Common mistake 1 with Spatial regression

Estimating the lag model by OLS on Wy. Wy is correlated with the error by construction - your neighbours are influenced by you - so the estimate is biased and the bias does not shrink with n. That is why this is done by maximum likelihood, and why the log-determinant term in the likelihood cannot be dropped.

Common mistake 2 with Spatial regression

Treating a significant ρ as proof of a spillover. Spatially correlated omitted variables produce the same symptom, which is exactly what the error model is for.

Common mistake 3 with Spatial regression

Choosing k or the distance band after seeing the results.

Common mistake 4 with Spatial regression

Reading the spatial models' standard errors as exact. They are asymptotic, and at a hundred-odd units they are optimistic.

EconStatsCraft › Spatial regression

Spatial regression

When a place's outcome depends on its neighbours' outcomes, OLS is biased and no amount of robust standard errors fixes it. Fits the spatial lag and spatial error models by maximum likelihood, runs the LM tests that say which one the data wants, and reports the spillover multiplier - the reason the total effect of a policy is larger than its coefficient.

What it needs

Dependent variable
Explanatory variables

The intuition

If neighbouring districts influence each other, OLS is wrong twice over: the standard errors are understated because observations are not independent, and if the influence is genuinely causal the coefficient is biased too. Spatial econometrics offers two models with very different meanings - one where a neighbour's OUTCOME affects yours, and one where only unobserved shocks are spatially shared - and choosing between them is the substantive decision.

When to use it

Whenever the units are places and the residuals from OLS are spatially clustered. Run Spatial autocorrelation on the residuals first; if it finds nothing, you do not need this.

Where the estimator comes from

The spatial lag (SAR) model is y=\rho Wy+X\beta+\varepsilon: a neighbour\'s outcome enters directly, so a change in x_i affects y_i, which affects y_j, which feeds back. Solving, y=(I-\rho W)^{-1}(X\beta+\varepsilon), and the expansion (I-\rho W)^{-1}=I+\rho W+\rho^2W^2+\dots shows the total effect of x_i exceeds \beta. The spatial error (SEM) model is y=X\beta+u with u=\lambda Wu+\varepsilon: only the disturbances are correlated, \beta keeps its usual meaning, and OLS remains unbiased but inefficient with wrong standard errors. OLS is inconsistent for SAR because Wy is endogenous by construction, so estimation is by maximum likelihood or by instrumenting Wy with WX and W^2X.

The model

\text{lag: } y=\rho Wy+X\beta+\varepsilon \qquad \text{error: } y=X\beta+u,\; u=\lambda Wu+\varepsilon

What it assumes

The weights matrix is right, and ρ or λ is inside the stable range. Outside it the reduced form (I − ρW)⁻¹ does not converge and the model describes nothing.
Every unit has every variable. A gap breaks the neighbour structure rather than costing one row.
The lag model assumes the spillover runs through the outcome and not through the regressors. If you think a neighbour's literacy affects your income directly, that is a third model (SLX) and the lag term will absorb it misleadingly.
The spatial structure is stable across the map. One weights matrix imposes the same neighbourhood definition on dense cities and empty districts, and five nearest neighbours means something quite different in each.

Statistical properties

In a SAR model beta is NOT the marginal effect. The direct effect includes feedback through neighbours and back, and the total effect summed over all units is beta/(1-rho). Reporting beta as if it were a marginal effect is the single commonest error in applied spatial work.
SAR and SEM have different economics. SAR says your neighbour's outcome affects you - policy diffusion, tax competition. SEM says you and your neighbour share unobserved shocks - the same weather, the same regional shock. These are not interchangeable.
The Lagrange multiplier tests of Anselin distinguish them. The ROBUST versions are the ones to read, because the plain LM tests each reject in the presence of the other form of dependence.
Under SEM, OLS coefficients remain unbiased. The loss is efficiency and correct standard errors, so if SEM is the truth the coefficient story survives OLS and the inference does not.
rho is bounded by the inverse of the largest eigenvalue of W, and for a row-standardised W that means rho below 1. An estimate near the bound indicates near-explosive spatial feedback and should be distrusted.
Omitted spatially-patterned variables produce apparent spatial dependence. Spatial models can absorb the symptom while leaving the misspecification in place.

How to read the output

Read the LM tests first, then choose. LM-lag and LM-error are both chi-squared with one degree of freedom; the larger statistic points at the model the data prefers. Both being large is common and means the two are hard to separate at this sample size.
ρ is not the effect. Because a change feeds into the neighbours and back, the total effect of a unit change in a regressor is about 1/(1−ρ) times its coefficient. At ρ = 0.5 that doubles it. Quoting the coefficient as the effect is the commonest error made with this model.
Compare the two spatial models by log-likelihood - they have the same number of parameters, so it is a fair comparison. Neither can be compared with OLS that way.
Check Moran's I on the fitted model's residuals. If dependence is still there, the spatial term has not solved it and something else is missing.

A worked reading

A district-level regression where OLS residuals show Moran's I significantly positive. The robust LM tests point at the lag form, so SAR is fitted and rho comes out around 0.56 - which is where the interpretation trap opens. The coefficient on the regressor is no longer its marginal effect: with rho = 0.56 the total effect is roughly beta/(1-0.56) = 2.3 times beta, spread across the district itself and its neighbours. A write-up quoting beta as "the effect" would understate it by more than half, and the software reports beta by default.

Symptom → diagnosis → what to do

If you see this	It means	Do this
Both LM tests rejecting	Common; the plain versions cross-react	Read the robust LM tests instead
Reporting beta as the marginal effect in SAR	Wrong; feedback is excluded	Report direct, indirect and total effects
rho near its upper bound	Near-explosive feedback	Distrust the estimate; check W
SAR and SEM giving similar betas	The dependence is weak either way	The choice matters little here; say so
Spatial dependence disappearing after adding a regressor	It was an omitted variable, not spillover	The better outcome. Prefer the simpler model
Results sensitive to W	Expected	Report the sensitivity rather than one specification

Common mistakes

Estimating the lag model by OLS on Wy. Wy is correlated with the error by construction - your neighbours are influenced by you - so the estimate is biased and the bias does not shrink with n. That is why this is done by maximum likelihood, and why the log-determinant term in the likelihood cannot be dropped.
Treating a significant ρ as proof of a spillover. Spatially correlated omitted variables produce the same symptom, which is exactly what the error model is for.
Choosing k or the distance band after seeing the results.
Reading the spatial models' standard errors as exact. They are asymptotic, and at a hundred-odd units they are optimistic.

Where to read this properly

Anselin (1988), Spatial Econometrics: Methods and Models.
Anselin, Bera, Florax and Yoon (1996), "Simple Diagnostic Tests for Spatial Dependence", Regional Science and Urban Economics 26(1).
LeSage and Pace (2009), Introduction to Spatial Econometrics - on direct and indirect effects.
Gibbons and Overman (2012), "Mostly Pointless Spatial Econometrics?", Journal of Regional Science 52(2), for the sceptical case.

All analyses

Data & variables

Each example is simulated from a known model, so you can check whether the estimator recovers the truth - and nothing here is anyone's private data.

Filter variables

Drag a variable into a slot on the right - or just click it.

Choose an example dataset above to begin.

Choose a transformation - it opens in the data editor, with a live preview.

Apply to

Multiply by

Columns ctrl-click for several

Group by ctrl-click for several

Within groups

Statistic

Bands

How many

Cut points

Keep rows where

Value

and

Fill with

Value

Also mark which rows were filled

Rules

Base period = 100

Which part

Percent at each tail

Threshold

Survey weight optional

Direction

Names column

Values column

Largest first

Delete them, not just count them

Read this column as

Tidy the text

New name optional

EconStatsCraft rigorous, plain-spoken econometrics

Econometrics that explains itself.

Regression, panel data, time series and the whole battery of tests. Every result is followed by a verdict you can defend.

Load a dataset to see it here.

Spatial regression

What it needs

The intuition

When to use it

Where the estimator comes from

The model

What it assumes

Statistical properties

How to read the output

A worked reading

Symptom → diagnosis → what to do

Common mistakes

Where to read this properly

See also

Related Causal

All analyses

Charts

Maps

Explore

Regression

Time series - one series

Time series - several series

Panel data

Multivariate statistics

Marketing research

Causal

Finance & risk

Actuarial

Tests & breaks

Machine learning

Econometrics that explains itself.