Spatial Lag Model (SAR)

Quick Reference

Class: panelbox.models.spatial.SpatialLag Import: from panelbox.models.spatial import SpatialLag Stata equivalent: xsmle y x1 x2, wmat(W) model(sar) fe R equivalent: splm::spml(..., model="within", lag=TRUE)

Overview

The Spatial Lag model (also called the Spatial Autoregressive model, SAR) captures outcome spillovers — situations where one unit's outcome directly affects its neighbors' outcomes. The key feature is the spatially lagged dependent variable \(Wy\) on the right-hand side of the equation.

The model is specified as:

\[y = \rho W y + X\beta + \alpha + \varepsilon\]

where:

• \(y\) is the \(NT \times 1\) vector of dependent variable observations
• \(W\) is the \(N \times N\) spatial weight matrix (row-standardized)
• \(\rho\) is the spatial autoregressive parameter measuring the strength of spatial spillovers
• \(X\) is the \(NT \times K\) matrix of explanatory variables
• \(\beta\) is the \(K \times 1\) vector of coefficients
• \(\alpha\) captures individual (entity) effects
• \(\varepsilon\) is the error term

The parameter \(\rho\) has a direct economic interpretation: it captures how much a neighbor's outcome affects your own. A positive \(\rho\) indicates spatial clustering of similar values, while a negative \(\rho\) indicates spatial dispersion.

Quick Example

import numpy as np
from panelbox.models.spatial import SpatialLag, SpatialWeights

# Load your panel data (must be balanced)
# data has columns: y, x1, x2, region, year

# Create weight matrix (e.g., from contiguity)
W = SpatialWeights.from_contiguity(gdf, criterion='queen')

# Fit SAR model with fixed effects
model = SpatialLag("y ~ x1 + x2", data, "region", "year", W=W.matrix)
results = model.fit(effects='fixed', method='qml')

# View results
print(results.summary())

# Spatial parameter
print(f"rho = {results.rho:.4f}")  # e.g., 0.35

# Spillover effects
effects = results.spillover_effects
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}")

When to Use

• Outcome-based interactions: neighbors' outcomes directly influence your own outcome
  • Regional unemployment spillovers (labor mobility)
  • Technology adoption cascading across firms
  • Crime contagion between neighborhoods
  • Housing price effects between neighboring areas
• Trade and migration: flows between regions create interdependence in economic outcomes
• Policy diffusion: policy adoption in one jurisdiction influences neighbors
• Epidemiology: disease prevalence in one area affects neighboring areas

Key Assumptions

• Balanced panel: all entities must have observations for all time periods
• Stationarity: \(|\rho| < 1\) (spatial process is stable)
• Exogeneity of \(W\): the weight matrix is exogenous and correctly specified
• No spatial error correlation: errors are independent across units (otherwise use SDM or SEM)
• Panel structure: \(W\) matches the cross-sectional dimension (\(N \times N\))

Detailed Guide

Data Preparation

Spatial models in PanelBox require balanced panels. Ensure every entity has observations for every time period.

import pandas as pd

# Check balance
counts = data.groupby("region").size()
print(f"Entities: {counts.nunique()}, Periods per entity: {counts.unique()}")

# If unbalanced, balance it
complete_periods = data.groupby("year")["region"].nunique()
valid_periods = complete_periods[complete_periods == complete_periods.max()].index
data_balanced = data[data["year"].isin(valid_periods)]

Estimation Methods

The SAR model offers three estimation approaches depending on the effects specification:

Fixed Effects (QML)Random Effects (ML)Pooled (QML)

The default and most common approach. Uses within-transformation to eliminate entity effects, then maximizes a concentrated quasi-log-likelihood over \(\rho\).

results = model.fit(effects='fixed', method='qml')

The concentrated log-likelihood is:

\[\ell_c(\rho) = -\frac{NT}{2}\ln(2\pi) - \frac{NT}{2}\ln(\hat{\sigma}^2(\rho)) + T \ln|I_N - \rho W|\]

where \(\hat{\sigma}^2(\rho)\) is the concentrated variance for a given \(\rho\), obtained by OLS of \(\tilde{y} - \rho W\tilde{y}\) on \(\tilde{X}\) (tilde denotes within-transformed).

Full maximum likelihood estimation with random individual effects.

results = model.fit(effects='random', method='ml')

Estimates the variance components \(\sigma_\alpha^2\) (between) and \(\sigma_\varepsilon^2\) (within) alongside \(\rho\) and \(\beta\).

Ignores individual effects entirely. Useful as a baseline or when entity effects are not relevant.

results = model.fit(effects='pooled', method='qml')

Interpreting Results

The SAR model requires careful interpretation because of the spatial multiplier effect.

The Spatial Parameter \(\rho\)

• \(\rho > 0\): positive spatial dependence (common in most applications)
• \(\rho < 0\): negative spatial dependence (competition effects)
• \(\rho = 0\): no spatial dependence (reduces to standard panel model)
• The spatial multiplier is \(\frac{1}{1 - \rho}\). For \(\rho = 0.3\), the multiplier is \(\approx 1.43\)

Why Coefficients Are Not Direct Effects

In a standard OLS model, \(\beta_k\) is the marginal effect of \(x_k\) on \(y\). In the SAR model, this is not the case because of spatial feedback:

\[y = (I - \rho W)^{-1} X\beta + (I - \rho W)^{-1} \varepsilon\]

The matrix \((I - \rho W)^{-1}\) creates a spatial multiplier that distributes the effect of \(x_k\) across all units. See the Spatial Effects page for a full treatment of direct, indirect, and total effects.

Summary Output

print(results.summary())

The summary includes:

• Coefficient table: estimates, standard errors, t-statistics, p-values
• Spatial parameter \(\rho\) with standard error and significance
• Model fit: log-likelihood, AIC, BIC, pseudo R-squared
• Spatial information: number of units, weight matrix density

Configuration Options

fit() Parameters

Parameter	Type	Default	Description
effects	str	'fixed'	'fixed', 'random', or 'pooled'
method	str	'qml'	'qml' (fixed/pooled) or 'ml' (random)
rho_grid_size	int	20	Grid search resolution for \(\rho\) (QML)
optimizer	str	'brent'	'brent' (1D) or 'l-bfgs-b'
maxiter	int	1000	Maximum optimization iterations
tol	float	1e-6	Convergence tolerance
verbose	bool	False	Print optimization progress

Results Attributes

Attribute	Type	Description
params	pd.Series	All estimated parameters (including \(\rho\))
bse	pd.Series	Standard errors
tvalues	pd.Series	t-statistics
pvalues	pd.Series	Two-tailed p-values
rho	float	Spatial autoregressive parameter
llf	float	Log-likelihood at convergence
aic	float	Akaike Information Criterion
bic	float	Bayesian Information Criterion
rsquared_pseudo	float	Pseudo R-squared
resid	np.ndarray	Residuals
fitted_values	np.ndarray	Fitted values
method	str	Estimation method used
effects	str	Effects specification
spillover_effects	dict	Direct, indirect, total effects

Predictions

# In-sample prediction
y_hat = results.predict()

# Out-of-sample prediction (requires W for new spatial structure)
y_new = results.predict(new_data=new_df, W=W_new)

For SAR models, prediction uses the spatial multiplier:

\[\hat{y} = (I - \hat{\rho} W)^{-1} (X\hat{\beta} + \hat{\alpha})\]

This is computed separately for each time period.

Diagnostics

After fitting a SAR model, check:

1. Significance of \(\rho\): if not significant, a standard panel model may suffice
2. Residual spatial autocorrelation: Moran's I on residuals should be insignificant
3. Model comparison: compare AIC/BIC with SEM and SDM alternatives

# Check if spatial dependence is captured
from panelbox.diagnostics.spatial import MoranIPanelTest

moran = MoranIPanelTest(results.resid, W.matrix)
moran_result = moran.run()
print(f"Moran's I on residuals: {moran_result.statistic:.4f}")
print(f"p-value: {moran_result.pvalue:.4f}")
# Should be non-significant if SAR adequately captures spatial dependence

See Spatial Diagnostics for the full diagnostic workflow.

Tutorials

Tutorial	Description	Links
Spatial Econometrics	Complete SAR, SEM, SDM workflow

See Also

• Spatial Weight Matrices — How to construct \(W\)
• Spatial Error (SEM) — When dependence is in errors, not outcomes
• Spatial Durbin (SDM) — Generalization with spatially lagged covariates
• Direct, Indirect, and Total Effects — Proper effect interpretation
• Choosing a Spatial Model — When to use SAR vs SEM vs SDM
• Spatial Diagnostics — Tests for spatial dependence

References

1. Anselin, L. (1988). Spatial Econometrics: Methods and Models. Kluwer Academic.
2. Lee, L.F. and Yu, J. (2010). Estimation of spatial autoregressive panel data models with fixed effects. Journal of Econometrics, 154(2), 165-185.
3. LeSage, J. and Pace, R.K. (2009). Introduction to Spatial Econometrics. Chapman & Hall/CRC.
4. Elhorst, J.P. (2014). Spatial Econometrics: From Cross-Sectional Data to Spatial Panels. Springer.