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Lecture Advanced Econometrics (Part II) - Chapter 11: Seemingly unrelated regressions

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Lecture "Advanced Econometrics (Part II) - Chapter 11: Seemingly unrelated regressions" presentation of content: Model, generalized least squares estimation of sur model, kronecker product, two case when sur provides no eficiency gain over, hypothesis testing.

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Nội dung Text: Lecture Advanced Econometrics (Part II) - Chapter 11: Seemingly unrelated regressions

  1. 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 β1, β2, …βM, such that = Y1 X β + ε1 1 1 country 1 (T ×1) (T ×k ) ( k ×1) (T ×1) = Y2 X β + ε2 2 2 country 2 (T ×1) (T ×k ) ( k ×1) (T ×1) … YM X M β M + ε M = country M (T ×1) (T ×k ) ( k ×1) (T ×1) • Assume each 𝜀⃗𝑖 (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= iε i ) ' σ=2 i IT σ ii IT E (ε iε 'j ) = σ ij IT Where σij: contemporaneous covariance between errors of equations i and j Nam T. Hoang University of New England - Australia 1 University of Economics - HCMC - Vietnam
  2. Advanced Econometrics Chapter 11: Seemingly unrelated regressions  X1 0  0   Y1   (T ×k )   β1   ε1   (T ×1)   0 X2  0   ( k ×1)   (T ×1)  Y  = (T ×k )  β  ε   2     2 + 2                 Y   0 0  XM  β  ε   (TM×1)   (T ×k )   ( k ×N1)   (T ×N1)        ( MT ×1) (TM ×kM ) ( kM ×1) ( NT ×1) Assumption: there is a β such that: Y Xβ +ε (1) ↔=  σ 11 I σ 12 I  σ 1M I  σ I σ 22 I  σ 2 M I  E (εε ′) =  21 = Σ⊗I ( MT × MT )         σ M 1 I σ M 2 I  σ MM I   σ 11 σ 12  σ 1M  σ σ 22  σ 2 M   21 Where: Σ =        σ M 1 σ M 2  σ MM  II. 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[ X '(Σ ⊗ I ) −1Y ] GLS is the best linear unbiased estimator: ( ) VarCov βˆSUR = [ X '( E (εε ')) −1 X ]−1 Advantages of SUR over single-equation OLS 1. Gain in efficiency: Because βˆSUR will have smaller varriance than βˆOLS Nam T. Hoang University of New England - Australia 2 University of Economics - HCMC - Vietnam
  3. Advanced Econometrics Chapter 11: Seemingly unrelated regressions  βˆ1( OLS )   ( k ×1)    β βˆOLS =  2( OLS )        β M ( OLS )  ( k ×1)  (TM ×1) Note that βˆi (OLS ) is efficient estimator for βi, but βˆOLS is not efficient estimator for β, and βˆSUR 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 ei , i = 1, 2, …, M. (T ×1) 2. Compute sample variances and covariances T ∑e e it jt σˆ ij = t =1 all ij pairs T −k  σˆ11 σˆ12  σˆ1M   e1/ e1 e1/ e2  e1/ eM   σˆ  /  σˆ 22  σˆ 2 M  1 /  e2 e1 e2 e2  e2/ eM  Σ = 21 =       T −k          /  σˆ M 1 σˆ M 2  σˆ MM  eM e1 eM e2  eM/ eM  / E (εε ′) = Σˆ ⊗ I ( MT × MT ) ( M ×M ) (T ×T ) 3. βˆFGLS = [ X '(Σˆ ⊗ I ) −1 X ]−1[ X '(Σˆ ⊗ I ) −1Y ] → 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, 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 3 University of Economics - HCMC - Vietnam
  4. Advanced Econometrics Chapter 11: Seemingly unrelated regressions Where 𝜑 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   ∑ (a1 j c j1 ) BD ∑ (a 1j c j 2 ) BD   a B a B  c D c D  =    21 22   21 22   ∑ (a2 j c j1 ) BD ∑ (a = 2 j c j 2 ) BD   AC ⊗ BD              ( A ⊗ B) −1 (2) =A−1 ⊗ B −1 if inverses are defined. Because: ( A ⊗ B ) ( A−1 ⊗ B −1 ) = ( AA −1 ) ⊗ BB −1 = I → ( A ⊗ B ) =A−1 ⊗ B −1 −1 ( A ⊗ B) / (3) =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 βˆOLS = βˆSUR VarCov βˆSUR ( ) = [ X '(Σ ⊗ I ) −1 X ]−1 Nam T. Hoang University of New England - Australia 4 University of Economics - HCMC - Vietnam
  5. Advanced Econometrics Chapter 11: Seemingly unrelated regressions  1  σ I 0   0  11   1  0 I  0  (Σ ⊗ I ) −1 = Σ −1 ⊗ I = =  σ 22           0 1  0  I  σ MM  ( VarCov βˆSUR = ) −1   1     X 1/ 0  0  σ I 0  0     (T × k )   11   X1 0  0   0 0  0   0   /  1  X  I  0  0 X2  =  2 (T × k ) σ 22                        X M/    0 0  X M   0 0 1    0 (T × k )  0  I    σ MM   −1  X 1/ X 1   0  0   σ 11   X 2/ X 2   0  0  = σ 22           X M/ X 1   0  σ MM  0  ( X 1/ X 1 ) −1σ 11 0  0    0 ( X 2 X 2 ) −1σ 22 /  0 =           0 0  ( X M/ X M ) −1σ MM  ( ) VarCov βˆiOLS = ( X i/ X i ) −1σ ii ( ) ( ) → VarCov βˆSUR (i ) = VarCov βˆi → no efficiency gains at all.  βˆ1OLS     βˆ2OLS  Exercise: Show: βˆSUR =  in this case.    βˆ   MOLS  Nam T. Hoang University of New England - Australia 5 University of Economics - HCMC - Vietnam
  6. Advanced Econometrics Chapter 11: Seemingly unrelated regressions 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 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  0    σ ( X / X ) −1 σ 22 ( X / X ) −1  0 =  21  → no efficiency gain.          0  σ MM ( X / X ) −1  ( ) VarCov βˆiOLS = σ ii ( X / X ) −1 X 0  0 0 X  0  X=        0 0  X V. HYPOTHESIS TESTING: 1. Contemporaneous correlation (spatial correlation): σ ij 0  0 0 σ  0 E (ε iε j ) =   / ij        0 0  σ ij  H0: σ ij = 0 for all i≠j HA: H0 false. M i −1 LM test statistic: λ = T ∑∑ rij2  χ M2 ( M −1) =i 2=j 1 2 Where rij is calculated using OLS residuals: Nam T. Hoang University of New England - Australia 6 University of Economics - HCMC - Vietnam
  7. Advanced Econometrics Chapter 11: Seemingly unrelated regressions ei/ e j rij = (ei/ ei )(e /j e j ) Under H0 → λ  χ M2 ( M −1) . If accept H0 → no efficiency gain. 2 2. Restrictions on coefficients: H0: R β = 0 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 2 ,..., 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×1) ( m×k ) k ×k ) ( k ×m )     (  ( 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 11Σ11 σ 12 Σ12  σ 1M Σ1M  σ Σ σ 22 Σ 22  σ 2 M Σ MM  E (εε ′) =  21 21         σ M 1Σ M 1 σ M 2 Σ M 2  σ MM Σ MM  MT ×MT  1 ρj  ρ Tj −1     ρj 1  ρ Tj −1  Where: Σij =       T −1   ρ j ρ Tj − 2  1  T ×T Nam T. Hoang University of New England - Australia 7 University of Economics - HCMC - Vietnam
  8. Advanced Econometrics Chapter 11: Seemingly unrelated regressions  ε i1  ε  E (ε iε j ) =  i 2  ε j1 ε j 2  ε jT  /      ε iT  Estimation: 1. Run OLS equation by equation by equation. Compute consistent estimate of ρi: T ∑e e it it −1 ρˆ i = t =2 T ∑e t =1 2 it 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. Nam T. Hoang University of New England - Australia 8 University of Economics - HCMC - Vietnam
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