Model Selection

CommonFactorTestResult

Test of the common-factor restriction, returned by common_factor_test.

Fields: stat, df, p_value, reject.

A Spatial Durbin model reduces to a spatial error model exactly when theta = -rho * beta. Failing to reject means the simpler error specification is adequate; rejecting means the Durbin terms carry information the error model cannot represent.

See also SDM, fit_spatial_error.

LRTestResult

Likelihood-ratio test between nested models, returned by lr_test.

Fields: stat, df, p_value, and both log-likelihoods (ll_restricted, ll_unrestricted).

VuongTestResult

Vuong test between two non-nested models, returned by vuong_test.

Fields: stat, p_value, aic_correction, and preference, which is :model1, :model2 or :indistinguishable.

Use this where lr_test does not apply — comparing a spatial lag against a spatial error specification, for instance, since neither nests the other. :indistinguishable is a genuine outcome, not a failure: the data may simply not separate the two.

WaldTestResult

Wald test of a linear restriction R*beta = r, returned by wald_test.

Fields: stat, df, p_value. Needs only the fitted model's coefficients and covariance, so unlike lr_test it does not require refitting under the restriction.

common_factor_test(coef_beta, coef_theta, rho, vcov_full)

Tests whether SDM simplifies to SEM, i.e. H₀: θ + ρβ = 0.

• coef_beta: vector of K slope coefficients (excluding intercept)
• coef_theta: vector of K spatial lag coefficients (WX terms)
• rho: spatial autoregressive parameter
• vcov_full: full variance-covariance matrix of [β; θ; ρ]

Uses the Wald test formulation from Burridge (1981).

lr_test(m_restricted, m_unrestricted)

Likelihood ratio test for nested models. The unrestricted model must have more parameters than the restricted model.

model_comparison_table(models...; names=nothing)

Print a formatted comparison table of spatial models.

spatial_aic(m)

Akaike Information Criterion: AIC = -2 ln L + 2k where k is the number of estimated parameters.

spatial_bic(m)

Bayesian Information Criterion: BIC = -2 ln L + k ln(n)

vuong_test(m1, m2)

Vuong (1989) test for non-nested model comparison. Uses AIC-corrected version. Positive statistic favors model 1.

wald_test(m, R, r)

General Wald test: H₀: Rβ = r.

• R: Q×K restriction matrix
• r: Q-vector of restriction values