Quasi-Likelihood Inference Under Spatial Misspecification

Why Quasi-Likelihood Inference Needs a Sandwich

Likelihood estimation for a spatial autoregressive model uses the spatial structure to account for simultaneous feedback. In practice, its Gaussian likelihood is often used even when the disturbances are not normally distributed. This is quasi-maximum likelihood: the Gaussian objective is an estimation criterion, not necessarily a complete description of the disturbance distribution.

That distinction matters for uncertainty. If the model’s conditional mean and spatial structure are correctly specified, Gaussian quasi-maximum likelihood can remain consistent under nonnormal innovations, subject to regularity conditions. But the Gaussian likelihood’s curvature alone generally does not give the correct sampling variance. Sandwich inference replaces that curvature-only variance with one that also accounts for the actual variance of the score.

Model and Assumptions

Consider the spatial autoregressive (SAR) model

Here \(y\) is an \(n\)-vector of outcomes, \(X\) is an \(n\times k\) matrix of exogenous regressors, \(\beta\) is a \(k\)-vector, \(W\) is the specified spatial-weights matrix, and \(\rho\) is the spatial-lag coefficient. The disturbance vector is \(u\). The model requires \(A(\rho)\) to be invertible over the parameter space under consideration.

For the distributional-robustness result discussed here, assume the conditional mean is correctly specified, \(E(u\mid X,W)=0\), and the innovations have common variance \(\sigma^2\). Assume also that the innovations are independent across spatial units, although they need not be normally distributed, and have finite fourth moments. The regressors are exogenous, and standard large-sample regularity conditions hold: the relevant parameters are interior, the model is identified, the normalized Hessian has a nonsingular limit, and the spatial design does not become unstable as \(n\) grows. These conditions are important: nonnormality is allowed, but unexplained spatial correlation in \(u\) is not.

The Gaussian quasi-log-likelihood, up to constants that do not affect its maximizer, is

The determinant term is essential: it accounts for the change of variables from \(y\) to the spatial innovations \(e\). Omitting it changes the likelihood and, in general, the estimate of \(\rho\).

Score, Curvature, and Score Variance

Let \(\eta=(\beta',\rho,\sigma^2)'\) collect the parameters, and let \(s_n(\eta)=\partial\ell_n(\eta)/\partial\eta\) be the full-sample score. Writing \(e=A(\rho)y-X\beta\), the score components are

For example, \(\partial e/\partial\rho=-Wy\), so differentiating \(-e'e/(2\sigma^2)\) with respect to \(\rho\) gives \(e'Wy/\sigma^2\). Differentiating \(\log|A(\rho)|\) gives \(-\operatorname{tr}(A(\rho)^{-1}W)\). At the true parameter, the expectation of the \(\rho\)-score is zero under the stated assumptions. In particular, when \(y=A^{-1}(X\beta+u)\), the disturbance part satisfies

This cancels the expected log-determinant contribution. The calculation illustrates why the spatial score is not merely a collection of independent observation-level regression scores: it contains a quadratic form in the full disturbance vector.

Define the expected negative Hessian \(H_n=-E[\partial s_n(\eta_0)/\partial\eta']\) and the score covariance \(J_n=\operatorname{Var}(s_n(\eta_0))\), evaluated at the true parameter \(\eta_0\). Under the regularity conditions,

Both \(H_n\) and \(J_n\) typically grow with \(n\), so this expression has the scale of a parameter covariance. Equivalently, if \(H_n/n\) and \(J_n/n\) converge to \(H\) and \(J\), then \(\sqrt n(\hat\eta-\eta_0)\) has limiting covariance \(H^{-1}JH^{-1}\). When the Gaussian likelihood is correctly specified, the information identity gives \(J=H\). Under nonnormal innovations, that identity need not hold; using only the inverse Hessian can therefore give incorrect standard errors.

Worked Examples

What Is Robust, and What Is Not?

Under nonnormal but independent innovations with mean zero and common variance, the Gaussian criterion can estimate the model parameters consistently while the sandwich adjusts the estimated covariance for the difference between the score variance and the Gaussian curvature. In the SAR case, estimating the score variance requires care. The \(\rho\)-score involves \(Wy\) and a quadratic form in disturbances, so spatially connected observations enter it together. A valid estimate of \(J_n\) must account for the resulting score covariance, including the relevant fourth-moment contribution. Simply treating every row of the full spatial score as an independent observation, or applying an ordinary regression HC formula to all SAR parameters, does not automatically do this.

The distinction from residual spatial dependence is fundamental. If \(E(uu'\mid X,W)\) is not \(\sigma^2 I\), then the innovations are spatially correlated or otherwise have a non-spherical covariance. The independence-based score covariance is no longer the right \(J_n\). A sandwich formula is not automatically robust to that misspecification: it works only when its “meat” consistently estimates the score covariance that actually arises. Accounting for residual spatial dependence requires a dependence-aware covariance method or a correctly specified disturbance structure; it is not achieved merely by allowing nonnormal marginal distributions.

Practical Inference and Common Pitfalls

A Wald statistic for a scalar component \(\eta_j\) is \(z=(\hat\eta_j-\eta_{j,0})/\widehat{SE}(\hat\eta_j)\). Under the null and the relevant large-sample conditions, it is approximately standard normal. For example, a two-sided test at level \(\alpha=0.05\) rejects when \(|z|>1.96\). For a vector restriction, use the corresponding Wald quadratic form with the sandwich covariance of the restricted parameter vector. These are asymptotic procedures; small samples or weak identification can make their approximation poor.

In implementation, form the full score and Hessian for the SAR quasi-likelihood, estimate the score covariance under the intended disturbance assumptions, assemble the sandwich, and report the appropriate parameter block. If \(\sigma^2\) is estimated, retaining it as a nuisance parameter in the joint calculation accounts for its estimation; the final covariance for \(\beta\) and \(\rho\) is obtained from the corresponding block. State whether the reported covariance assumes independent innovations or allows a specified form of dependence.

• Using only inverse curvature. The inverse Hessian is justified by the information identity under a correctly specified likelihood. Under nonnormality, use the score-variance component as well.
• Assuming “robust” means robust to all misspecification. Distributional robustness to nonnormal innovations does not automatically cover spatial correlation in the disturbances, an incorrect mean, or an incorrect \(W\).
• Dropping the log-determinant. The determinant contributes directly to the \(\rho\)-score and is part of the spatial likelihood.
• Applying an observation-wise formula without checking the score structure. Spatial feedback makes the likelihood score involve multiple locations. The covariance calculation must preserve the relevant dependence among score contributions.