Urban Sprawl & Public Service Costs: Spec Ladder Results

Summary Statistics

394

Unique Counties

3

Years (2012, 2017, 2022)

1,148

Total Observations

72

Models (12 cat. x 6 specs)

HC1 + Clustered

Standard Errors

Real 2022

Dollars (CPI-adjusted)

Note: Counties classified as metropolitan under 2023 CBSA delineations, applied uniformly to all panel years. Time-varying urban land area uses 2010 UA vintage for 2012/2017 and 2020 UA vintage for 2022.

Specification Ladder

The spec ladder shows how the density coefficient changes across 6 progressively different model specifications. Spec 1 matches the original paper; subsequent specs add county fixed effects, clustered standard errors, additional controls, or alternative dependent variable transforms. Bold values indicate statistical significance at p < 0.05.

Expenditure Category:

Spec	FE Type	SE Type	DV Transform	Revenue	Extra Controls	Density Coeff	Density SE	p-value	N	R-squared

Density Coefficient Comparison

Comparing Original (1992) vs Updated Spec 1 (2022) density coefficients. Error bars show 95% confidence intervals. Original paper used 1982–92 data; updated results use 2012–2022 panel in real 2022 dollars.

Standardized Effects: +1 SD Density Change

For each expenditure category, the standardized effect shows the percentage change in per capita spending associated with a one standard-deviation increase in density. Values from Spec 1 (baseline). In the semilog model, the standardized coefficient approximates the percentage change.

Specification:

Zero-Value Expenditure Counts

Number and percentage of county-year observations with zero expenditure in each category. This justifies the ln(1+y) specification in Spec 4: categories with substantial zeros (e.g., Other Transportation at 17.9%) lose observations under ln(y), potentially biasing results.

Expenditure Category	Zero Obs	% Zero	Total Obs

Coefficient Comparison: Original vs Updated

Original (1992) vs Updated Spec 1 (2022) coefficient comparison for all 8 key variables.

Expenditure Model:

Same sign & significant

Same sign, not significant

Sign flipped

Variable	Original Coef.	Orig. t-stat	Updated Coef.	Upd. t-stat	Upd. p-value	Status

Robustness Checks

Four robustness checks re-estimate Specs 1 and 3 on four key expenditure categories (Total Direct, Roadways, Sewerage, Police Protection) under alternative sample restrictions or variable definitions. Use the dropdown to view results from each check.

Robustness Check:

Category	Spec	Density Coeff	Density SE	p-value	N

Key Findings

Density is not consistently associated with lower per capita costs. The original paper found that higher density was associated with significantly lower per capita costs for most expenditure categories (negative density coefficients in 9 of 12 models). In the updated 2012–2022 panel, density coefficients are near zero or positive in most categories under the baseline specification. The conditional correlation between density and spending is sensitive to specification choice.
The density effect depends on specification. The spec ladder reveals that the density coefficient is sensitive to the choice of fixed effects (state vs. county), standard error clustering, and inclusion of revenue controls. This highlights the importance of robustness testing rather than relying on a single specification.
Revenue variables remain robustly associated with spending — the most consistent finding across both time periods and all specifications. Per capita local tax revenue and intergovernmental revenue are highly significant and positive in nearly every model. Note: revenue controls are endogenous — Spec 5 shows results without them.
Other Transportation: the one category where the original finding replicates. The density coefficient remains positive and is associated with higher spending, consistent with the original paper. This includes transit, airports, and ports — services where economies of density favor more spending, not less.
Dollars are now inflation-adjusted. All expenditure and revenue variables are deflated to real 2022 dollars using CPI-U annual averages. The 2012 panel year was added for improved identification (T=3 vs. the original T=2).
The spatial extent effect persists. Urbanized land area remains positively associated with per capita costs in most models under State FE specifications, consistent with the original finding that counties covering more geographic area face higher per-capita costs for service delivery.