Space-Time Moran's I

Running Moran's I at each period gives snapshots of spatial autocorrelation but ignores temporal dependence between periods. Space-Time Moran's I uses a combined weight matrix that considers neighbors in both dimensions simultaneously, detecting patterns where clustering occurs across both space and time — something a set of cross-sectional statistics cannot capture.

The most common choice is a first-order temporal contiguity matrix, where periods t and t−1 (and t+1) are neighbors. Some applications use a bandwidth-based decay (e.g., exponential decay with distance in time). The choice should reflect the substantive temporal scale of the process; sensitivity analysis across at least two specifications is recommended.

Permutation testing is generally preferred because it is assumption-free about the distribution of I under the null. The normal approximation is faster but relies on asymptotic arguments that may not hold for small or irregular panels.

The standard formulation assumes regular intervals or at least a meaningful neighbor definition in time. For irregular intervals, the temporal weight matrix must be adapted (e.g., using inverse time-distance rather than first-order contiguity), and interpretation requires extra care.

Use local space-time LISA (Local Indicators of Spatial Association extended to the space-time domain) to decompose the global statistic into unit-level contributions. Each space-time observation receives a local I_{ST,i} score, enabling cluster maps that highlight high-high, low-low, and spatial outlier space-time units.