Spatiotemporal Autoregressive Models for Areal Compositional Data

Abstract

Compositional data, such as regional shares of economic sectors or property transactions, are central to understanding structural change in economic systems across space and time. This paper introduces a spatiotemporal multivariate autoregressive model tailored for panel data with composition-valued responses at each areal unit and time point. The proposed framework enables the joint modelling of temporal dynamics and spatial dependence under compositional constraints, and is estimated via a quasi-maximum likelihood approach. We build on recent theoretical advances to establish the identifiability and asymptotic properties of the estimator as both the number of regions and the number of time points grow. The utility and flexibility of the model are demonstrated through two applications: analysing property transaction compositions in an intra-city housing market (Berlin), and regional sectoral compositions in Spain's economy. These case studies highlight how the proposed framework captures key features of spatiotemporal economic processes that are often missed by conventional methods.

Figures (6)

• Figure 1: Overview of the Berlin real-estate market composition data set. First row: Compositions of real estate transactions (U: Undeveloped land, D: Developed land, C: Condominium) in January 1995 (left) and December 2014 (centre). The points are coloured according to the spatial location as shown in the map legend (right). Middle row: Composition of two selected regions, where the points are coloured according to the time points. The colours correspond to the colour of the marks on the map. Bottom row: Temporal dynamics of closed 3-part composition for two selected locations over time. Area under black line corresponds to the first, area between black and red to the second, and area above the red line to the third compositional part.
• Figure 2: Finite-sample bias under strong spatial dependence (setting A, left) and multi-lag dependence (setting B, right).
• Figure 3: Relative estimation efficiency across sample sizes for settings A (left) and B (right).
• Figure 4: Finite-sample coverage of naïve, OPG, and HAC variance estimators.
• Figure 5: Impulse response functions on the simplex for a unit innovation in ilr coordinate $k=1$ (left) and $k = 2$ (right) at a single spatial unit. The figure reports direct effects (own-unit), indirect effects (average spillovers), and total effects for each compositional component over time.

Theorems & Definitions (4)

• Theorem 3.1
• Theorem 3.2
• proof : Proof of Theorem \ref{['th:consistency']}
• proof : Proof of Theorem \ref{['thm:AN']}