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Static Spatial Panels

Static spatial panels with fixed effects.

SpatialEconometrics.SpatialPanelEconometrics.DirectQML — Type
DirectQML(; bias_correct = true)

Estimate a spatial panel by quasi-maximum likelihood on the demeaned data, optionally applying an analytical bias correction.

The straightforward approach: demean, then maximize the likelihood. With bias_correct = true the known O(1/T) bias in the variance and spatial parameter is corrected analytically after the fact; setting it to false leaves the raw QML estimates, which is useful mainly for showing what the correction does.

LeeYuTransformation avoids the bias by construction and is preferred unless you want this estimator specifically.

References

Lee, L.-F. & Yu, J. (2010). Journal of Econometrics, 154(2).

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SpatialEconometrics.SpatialPanelEconometrics.LeeYuTransformation — Type
LeeYuTransformation()

Estimate a spatial panel using the Lee-Yu orthogonal transformation.

Sweeping out fixed effects by demeaning leaves the variance estimate inconsistent as the number of periods grows slowly, because the demeaned errors are no longer independent. The Lee-Yu transformation instead projects onto an orthonormal basis of the demeaning operator's complement, which removes the fixed effects while keeping the errors independent, so no bias correction is needed afterwards.

The default, and the right choice for short panels.

See also DirectQML.

References

Lee, L.-F. & Yu, J. (2010). Estimation of spatial autoregressive panel data models with fixed effects. Journal of Econometrics, 154(2).

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SpatialEconometrics.SpatialPanelEconometrics.SpatialPanelModel — Type
SpatialPanelModel

A fitted static spatial panel, returned by fit_spatial_panel. Implements the StatsAPI.RegressionModel interface.

Carries the fitted variant, fe_spec and estimation approach alongside coef, vcov, the spatial parameter rho with its standard error, sigma2, loglik, the weight matrix, the transformed design X_tilde, residuals and coefnames, plus the panel dimensions N and T_periods.

As in the cross-section, coef holds structural parameters rather than marginal effects — use spatial_effects.

See also fit_spatial_panel, robust_vcov.

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SpatialEconometrics.SpatialPanelEconometrics.fit_spatial_panel — Function
fit_spatial_panel(variant, fe_spec, W, X, y, N, T_per; approach, method,
                  varnames, rho_bounds)

Fit a static spatial panel by quasi-maximum likelihood with fixed effects.

Arguments

  • variant: SAR or SDM.
  • fe_spec: a FixedEffectsSpec — NoFE, IndivFE, TimeFE or TwoWayFE.
  • W: N x N sparse weight matrix for the cross-section, not the stacked NT x NT matrix.
  • X: (N*T_per) x K covariates, no intercept.
  • y: stacked response of length N*T_per.
  • N, T_per: units and periods.

Keywords

Returns

A SpatialPanelModel.

Stacking order

X and y must be stacked with units varying fastest within each period: [unit_1 t_1, unit_2 t_1, ..., unit_N t_1, unit_1 t_2, ...]. The wrong order will not error; it will silently misalign every unit with its neighbours.

Examples

m = fit_spatial_panel(SAR(), TwoWayFE(), W, X, y, N, T_per;
                      varnames = ["income", "density"])
m.rho
robust_coeftable(m, DriscollKraayVCOV())

See also fit_dynamic_spatial_panel, panel_lm_tests, spatial_hausman_test.

References

Lee, L.-F. & Yu, J. (2010). Journal of Econometrics, 154(2).

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