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Spatial Error Models

Models placing the dependence in the disturbance rather than in y.

SpatialEconometrics.SpatialErrorModel.SpatialErrorRegressionModel — Type
SpatialErrorRegressionModel

A fitted cross-sectional spatial error model, returned by fit_spatial_error. Implements the StatsAPI.RegressionModel interface.

Carries coef and vcov for the regression parameters, the spatial error parameter lambda with its standard error, sigma2, loglik, the weight matrix, residuals and coefnames.

Note

Unlike SpatialRegressionModel, coef is interpretable directly: dependence sits in the error, so there is no feedback through y and no need for spatial_effects. What the model buys you is correct standard errors, not different point estimates.

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SpatialEconometrics.SpatialErrorModel.fit_spatial_error — Function
fit_spatial_error(W, X, y; method, varnames, lambda_bounds)
fit_spatial_error(fe_spec, W, X, y, N, T_per; method, varnames, lambda_bounds)

Fit a spatial error model by concentrated maximum likelihood.

\[y = X*beta + u, u = lambda*W*u + e\]

Dependence is in the disturbance rather than in y. OLS stays unbiased under this process but its standard errors are wrong, so the gain is inferential: beta is interpretable as it stands, with no spillover decomposition needed.

The likelihood is concentrated by spatially filtering y and X with (I - lambda*W) at each trial lambda, then maximizing over lambda alone.

The first method fits a cross-section; the second fits a panel, taking a FixedEffectsSpec and the panel dimensions, with X and y stacked units-fastest.

Arguments

  • W: N x N sparse weight matrix.
  • X: covariates without an intercept — one is added.
  • y: response.

Keywords

  • method: a LogDetMethod.
  • varnames: covariate names.
  • lambda_bounds: search interval for lambda, default (-0.995, 0.995).

Returns

A SpatialErrorRegressionModel or SpatialErrorPanelModel.

Examples

m = fit_spatial_error(W, X, y; varnames = ["income", "density"])
m.lambda
coef(m)                              # directly interpretable

fit_spatial_error(TwoWayFE(), W, X, y, N, T_per)

Use lm_tests to decide between this and fit_spatial before fitting either.

References

Anselin, L. (1988). Spatial Econometrics: Methods and Models. Kluwer.

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