Advanced Econometrics Chapter 11: Seemingly unrelated regressions Chapter 11 SEEMINGLY UNRELATED REGRESSIONS I MODEL Seemingly unrelated regressions (SUR) are often a set of equations with distinct dependent and independent variables, as well as different coefficients, are linked together by some common immeasurable factor Consider the following set of equations: there are = Y1 (T ×1) = Y2 (T ×1) X β + ε1 country X β + ε2 country 1 (T ×k ) ( k ×1) 2 (T ×k ) ( k ×1) 1, 2, … M, such that (T ×1) (T ×1) … = YM X M M + M (T ì1) (T ×k ) ( k ×1) Assume each country M (T ×1) (i = 1, 2, …, M) meets classical assumptions so OLS on each equation separately in fine • Although each of M equations may seem unrelated, the system of equations may be linked through their mean – zero error structure • We use cross-equation error covariance to improve the efficiency of OLS M equations are estimated as a system ' E (ε= σ= σ ii IT iε i ) i IT E (ε iε 'j ) = σ ij IT Where ij: contemporaneous covariance between errors of equations i and j Nam T Hoang University of New England - Australia University of Economics - HCMC - Vietnam Advanced Econometrics Chapter 11: Seemingly unrelated regressions X1 Y1 (T ×k ) (T ×1) X2 Y = (T ×k ) Y 0 XM (TM×1) (T ×k ) (TM ×kM ) ( MT ×1) β1 ε1 ( k ×1) (T ×1) β ε + β ε ( k ×N1) (T ×N1) ( kM ×1) ( NT ×1) Assumption: there is a such that: (1) ↔= Y Xβ +ε σ 11 I σ I E (εε ′) = 21 ( MT × MT ) σ M I σ 11 σ 21 Where: Σ = σ M II σ 12 I σ 22 I σ 1M I σ M I = Σ⊗ I σ M I σ MM I σ 12 σ 1M σ 22 σ M σ M σ MM GENERALIZED LEAST SQUARES ESTIMATION OF SUR MODEL (GLS) The equation (1) can be estimated by GLS if E(εε’) is known: βˆSUR = [ X '( E (εε ')) −1 X ]−1[ X '( E (εε ')) −1Y ] βˆSUR = [ X '(Σ ⊗ I ) −1 X ]−1[ΣX⊗'( I ) −1Y ] GLS is the best linear unbiased estimator: ( ) VarCov βˆSUR = [ X '( E (εε ')) −1 X ]−1 Advantages of SUR over single-equation OLS Gain in efficiency: Because βˆSUR will have smaller varriance than βˆOLS Nam T Hoang University of New England - Australia University of Economics - HCMC - Vietnam Advanced Econometrics βˆOLS Chapter 11: Seemingly unrelated regressions βˆ1( OLS ) ( k ×1) β = 2( OLS ) β M ( OLS ) k ×1) ( (TM ×1) Note that βˆi (OLS ) is efficient estimator for i, but βˆOLS is not efficient estimator for , and βˆSUR is efficient estimator for Test or impose cross-section restriction (Allowing to test or impose) Usually E(εε’) unknown Feasible GLS estimation Estimate each equation by OLS, save residuals ei , i = 1, 2, …, M (T ×1) Compute sample variances and covariances T σˆ ij = ∑e e it t =1 jt σˆ11 σˆ12 σˆ σˆ 22 Σ = 21 σˆ M σˆ M E (εε ′) = ( MT × MT ) all ij pairs T −k σˆ1M σˆ M = T −k σˆ MM e1/ e1 e1/ e2 / / e2 e1 e2 e2 / / eM e1 eM e2 e1/ eM e2/ eM eM/ eM Σˆ ⊗ I ( M ×M ) (T ×T ) βˆFGLS = [ X '(Σˆ ⊗ I ) −1 X ]−1[ΣX⊗'( ˆ I ) −1Y ] → Σˆ is a consistent estimator of It is also possible to interate & until convergence which will produce the maximum likelihood estimator under multivariate normal errors In other words, βˆFGLS and βˆML will have the same limiting distribution such that: asy βˆML , FGLS N ( β , ϕ ) Nam T Hoang University of New England - Australia University of Economics - HCMC - Vietnam Advanced Econometrics Where Chapter 11: Seemingly unrelated regressions is consistently estimated by = ϕˆ [ X '(Σˆ ⊗ I ) −1 X ]−1 III KRONECKER PRODUCT: Definition: For any two matrices A,B A ⊗ B is defined by the matrix consisting of each element of A time the entire second matrix B Propositions: (1) ( A ⊗ B )( C ⊗ D ) = AC ⊗ BD a11 B a12 B c11 D c12 D a B a B c D c D = 22 22 21 21 (2) ( A ⊗ B) −1 ∑ (a ∑ (a c j ) BD = j c j ) BD AC ⊗ BD 1j =A−1 ⊗ B −1 if inverses are defined ( A ⊗ B ) ( A−1 ⊗ B −1 ) = Because: ∑ (a1 j c j1 ) BD ∑ (a2 j c j1 ) BD ( AA −1 ) ⊗ BB −1 = I → ( A ⊗ B ) =A−1 ⊗ B −1 −1 (3) IV ( A ⊗ B) / =A/ ⊗ B / (you show) TWO CASE WHEN SUR PROVIDES NO EFFICIENCY GAIN OVER SINGLE OLS: When ij = for all i≠j: the equations are not linked in any fashion and GLS does not provide any efficiency gains → we can show that βˆOLS = βˆSUR ( ) VarCov βˆSUR = [ X '(Σ ⊗ I ) −1 X ]−1 Nam T Hoang University of New England - Australia University of Economics - HCMC - Vietnam Advanced Econometrics Chapter 11: Seemingly unrelated regressions σ I 11 (Σ ⊗ I ) −1 = Σ −1 ⊗ I = = ( I σ 22 I σ MM 0 ) VarCov βˆSUR = X 1/ (T × k ) = X 1/ X σ 11 = / (T × k ) X σ I 11 X M/ (T × k ) X 2/ X σ 22 X M/ X σ MM ( X 1/ X ) −1σ 11 / ( X X ) −1σ 22 = 0 ( I σ 22 σ MM X1 I X2 X M −1 −1 ( X M/ X M ) −1σ MM ) VarCov βˆiOLS = ( X i/ X i ) −1σ ii ( ) ( ) → VarCov βˆSUR (i ) = VarCov βˆi → no efficiency gains at all Exercise: Show: βˆSUR Nam T Hoang University of New England - Australia βˆ1OLS βˆ2OLS = in this case βˆ MOLS University of Economics - HCMC - Vietnam Advanced Econometrics Chapter 11: Seemingly unrelated regressions Note: The greater is the correlation of disturbance, the greater is the gain in efficiency in using SUR & GLS The less correlation then is in between the X matrices, the greater is gain in using GLS When X 1= X 2= = X M= X ( ) VarCov βˆSUR = [ X '(Σ −1 ⊗ I ) X ]−1 = [( I ⊗ X ) / (Σ −1 ⊗ I )( I ⊗ X )]−1 = [Σ −1 ⊗ ( X / X )]−1 = Σ ⊗ ( X / X ) −1 σ 11 ( X / X ) −1 σ 12 ( X / X ) −1 σ ( X / X ) −1 σ 22 ( X / X ) −1 = 21 ( → no efficiency gain σ MM ( X / X ) −1 0 ) VarCov βˆiOLS = σ ii ( X / X ) −1 X 0 X= 0 V X 0 X HYPOTHESIS TESTING: Contemporaneous correlation (spatial correlation): σ ij 0 σ ij / E (ε iε j ) = 0 σ ij H0: σ ij = for all i≠j HA: H0 false M i −1 LM test statistic: λ = T ∑∑ rij2 χ M2 ( M −1) =i 2=j Where rij is calculated using OLS residuals: Nam T Hoang University of New England - Australia University of Economics - HCMC - Vietnam Advanced Econometrics rij = Chapter 11: Seemingly unrelated regressions ei/ e j (ei/ ei )(e /j e j ) Under H0 → λ χ M2 ( M −1) If accept H0 → no efficiency gain 2 Restrictions on coefficients: H0: R β = HA: H0 false The general F test can be extended to the SUR system However, since the statistic requires using Σˆ , the test will only be valid asymptotically Where β = ( β1 , β , , β M ) Within SUR framework, it is possible to test coefficient restriction across equations One possible test statistic is: ( R βˆFGLS q ) / [ R VarCov( βˆFGLS ) R /−]−1 ( RβˆFGLS q ) W= ( m×k ) ( k ×1) ( m×k ) ( k ×m ) ( m×1) k ×k ) ( ( m×1) ((1×m ) ( m×m ) asy W χ m2 under H0 VI AUTOCORRELATION: Heteroscedasticity and autocorrelation are possibilities within SUR framework I will focus on autocorrelation because SUR systems are often comprised of time series observations for each equation Assume the errors follow: ε i ,t ρiε i ,t −1 + uit = Where uit is white noise The overall error structure will now be: σ 12 Σ12 σ 11Σ11 σ Σ σ 22 Σ 22 E (εε ′) = 21 21 σ M 1Σ M σ M Σ M ρj ρj Where: Σij = T −1 T −2 ρj ρ j Nam T Hoang University of New England - Australia σ 1M Σ1M σ M Σ MM σ MM Σ MM MT ×MT ρ Tj −1 ρ Tj −1 T ×T University of Economics - HCMC - Vietnam − Advanced Econometrics Chapter 11: Seemingly unrelated regressions ε i1 ε / E (ε iε j ) = i ε j1 ε j ε jT ε iT Estimation: Run OLS equation by equation by equation Compute consistent estimate of ρi: T ρˆ i = ∑e e it it −1 t =2 T ∑e t =1 it Transform the data, using Cochrane-Orcutt, to remove the autocorrelation Calculate FGLS estimates using the transformed data • Estimate Σ using the transformed data as in GLS • Use Σˆ to calculate FGLS Nam T Hoang University of New England - Australia University of Economics - HCMC - Vietnam ...Advanced Econometrics Chapter 11: Seemingly unrelated regressions X1 Y1 (T ×k ) (T ×1) X2 Y = (T ×k ) ... Australia University of Economics - HCMC - Vietnam Advanced Econometrics βˆOLS Chapter 11: Seemingly unrelated regressions βˆ1( OLS ) ( k ×1) β = 2( OLS ) β M ( OLS... Australia University of Economics - HCMC - Vietnam Advanced Econometrics Where Chapter 11: Seemingly unrelated regressions is consistently estimated by = ϕˆ [ X '(Σˆ ⊗ I ) −1 X ]−1 III KRONECKER