SUR
What is SUR
Seemingly unrelated regressions: although each equation in the system has its own coefficient vector (seems unrelated)
correlation across the error terms in different equations can provide some links in estimation (actually related)
SUR formulas
Robust Variance estimator
The unconditional variance of errors, E(uiu′i), is entirely unrestricted It allows cross equation correlation as well as different error variances in each equation
The conditional variance of errors, E(uiu′i|Xi), can depend on Xi in an arbitrary unknown fashion
How to perform test on SOLS coefficients
How to impose cross equation restrictions
OPTION 1: Write the regressor and coefficient matrix in the usual way
OPTION 2: We can directly incorporate the restriction when constructing the regressor matrix and coefficient vector
How to calculate SGLS/FGLS estimator and its variance estimator
Assumptions
Each element of ui is uncorrelated with each element of Xi, i.e. errors and regressors are uncorrelated within and across equations
(SOLS.A1 only requires no correlation within each equation)
SGLS uses the transformed data, so the transformed Xi and ui should also be uncorrelated, i.e. E(Xi′Ω−1ui) = 0
If one additional assumption holds, then FGLS is asymptotically more efficient than SOLS and others.
Difficult to interpret, but if errors are heteroscedastic within each equation, e.g. var(ui1) = 0.3x13,i, then SGLS.A3 fails
Formulas
, In FGLS we replace the unknown matrix Ω with a consistent estimator
Under SGLS.A1–SGLS.A3, the asymptotic variance of βFGLS now becomes
Implication:
FGLS estimator is consistent as SGLS
FGLS estimator follows normal distribution asymptotically
In finite sample, especially with small sample size N, the actual distribution of FGLS is likely to be non-normal
Under which situation is there singular variance matrix
In which cases SOLS of FGLS do not work
When the weakest orthogonal condition is not satisfied
How to calculate the 2SLS/GMM estimator and variance
Identification of the SUR system is equivalent to identification equation by equation if β are unrestricted across equations
What is SUR
Seemingly unrelated regressions: although each equation in the system has its own coefficient vector (seems unrelated)
correlation across the error terms in different equations can provide some links in estimation (actually related)
SUR formulas
Robust Variance estimator
The unconditional variance of errors, E(uiu′i), is entirely unrestricted It allows cross equation correlation as well as different error variances in each equation
The conditional variance of errors, E(uiu′i|Xi), can depend on Xi in an arbitrary unknown fashion
How to perform test on SOLS coefficients
How to impose cross equation restrictions
OPTION 1: Write the regressor and coefficient matrix in the usual way
OPTION 2: We can directly incorporate the restriction when constructing the regressor matrix and coefficient vector
How to calculate SGLS/FGLS estimator and its variance estimator
Assumptions
Each element of ui is uncorrelated with each element of Xi, i.e. errors and regressors are uncorrelated within and across equations
(SOLS.A1 only requires no correlation within each equation)
SGLS uses the transformed data, so the transformed Xi and ui should also be uncorrelated, i.e. E(Xi′Ω−1ui) = 0
If one additional assumption holds, then FGLS is asymptotically more efficient than SOLS and others.
Difficult to interpret, but if errors are heteroscedastic within each equation, e.g. var(ui1) = 0.3x13,i, then SGLS.A3 fails
Formulas
, In FGLS we replace the unknown matrix Ω with a consistent estimator
Under SGLS.A1–SGLS.A3, the asymptotic variance of βFGLS now becomes
Implication:
FGLS estimator is consistent as SGLS
FGLS estimator follows normal distribution asymptotically
In finite sample, especially with small sample size N, the actual distribution of FGLS is likely to be non-normal
Under which situation is there singular variance matrix
In which cases SOLS of FGLS do not work
When the weakest orthogonal condition is not satisfied
How to calculate the 2SLS/GMM estimator and variance
Identification of the SUR system is equivalent to identification equation by equation if β are unrestricted across equations
