📈 economics

A Spatial Paradox in U.S. Metropolitan Growth: Spatially Clustered Growth Rates, Spatially Independent Levels

Despite strong spatial clustering in annual metropolitan GDP growth rates across 376 U.S. areas from 2001 to 2023, the accumulated GDP levels exhibit no spatial autocorrelation because small, mean-reverting growth shocks cause spatial clusters to shift locations over time rather than persist in the same places.

Original authors: Michael L. Chohaney, Oleg A. Smirnov

Published 2026-09-30

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Michael L. Chohaney, Oleg A. Smirnov

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Economic geography often operates on a simple intuition: if a place grows faster than its neighbors, it should eventually become richer than them, and if a whole region grows together, that region should stand out as a distinct, wealthy, or poor block on the map. This expectation relies on the idea that growth is a steady, accumulating force. When a city adds a layer of economic activity every year, those layers should stack up to create a permanent shape, much like sediment building a canyon over millennia. For decades, researchers have studied how cities cluster, looking for patterns where prosperous areas pull in more prosperity and struggling areas fall further behind. The standard view suggests that these spatial patterns in growth rates should eventually leave a permanent mark on the total wealth of a region.

However, a new analysis of the United States reveals a surprising disconnect between how cities grow and how rich they become. Researchers examined the economic performance of 376 metropolitan areas across the country over a twenty-two-year period, spanning from 2001 to 2023. This timeframe included two of the most significant economic disruptions in modern history: the financial crisis of 2008 and the pandemic-induced shock of 2020. By tracking the annual growth rates of real gross domestic product for each city and comparing them to the total accumulated wealth of those same cities, the team uncovered a paradox. While the speed at which cities grew was highly clustered—meaning fast-growing cities tended to sit next to other fast-growing cities, and slow-growing cities sat next to other slow-growing ones—the total wealth of these cities showed no such pattern. The map of growth was full of hotspots and cold spots, but the map of accumulated wealth looked completely random.

The researchers set out to explain why this dissociation exists. They found that the answer lies in two specific behaviors of the growth process itself: the size of the yearly changes and the fleeting nature of the clusters. First, the annual growth rates, even when they are clustered, are very small compared to the massive amount of economic activity already accumulated in a city. Imagine a city with a huge existing economy; a year of strong growth adds only a thin, new layer on top of that vast foundation. Because this new layer is so thin relative to the whole, it takes an incredibly long time for the pattern of that layer to change the overall shape of the city's wealth. In fact, the researchers used computer simulations to show that at the actual rates observed in the real economy, it would take centuries for the spatial patterns of growth to become visible in the total wealth levels.

The second, and perhaps more critical, factor is that these clusters of growth do not stay in one place. The study tracked where the high-growth and low-growth areas appeared year after year and found that they were constantly on the move. A cluster of booming cities in the Midwest one year might dissolve and re-form in Texas or the Great Plains the next. The researchers used a statistical method to show that a city's growth state is highly unstable; a city experiencing a boom is likely to return to a normal growth rate within about two years, and the same is true for cities in a slump. Because the clusters shift location so quickly and the extreme growth periods are so short-lived, the "layers" of growth never stack up in the same spot long enough to create a permanent pattern. The high-growth areas of 2005 are not the same as those of 2010, so the economic "sediment" never settles into a recognizable shape.

To confirm this, the team ran simulations where they kept the size of the growth shocks the same but forced the clusters to stay in the exact same locations every year. In that scenario, the accumulated wealth did eventually show a clear spatial pattern, proving that the randomness of the real-world data was not due to a lack of connection between neighbors, but rather to the fact that the connections kept moving. The study also ruled out the idea that this phenomenon was simply a result of the major economic crises during the period. While the clustering of growth was indeed stronger during the financial crisis and the pandemic, the dissociation between growth patterns and wealth levels persisted even in the years between these shocks. The pattern of itinerant growth is a structural feature of the US economy, not just a temporary glitch caused by bad times.

Ultimately, the research suggests that the geography of economic growth is fundamentally different from the geography of economic wealth. The clusters of growth are itinerant, forming, dissolving, and re-forming in shifting locations, leaving no lasting spatial imprint on the total levels of economic activity. This finding challenges the standard assumption that spatial patterns in growth rates will naturally translate into spatial patterns in wealth. It implies that for regional economies, the story of where growth happens is a story of constant motion and rapid change, where the momentum of the past is too heavy to be quickly reshaped by the fleeting waves of the present. The map of who is growing fast today is not a reliable predictor of who will be rich in a decade, because the growth itself refuses to stay put.

Technical Summary: A Spatial Paradox in U.S. Metropolitan Growth

Problem Statement
The paper addresses a fundamental paradox in the spatial dynamics of U.S. metropolitan economies. While annual growth rates of real metropolitan gross domestic product (MGDP) exhibit strong spatial clustering (positive spatial autocorrelation), the accumulated levels of MGDP show no detectable spatial autocorrelation. Intuitively, if growth rates are persistently clustered in space, the running sum of those rates (the levels) should eventually imprint the same spatial pattern. However, empirical data from 2001 to 2023 reveals a dissociation: growth rates are spatially dependent in every year, yet the resulting economic levels are spatially independent. The paper seeks to explain why this spatial structure in growth fails to accumulate into a lasting spatial structure in levels.

Methodology
The study utilizes a balanced panel of 376 U.S. Metropolitan Statistical Areas (MSAs) from 2001 to 2023, sourced from the Bureau of Economic Analysis. Spatial relationships are defined using a Thiessen (Voronoi) polygon tessellation based on 2018 OMB delineations, employing a queen-contiguity criterion to generate a symmetric, row-standardized spatial weights matrix ( $W$ ).

The authors employ a three-pronged analytical approach:

Descriptive Spatial Statistics: Calculation of Moran's $I$ for both growth rates and levels annually, alongside Local Indicators of Spatial Association (LISA) to track cluster stability.
Spatial Autoregressive Accumulation Simulation: A data-generating process where growth rates ( $g_t$ ) are modeled as a spatial autoregressive (SAR) process driven by shocks ( $\epsilon_t$ ). These rates are accumulated over time to simulate levels ( $y_t$ ). The authors calibrate this model using empirical moments (mean and standard deviation of observed growth) and spatial dependence parameters ( $\rho$ ) to test under what conditions spatial dependence transfers from growth to levels. They compare scenarios with varying shock scales and persistence (relocating vs. fixed patterns).
Spatial Markov Chain Analysis: The authors classify MSAs into growth states (low, neutral, high) based on quartiles of the annual growth distribution. They estimate transition matrices to analyze mean reversion and mixing times. Furthermore, they employ spatial Markov chains to condition transition probabilities on the growth context of neighboring MSAs (using the spatial lag of neighbors' growth).

Key Results

The Paradox Confirmed: Growth rates are positively and significantly spatially autocorrelated in all 22 years of the panel (mean Moran's $I \approx 0.17$ ), particularly during economic downturns (2008–2009, 2020–2021). Conversely, MGDP levels show no significant spatial autocorrelation in any year (mean Moran's $I \approx 0.00$ ).
Itinerant Clusters: Spatial clusters of growth are highly transient. Local high-high and low-low clusters rarely persist in the same location from one year to the next; only about 20% of metros in a high-growth cluster remain in a cluster the following year.
Mechanism 1: Small Shock Scale: Simulation results indicate that because real-world growth shocks are small relative to the accumulated stock of economic activity, each year adds only a "thin" spatially structured layer. Theoretically, spatial dependence in levels would eventually emerge, but the convergence rate is extremely slow (taking centuries) given the empirical scale of shocks.
Mechanism 2: Mean Reversion and Transience: The spatial Markov analysis reveals that growth states are strongly mean-reverting. High-growth and low-growth spells are short-lived (approx. 1.6–1.7 years), and the cross-sectional distribution of states re-balances within two years. Crucially, clusters relocate rather than recur in the same places.
The Interaction of Scale and Transience: The simulation demonstrates that while small shocks slow accumulation, the relocation of clusters is the primary factor preventing spatial dependence in levels. If the same spatial pattern were held fixed (non-relocating) at the empirical shock scale, significant spatial dependence in levels would emerge within the 22-year panel. However, because clusters shift annually, the structured variance does not compound in specific locations, keeping levels spatially independent.
Spatial Conditioning: Transition probabilities between growth states are significantly conditioned by the growth states of neighbors. High-growth metros are more likely to remain high if neighbors are high, and low-growth metros are more likely to remain low if neighbors are low. This spatial conditioning is robust across different weight definitions, state categorizations, and time periods (including non-crisis years).

Contributions
The paper provides a descriptive resolution to the paradox of clustered growth rates versus independent levels. It demonstrates that the spatial structure of growth is "itinerant"—forming, dissolving, and re-forming in shifting locations—leaving no lasting spatial imprint on accumulated levels.
Methodologically, the authors argue that the transience of clustering is a feature of the data-generating process (mean reversion and rapid mixing), not an artifact of the spatial weights matrix. Consequently, they suggest that time-varying weight matrices ( $W_t$ ) are not the appropriate solution for this specific phenomenon, as the underlying connections are stable; rather, the non-stationarity lies in the shifting location of the clustering itself.

Significance and Claims
The authors position this work as a descriptive characterization of the spatial-temporal structure of U.S. metropolitan growth, explicitly avoiding the identification of specific economic mechanisms (e.g., spillovers vs. common shocks) that drive the co-movement.

Empirical Regularity: The paper documents a regularity where the geography of growth and the geography of accumulated activity are decoupled due to the transient and mean-reverting nature of growth shocks.
Methodological Caution: It highlights a pitfall in spatial panel modeling: assuming that spatial dependence in levels is the primary object of study when the dependence exists only in the changes (growth rates) and is transient. The authors caution that standard spatial models with fixed weights may fail to capture this dynamic if the clustering itself is the object of study.
Generalizability: While focused on MGDP, the authors note that the arithmetic identity between levels and first differences implies this dissociation could apply to other stock-flow relationships (e.g., population/migration, employment/job flows), suggesting that spatial dependence in flows does not necessarily imply spatial dependence in stocks.

The paper concludes that understanding regional growth requires investigating the dynamics of where clusters appear and how long they persist, not just fitting spatial models to static levels. The observed dissociation is the expected outcome of accumulating small, spatially dependent, yet transient and mean-reverting shocks.

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