SUR_REGRESSION

SUR_REGRESSION estimates a Seemingly Unrelated Regression system from a previously created SIMU_REGRESSION object. It compares ordinary least squares estimates with SUR estimates and reports whether cross-equation residual relationships improve coefficient efficiency.

Syntax

sur_regression(simuRegression)

Example

r1 = REGRESSION(LOGDIFF(LIST(CPIAUCSL, PPIACO, M2SL)))
r2 = REGRESSION(LOGDIFF(LIST(PCE, CPIAUCSL, PAYEMS)))
r3 = REGRESSION(LOGDIFF(LIST(REALLN, PAYEMS, M2SL)))

s = SIMU_REGRESSION(r1, r2, r3)
sur = SUR_REGRESSION(s)
slideshow(sur)

What it does

SUR starts with a system of separate regression equations. Each equation has its own dependent variable and explanatory variables, but the unexplained portions of the equations may be correlated. When those residual correlations are meaningful, SUR can improve standard errors and t-statistics compared with estimating each equation separately.

RainbowStats uses a memory-efficient sparse SUR formulation. Instead of constructing the full system covariance matrix, it works directly with the smaller residual covariance matrix across equations. This allows larger time-series systems to be estimated without creating a massive Omega matrix.

Output

• OLS beta estimates
• OLS standard errors
• SUR beta estimates
• SUR standard errors
• standard-error efficiency gain
• residual covariance matrix
• residual correlation matrix
• system diagnostic summary

System summary

SUR System Summary
------------------
Method                    Feasible GLS (Sparse SUR)
Equations                 3
Observations              807
Average residual corr.     0.094
Maximum residual corr.     0.191
Residual dependence        Weak
System SSR improvement     1.015

Interpretation

The command is most useful when the equations have different explanatory variables and their residuals are correlated. If residual correlations are weak, the SUR estimates will usually be close to OLS. If residual correlations are moderate or strong, SUR may produce smaller standard errors and stronger t-statistics.

The Efficiency Gain column reports the ratio of the OLS standard error to the SUR standard error. A value above 1.00 means the SUR estimate has a smaller standard error than the corresponding OLS estimate.

Notes

• The input must be a SIMU_REGRESSION object.
• Equations are aligned to a common observation count before SUR estimation.
• SUR improves coefficient efficiency; it is not primarily an R-squared improvement tool.
• If every equation uses exactly the same regressors, SUR will usually be very close to OLS.