Spatial Econometrics FAQ

Questions specific to spatial panel models in PanelBox — when to use spatial models, how to choose between SAR/SEM/SDM, weight matrix construction, parameter interpretation, and performance.

Looking for other topics?

• General questions: General FAQ
• Advanced methods (GMM, VAR, Heckman): Advanced FAQ
• Error messages and debugging: Troubleshooting

---

When to Use Spatial Models

When should I use spatial models instead of standard panel models?

Consider spatial models when:

• Your data has a geographic or network structure (regions, countries, firms in supply chains)
• Moran's I test shows significant spatial autocorrelation (p < 0.05)
• Economic theory suggests spillovers or interactions between units
• You observe clustering patterns in OLS residuals
• Cross-sectional dependence tests indicate spatial patterns

Run diagnostics first:

from panelbox.diagnostics import MoranIPanelTest

moran = MoranIPanelTest(data, variable="y", W=W)
result = moran.run()

if result.pvalue < 0.05:
    print("Significant spatial autocorrelation — consider spatial models")

How do I test for spatial autocorrelation?

The Moran's I test is the standard diagnostic:

from panelbox.diagnostics import MoranIPanelTest

moran = MoranIPanelTest(data, variable="y", W=W)
result = moran.run()
print(f"Moran's I = {result.statistic:.4f}, p-value = {result.pvalue:.4f}")

• Significant Moran's I (p < 0.05): spatial autocorrelation exists
• Positive I: similar values cluster together (most common)
• Negative I: dissimilar values cluster together (checkerboard pattern)

For more detail, use Local Moran's I (LISA) to identify spatial clusters:

from panelbox.diagnostics import LocalMoranI

lisa = LocalMoranI(data, W=W)
lisa_result = lisa.run()
# Identifies hot spots, cold spots, and spatial outliers

---

Model Selection (SAR vs SEM vs SDM)

How do I choose between SAR, SEM, and SDM?

Use the Lagrange Multiplier (LM) test decision tree:

LM Test Results	Recommended Model	Interpretation
Only LM-lag significant	SAR (Spatial Lag)	Spatial dependence in dependent variable
Only LM-error significant	SEM (Spatial Error)	Spatial dependence in error term
Both significant → check robust versions	See below	Need more specific tests
Robust LM-lag significant	SAR	Spatial lag dominates
Robust LM-error significant	SEM	Spatial error dominates
Both robust tests significant	SDM or GNS	Both types of dependence

from panelbox.diagnostics import lm_lag_test, lm_error_test, run_lm_tests

# Run all LM tests at once
lm_results = run_lm_tests(result, W)
print(lm_results)

When in doubt: LeSage & Pace (2009) recommend starting with SDM (Spatial Durbin Model). SDM nests both SAR and SEM as special cases and avoids bias from omitting relevant spatial lags.

What are the differences between all the spatial models?

Model	Equation	Parameters	Use case
SAR (Spatial Lag)	\(y = \rho Wy + X\beta + \varepsilon\)	\(\rho\)	Spillovers in outcomes
SEM (Spatial Error)	\(y = X\beta + u\), \(u = \lambda Wu + \varepsilon\)	\(\lambda\)	Spatially correlated omitted variables
SDM (Spatial Durbin)	\(y = \rho Wy + X\beta + WX\theta + \varepsilon\)	\(\rho, \theta\)	Both outcome and covariate spillovers
GNS (General Nesting)	\(y = \rho Wy + X\beta + WX\theta + u\), \(u = \lambda Wu + \varepsilon\)	\(\rho, \theta, \lambda\)	All types of spatial dependence

from panelbox.models.spatial import SpatialLag, SpatialError, SpatialDurbin

sar = SpatialLag(formula, data, entity_col, time_col, W=W).fit()
sem = SpatialError(formula, data, entity_col, time_col, W=W).fit()
sdm = SpatialDurbin(formula, data, entity_col, time_col, W=W).fit()

My LM tests are both significant — what should I do?

When both LM-lag and LM-error are significant, use the robust versions:

from panelbox.diagnostics import run_lm_tests

lm_results = run_lm_tests(result, W)
# Check robust LM-lag and robust LM-error

• If robust LM-lag is significant but robust LM-error is not → SAR
• If robust LM-error is significant but robust LM-lag is not → SEM
• If both robust tests are significant → SDM (Spatial Durbin Model) or GNS

This follows the Anselin (2005) testing strategy.

---

Spatial Weight Matrices

How do I create a spatial weight matrix?

PanelBox supports multiple methods:

1. Contiguity-based (shared borders):

import libpysal
W = libpysal.weights.Queen.from_dataframe(gdf)
W_array = W.full()[0]

2. Distance-based (within threshold):

import libpysal
coords = data[["longitude", "latitude"]].values
W = libpysal.weights.DistanceBand.from_array(coords, threshold=500)

3. k-Nearest Neighbors:

W = libpysal.weights.KNN.from_dataframe(gdf, k=5)

4. Custom matrix (e.g., trade flows, input-output):

import numpy as np
W_matrix = np.array([[0, 1, 0], [1, 0, 1], [0, 1, 0]], dtype=float)
# Row-standardize manually
row_sums = W_matrix.sum(axis=1, keepdims=True)
W_matrix = W_matrix / row_sums

Should I row-standardize my weight matrix?

Yes, in almost all cases. Row-standardization means each row sums to 1, so the spatial lag \(Wy\) becomes a weighted average of neighbors.

Benefits:

• Facilitates interpretation (weighted average of neighbors)
• Ensures \(\rho\) is bounded between -1 and 1
• Makes results comparable across different W specifications
• Required by most spatial model implementations

When not to row-standardize:

• When using inverse distance weights with specific interpretation
• Some network models with meaningful absolute edge weights
• When preserving absolute distance is important for the research question

How does the cutoff distance affect results?

The choice of distance threshold for distance-based weight matrices can affect results. Perform a sensitivity analysis:

from panelbox.models.spatial import SpatialLag

for threshold in [100, 200, 500, 1000]:
    W_t = create_distance_weights(coords, threshold=threshold)
    model = SpatialLag(formula, data, entity_col, time_col, W=W_t)
    result = model.fit()
    print(f"Threshold={threshold}km: rho={result.rho:.3f}, "
          f"beta={result.params['x1']:.3f}")

If results are highly sensitive to the threshold, this suggests the spatial structure is not well-defined and you should explore alternative specifications.

How do I handle islands (units with no neighbors)?

Islands (isolated units with no neighbors) cause problems in spatial models because their row in W is all zeros.

Options:

1. Remove islands from the analysis
2. Connect to nearest neighbor regardless of threshold
3. Increase the distance threshold to include more connections

import numpy as np

# Check for islands
W_array = W.full()[0]
islands = np.where(W_array.sum(axis=1) == 0)[0]
if len(islands) > 0:
    print(f"Warning: {len(islands)} islands found")
    # Option 1: Remove them
    data_no_islands = data[~data.index.isin(islands)]

---

Parameter Interpretation

What does rho (spatial lag parameter) mean?

The spatial lag parameter \(\rho\) in SAR/SDM models measures the strength of spatial spillovers in outcomes:

• \(\rho > 0\): positive spatial dependence (similar values cluster together)
• \(\rho < 0\): negative spatial dependence (dissimilar neighbors)
• \(\rho = 0\): no spatial dependence
• \(|\rho| < 1\): required for model stability

Interpretation example: \(\rho = 0.3\) means a 10% increase in neighbors' average \(y\) is associated with a 3% increase in own \(y\). The spatial multiplier is \(1/(1-\rho)\), so \(\rho = 0.3\) gives a total multiplier of approximately 1.43.

What does lambda (spatial error parameter) mean?

The spatial error parameter \(\lambda\) in SEM models captures spatial correlation in unobserved factors:

• \(\lambda > 0\): positive spatial correlation in errors (spatially correlated omitted variables)
• \(\lambda < 0\): negative spatial correlation in errors
• \(|\lambda| < 1\): required for stability

Unlike \(\rho\), \(\lambda\) does not have a direct economic interpretation for spillovers. It indicates that omitted variables have a spatial pattern and that ignoring this would lead to inefficient estimates and biased standard errors.

How do I interpret direct vs indirect effects?

In spatial models (especially SAR and SDM), the coefficients \(\beta\) do not directly give the marginal effects because of the spatial multiplier \((I - \rho W)^{-1}\).

• Direct effect: impact of changing \(X_i\) on \(y_i\) (own-unit effect)
• Indirect effect: impact of changing \(X_i\) on \(y_j\) for \(j \neq i\) (spillover)
• Total effect: direct + indirect

from panelbox.effects import compute_spatial_effects

effects = compute_spatial_effects(result, W)
print(f"Direct effect of x1: {effects.direct['x1']:.4f}")
print(f"Indirect effect of x1: {effects.indirect['x1']:.4f}")
print(f"Total effect of x1: {effects.total['x1']:.4f}")

Example: If direct effect of education = 0.5 and indirect = 0.2:

• 1% increase in own education → 0.5% increase in own outcome
• 1% increase in own education → 0.2% increase in neighbors' outcomes
• Total impact = 0.7%

Do not interpret raw coefficients as marginal effects

In SAR/SDM models, always compute and report the effects decomposition rather than the raw \(\beta\) coefficients.

What should I report in my paper?

For spatial panel results, report:

1. Diagnostics: Moran's I test, LM tests (justifying model choice)
2. Model comparison: AIC, BIC, Log-likelihood across OLS / SAR / SEM / SDM
3. Spatial parameters: \(\rho\) and/or \(\lambda\) with standard errors
4. Effects decomposition: direct, indirect, and total effects (for SAR/SDM)
5. Post-estimation: residual Moran's I (should be non-significant)
6. Weight matrix: description, summary statistics (avg. neighbors, sparsity)

---

Dynamic Spatial Panels

When should I add a time lag to a spatial model?

Add a time lag (\(y_{i,t-1}\)) when there is persistence over time — the outcome depends on its own past values, not just spatial neighbors.

from panelbox.models.spatial import DynamicSpatialPanel

dsp = DynamicSpatialPanel(
    formula, data, entity_col, time_col, W=W,
    lags=1  # Include y_{t-1}
)
result = dsp.fit()

Examples requiring dynamic spatial models:

• Regional GDP growth with both spatial spillovers and persistence
• House prices influenced by neighbors and lagged own prices
• Pollution levels with spatial diffusion and temporal inertia

How do I interpret the space-time lag (phi * W * y_{t-1})?

The space-time lag \(\phi W y_{t-1}\) captures diffusion effects — how neighbors' past outcomes affect current own outcomes.

• \(\phi > 0\): neighbors' past high values pull your current value up
• \(\phi < 0\): neighbors' past high values push your current value down

This differs from the contemporaneous spatial lag \(\rho W y_t\), which captures instantaneous feedback.

---

Diagnostics

How do I interpret Local Moran's I (LISA) results?

Local Moran's I identifies four types of spatial clusters:

Cluster Type	Meaning	Local I
High-High (hot spot)	High values surrounded by high neighbors	Positive
Low-Low (cold spot)	Low values surrounded by low neighbors	Positive
High-Low (spatial outlier)	High value surrounded by low neighbors	Negative
Low-High (spatial outlier)	Low value surrounded by high neighbors	Negative

from panelbox.diagnostics import LocalMoranI

lisa = LocalMoranI(data, W=W)
result = lisa.run()
# Plot cluster map if geopandas + shapefile available

Significant LISA statistics (after multiple-testing correction) identify locations that deviate from spatial randomness.

How do I test if the spatial model improved over OLS?

Use multiple criteria:

1. Likelihood Ratio Test:

from scipy import stats

lr_stat = 2 * (spatial_result.llf - ols_result.llf)
p_value = 1 - stats.chi2.cdf(lr_stat, df=1)
print(f"LR test p-value: {p_value:.4f}")

2. Information Criteria:

print(f"OLS AIC: {ols_result.aic:.2f}")
print(f"SAR AIC: {sar_result.aic:.2f}")
# Lower AIC is better

3. Residual Moran's I (should be non-significant after spatial model):

moran_resid = MoranIPanelTest(spatial_result.resid, W=W)
result = moran_resid.run()
# p > 0.05: no remaining spatial autocorrelation

---

Inference and Standard Errors

What is Spatial HAC and when should I use it?

Spatial HAC (Conley, 1999) standard errors are robust to both spatial and temporal correlation. Use them when:

• You have both spatial and temporal dependence
• Standard errors seem too small (overly optimistic)
• Cross-sectional dependence persists in residuals

from panelbox.standard_errors import SpatialHAC

shac = SpatialHAC(
    result,
    coords=data[["lon", "lat"]].values,
    spatial_cutoff=500,   # km for spatial correlation
    temporal_cutoff=2     # periods for temporal correlation
)
robust_se = shac.compute()

Driscoll-Kraay vs Spatial HAC — which should I use?

Feature	Spatial HAC	Driscoll-Kraay
Requires W	No (uses distance cutoff)	No
Spatial structure	Explicit (distance-based)	General cross-sectional dependence
Flexibility	More specific	More agnostic

Use Spatial HAC when you know the spatial structure and can specify a distance cutoff.

Use Driscoll-Kraay when the spatial structure is unknown or you want to be agnostic about the form of cross-sectional dependence.

from panelbox.standard_errors import DriscollKraayStandardErrors

dk = DriscollKraayStandardErrors(result, lag_cutoff=2)
robust_se = dk.compute()

---

Performance

Spatial estimation is slow. How can I speed it up?

Spatial ML estimation requires computing the log-determinant of \((I - \rho W)\) for each optimization step, which is expensive for large N.

Performance guidelines:

N (entities)	Estimation Time	Recommended Approach
< 500	< 10 seconds	Standard ML
500 - 1,000	10 - 60 seconds	Standard ML, sparse W
1,000 - 5,000	1 - 5 minutes	Sparse matrices required
5,000 - 10,000	5 - 30 minutes	Chebyshev approximation
> 10,000	> 30 minutes	Consider GMM/2SLS

Tips:

# Use sparse weight matrix
from scipy import sparse
W_sparse = sparse.csr_matrix(W_array)

# Pre-compute eigenvalues (speeds up repeated estimation)
eigenvalues = np.linalg.eigvalsh(W_array)

# Use Chebyshev approximation for log-determinant
result = model.fit(method="chebyshev", order=50)

Can I use spatial models with very large panels (N > 10,000)?

For very large N, ML estimation becomes prohibitively slow. Alternatives:

1. GMM/2SLS estimation — faster but less efficient
2. Chebyshev log-determinant approximation — reduces computation cost
3. Sparse weight matrices — critical for distance-based W
4. Subsample for exploration — estimate on subset, then run on full data

Monitor sparsity of your weight matrix:

sparsity = (W_array == 0).sum() / W_array.size
print(f"Weight matrix sparsity: {sparsity:.1%}")
# If > 90% sparse, use sparse matrix format for significant speedup

---

See Also

• General FAQ — getting started, model selection, results interpretation
• Advanced FAQ — GMM, VAR, Heckman, cointegration
• Troubleshooting — error messages and debugging
• Spatial API Reference
• Spatial Theory
• Spatial Tutorial