Spatial Lag vs Spatial Error Model: How to Choose

TL;DR — Fit spreg.OLS(y, X, w, spat_diag=True) and read the Lagrange Multiplier tests. If only LM-Lag is significant, use a spatial lag model (spreg.GM_Lag); if only LM-Error is significant, use a spatial error model (spreg.GM_Error). If both are significant, compare the robust variants and follow the more significant one. Lag = neighbours’ outcomes drive yours (spillover); error = omitted spatial factors correlate the residuals (nuisance).

Why this matters #

Both models correct the same visible symptom — spatially autocorrelated OLS residuals — but they encode completely different data-generating processes, and choosing the wrong one gives you either biased coefficients or a misread of the mechanism. The decision is not a matter of taste: it follows a well-defined diagnostic rule. This page is the decision framework for the spatial regression models workflow, within the broader Python workflows for spatial modeling and regression pipeline. For the mechanics of fitting a lag model once you have chosen it, see implementing spatial lag models in Python.

Environment #

pip install "geopandas>=0.14" "libpysal>=4.8" "spreg>=1.5" \
            "numpy>=1.24" "scipy>=1.10"

import geopandas as gpd           # >= 0.14
import numpy as np                # >= 1.24
import libpysal                   # >= 4.8
import spreg                      # >= 1.5

The two models side by side #

Spatial Lag Model (SAR). The outcome depends directly on neighbours’ outcomes through a feedback term:

$$\mathbf{y} = \rho \mathbf{W}\mathbf{y} + \mathbf{X}\boldsymbol{\beta} + \boldsymbol{\varepsilon}$$

$\rho$ is the spatial autoregressive coefficient. Because $\mathbf{W}\mathbf{y}$ is on the right-hand side, a shock anywhere propagates through the whole network, so effects spill over.

Spatial Error Model (SEM). The outcome does not depend on neighbours’ outcomes; instead the disturbance is spatially structured:

$$\mathbf{y} = \mathbf{X}\boldsymbol{\beta} + \mathbf{u}, \qquad \mathbf{u} = \lambda \mathbf{W}\mathbf{u} + \boldsymbol{\varepsilon}$$

$\lambda$ is the spatial error coefficient. Here spatial dependence is a nuisance caused by omitted, spatially patterned covariates. OLS stays unbiased but inefficient with wrong standard errors.

The decision rule #

The choice is made empirically with the Anselin–Florax Lagrange Multiplier sequence, then confirmed with theory. The logic:

• Only LM-Lag significant → spatial lag model.
• Only LM-Error significant → spatial error model.
• Both significant → the standard tests are contaminated by each other; switch to the robust variants and follow whichever robust statistic is more significant.
• Neither significant, but residual Moran’s I is → an omitted spatially structured covariate; investigate specification before adding a spatial parameter.
• Neither significant → OLS is adequate.

Running the diagnostic that decides #

The single call below produces every statistic the decision rule needs. Build a validated spatial weight matrix first, then run OLS with spat_diag=True.

# Weights: row-standardised, no islands
gdf = gpd.read_file("data/tracts.gpkg").to_crs("EPSG:32633").reset_index(drop=True)
w = libpysal.weights.Queen.from_dataframe(gdf)
w.transform = "r"
assert len(w.islands) == 0, f"islands: {w.islands}"

y = gdf["log_price"].values.reshape(-1, 1)
X = gdf[["sqft", "age", "dist_cbd"]].values
names = ["sqft", "age", "dist_cbd"]

ols = spreg.OLS(y, X, w, spat_diag=True, moran=True, name_y="log_price", name_x=names)

# The four decision statistics (each is a (statistic, p-value) pair)
lm_lag,  lm_err  = ols.lm_lag,        ols.lm_error
rlm_lag, rlm_err = ols.lm_lag_robust, ols.lm_error_robust
print(f"LM-Lag        stat={lm_lag[0]:.2f}  p={lm_lag[1]:.4f}")
print(f"LM-Error      stat={lm_err[0]:.2f}  p={lm_err[1]:.4f}")
print(f"Robust LM-Lag stat={rlm_lag[0]:.2f}  p={rlm_lag[1]:.4f}")
print(f"Robust LM-Err stat={rlm_err[0]:.2f}  p={rlm_err[1]:.4f}")

Encode the decision rule directly so the choice is reproducible:

def choose_spatial_model(lm_lag, lm_err, rlm_lag, rlm_err, alpha=0.05):
    lag_sig,  err_sig  = lm_lag[1]  < alpha, lm_err[1]  < alpha
    if lag_sig and not err_sig:
        return "lag"
    if err_sig and not lag_sig:
        return "error"
    if lag_sig and err_sig:
        # both significant -> robust tests break the tie
        return "lag" if rlm_lag[1] < rlm_err[1] else "error"
    return "ols"   # neither significant

choice = choose_spatial_model(lm_lag, lm_err, rlm_lag, rlm_err)
print("Selected specification:", choice)

Then fit the chosen specification and compare fit:

if choice == "lag":
    model = spreg.GM_Lag(y, X, w, name_y="log_price", name_x=names)
elif choice == "error":
    model = spreg.GM_Error(y, X, w, name_y="log_price", name_x=names)
else:
    model = ols
print(model.summary)

Interpreting the output #

Read the four LM statistics as pairs of (statistic, p-value). The standard LM tests answer “is there any lag/error dependence?”, but each is inflated when the other process is present. The robust variants partial out that contamination, which is exactly why they are the tie-breaker when both standard tests fire. Follow the robust test with the smaller p-value.

After fitting, confirm the choice two ways. First, theory: a lag model is only defensible when a genuine outcome-to-outcome spillover mechanism exists (prices, contagion, diffusion). If no such feedback is plausible, prefer the error model even when Robust LM-Lag edges it. Second, fit: compare model.aic between a fitted lag and error model; a large AIC gap corroborates the diagnostic. Finally, refit Moran’s I on the model residuals — if autocorrelation persists, neither simple specification is enough and you should consider a Spatial Durbin Model or geographically varying coefficients.

Best practices #

Never skip the robust tests when both fire #

Standard LM-Lag and LM-Error are each biased upward in the presence of the other process. When both are significant, the non-robust statistics cannot discriminate; only the robust variants can. Building the decision on the standard tests alone is the classic mistake.

Let theory veto the statistics #

The LM sequence is empirical, not causal. If your domain has no mechanism by which one location’s outcome changes another’s, a significant Robust LM-Lag more likely reflects an omitted spatial covariate — favour the error model. Statistics narrow the choice; substance confirms it.

Fix the weights matrix before trusting any test #

Islands and disconnected components corrupt the LM statistics and can make $\rho$ or $\lambda$ escape $(-1, 1)$. Validate w.islands == [] and w.n_components == 1 before reading diagnostics, as detailed in spatial weight matrices.

Remember the coefficients mean different things #

Error-model $\beta$ are ordinary marginal effects; lag-model $\beta$ are direct effects only and need a $(I - \rho W)^{-1}$ decomposition for the total impact. Do not compare the raw coefficient tables across the two model types as if they were on the same footing.

Troubleshooting #

Next steps #

Once you have selected the lag specification, work through the full fit, rho interpretation, and effect decomposition in implementing spatial lag models in Python. If diagnostics point to relationships that vary across space rather than a single global spillover, move to geographically weighted regression.

Frequently Asked Questions #

What is the core difference between a spatial lag and a spatial error model?

A spatial lag model (SAR) adds a spatially lagged dependent variable, $\rho \mathbf{W}\mathbf{y}$, to the equation: an outcome at one location is directly influenced by neighbours’ outcomes, so effects spill over across the map. A spatial error model (SEM) instead places spatial structure in the disturbance, $\mathbf{u} = \lambda \mathbf{W}\mathbf{u} + \boldsymbol{\varepsilon}$: the outcome is not affected by neighbours’ outcomes, but omitted spatially patterned factors correlate the errors. Lag models a substantive feedback; error models a nuisance.

Which Lagrange Multiplier test decides between lag and error?

Run OLS with spat_diag=True and read LM-Lag and LM-Error. If only one is significant, pick the matching model. If both are significant, the standard tests are contaminated by the other process, so compare the robust variants Robust LM-Lag and Robust LM-Error and follow whichever is more significant. This is the Anselin–Florax decision sequence.

Can I interpret coefficients the same way in both models?

No. In a spatial error model the $\beta$ coefficients are ordinary marginal effects, read exactly like OLS but with corrected standard errors. In a spatial lag model the $\beta$ are only direct effects; because a change propagates through the network via $(I - \rho W)^{-1}$, the total effect includes an indirect spillover and must be decomposed. Reading lag betas as if they were OLS coefficients understates the true impact.

Related

• Spatial Regression Models — the full lag, error, and Durbin workflow this decision sits inside
• Implementing Spatial Lag Models in Python — fitting and effect decomposition once you have chosen the lag model
• Spatial Weight Matrices — the validated W that every LM diagnostic depends on

← Back to Spatial Regression Models