En [xG xG ] ˆ ˆ ˆ ˆ ˆ Therefore SOLS can be written as β n = β n1 , β n2 , . . . , β nG where each β ng is the single-equation OLS estimator of the g-equation: System OLS of a SUR model is equivalent to OLS equation by equation (without extra restrictions on the parameter vector). EX. 2 (Panel Data Models) Here T T En [XX ] = En xt xt , En [Xy] = En xt yt , t=1 t=1 so n T −1 n T ˆ βn = xit xit xit yit , i=1 t=1 i=1 t=1 which is called the Pooled Ordinary Least Squares (POLS) Estimator because is equivalent to run OLS for all observations running in both indexes, i and t (pooling or staking all observations of, e.g., yit in a single vector of dimension T n × 1). The POLSE is consistent under the orthogonality condition (1.4) and that T rank E xt xt = K. t=1 In the general system (1.5), System OLS may have not an interpretation in terms of equation by equation or pooled OLSE, e.g. when we impose in the SUR model cross- equation restrictions. Unbiasedness (conditional on X) follows under the additional assumption that rank(En [XX ]) = K, together with E[u| X] = 0 (which implies ASS. 1.) 7 Econometrics II-1. Systems of Equations. 2009/10 UC3M. Master in Economic Analysis THM. 2 (CAN of SOLS) Under ASS. 1 and 2 √ ˆ n β n −β →d N(0, A−1 BA−1 ), where A := E [XX ] B := E [Xuu X ] if the elements of Xuu X have ﬁnite expected absolute value. AVar Estimation. Consistent estimation of A is simple by means of ˆ An := En [XX ] , while a consistent estimate of B can developed by the analogy principle, since En [Xuu X ] →p B := E [Xuu X ] . Therefore, because the u are not observed, we use instead the SOLS residuals: ˆ ˆ un = y − X β n = u − X β n −β , ˆ and given that the expectation of Xuu X is ﬁnite, then ˆ Bn := En [Xˆ u X ] →p B. uˆ √ ˆ Therefore AVar n βn − β is consistently estimated by ˆ ˆn ˆ ˆn Vn := A−1 Bn A−1 , (1.8) which is a robust variance matrix estimator, because does not require particular assumptions on the second moments of u : • The unconditional variance matrix Ω := V [u] = E [uu ] is unrestricted, allowing: - in a SUR system: for cross equation restrictions and diﬀerent variances in each equation. - in Panel data models: for arbitrary serial correlation and time-varying variances in the disturbances. • The conditional variance matrix V [u|X] can depend on X in any way. 8 Econometrics II-1. Systems of Equations. 2009/10 UC3M. Master in Economic Analysis In some cases it can be desirable to impose more structure on the conditional and unconditional covariance matrix of u to simplify its estimation, such as Ω := E [uu ] = V [u|X] . For testing H0 : R β = r q×K we can use the Wald statistic −1 ˆ W n = n Rβ n − r ˆ RV n R ˆ Rβ n − r which under H0 , converges in distribution to a χ2 . q In the SUR model this is the most general form of testing cross equation restrictions among the parameters in diﬀerent equations. 9 Econometrics II-1. Systems of Equations. 2009/10 UC3M. Master in Economic Analysis 1.2.1 GLS and FGLS Estimation of Linear Systems If we strengthen ASS. 1 and add assumptions on the conditional variance matrix of u, V [u|X] , we can do better than OLS by means of GLS. As in single equation GLS, the idea is to transform the model into a system of equations where the error has a scalar variance-covariance matrix, multiplying (1.5) by Ω−1/2 : Ω−1/2 y = Ω−1/2 X β + Ω−1/2 u, or y∗ = X∗ β + u∗ , where E [u∗ u∗ ] = IG . Then the Generalized Least Squares (GLS) Estimator of β is En [X∗ X∗ ]−1 En [X∗ y∗ ] , i.e., ˆ GLS β n := En XΩ−1 X −1 En XΩ−1 y . For consistent GLS estimation we need that each element of u is uncorrelated with each element of X : ASS. 3 (GLS Orthogonality) E [X ⊗ u] = 0. In practice, one element of X is 1, so ASS. 3 implies that E [u] = 0. This is stronger than ASS. 1 and a suﬃcient condition is the zero mean conditional expectation (1.6). For GLS the key element is the second-moment matrix of u, Ω := E [uu ] , so in place of ASS. 2 we impose a not too restrictive weighted version: ASS. 4 (GLS Rank Condition) Ω is positive deﬁnite and E XΩ−1 X is nonsingu- lar (has rank K). 10 Econometrics II-1. Systems of Equations. 2009/10 UC3M. Master in Economic Analysis Consistency of GLS under ASS. 3 and ASS. 4. We can write that ˆ GLS β n − β := En XΩ−1 X −1 En XΩ−1 u . By the WLLN, En XΩ−1 X →p E XΩ−1 X . By ASS. 4 and Slutsky’s Theorem −1 En XΩ−1 X →p A−1 , where A := E XΩ−1 X . To show that En XΩ−1 u →p 0, we can use the WLLN and that E XΩ−1 u = 0, because since each element of u is uncorrelated with each element of X, so is any linear combination of X, such as XΩ−1 : vec En XΩ−1 u = En [u ⊗ X] vec Ω−1 = 0. However consistency may fail under ASS. 1, because E [Xu] = 0 does not imply E XΩ−1 u = 0, except for particular Ω, because the transformation by Ω−1/2 induces some correlation between X∗ and u∗ . Asymptotic Normality of GLS: we need ASS. 3 and 4 and some extra moment conditions: √ −1 √ ˆ GLS n β n − β := En XΩ−1 X nEn XΩ−1 u . By the CLT, √ nEn XΩ−1 u →d N (0, B) where B := E XΩ−1 uu Ω−1 X , (given the expectation is OK) so it is immediate to obtain that √ √ ˆ GLS n βn − β = A−1 nEn XΩ−1 u + op (1) →d N 0, A−1 BA−1 . In general A = B, so we do not obtain the usual GLS AVar, A−1 . 11 Econometrics II-1. Systems of Equations. 2009/10 UC3M. Master in Economic Analysis Feasible GLS. The GLS estimate requires knowing Ω up to scale, Ω = σ 2 C, where C is a known G × G pd matrix and σ 2 can be an unknown constant. Since generally C is ˆ unknown we need some feasible procedure, replacing Ω by some consistent estimate Ωn . First-order asymptotic properties of FGLS would be equivalent to those of GLS under ASS. 3 and 4. Given that ˆ Ωn →p Ω as n → ∞, the Feasible GLS (FGLS) Estimate is −1 ˆ F GLS := En XΩ−1 X βn ˆn ˆ −1 En XΩn y . This estimate is generally known as the SUR Estimate, because it exploits the possible correlation among the componentes of u. ˆ For Ωn we can use the residuals of a ﬁrst estimation by SOLS, ˆ u ˆ Ωn := En [ˆ n un ] (1.9) ˆ ˆ where un := y −X β n and β n is the OLSE, consistent under ASS. 1 and 2 (and therefore ˆ under ASS. 3 and 4). ˆ Sometimes the elements of Ω are restricted, so Ωn can exploit these restrictions, but if false the estimates would be inconsistent in general. THM. 3 (CAN of FGLS) Under ASS. 3 and 4 √ ˆ F GLS −β →d N(0, A−1 BA−1 ), n βn where A := E XΩ−1 X , B := E XΩ−1 uu Ω−1 X if the elements of Xuu X have ﬁnite expected absolute value. ˆ F GLS and β GLS are equivalent for asymptotic inference: it does not matter Therefore β n ˆn that Ω has to be estimated, though, undoubtedly, it will aﬀect ﬁnite sample performance. The estimate of the AVar of β n ˆ F GLS is n−1 A−1 Bn A−1 , which is valid under ASS. 3 and 4, ˆn ˆ ˆn F GLS where, un := y − X β n ˜ ˆ , ˆ ˆ −1 An := En XΩn X ˆ ˆ −1 ˜ ˜ ˆ n Bn := En XΩn un un Ω−1 X , 12 Econometrics II-1. Systems of Equations. 2009/10 UC3M. Master in Economic Analysis Asymptotic Variance of FGLS. We have not showed yet that FGLS is better in any sense compared to SOLS, and it is less robust, since needs more assumptions for consistency and asymptotic normality. However, under some additional system ho- moskedasticity assumptions it is more eﬃcient than SOLS : ASS. 5 (System Homoskedasticity) E XΩ−1 uu Ω−1 X = E XΩ−1 X , where Ω := E [uu ]. When G = 1, ASS. 5 is equivalent to the usual conditional homoscedasticity assumption for single equation OLS. If Ω is diagonal and X has the structure of SUR or panel data, ASS. 5 implies a kind of conditional homoskedasticity in each equation. A suﬃcient, but not necessary, condition is that E [uu |X] = E [uu ] = Ω. THM. 4 (Eﬃciency of FGLS) Under ASS. 3-5 the asymptotic variance of the FGLS estimator is A−1 . • This is the usual formula for the asymptotic variance of FGLS, and in this case we can ˆ use the previous estimate An (but this is non robust to cond. heteroskedasticity in u.) • Also, under the previous assumptions, the FGLS estimator is more eﬃcient than the SOLS estimator: and in fact, FGLS is more eﬃcient than any other estimator that uses the orthogonality conditions E [X ⊗ u] = 0. Testing. We can use robust or nonrobust versions of the asymptotic variance to construct t statistics, conﬁdence intervals or more general Wald statistics, with chi- square limit distributions to test H0 : R β = r. q×K • Pseudo-LR: If ASS. 5 holds under H0 , then we can deﬁne a test statistic based on the weighted sum of squared residuals, estimating the model with and without restrictions imposed on β, where the same estimate of Ω are used in both cases (so that ˆ is consistent under H0 and H1 , e.g. that based on the unrestricted SOLS residuals un ). r ˜ Thus under ASS. 3-5, if un are the residuals from constrained FGLS (with q restrictions ˜ ˆ on β n ) using variance Ωn , ˜n ˆ n ˜n ˜ ˆn ˆ LRn = n En ur Ω−1 ur − En un Ω−1 un →d χ2 . q 13 Econometrics II-1. Systems of Equations. 2009/10 UC3M. Master in Economic Analysis 1.3 The SUR model OLS equation by equation is simple and leads to standard inference under the OLS homoskedasticity assumption E u2 |xg = σ 2 . By contrast, a suﬃcient condition for g g consistency of FGLS for β g requires that E [xg uh ] = 0, g, h = 1, 2, . . . , G, which is ASS. 3 for the SUR model. However we may be interested in running FGLS because: • Eﬃciency. If we can maintain that E [uu |X] = E [uu ] , in addition to ASS. 3-4, the FGLS is asymptotically at least as eﬃcient as SOLS. • Testing. SOLS does not provide an easy way to test cross-equation hypothesis (unless we use AVar estimates such as (1.8)). OLS vs FGLS. There are two cases were FGLS is equivalent to OLS: ˆ ˆ • Ωn is diagonal. In applications Ωn should not be diagonal unless we impose such ˆ √ restriction. If Ω is diagonal, then consistency of Ωn will lead to the n-asymptotic equivalence of FGLS and OLS -though they are not algebraically equivalent. • If x1 = x2 = · · · = xG -same regressors in all equations. This means that FGLS improves eﬃciency by using exclusion restrictions in some equations (when e.g. x1 = x2 ). Without these restrictions there are no eﬃciency gains Even in this case, it is interesting to use SUR subroutines to estimate such models, in ˆ ˆ order to obtain estimates of joint covariance matrix of β n , no only that of each β ng via equation by equation OLS. Cross equation restrictions in SUR: In some models there are cross equation restrictions among the β g . These models can still be written in the general form, and therefore, are amenable for OLS and FGLS. These models are often termed ”SUR”, though now the equations are eﬀectively related. The methods rely on appropriately deﬁning β and x accordingly. Then it is quite simple to estimate and to test the restrictions, using e.g. sum of squared residuals test statistics. 14 Econometrics II-1. Systems of Equations. 2009/10 UC3M. Master in Economic Analysis 1.4 Panel Data Model We study the dynamic relationships in the model, for which we need the data ordered over time for each cross section unit in yt = xt β + ut , t = 1, 2, . . . , T • β is the same for all periods, but with particular choices of xt we can allow for parameters changing over time (e.g. period dummies). • It can be that xt is not changing over time, describing characteristic not changing over time (e.g. gender dummies). Thus it can be interesting to allow for diﬀerent intercepts for each time period if T is small and n large. Suﬃcient assumptions for Pooled OLS to estimate consistently β : ASS. 6 (Orthogonality POLS) E [xt ut ] = 0, t = 1, 2, . . . , T ASS. 7 (Rank POLS) T rank E [xt xt ] = K. t=1 • ASS. 6 says nothing about the relationship between xt and us for t = s. • ASS. 7 rules out perfect linear dependencies among explanatory variables (for all periods). 15 Econometrics II-1. Systems of Equations. 2009/10 UC3M. Master in Economic Analysis Use of OLS statistics from the POLS regression across i and t, requires additional homoskedasticity and no serial correlation assumptions: ASS. 8 (Cond. Homosked. No autocorrelation. POLS) a) E u2 xt xt t = σ 2 E [xt xt ] , t = 1, 2, . . . , T, σ 2 = E u2 , ∀t. t b) E [ut us xt xs ] = 0, t = s, t, s = 1, 2, . . . , T ASS. 8.a) holds if E [u2 |xt ] = σ 2 for all t : so the conditional variance does not depend t on xt and is constant for each period. ASS. 8.b) sets the conditional covariances equal to 0. A suﬃcient condition is that E [ut us |xt , xs ] = 0, t = s, t, s = 1, 2, . . . , T and a necessary condition is E [ut us ] = 0, t = s, if xt includes a constant. Therefore, ASS. 8 imposes a particular unconditional covariance matrix for u, E [uu ] = σ 2 IT , but also restricts the conditional covariances. Theorem 1 (Asymptotic Normality of POLS) Under ASS. 6-7 the POLS estimate is CAN. If ASS. 8 also holds then T −1 ˆ −1 AVar β n = n−1 σ 2 (E [XX ]) = n−1 σ 2 E [xt xt ] , t=1 which is equal to (nT )−1 σ 2 (E [xt xt ])−1 if E [xt xt ] is constant, and an estimate of AVar β n ˆ is n T −1 1 2 −1 1 2 1 ˆ σ (En [XX ]) = σ n ˆ xit xit , n n n n i=1 t=1 where σ 2 is the usual OLS residual-variance estimator of σ 2 from the pooled regression ˆn with nT observations. Therefore the usual t and F statistics are valid asymptotically. Note that under ASS. 8, B = σ 2 A, where now T T T A= E [xt xt ] , B= E [ut us xt xs ] , t=1 t=1 s=1 are the matrices appearing in the AVar of THM. 2 for the Panel data case. 16 Econometrics II-1. Systems of Equations. 2009/10 UC3M. Master in Economic Analysis Dynamic Completeness. Not always ASS. 8 can be maintained, but it must hold if we wish to use results from standard OLS asymptotics. A suﬃcient condition is E [yt |xt , yt−1 , xt−1 , . . . , y1 , x1 ] = E [yt |xt ] . (1.10) This establishes the dynamic completeness of the conditional mean: xt contains suﬃcient lags of all variables such that additional lagged variables have no partial eﬀect on yt . If additionally, the homoskedasticity assumption V [yt |xt ] = σ 2 holds, then ASSs. 6 and 8 both hold and standard OLS inference is valid. If (1.10) does not hold, then care must be taken when estimating the asymptotic variance of the POLS Estimator to be robust to serial correlation (and also to heteroskedasticity). However POLS is consistent in any case. Robust Asymptotic Variance Matrix. We need a consistent estimate of AVar(β) which is valid in absence of the restrictive ASS. 8. The general form of the estimator is ˆ 1ˆ ˆ ˆ Vn := A−1 Bn A−1 , n n n ˆ ˆ where An := En [XX ] and Bn := En [Xˆ n un X ] , using the T × 1 POLS residuals u ˆ ˆ un,i = yi − Xi β n for cross section observation i. In any case the data have to be stored ˆ in such a way that (yi , Xi ) are stacked on the top of one another for i = 1, . . . , n. Testing for Heteroskedasticity. The basic issue is the checking of ASS. 8 (apart from the serial correlation problem). Suppose that E [ut |xt ] = 0, t = 1, 2, . . . , T, which is slightly stronger that ASS. 6 but is weaker than strict exogeneity. Then, the null of homoskedasticity can be stated as H0 : E u2 |xt = σ 2 , t t = 1, 2, . . . , T Under H0 , u2 is uncorrelated with any function of xt : let ht be a Q × 1 vector of t nonconstant functions of xt . It can include dummy variables for diﬀerent time periods. The usual procedure is to use a LM test regressing the squares of the POLS residuals, u2 , on hi,t , ˆn,t u2 |1, hi,t , t = 1, . . . , T ; i = 1, 2, . . . , n, ˆn,i,t so the test statistic nT R2 is asymptotically χ2 under H0 . Q 17 Econometrics II-1. Systems of Equations. 2009/10 UC3M. Master in Economic Analysis 1.5 A General Linear System of Equations with En- dogenous Regressors We consider systems of equations where the explanatory variables may not satisfy the exogeneity assumptions necessary for the consistency of SOLS and GLS procedure: In- strumental Variables methods are needed. The current approach to System IV is by means of the Generalized Method of Moments (GMM). The asymptotic properties of such methods can be deduced in a similar way to the single equation framework. The most well-known application of SIV estimation is to Simultaneous Equation Models (SEM), but the methods go beyond, including the analysis of panel data models. EX. 3 (SEM: Labor Supply and Wage Oﬀer Functions) Consider a labor supply function for the hours of labor supply, hs , at any wage, ω, for a given individual: hs (ω) = γ 1 ω + z1 δ 1 + u1 where z1 is a vector of observed labor supply shifters (education, age, experience, children, etc.). Though this equation describes the utility-maximizing behaviour of an individual, we can only observe equilibrium values. A wage oﬀer function gives the hourly wage that the market oﬀers as a function of hours worked ω o (h) = γ 2 h + z2 δ 2 + u2 where z2 are productivity measures (education, experience, etc.). If we assume that the observed hours and wage are such that both equations are satisﬁed, then the equilibrium values (h, ω) satisfy h = γ 1 ω + z1 δ 1 + u1 ω = γ 2 h + z2 δ 2 + u 2 . Under restrictions on the parameters, the equations can be solved uniquely for (h, ω) as functions of z1 , z2 , u1 , u2 and the parameters. If further z1 , z2 are exogenous, such that E [u1 |z1 , z2 ] = E [u2 |z1 , z2 ] = 0, then we can estimate consistently the parameters (with the usual identiﬁcation assump- tions). Note that in general ω will be correlated with u1 in the ﬁrst equation, while h will be correlated with u2 : ω is endogenous in the ﬁrst equation, and h is endogenous in the second. 18 Econometrics II-1. Systems of Equations. 2009/10 UC3M. Master in Economic Analysis EX. 4 (Omitted Variables: student performance) Consider a model to test the eﬀect of Head Start participation (measured as the binary variable HeadStart) on sub- sequent student performance score = γ 1 HeadStart + z1 δ 1 + u1 . z1 contains other observed factors (income, education, etc.). u1 contains unobserved factors that aﬀect score, such as child’s ability, that might be correlated with HeadStart. To capture the possible endogeneity of Headstart we may set HeadStart = z δ 2 + u2 , where z should contain at least one factor aﬀecting HeadStart participation, but which does not have a direct eﬀect on score (e.g. distance): we only want to assume that E [zu2 ] = 0. Correlation between u1 and u2 indicate that Head Start is endogenous in the ﬁrst equation. The previous examples can be written as y1 = x1 β 1 + u1 y2 = x2 β 2 + u2 which is like a SUR system, but where x1 and x2 can contain also endogenous regres- sors. Since x1 and x2 are generally correlated with u1 and u2 , System OLS and GLS estimation of these equations will be inconsistent. We could apply single equation methods to each of them like 2SLS, but we could exploit the joint information of all the system variables to improve the eﬃciency. 19 Econometrics II-1. Systems of Equations. 2009/10 UC3M. Master in Economic Analysis We study the following model y = X β + u, with y , X , β . G×1 K×G K×1 The rows can represent diﬀerent time periods for the same unit or diﬀerent variables, so we cover also panel data models. ASS. 9 (Orthogonality) E[Zu] = 0 L×1 where Z is a L × G matrix of observable instrumental variables. We may assume that E [u] = 0, which would be true in most cases. ASS. 9 is not enough to identify β. A suﬃcient condition is a rank condition that generalizes the rank condition for single equations: ASS. 10 (Rank Condition) rank E[ZX ] = K. L×K A necessary condition for ASS. 10 to hold is the order condition L ≥ K. EX. 5 Consider a G equation system y1 = x1 β 1 + u1 (1.11) . . . yG = xG β G + uG where for each equation g, xg is Kg × 1 vector that may contain both exogenous and endogenous variables. This looks like the SUR system, except from the diﬀerent prop- erties of some elements of xg , which might be endogenous. Then, for each equation we have a set of Lg × 1 instrumental variables zg which are exogeneous E [zg ug ] = 0, g = 1, . . . , G. In most cases one element of zg is the intercept, so E [ug ] = 0. 20 Econometrics II-1. Systems of Equations. 2009/10 UC3M. Master in Economic Analysis If xg contains some elements correlated with ug , the zg must contain more than just the exogeneous variables appearing in each equation. In many cases the set instruments consistent of all exogenous variables in the system are valid for each equation, zg = z, g = 1, . . . , G. In some cases this is not possible. Then we have y = (y1 , y2 , . . . , yg ) , u = (u1 , u2 , . . . , ug ) , x1 0 0 ··· 0 β1 0 x2 0 0 β2 .. . . , β = X = 0 0 . . K×G . . K×1 . . . . 0 βG 0 0 0 · · · xG where X is K × G, with K = K1 + K2 + · · · + KG , and z1 0 0 · · · 0 0 z2 0 0 .. . Z = 0 0 . . . L×G . . . 0 0 0 0 · · · zG where Z is L × G, with L = L1 + L2 + · · · + LG . Then Zu = z1 u1 , z2 u2 , . . . , zg ug and E [z1 x1 ] 0 0 ··· 0 0 E [z2 x2 ] 0 0 .. . E [ZX ] = 0 0 . . . . . . 0 0 0 0 · · · E [zG xG ] where E zg xg is Lg × Kg . ASS. 10 requires that E [ZX ] is full column rank, where the number of columns is K. Since a block diagonal matrix has full column rank iﬀ each block of the matrix is full column rank: ASS. 10 holds iﬀ rank E zg xg = Kg , g = 1, . . . , G, which is exactly the rank condition needed to estimate each equation by 2SLS. There- fore identiﬁcation of the SUR system is equivalent to identiﬁcation of equation by equation. 21 Econometrics II-1. Systems of Equations. 2009/10 UC3M. Master in Economic Analysis 1.6 Generalized Method of Moments Estimation The orthogonality conditions in ASS. 9 suggest the estimation method: under ASS. 9 and ASS. 10 the vector β is the unique K × 1 vector that E [Z (y − X β)] = 0, in other words, β is identiﬁed. Therefore, by the analog principle we can estimate β by the sample analogue choosing an estimate that satisﬁes ˆ En Z y − X β n = 0, (1.12) ˆ which is a set of L linear equations in the K unknowns in β n . When K = L, so we have exactly IV for the explanatory variables in the system: then if En [ZX ] is not singular, ˆ IV βn = En [ZX ] −1 En [Zy] , (1.13) which is the System Instrumental Variables (SIV) Estimator. Consistency of SIV follows by the WLLN under ASS. 9 and 10. When L > K -so there are more columns in the IV matrix Z than we need for ˆ identiﬁcation- choosing β n is not straightforward. Except in special cases (1.12) will not ˆ have a solution. Instead we can take the solution β n to make the vector in equation (1.12) as small as possible: one solution is to minimize its norm, ˆ En Z y − X β n ˆ En Z y − X β n , or in general we can use a weighting matrix to produce the best estimator in some sense. ˆ If Wn is an L × L positive semideﬁnite matrix, possibly depending on data, a Gen- ˆ eralized Method of Moments (GMM) Estimate of β is a vector β n which solves ˆ min En [Z (y − X b)] Wn En [Z (y − X b)] . b Since this is a quadratic form in b, the solution has a closed form: −1 ˆ ˆ ˆ ˆ β n = β n Wn = En [XZ ] Wn En [ZX ] ˆ En [XZ ] Wn En [Zy] (1.14) ˆ assuming En [XZ ] Wn En [ZX ] is nonsingular. 22 Econometrics II-1. Systems of Equations. 2009/10 UC3M. Master in Economic Analysis ˆ To show that the GMM Estimate is consistent we need to assume that Wn has a non- singular probability limit: ASS. 11 (GMM weighting matrix) ˆ Wn →p W as n → ∞, where W is a nonrandom, symmetric, L × L positive deﬁnite matrix. In applications, the convergence in ASS. 11 follows by the law of large numbers, because ˆ Wn will be a function of samples averages, which will be positive deﬁnite with probability approaching one. THM. 5 (Consistency of GMM) Under ASS. 9-11 ˆ ˆ β n Wn → p β as n → ∞. PROOF. We can write −1 ˆ ˆ β n − β = En [XZ ] Wn En [ZX ] ˆ En [XZ ] Wn En [Zu] . Now under ASS. 10, C := E [ZX ] has rank K, and with ASS. 11, C WC has rank K and therefore is nonsingular. Therefore from the LLN ˆ −1 βn − β = (C WC) + op (1) (C W+op (1)) En [Zu] = Op (1)op (1) = op (1), because En [Zu] →p 0 by ASS. 9. • When K = L, the GMM estimate in (1.14) is equal to the IV estimate in (1.13), no ˆ matter the choice of W, because En [XZ ] is a K × K nonsingular matrix. 23 Econometrics II-1. Systems of Equations. 2009/10 UC3M. Master in Economic Analysis The GMM estimate is asymptotically normally distributed under the same assumptions: THM. 6 (CAN of GMM) Under ASS. 9-11 √ −1 −1 ˆ n β n − β →d N 0, (C WC) C WΛWC (C WC) , where Λ := E [Zuu Z ] = V [Zu] if the elements of Zuu Z have ﬁnite expected absolute value. PROOF. We only need to show that √ −1 √ ˆ n β n − β = (C WC) + op (1) (C W+op (1)) nEn [Zu] and that √ nEn [Zu] →d N (0, Λ) . by ASS. 9. AVar Estimation. Consistent estimation of the above asymptotic sandwich variance ˆ matrix is simple if a consistent estimate Λ is available by means of 1 −1 −1 ˆ ˆ ˆ ˆ −1 ˆ En [XZ ] Wn En [ZX ] En [XZ ] Wn Λn Wn En [ZX ] En [XZ ] Wn En [ZX ] . n ˆ This formula simpliﬁes when Wn is chosen optimally. A consistent estimate of Λ can be ˆ u ˆ Λn = En [Zˆ n un Z ] , ˆ ˆ where un = y − X β n are residuals computed using a consistent estimate β n . ˆ 24 Econometrics II-1. Systems of Equations. 2009/10 UC3M. Master in Economic Analysis 1.6.1 The System 2SLS Estimator ˆ The choice of Wn ˆ −1 Wn = En [ZZ ] leads to a familiar estimator. ASS. 11 simply requires that E [ZZ ] exists and is nonsin- ˆ gular. When we plug this choice of W in (1.14), we obtain −1 ˆ −1 −1 β n = En [XZ ] En [ZZ ] En [ZX ] En [XZ ] En [ZZ ] En [Zy] , which looks like the single-equation 2SLS, so can be termed as System 2SLS Estimate. • It can be showed that for system (1.11) in EX. 5, S2SLS produces 2SLS equation by equation. • For a particular choice of Z in a panel data set up, S2SLS produces a Pooled 2SLS estimate. However S2SLS is not necessarily the asymptotically eﬃcient estimator. 1.6.2 Optimal Estimates There is a choice of W that produces the GMM estimator of minimum variance. As in a single equation set up, if we set W = Λ−1 , the AVar of the GMM estimate simpliﬁes −1 to (C Λ−1 C) , and it can be shown that −1 −1 −1 (C WC) C WΛWC (C WC) − C Λ−1 C is positive semideﬁnite for any L × L positive deﬁnite matrix W. ASS. 12 (Optimal Weighting) −1 W = Λ−1 := E [Zuu Z ] . THM. 7 (Eﬃcient GMM) Under ASS. 9-12, the resulting GMM estimate is eﬃcient among all GMM estimators of the form (1.14). 25 Econometrics II-1. Systems of Equations. 2009/10 UC3M. Master in Economic Analysis If we can estimate consistently Λ, we can obtain an estimate which has the same ﬁrst order asymptotic properties of the eﬃcient GMM estimate, so is asymptotically eﬃcient: ˆ 1. Let β n be an initial consistent estimate of β (e.g. S2SLS). 2. Obtain the G × 1 residual vectors ˆ ˆ un = y − X β n = u − X β n −β , ˆ i = 1, . . . , n. 3. Compute a consistent estimate of Λ such that ˆ u ˆ Λn = En [Zˆ n un Z ] . ˆ ˆn 4. Choose Wn = Λ−1 to obtain the asymptotically optimal GMM estimate. ˆ The estimate Λn is consistent for E [Zuu Z ] , even in the presence of conditional het- eroskedasticity or serial correlation (because n → ∞, with T ﬁxed). ˆ ˆn The asymptotic variance of the eﬃcient GMM estimate β n Λ−1 is estimated as 1 −1 ˆ Vn := ˆn En [XZ ] Λ−1 En [ZX ] (1.15) n ˆ ˆ where Λn can be obtained using the ﬁrst step residuals, un = y − X β n , or the second ˆ ˆ ˆn step residuals uEF F = y − X β n Λ−1 . ˆn This estimate is labelled as a Minimum Chi-Square Estimator. • When Z = X, and the un are the System OLS residuals, then the estimate (1.15) ˆ becomes the robust variance estimate of SOLS. • The estimate reduces to the robust variance estimate for FGLS when Z = XΩ−1 and ˆ the un are the FGLS residuals. • If it is known that E [Zuu Z ] = E [ZΩZ ] , where Ω := E [uu ] , this can be used to ˆ −1 estimate Λ by En ZΩn Z (3SLS or FIV estimate). 26 Econometrics II-1. Systems of Equations. 2009/10 UC3M. Master in Economic Analysis 1.7 Testing using the GMM 1.7.1 Testing Classical Hypothesis H0 : R β = r. q×K Wald tests for linear restrictions: it can be used an optimal GMM Estimate or a 3SLS Estimate if ASS. ?? is assumed then −1 ˆ W n = n Rβ n − r ˆ RV n R ˆ Rβ n − r →d χ2 q under H0 . In general 2SLS should not be used to test System hypothesis because its AVar is much more complicated that in those cases. Pseudo-LR Tests. Other method consists on using the GMM objective function with and without the restrictions imposed. It is necessary that we use an optimal GMM estimate so Wn estimates consistently V [Zu]−1 = Λ−1 . Then ˆ ˆ nEn [Zu] Wn En [Zu] →d χ2 L since Zu is an L × 1 vector with zero mean and variance Λ. ˆ ˜ Denote by un := y−X β n the unrestricted residuals and by ur := y−X β n the residuals ˆ ˜n obtained from the restricted model (imposing the q restrictions in H0 ). Then the GMM distance statistic has a limiting chi-square distribution: un ˆ u ˆ LRn = n En [Z˜ r ] Wn En [Z˜ r ] − En [Zˆ n ] Wn En [Zˆ n ] →d χ2 un u q when H0 is true. This is only the diﬀerence of the GMM criterion multiplied by n (why is non negative?). 27 Econometrics II-1. Systems of Equations. 2009/10 UC3M. Master in Economic Analysis 1.7.2 Testing Overidentifying Restrictions There are overidentifying restrictions when L > K. In this case and under H0 : E[Zu] = 0, L×1 that all the restrictions are true u ˆ nEn [Zˆ n ] Wn En [Zˆ n ] →d χ2 u L−K ˆ ˆn if Wn = Λ−1 is an asymptotically optimal weighting matrix. Replacing u by un reduces ˆ the degrees of freedom from L to L − K (note that when L = K, the lhs is 0), because we have estimated K parameters. ˆ If Wn is not optimal then the result does not hold. If the null hypothesis is rejected, but not in case of single equation analysis, 2SLS should be preferred. Hausman’s test compares directly the 2SLS and 3SLS estimates directly, assuming that under the null, the 3SLS is more eﬃcient. 28 Econometrics II-1. Systems of Equations. 2009/10 UC3M. Master in Economic Analysis 1.8 Optimal Instruments How many instruments in Z? In principle we should use all instruments available, given that the initial set satisﬁes the identiﬁcation assumptions and that we use in any optimal weighting matrices. If we have Z := (Z1 , Z2 ) , then √ √ −1 −1 AVar ˆ nβ n (Z1 ) − AVar ˆ nβ n (Z) = C1 Λ−1 C1 1 − C Λ−1 C where C1 := E [Z1 X ] . Then it is easy to check that C Λ−1 C − C1 Λ−1 C1 is psd (White 1 1984, Prop 4.49). Then we cannot do worse asymptotically by adding instruments to Z1 . However we might not improve when C2 = E [Z2 uu Z1 ] Λ−1 C1 , 1 (1.16) where C2 := E [Z2 X ] (White, 1984). • Under conditional homoskedasticity so that we assume E [Zuu Z ] = σ 2 E [ZZ ] (so 2SLS is optimal), this condition is −1 E [Z2 X ] = E [Z2 Z1 ] (E [Z1 Z1 ]) E [Z1 X ] = 0 −1 ⇔ 0 = E Z2 − E [Z2 Z1 ] (E [Z1 Z1 ]) Z1 X ⇔ 0 = E [(Z2 − L [Z2 |Z1 ]) X ] . This means that X is orthogonal to the part of the (linear) information in Z2 that was not already in Z1 : in this case Z2 gives no additional information (on X) and estimation can not be improved. • In the general case, where the errors u can be correlated or there is conditional het- eroskedasticity, it is quite unlikely that the original condition (1.16) is satisﬁed. 29 Econometrics II-1. Systems of Equations. 2009/10 UC3M. Master in Economic Analysis If the errors satisfy a zero conditional expectation assumption, E [u|Z] = 0, then there are unlimited IV available. • In the general regression case E [y − x β|x] = 0 and the OLSE is the IVE with IV z = x. ⇒ If V [u] = V [u|x] , (Cond. heteroscedasticity) there are inﬁnite IVE that can improve on OLS because any h(z) is a valid instrument since E [uh(x)] = E [h(x)E [u|h(x)]] = 0. Then the minimum chi-square estimate with IV z = x , h(x) is generally more eﬃcient than OLS (Chamberlain, 1982). ⇒ If V [u|x] is constant (Cond. homoscedasticity), adding functions to the IV list results in no asymptotic improvement because the linear projection of x onto x and h(x) does not depend on h(x). Therefore under homoskedasticity, adding moment conditions does not improve neither reduce the asymptotic eﬃciency of the OLSE. However ﬁnite sample performance with many overidentifying restrictions can be quite poor. 30 Econometrics II-1. Systems of Equations. 2009/10 UC3M. Master in Economic Analysis Is possible instead to obtain a small set of optimal IV? If we replace ASS. 9 by E [ug |z] = 0, g = 1, . . . , G for some vector z (which is a valid set of IV for any equation) then it can be showed that the optimal choice of instruments is Z∗ = E [X|z] Ω (z)−1 if rank{E [Z∗ X ]} = K, where Ω (z) := E [uu |z] , and we can forget about any other function of z. • If E [ug |z] = 0, E [uu |z] = E [uu ] , (conditional homoscedasticity) and E [X1 |z] = L [X1 |z] = Π Z (linearity of the conditional expectation) then the 3SLS estimator is the eﬃcient among the SIV estimators based on the orthogonality condition E [ug |z] = 0. • If E [u|X] = 0 and E [uu |X] = Ω, (exogeneous regressors) then the optimal IV are XΩ−1 , which gives the GLS estimator. • Without further assumptions, E [X|z] and Ω (z) can be arbitrary functions of z, in which case the optimal IV estimate is not easy to obtain. However these functions could be estimated (non)parametrically (Robinson, 1991; Newey, 1990). RECOMMENDED READINGS: Wooldridge (2002, Ch. 7-8). Hayashi (2000, Ch. 4). Ruud (2000, Ch 26.2). Mittelhammer et al. (2000, Ch. 15.2.1) 31 Econometrics II-1. Systems of Equations. 2009/10 UC3M. Master in Economic Analysis Problem Set 1 1. Consider the SUR model under (1.2) and ASS. 1, 2 and 5 with Ω = diag (σ 2 , . . . , σ 2 ) . 1 G (a) Show that ASS. 3 and 4 hold. (b) Show that GLS and OLS estimation equation by equation are the same. (c) Show that single-equation OLS estimators for any two equations, say, β ng ˆ ˆ ˆ and β nh are asymptotically uncorrelated, i.e., the asymptotic variance of β n is block diagonal. (d) Under the same assumptions, explain how you would test H0 : β ng = β nh against H1 : β ng = β nh if they have the same dimension. (e) Now drop ASS. 5, but maintain everything else. Suppose that Ω is estimated in an unrestricted way. Are FGLS and OLS algebraically equivalent? Show √ ˆ ˆ F GLS = op (1). that n β n − β n √ ˆ ˆ 2. Using the n-consistency of the SOLS estimator β n,OLS , for Ωn in (1.9) show that n √ vec ˆ n Ωn − Ω = vec n−1/2 ( ui ui − Ω) + op (1) i=1 under ASS 1 and 4. State the moment conditions you need. 3. Show the equivalency OLS=FGLS for SUR models when: ˆ (a) Ωn is diagonal. (b) All equations have the same regressors, X = IG ⊗ x, so En XΩ−1 X = Ω−1 ⊗ En [xx ] = (IG ⊗ En [xx ]) Ω−1 ⊗ IK . 4. Consider the panel data model yt = xt β + ut t = 1, . . . , T E [ut |xt , ut−1 , xt−1 , . . .] = 0 E u2 |xt t = E u2 = σ 2 , t t t = 1, . . . , T Note that σ 2 may vary in each time period. t 1 Econometrics II-1. Systems of Equations. 2009/10 UC3M. Master in Economic Analysis (a) Show that Ω := E [uu ] is diagonal. (b) Write down the GLS estimator when Ω is known. (c) Show that ASS 3 does not necessarily hold under the assumptions made, taking xt = yt−1 . (d) Show that the GLS estimator of b) is consistent for β by showing that E XΩ−1 u = 0. Is ASS 1 necessary or suﬃcient for consistency of GLS? (e) Explain how to estimate consistently each σ 2 (n → ∞). t (f) Justify that, under the previous assumptions, valid inference can be obtained by weighting each observation (yt , xt ) by 1/σ t and then running Pooled OLS. (g) What happens if we assume that σ 2 = σ 2 for all t = 1, . . . , T . t 5. Consider the FRINGE.RAW dataset from Wooldridge (2002, p. 165) on wages and fringe beneﬁts for 616 workers. Estimate a two-equation system for hourly wage (hrearn) and hourly beneﬁts (hrbens) . Include in the regressors (the same for each equation) variables such as education, experience, tenure and dummies on belong- ing to a union, south, nrtheast, nrthcen, married, white and male. [EVIEWS: objects/new objects/system/spec and write down the equations.] (a) Estimate the System by OLS. Check that the results are the same than when estimating by OLS each equation separately. Why is that? (b) Check the correlation between the residuals of the two equations. [view/ residuals/ correlation] (c) Estimate both equations by FGLS using the SU R option in Eviews. What would have been the result of FGLS (Eviews SUR) if the previous residual correlation were zero? (d) Investigate the sign of the regression coeﬃcients. (e) Test the joint signiﬁcance on both equations of married and white (four restrictions) by means of a Wald test. [view/Wald coeﬃcient tests]. (f) Disaggregate the beneﬁts categories into value of vacation days, value of sick leave, value of employer-provided insurance and value of pension. Use hourly measures of these variables along with hrearn and estimate a 5-equation SUR model. Does marital status appear to aﬀect any form of compensation? Test whether another year of education increases expected pension value and expected insurance by the same amount. 2 Econometrics II-1. Systems of Equations. 2009/10 UC3M. Master in Economic Analysis 6. Consider the panel data model to explain annual family saving over a ﬁve year span: savt = β 0 + β 1 inct + β 2 aget + β 3 educt + ut , t = 1, . . . , 5 where inct is annual income, educt is years of education of the household head, and aget is age of household head. If we add ”wealth at the beginning of year t” to the saving equation, is the strict exogeneity assumption likely to hold? 7. Use the GPA3.RAW data set from Wooldridge (2002, p. 173) to investigate the eﬀect of being in season on grade point average. The data are on 366 students- athletes at a large university. There are two semesters of data for each students (T = 2). Of primary interest is the in-season eﬀect on athletes’ GPA: trmgpait = β 0 + β 1 springt + β 2 cumgpait + β 3 crsgpait + β 4 frstsemit +β 5 seasonit + β 6 SATi + β 7 verbmathi + β 8 hspersci +β 9 hssizei + β 10 blacki + β 11 femalei + uit . The variable cumgpait is cumulative GPA at the beginning of the term, and this clearly depends on past-term GPA, which introduces something similar to a lagged dependent variable. There are variables that change over time (season) and other other that do not (SAT). Assume that uit is uncorrelated with all variables on the right hand side. [Note that data are stacked, so the two observations for each period of a given student are consecutive: stacked by cross section.] (a) Estimate the equation by (pooled) OLS on the stacked data ﬁle. Is the in season eﬀect signiﬁcative? (b) Under which conditions are the standard errors provided valid? Will the absence of these conditions aﬀect the consistency of OLS estimates? (c) Describe how would you test such assumptions given the structure of the data set (and the residuals). 8. Show that S2SLS produces 2SLS equation by equation in EX. 5. 9. Consider the standard panel data model yt = xt β + ut (1.17) 1×K and xt might have some elements correlated with ut . Let zt be a L × 1 vector of instruments, L ≥ K, such that E [zt ut ] = 0, t = 1, 2, . . . , T. 3 Econometrics II-1. Systems of Equations. 2009/10 UC3M. Master in Economic Analysis (a) Give an expression for the S2SLS estimator if the instrument matrix is Z = (z1 , z2 , . . . , zT ) . L×T Show that this is the Pooled 2SLS estimator obtained by 2SLS estimation of (1.17) using instruments zt , pooled across all t. (b) What is the rank condition for the pooled 2SLS estimator? (c) Propose a consistent estimate of the asymptotic variance of the Pooled 2SLS estimate without using further assumptions. (d) Show that under E [ut |zt , ut−1 , zt−1 , . . . , u1 , z1 ] = 0, t = 1, 2, . . . , T (1.18) E u2 |zt t = σ 2 , t = 1, 2, . . . , T (1.19) the usual standard errors and test statistics from the Pooled 2SLS estimation are valid. (e) What estimator would you use under condition (1.18) and relaxing condition (1.19) to E u2 |zt = E u2 = σ 2 , t = 1, 2, . . . , T. t t t (You would probably need a ﬁrst step using an initial Pooled 2SLS estima- tion). 10. Let x be a K × 1 random vector and let z be a M × 1 random vector. Suppose that the following linear condition expectation condition holds, E [x|z] = L [x|z] . Let h(z) be any q × 1 matrix on linear functions of z, and deﬁne an expanded list of IV w := [z , h(z) ] . (a) Show that rankE [zx ] = rankE [wx ] . (b) Consider the system of equations y1 = x1 β 1 + u1 . . . yg = xg β G + ug 4 Econometrics II-1. Systems of Equations. 2009/10 UC3M. Master in Economic Analysis and let z be a vector of exogenous variables in every equation so that E [ug |z] = 0, g = 1, . . . , G, allowing for any nonlinear function of z to be a valid instrument in every equation. Suppose that E [xg |z] is linear in z for all g. Show that adding nonlinear functions of z to the instrument list cannot help in satisfying the rank condition. (c) What happens when E [xg |z] is a nonlinear function of z for some g? 11. Describe situations where System GMM procedures are equivalent to GMM equa- tion by equation procedures in overidentiﬁed systems. 12. Obtain the asymptotic distribution of the 3SLS Estimate when nor ASS. ?? (nei- ther ASS. 12) hold. 13. Consider the system of equations y1 = x1 β 1 + u1 y2 = x2 β 2 + u2 with the following instrument matrix z1 0 Z= 0 z2 and where the covariance matrix Ω of u := (u1 , u2 ) has inverse σ 11 σ 12 Ω−1 = . σ 21 σ 22 (a) Find E [ZΩ−1 u] and show that is not necessarily 0 under the orthogonality conditions E [z1 u1 ] = 0 and E [z2 u2 ] = 0. (b) What happens when Ω is diagonal? (c) What if z1 = z2 for a general Ω? 5