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

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. | 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 1 2 M such that Y1 - X1 P1 S1 T X1 T Xk kX1 T X1 Y2 - X2 P2 s2 T X1 T xk kx1 T X1 country 1 country 2 YM - XM Pm Zm country M T X1 T xk kx1 T X1 Assume each El 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 eft - vfIT - vJt E st j - ơt ỈT Where Oj. contemporaneous covariance between errors of equations i and j Nam T. Hoang University of New England - Australia 1 University of Economics - HCMC - Vietnam Advanced Econometrics Chapter 11 Seemingly unrelated regressions Assumption there is a p such that 1 o Y XT s a 1 a12I ơ I ƠV I E ss 21 22 MT xMT ỉ ỉ _ơM1I Tt 2I Ơ1MI ơ2 MI ƠMM S I a11 a12 Ơ1M Where S CT2- ỉ a 22 ơ2 M ỉ M 1 ƠM 2 ƠMM _ II. GENERALIZED LEAST SQUARES ESTIMATION OF SUR MODEL GLS The equation 1 can be estimated by GLS if E ss is known Tsur X E ss -1 X -1 X E ss -1 Y Tur X S I -1 X -1 X S I -1 Y GLS is the best linear unbiased estimator VarCov Tur X E ss -1 X -1 Advantages of SUR over single-equation OLS 1. Gain in efficiency Because B m will have smaller varriance than Tnts sUR OLs Nam T. Hoang University of New England - Australia 2 University of Economics - HCMC - Vietnam Advanced Econometrics Chapter 11 Seemingly unrelated regressions Pols à P1 OLS kx1 P2 OLS Pm OLS kx1 V TM xl Note that p OLS is efficient estimator for p but Pols is not efficient estimator for p and Psur is efficient estimator for p. 2. Test or .