State-Building through Public Land Disposal? An Application of Matrix Completion for Counterfactual Prediction

Figures & data

Fig. 1 Log per-capita cumulative homesteads in 1870 and 1900, overlaid on 1911 county borders (Long 1995). White-colored counties have no homestead entries. States bordered in blue (⊡) are state land states; yellow (⊡) denote western public land states; orange (⊡) denote southern public land states.

Fig. 2 Absolute bias and bootstrap variance in generated data, varying the rank of L_it . Estimator: ▀ DID; ▀ MC; ▀ MC-W; ▀ SCM; ▀, SCM-L1.

Fig. 3 Matrix completion estimates of the effect of the year of initial homestead entry (1869; dashed vertical line) on state government expenditure, 1809 to 1982: ━━━, factual treated; ┅, factual control; ╍╍, counterfactual treated; ━╻━, ${\widehat{\tau }}_{t,\infty a_i^{\prime }}$.

Table 1 Estimates of the ATT averaged over the counterfactual period of 1869–2008 and bootstrap standard errors (in parentheses).

	All PLS		Southern PLS		Western PLS
	Expenditure	Revenue	Expenditure	Revenue	Expenditure	Revenue
DID	–0.90 (0.32)	–0.95 (0.27)	–0.69 (0.39)	–0.55 (0.35)	–0.94 (0.45)	–1.04 (0.51)
MC–W	–0.84 (0.24)	–0.79 (0.25)	–0.63 (0.30)	–0.42 (0.30)	–0.88 (0.37)	–0.86 (0.39)
SCM	0.13 (0.12)	0.17 (0.14)	–0.09 (0.21)	–0.14 (0.23)	0.17 (0.26)	0.23 (0.27)
SCM–L1	0.13 (0.17)	0.17 (0.19)	–0.08 (0.23)	–0.11 (0.21)	0.17 (0.27)	0.22 (0.29)

Table 2 Placebo ATT estimates and bootstrap standard errors (in parentheses).

	Expenditure			Revenue
	Δ = 1	Δ = 10	Δ = 25	Δ = 1	Δ = 10	Δ = 25
DID	–0.61 (0.43)	–0.38 (0.24)	–0.39 (0.22)	–0.63 (0.48)	–0.70 (0.42)	–0.67 (0.34)
MC–W	–0.26 (0.32)	–0.44 (0.16)	–0.47 (0.15)	–0.33 (0.38)	–0.26 (0.21)	–0.28 (0.16)
SCM	0.94 (0.45)	0.86 (0.24)	0.76 (0.19)	0.66 (0.47)	0.52 (0.28)	0.64 (0.22)
SCM–L1	0.48 (0.41)	0.34 (0.22)	0.48 (0.16)	0.61 (0.39)	0.39 (0.26)	0.20 (0.21)

Table 3 Continuous DID estimates of the effect of (log) per–capita homestead patents on state size or land inequality.

	Expenditure	Revenue	Land inequality
Treatment effect ($\widehat{\varphi }$)	–0.04	–0.04	–0.0007
	(0.002)	(0.002)	(0.0003)
Adjusted ${\mathrm{R}}^2$	0.538	0.540	0.801
$\mathrm{N}$	8,618	8,618	463

Long, J. H. (1995), “Atlas of Historical County Boundaries,” The Journal of American History, 81, 1859–1863. DOI: 10.2307/2081880.

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Data Availability Statement

Data and R code to reproduce the results of the article are available at https://github.com/jvpoulos/homesteads.