Spatial Autocorrelation

Definition

Spatial autocorrelation is the tendency of values observed at nearby locations to be more (positive autocorrelation) or less (negative) similar than under spatial randomness — "everything is related to everything else, but near things are more related than distant things" (Tobler's first law). It is measured relative to a spatial-weights matrix $W$ (with $w_{ij}>0$ when $j$ is a "neighbor" of $i$, usually row-standardized), and quantified by global indices such as Moran's I and by local indicators (LISA) that map where dependence concentrates.

Key Ideas

• Spatial weights $W$. The analyst encodes "neighborhood" — contiguity (shared border), $k$-nearest-neighbors, or distance bands — as $w_{ij}$; every autocorrelation statistic is defined relative to this choice, so results are conditional on $W$.
• Global Moran's I. $I = \dfrac{n}{S_0}\dfrac{\sum_i\sum_j w_{ij} z_i z_j}{\sum_i z_i^2}$ with $z_i$ the mean-centered values and $S_0=\sum_{ij}w_{ij}$; positive $I$ indicates clustering of similar values, near-zero indicates spatial randomness. (Geary's C is a related, more local-differences-based alternative.)
• The global average hides local structure. A single global $I$ can be near zero while strong clusters and anti-clusters coexist and cancel — motivating local decomposition.
• LISA and the local Moran (Anselin 1995): a local indicator of spatial association measures, per location, significant clustering of similar values and sums (over locations) to a quantity proportional to the global statistic. The local Moran $I_i = z_i\sum_j w_{ij} z_j$ satisfies $\sum_i I_i \propto I$; it flags High-High / Low-Low clusters ("hot/cold spots") and High-Low / Low-High spatial outliers. Related local statistics are the Getis-Ord $G_i/G_i^*$.
• Moran scatterplot. Plotting each $z_i$ against its spatial lag $\sum_j w_{ij} z_j$ gives four quadrants (HH, LL, HL, LH) classifying local association; the OLS slope equals global Moran's I, and high-leverage points are influential locations.
• Inference. Under the null of spatial randomness, moments of $I$ are known (normal or randomization assumption); for local statistics — whose neighborhoods overlap — significance is judged by conditional permutation, with a serious multiple-comparison problem (one test per location).

Why It Matters

• Exploratory spatial data analysis (ESDA). LISA maps turn "is there dependence?" into "where are the clusters and anomalies?", the workhorse of applied spatial epidemiology, criminology, regional economics, and ecology.
• Model diagnostics. A significant Moran's I on regression residuals signals omitted spatial structure, motivating spatial-lag / spatial-error models and the CAR/areal modeling machinery.
• LM specification tests (Anselin-Bera-Florax-Yoon 1996): OLS-residual-based Lagrange-multiplier tests choose which spatial specification is needed. The one-directional $LM_{err}$ (spatial error) and $LM_{lag}$ (spatial lag) over-reject when the other form is present, so their robust counterparts $RLM_{err}$ and $RLM_{lag}$ — each adjusted (via Bera–Yoon 1993) to be valid under a locally present nuisance parameter — are used to discriminate lag from error dependence. This turns "residuals are spatially dependent" (a significant Moran's I) into an actionable "fit a spatial-lag vs spatial-error model" decision rule.
• Complement to spatial modeling. Autocorrelation statistics measure and test dependence; CAR models and point processes model it.

Open Questions

• Sensitivity of all conclusions to the (arbitrary) choice of spatial-weights matrix $W$.
• Valid multiple-testing correction for LISA maps given the strong dependence across overlapping local tests.
• Distinguishing true clustering from spatial heterogeneity (nonstationary mean) — the two can masquerade as each other.
• The identification critique (Gibbons-Overman 2012). Fitting spatial-lag/spatial-error regressions to explain spatial correlation runs into Manski's reflection problem: endogenous interaction (outcomes affecting neighbours' outcomes), exogenous/contextual effects (neighbours' characteristics), and correlated effects (spatially correlated unobservables) cannot be separately identified from the reduced form — only their composite is. Spatial econometrics "identifies" them only by assuming the functional form and the weights matrix $W$ and letting model comparison choose a specification; Gibbons-Overman argue this yields, at best, very weak identification, so such regressions are "pointless" for recovering causal economic mechanisms (though fine for description). They advocate the experimentalist paradigm — real or natural experiments, IV, RD — that puts identification and causality first. The moral for this page: a significant Moran's I on residuals is a sound diagnostic, but modelling the dependence away is not the same as identifying its cause.

Related

• Conditional Autoregressive Model
• Spatial Point Process
• Spatial VAR
• Anselin (1995) Local Indicators of Spatial Association — LISA
• Anselin-Bera-Florax-Yoon (1996) Simple Diagnostic Tests for Spatial Dependence
• Gibbons-Overman (2012) Mostly Pointless Spatial Econometrics?
• Luc Anselin
• Anil K. Bera
• Stephen Gibbons
• Henry G. Overman