Advanced Econometrics - Chapter 11: Seemingly unrelated regressions

Advanced Econometrics Chapter 11: Seemingly unrelated regressions Nam T. Hoang University of New England - Australia 1 University of Economics - HCMC - Vietnam 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 β1, β2, βM, such that 1 1 1 1 ( 1) ( ) ( 1) ( 1)T T k k T Y X β ε × × × × = + country 1 2 2 2 2 ( 1) ( ) ( 1) ( 1)T T k k T Y X β ε × × × × = + country 2 ( 1) ( ) ( 1) ( 1) M M M M T T k k T Y X β ε × × × × = + country M • Assume each 𝜀𝑖 (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. ' 2( )i i i T ii TE I Iε ε σ σ= = '( )i j ij TE Iε ε σ= Where σij: contemporaneous covariance between errors of equations i and j Advanced Econometrics Chapter 11: Seemingly unrelated regressions Nam T. Hoang University of New England - Australia 2 University of Economics - HCMC - Vietnam  1 ( 1) 2 ( 1) ( 1) T M T MT Y Y Y × × ×                 1 ( ) 2 ( ) ( ) ( ) 0 0 0 0 0 0 T k T k M T k TM kM X X X × × × ×        =                    1 1 ( 1) ( 1) 2 2 ( 1) ( 1) ( 1) ( 1) k T N N k T kM NT β ε β ε β ε × × × × × ×                +                  Assumption: there is a β such that: (1) ↔ Y X β ε= + ( ) ( ) MT MT E εε × ′ 11 12 1 21 22 2 1 2 M M M M MM I I I I I I I I I I σ σ σ σ σ σ σ σ σ      = = Σ ⊗              Where: 11 12 1 21 22 2 1 2 M M M M MM σ σ σ σ σ σ σ σ σ      Σ =              II. GENERALIZED LEAST SQUARES ESTIMATION OF SUR MODEL (GLS) The equation (1) can be estimated by GLS if E(εε’) is known: 1 1 1ˆ [ '( ( ')) ] [ '( ( ')) ]SUR X E X X E Yβ εε εε − − −= 1 1 1ˆ [ '( ) ] [ '( ) ]SUR X I X X I Yβ − − −= Σ ⊗ Σ ⊗ GLS is the best linear unbiased estimator: ( ) 1 1ˆ [ '( ( ')) ]SURVarCov X E Xβ εε − −= Advantages of SUR over single-equation OLS 1. Gain in efficiency: Because ˆSURβ will have smaller varriance than ˆOLSβ Advanced Econometrics Chapter 11: Seemingly unrelated regressions Nam T. Hoang University of New England - Australia 3 University of Economics - HCMC - Vietnam 1( ) ( 1) 2( ) ( ) ( 1) ( 1) ˆ ˆ OLS k OLS OLS M OLS k TM β ββ β × × ×        =           Note that ( )iˆ OLSβ is efficient estimator for βi, but ˆOLSβ is not efficient estimator for β, and ˆ SURβ is efficient estimator for β. 2. Test or impose cross-section restriction (Allowing to test or impose) Usually E(εε’) unknown Feasible GLS estimation 1. Estimate each equation by OLS, save residuals ( 1) i T e × , i = 1, 2, , M. 2. Compute sample variances and covariances 1ˆ T it jt t ij e e T k σ == − ∑ all ij pairs 11 12 1 21 22 2 1 2 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ M M M M MM σ σ σ σ σ σ σ σ σ      Σ =              / / / 1 1 1 2 1 / / / 2 1 2 2 2 / / / 1 2 1 M M M M M M e e e e e e e e e e e e T k e e e e e e      =  −             ( ) ( ) MT MT E εε × ′ ( ) ( ) ˆ M M T T I × × = Σ ⊗ 3. 1 1 1ˆ ˆ ˆ[ '( ) ] [ '( ) ]FGLS X I X X I Yβ − − −= Σ ⊗ Σ ⊗ → Σˆ is a consistent estimator of ∑ It is also possible to interate 2 & 3 until convergence which will produce the maximum likelihood estimator under multivariate normal errors. In other words, ˆFGLSβ and ˆMLβ will have the same limiting distribution such that: ,ˆ ( , ) asy ML FGLS Nβ β ϕ Advanced Econometrics Chapter 11: Seemingly unrelated regressions Nam T. Hoang University of New England - Australia 4 University of Economics - HCMC - Vietnam Where 𝜑 is consistently estimated by 1 1ˆˆ [ '( ) ]X I Xϕ − −= Σ ⊗ 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⊗ ⊗ = ⊗ 11 12 21 22 a B a B a B a B               11 12 21 22 c D c D c D c D     =          1 1 1 2 2 1 2 2 ( ) ( ) ( ) ( ) j j j j j j j j a c BD a c BD a c BD a c BD AC BD     = ⊗     ∑ ∑ ∑ ∑      (2) ( ) 1 1 1A B A B− − −⊗ = ⊗ if inverses are defined. Because: ( )( ) ( )1 1 1 1A B A B AA BB I− − − −⊗ ⊗ = ⊗ = → ( ) 1 1 1A B A B− − −⊗ = ⊗ (3) ( )/ / /A B A B⊗ = ⊗ (you show). IV. TWO CASE WHEN SUR PROVIDES NO EFFICIENCY GAIN OVER SINGLE OLS: 1. When σij = 0 for all i≠j: the equations are not linked in any fashion and GLS does not provide any efficiency gains → we can show that ˆOLSβ = ˆ SURβ ( ) 1 1ˆ [ '( ) ]SURVarCov X I Xβ − −= Σ ⊗ Advanced Econometrics Chapter 11: Seemingly unrelated regressions Nam T. Hoang University of New England - Australia 5 University of Economics - HCMC - Vietnam 1 1( )I I− −Σ ⊗ = Σ ⊗ = 11 22 1 0 0 10 0 10 0 MM I I I σ σ σ          =                   ( )ˆSURVarCov β = 1 / 1 11( ) 1 / 2 2( ) 22 / ( ) 1 0 00 0 0 0 10 0 0 0 0 0 0 00 0 10 0 T k T k M M T k MM IX X X I X XX I σ σ σ − × × ×                         =                                              1/ 1 1 11 / 2 2 22 / 1 0 0 0 0 0 0 M MM X X X X X X σ σ σ −           =                    / 1 1 1 11 / 1 2 2 22 / 1 ( ) 0 0 0 ( ) 0 0 0 ( )M M MM X X X X X X σ σ σ − − −      =               ( ) / 1ˆ ( )iOLS i i iiVarCov X Xβ σ−= → ( )( )ˆSUR iVarCov β = ( )iˆVarCov β → no efficiency gains at all. Exercise: Show: 1 2 ˆ ˆˆ ˆ OLS OLS SUR MOLS β β β β       =          in this case. Advanced Econometrics Chapter 11: Seemingly unrelated regressions Nam T. Hoang University of New England - Australia 6 University of Economics - HCMC - Vietnam Note: 1. The greater is the correlation of disturbance, the greater is the gain in efficiency in using SUR & GLS. 2. The less correlation then is in between the X matrices, the greater is gain in using GLS. 3. When 1 2 ... MX X X X= = = = ( ) 1 1ˆ [ '( ) ]SURVarCov X I Xβ − −= Σ ⊗ / 1 1[( ) ( )( )]I X I I X− −= ⊗ Σ ⊗ ⊗ 1 / 1 / 1[ ( )] ( )X X X X− − −= Σ ⊗ = Σ ⊗ / 1 / 1 11 12 / 1 / 1 21 22 / 1 ( ) ( ) 0 ( ) ( ) 0 0 ( )MM X X X X X X X X X X σ σ σ σ σ − − − − −      =               → no efficiency gain. ( ) / 1ˆ ( )iOLS iiVarCov X Xβ σ −= X = 0 0 0 0 0 0 X X X                    V. HYPOTHESIS TESTING: 1. Contemporaneous correlation (spatial correlation): / 0 0 0 0 ( ) 0 0 ij ij i j ij E σ σ ε ε σ      =               H0: 0ijσ = for all i≠j HA: H0 false. LM test statistic: 1 2 2 ( 1) 2 1 2 M i ij M M i j T rλ χ − − = = = ∑∑  Where rij is calculated using OLS residuals: Advanced Econometrics Chapter 11: Seemingly unrelated regressions Nam T. Hoang University of New England - Australia 7 University of Economics - HCMC - Vietnam / / /( )( ) i j ij i i j j e e r e e e e = Under H0 → 2 ( 1) 2 M Mλ χ − . If accept H0 → no efficiency gain. 2. Restrictions on coefficients: H0: 0Rβ = 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 1 2( , ,..., )Mβ β β β= . Within SUR framework, it is possible to test coefficient restriction across equations. One possible test statistic is: / / 1 ( ) ( ) ( )( 1)( 1) ( ) ( 1) ((1 ) ( ) ˆ ˆ ˆ( ) [ ( ) ] ( )FGLS FGLS FGLSm k m k k mmk k k m m m m W R q R VarCov R R qβ β β− × × ××× × × × × = − −    2 asy mW χ 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: , , 1i t i i t ituε ρ ε −= + Where uit is white noise. The overall error structure will now be: ( )E εε ′ 11 11 12 12 1 1 21 21 22 22 2 1 1 2 2 M M M MM M M M M MM MM MT MT σ σ σ σ σ σ σ σ σ × Σ Σ Σ   Σ Σ Σ =    Σ Σ Σ         Where: 1 1 1 2 1 1 1 T j j T j j ij T T j j T T ρ ρ ρ ρ ρ ρ − − − − ×      Σ =               Advanced Econometrics Chapter 11: Seemingly unrelated regressions Nam T. Hoang University of New England - Australia 8 University of Economics - HCMC - Vietnam 1 2/ 1 2( ) i i i j j j jT iT E ε ε ε ε ε ε ε ε        =          Estimation: 1. Run OLS equation by equation by equation. Compute consistent estimate of ρi: 1 2 2 1 ˆ T it it t i T it t e e e ρ − = = = ∑ ∑ Transform the data, using Cochrane-Orcutt, to remove the autocorrelation. 2. Calculate FGLS estimates using the transformed data. • Estimate Σ using the transformed data as in GLS. • Use Σˆ to calculate FGLS.