Dynamic Spatial Panels

Panels carrying both a time lag and a spatial lag, with bias correction.

SpatialEconometrics.DynamicSpatialPanelEconometrics.DynamicSpatialEffectsResult — Type

DynamicSpatialEffectsResult

Short-run and long-run direct, indirect and total effects for a dynamic spatial panel, returned by dynamic_spatial_effects.

The short-run effects are the impact within the period of the shock; the long-run effects cumulate the time dynamics to the new steady state and are larger whenever the autoregressive coefficient is positive.

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SpatialEconometrics.DynamicSpatialPanelEconometrics.DynamicSpatialPanelModel — Type

DynamicSpatialPanelModel

A fitted dynamic spatial panel, returned by fit_dynamic_spatial_panel. Implements the StatsAPI.RegressionModel interface.

The specification carries three dependence parameters, all reported separately:

Field	Term	Meaning
`gamma`	`y_{t-1}`	pure time lag
`eta`	`W*y_{t-1}`	space-time lag: last period's neighbours
`rho`	`W*y_t`	contemporaneous spatial lag

Other fields

coef, vcov, coefnames, sigma2, loglik, residuals: as usual. coefnames runs "y_lag", "Wy_lag", the covariates, the W_-prefixed terms for an SDM, then "ρ".
coef_uncorrected, sigma2_uncorrected: the estimates before the bias correction, so you can see what it moved.
spectral_radius_A: stability diagnostic for the implied system. The long-run effects only exist if this is below 1.
W, Z_tilde, y_tilde, trace_powers: the weight matrix, the transformed design and response, and traces reused by dynamic_spatial_effects.
N, T_total, T_post: one period is consumed in forming the lag, so estimation uses T_post = T_total - 1.

Because dependence runs over both space and time, short-run and long-run effects differ — see dynamic_spatial_effects.

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SpatialEconometrics.DynamicSpatialPanelEconometrics.dynamic_spatial_effects — Function

dynamic_spatial_effects(model; max_power = 50, vcov_mat = model.vcov)
dynamic_spatial_effects(model, spec::AbstractRobustVCOV; max_power = 50)

Short-run and long-run marginal effects for a dynamic spatial panel.

The short-run effects are the within-period response through (I - rho*W)^{-1}, so the total is (beta + theta) / (1 - rho).

The long-run effects cumulate the time dynamics to the steady state, which brings in both lag parameters: the total becomes

\[(beta + theta) / (1 - gamma - rho - eta)\]

and the direct effect is evaluated at the long-run spatial parameter (rho + eta) / (1 - gamma), scaled by 1 / (1 - gamma). Each of the six quantities comes with a delta-method standard error.

Returns

A DynamicSpatialEffectsResult, whose fields are prefixed sr_ for short-run and lr_ for long-run.

Warning

The long-run effects exist only if gamma + rho + eta < 1. As that sum approaches 1 the denominator goes to zero, the effects diverge, and the delta-method standard errors become unreliable well before that point. Check model.spectral_radius_A first.

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SpatialEconometrics.DynamicSpatialPanelEconometrics.fit_dynamic_spatial_panel — Function

fit_dynamic_spatial_panel(variant, fe_spec, W, X, y, N, T_total; method,
                          varnames, rho_bounds, n_bias_probes, rng)

Fit a dynamic spatial panel — one carrying both a time lag and a spatial lag of the dependent variable — by bias-corrected quasi-maximum likelihood.

With fixed effects and a lagged dependent variable the QML estimator suffers the Nickell bias, of order 1/T. The Yu, de Jong and Lee analytical correction is applied, using stochastic probes to estimate the traces the correction needs.

Arguments

variant: SAR or SDM.
fe_spec: a FixedEffectsSpec; IndivFE or TwoWayFE are the meaningful choices here.
W: N x N cross-sectional weight matrix.
X, y: stacked over all T_total periods, units varying fastest.
N, T_total: units and periods including the one consumed by the lag; estimation uses T_total - 1.

Keywords

n_bias_probes, rng: probes used by the bias correction, and the random source for them. Results vary slightly with rng; pass a seeded one for reproducibility.
method, varnames, rho_bounds: as for fit_spatial_panel.

Returns

A DynamicSpatialPanelModel.

Warning

Same stacking requirement as fit_spatial_panel: units vary fastest within each period. The bias correction also assumes T is small relative to N; with long panels the correction matters less and the uncorrected estimates are already close.

Examples

m = fit_dynamic_spatial_panel(SAR(), IndivFE(), W, X, y, N, T_total;
                              rng = MersenneTwister(1))
dynamic_spatial_effects(m)

References

Yu, J., de Jong, R. & Lee, L.-F. (2008). Journal of Econometrics, 146(1).

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