SUR Estimation Guide¶

Key Takeaway

Use SUR when your panel entities have correlated errors and different regressors. If both conditions hold, SUR is more efficient than equation-by-equation OLS. If regressors are identical across entities, SUR gives the same point estimates as OLS.

When to Use SUR¶

Decision Checklist¶

Ask these questions in order:

1. Do entities have correlated errors?
   ├── NO  → OLS per entity is fine. SUR won't help.
   └── YES
        2. Do entities have different regressors?
           ├── NO  → SUR = OLS (no efficiency gain). Use FE/RE instead.
           └── YES
                3. Are coefficients entity-specific?
                   ├── YES → SUR is the right model.
                   └── NO (common β)
                        → Consider Pooled OLS + PCSE for common β
                          with cross-sectional correlation correction.

SUR vs Alternatives¶

Scenario	Best Model	Why
Correlated errors + different regressors	SUR	Exploits cross-equation structure
Correlated errors + same regressors	FE/RE or OLS+PCSE	SUR = OLS when X identical
Uncorrelated errors + different regressors	OLS per entity	No cross-equation info to exploit
Common \(\beta\) + cross-sectional dependence	Pooled OLS + PCSE	One coefficient vector, robust SEs
Slope heterogeneity + large N	Mean Group	MG averages, SUR estimates jointly
Dynamic model + endogeneity	GMM	SUR assumes exogenous regressors

Quick Start¶

Basic SUR (same formula, all entities)¶

import panelbox as pb

data = pb.load_grunfeld()
sur = pb.SUR("invest ~ value + capital", data, "firm", "year")
result = sur.fit()

# System-level summary with Sigma matrix and BP test
print(result.system_summary())

# Individual equation
print(result.equation_summary("General Motors"))

Heterogeneous Formulas (dict)¶

sur = pb.SUR(
    formula={
        "General Motors": "invest ~ value + capital",
        "Chrysler": "invest ~ value",
        "General Electric": "invest ~ capital",
    },
    data=data,
    entity_col="firm",
    time_col="year",
)
result = sur.fit()

Each entity gets its own regression specification. Entities not in the dict are excluded.

Iterated SUR (ISUR ≈ MLE)¶

sur = pb.SUR(
    "invest ~ value + capital", data, "firm", "year",
    iterate=True, max_iter=200, tol=1e-8,
)
result = sur.fit()
print(f"Converged: {result.converged} in {result.n_iterations} iterations")

Interpreting Results¶

System Summary¶

The system_summary() output includes:

Section	What to Look For
System R² (McElroy)	Overall fit across all equations jointly
Sigma matrix	Cross-equation error covariances — large off-diagonal → SUR gains
Correlation matrix	Standardized version of Sigma — magnitudes close to ±1 mean strong cross-equation dependence
Breusch-Pagan test	If \(p < 0.05\), reject independence → SUR is beneficial
Per-equation R²	Fit quality per entity
Efficiency gain	Ratio SE(SUR)/SE(OLS) — values < 1 mean SUR improved precision

Breusch-Pagan Independence Test¶

bp = result.bp_independence_test
print(f"LM stat: {bp['statistic']:.2f}, p-value: {bp['pvalue']:.4f}")

Result	Interpretation
\(p < 0.05\)	Errors are correlated across entities — SUR is worthwhile
\(p \geq 0.05\)	Cannot reject independence — SUR ≈ OLS, no efficiency gain

Efficiency Gain Table¶

# Values < 1 mean SUR is more efficient
for entity, gains in result.efficiency_gain.items():
    print(f"\n{entity}:")
    print(gains)

A gain ratio of 0.85 means SUR standard errors are 85% of OLS standard errors for that coefficient — a 15% improvement.

The Degenerate Case¶

Same Regressors = No Gain

When all entities share the exact same formula and data structure, SUR point estimates are identical to equation-by-equation OLS (Zellner 1962). The efficiency gain comes from different regressors across equations combined with correlated errors.

If your Breusch-Pagan test rejects independence but the efficiency gains are negligible, check whether your regressors are effectively the same across entities.

Iterated SUR vs Non-Iterated¶

Feature	Non-iterated (FGLS)	Iterated (ISUR)
Steps	2-step: OLS → GLS	Iterates until convergence
Equivalence	—	MLE under normality
Efficiency	Asymptotically efficient	Asymptotically efficient
Finite-sample	May differ slightly	Closer to MLE in small samples
Robustness	Less sensitive to \(\hat{\Sigma}\)	More sensitive to specification
Use when	Default choice	Want MLE-equivalent estimates

Practical Advice

Start with the non-iterated SUR (iterate=False, the default). If results are sensitive to switching to ISUR, this may indicate misspecification. In well-specified models, the two should be very close.

Connection to PanelVAR¶

PanelBox's PanelVAR module supports cov_type='sur', which uses the same cross-equation covariance estimation:

from panelbox.var import PanelVAR

var = PanelVAR(data, variables=["y1", "y2"], entity_col="firm", time_col="year", lags=2)
result = var.fit(cov_type="sur")

In a VAR system, each equation has different regressors (different lags appear on the RHS), so the SUR structure provides genuine efficiency gains. The underlying mechanics are the same:

Estimate each VAR equation by OLS
Build the cross-equation \(\hat{\Sigma}\) from residuals
Apply FGLS using Kronecker structure

Common Pitfalls¶

1. Too Many Entities, Too Few Periods¶

\(\hat{\Sigma}\) is an \(N \times N\) matrix estimated from \(T\) observations. When \(N\) is large relative to \(T\), \(\hat{\Sigma}\) is poorly estimated or singular.

Rule of thumb: SUR works best with small \(N\) (few entities) and moderate to large \(T\).

2. Unbalanced Panels¶

SUR handles unbalanced panels by computing \(\hat{\sigma}_{ij}\) from the \(T_{ij}\) common time periods between entities \(i\) and \(j\). If two entities share very few periods, their covariance estimate will be noisy.

3. Endogenous Regressors¶

SUR assumes all regressors are exogenous. If you suspect endogeneity, use GMM or IV estimation instead.

4. Expecting Gains with Identical Regressors¶

If all entities have the same formula and the same observations, SUR cannot improve on OLS. This is a mathematical identity, not a limitation.

Full Example¶

import panelbox as pb

# Load data
data = pb.load_grunfeld()

# Fit SUR with heterogeneous formulas
sur = pb.SUR(
    formula={
        "General Motors": "invest ~ value + capital",
        "Chrysler": "invest ~ value",
    },
    data=data,
    entity_col="firm",
    time_col="year",
    iterate=True,
)
result = sur.fit()

# System-level diagnostics
print(result.system_summary())

# Correlation heatmap
result.plot_correlation_matrix()

# Check if SUR was worth it
bp = result.bp_independence_test
if bp["pvalue"] < 0.05:
    print("Cross-equation correlation is significant — SUR is beneficial.")
else:
    print("No significant cross-equation correlation — OLS is sufficient.")