Replication with R

Estimating Marginal Returns to Education (Carneiro, Heckman and Vytlacil 2011)

Author

Hu Huaping

Published

May 1, 2026

Introduction

R replication of tables in (Carneiro, Heckman, and Vytlacil 2011) following the AER replication package in replication_aer/. For the narrative walkthrough (research questions, identification, reproducibility), start with the replication guide. For commands and env vars, see the minimum runnable checklist.

Shared modules:

chv2011-data-prep.R — data merge, propensity score, bootstrap samples
getboot.R — R port of Stata getboot.do (replication_aer/bootdata/)
chv2011-table4a-core.R + export_data_with_tuition.R — Table 4(a) input and test
chv2011-mte-core.R — semiparametric MTE and treatment parameters
chv2011-gt-quarto.R — gt table helpers

Data mode: bundled_aligned — Merging basicvariables.dta and localvariables.dta by row order (author-aligned replication bundle; N should be 1747).

P0 merge: Bundled files use row-order alignment (bundled_aligned). Core Table A-3 statistics match; local instruments are approximate until geocode merge (P1 dataset.dta).

P1 bootstrap: Run Rscript 04-topics/rep-chv2011/Rcode/getboot.R to create replication_aer/bootdata/.

P2 Table 4(a): Run export_data_with_tuition.R then Table_4a.R with CHV2011_BOOT_4A=1000 for author-matching bootstrap p-values.

P3 MTE / treatment parameters: Run Table_4b.R, Table_5.R, and run_all_figures.R with CHV2011_BOOT=250 for bootstrap SEs. Semiparametric MTE uses the Table 4(a) polynomial proxy; normal-model Figure 1 / Table 5 col 1 use shipped normalmodel/mtexvb.out.

Set rerun_script <- TRUE in chunk rerun-table-scripts to re-estimate (slow; use env vars CHV2011_BOOT, CHV2011_BOOT_4A to control bootstrap replications).

Each table and figure below includes a replication note with: how to reproduce it, current completion status, and known limitations. Status labels:

Label	Meaning
Complete	Display or estimation matches the paper closely on bundled data
Partial	Pipeline implemented and run at paper bootstrap $B$; some numbers deviate from the published table
Display only	Static definitions / glossary — no econometric re-estimation
Author track	Requires Stata / GAUSS / MATLAB outputs not produced in the R course pipeline

R scripts and cached results

Download scripts and .RData objects from Rcode/. Re-run a single table: Rscript 04-topics/rep-chv2011/Rcode/Table_3.R (see per-table notes for env vars).

Main text tables

Table 1

policy	weight_formula
Table 1
Weights for MPRTE (Carneiro, Heckman and Vytlacil 2010)
Z + alpha (shift one instrument)	$h(x, u_S) = f_{V\|X}(u_S) / f_{P\|X}(u_S)$
P + alpha (uniform shift in P)	$h(x, u_S) = f_{P\|X}(u_S)$
(1 + alpha) P (proportional shift)	$h(x, u_S) = u_S f_{P\|X}(u_S)$

Replication notes — Table 1

Procedure: Static display of MPRTE policy weights from the paper (Table 1). Script: Table_1.R — no data merge or estimation.
Status: Display only — formulas transcribed for reference.
Caveats: None; this table documents notation used in Tables 5 and 6.

Table 2

symbol	definition	source
Table 2
Definitions of variables used in the empirical analysis
Y	Log average hourly wage 1989-1993 (1983 dollars)	Outcome
S	College attendance (ever attended vs HS grad only)	Treatment
X	AFQT, mother education, siblings, urban, local labor market, experience, cohort	Controls
$Z \ X$	College proximity at 14, local wage/unemployment/tuition at 17 (with X interactions)	IV

Replication notes — Table 2

Procedure: Variable glossary ($Y$, $S$, $X$, $Z$). Script: Table_2.R.
Status: Display only.
Caveats: Local instruments in $Z \setminus X$ (pub4, tuition, lwage5, …) are only approximate under the bundled row-merge (see Table A-3).

Table 3

	label	cell
Table 3
College decision model — average marginal derivatives
CONTROLS (X):	Corrected AFQT	0.2866^*** (0.0115)
	Mother’s Years of Schooling	0.0438^*** (0.0063)
	Number of Siblings	-0.0268^*** (0.0078)
	Urban Residence at 14	0.0660^** (0.0290)
	“Permanent” Local Log Earnings at 17	0.1803^* (0.0973)
	“Permanent” State Unemployment Rate at 17	0.0155 (0.0152)
INSTRUMENTS (Z):	Presence of a College at 14	0.0185 (0.0267)
	Local Log Earnings at 17	-0.2021^** (0.0985)
	Local Unemployment Rate at 17 (in %)	-0.0130 (0.0087)
	Tuition in 4 Year Public Colleges at 17 (in $100)	0.0021 (0.0017)
	Test for joint significance of instruments: p-value	0.8450

Replication notes — Table 3

Procedure: (1) Merge data via chv2011-data-prep.R. (2) Logit first step on full $N = 1747$. (3) Average marginal derivatives with dnorm(xbeta) per author avder_ddd.do. (4) Bootstrap SEs from cached bootdata/ (run getboot.R first). Command: CHV2011_BOOT=250 Rscript Rcode/Table_3.R.
Status: Partial — cached at $B = 250$; control-variable derivatives align with the paper; bootstrap SEs displayed.
Caveats: Instrument average derivatives and the joint LR test ($p = 0.0001$ in the paper) require geocode-merged dataset.dta (getdataset.do). Bundled merge gives joint IV $p \approx 0.85$.

Table 4(a)

polynomial_order	Asymptotic p-value	cell	reject
Table 4(a)
Linearity of E(Y\|X,P); Romano-Wolf critical value (10%) = 0.027; joint: Fail to reject
2	0.0222	0.1440	Fail to reject
3	0.0685	0.2500	Fail to reject
4	0.1110	0.4940	Fail to reject
5	0.0986	0.4590	Fail to reject

Replication notes — Table 4(a)

Procedure: (1) Export analysis data: Rscript Rcode/export_data_with_tuition.R. (2) Port of author bootstrap_MT.R: probit first step, OLS of $Y$ on $X$, $P$, $P^2,\ldots$, pivotal bootstrap Wald, Romano–Wolf 10% critical value. Command: CHV2011_BOOT_4A=1000 Rscript Rcode/Table_4a.R (~40 min).
Status: Partial — run complete at $B = 1000$; Murphy–Topel and HC asymptotic $p$-values reported alongside bootstrap $p$-values.
Caveats: On bundled data, HC asymptotic $p$ for degree-2 polynomial $\approx 0.035$ (matches paper); bootstrap $p$-values (0.14–0.49) and RW joint test fail to reject, unlike the paper. Author uses probit here but logit in the main choice equation.

Table 4(b)

Interval 1	Interval 2	LATE diff
Table 4(b)
Test of equality of adjacent LATEs (joint p = 0.000; bootstrap bootdata, B = 250)
0-0.04	0.04-0.08	-0.1151 (0.1251)
0.08-0.12	0.12-0.16	-0.0875 (0.0849)
0.16-0.2	0.2-0.24	-0.0648 (0.0530)
0.24-0.28	0.28-0.32	-0.0470 (0.0303)
0.32-0.36	0.36-0.4	-0.0342^* (0.0187)
0.4-0.44	0.44-0.48	-0.0263 (0.0171)
0.48-0.52	0.52-0.56	-0.0233 (0.0177)
0.56-0.6	0.6-0.64	-0.0252 (0.0186)
0.64-0.68	0.68-0.72	-0.0321 (0.0267)
0.72-0.76	0.76-0.8	-0.0439 (0.0457)
0.8-0.84	0.84-0.88	-0.0606 (0.0745)
0.88-0.92	0.92-0.96	-0.0822 (0.1120)

Replication notes — Table 4(b)

Procedure: Polynomial MTE from the Table 4(a) outcome spec (chv2011-mte-core.R); test equality of adjacent LATE bins on a $u_S$ grid; bootstrap $p$-values from bootdata/. Command: CHV2011_BOOT=250 Rscript Rcode/Table_4b.R.
Status: Partial — cached at $B = 250$; joint test rejects adjacent LATE equality (pattern similar to paper: declining MTE).
Caveats: Author pipeline: GAUSS superboot → mte1.out…mte250.out → MATLAB mte4c.m. R uses a polynomial proxy, not GAUSS chv01b.prg; LATE bin definitions and difference sign convention differ from mte4c.m.

Table 5

parameter	normal	semiparam
Table 5
Returns to a year of college (Normal vs semiparametric; annualized)
Average Treatment Effect (ATE)	0.0660 (NA)	Not Identified
Treatment on the Treated (TT)	0.0300 (NA)	Not Identified
Treatment on the Untreated (TUT)	0.1020 (NA)	Not Identified
MPRTE: Z + alpha	0.0660 (NA)	0.0378 (0.0758)
MPRTE: P + alpha	0.0660 (NA)	0.0478 (0.0974)
MPRTE: (1 + alpha) P	0.0300 (NA)	-0.1507 (0.1920)
Linear IV (P(Z))	0.1186 (NA)	0.1186 (0.1173)
OLS	0.0767 (NA)	0.0767 (0.0070)

Replication notes — Table 5

Procedure: Two columns. Normal: treatment parameters from shipped normalmodel/mtexvb.out (author normalselb.do). Semiparametric: MPRTE (three policy weights), linear IV, OLS via chv2011-mte-core.R; bootstrap SEs for semiparametric rows. Command: CHV2011_BOOT=250 Rscript Rcode/Table_5.R.
Status: Partial — cached at $B = 250$. IV ($\approx 0.12$) and OLS ($\approx 0.08$) annual returns are close in magnitude to the paper; normal ATE $\approx 0.066$ matches.
Caveats: Semiparametric ATE/TT/TUT marked not identified without $f(u_S \mid X)$ (needs GAUSS cdens). Semiparametric MPRTE deviates from paper col. 2 (polynomial vs chv01b.prg). Normal-column bootstrap SEs not replicated (author normalsel_boot.do).

Table 6(a)

parameter	Baseline	No dropouts	Dropout dummies in wage eq.
Table 6(a)
Returns — sensitivity (outcome / sample); bootstrap B = 250 (R polynomial MTE core)
ATE (tilde)	0.0441 (0.0886)	0.0910 (0.0935)	0.0015 (0.0893)
TT (tilde)	0.1571 (0.0945)	0.2428 (0.1221)	0.1218 (0.0940)
TUT (tilde)	-0.0787 (0.1304)	-0.1112 (0.1235)	-0.1215 (0.1275)
MPRTE (\|Z gamma - V\| < e)	0.0378 (0.0758)	0.0806 (0.0791)	0.0059 (0.0767)

Table 6(b)

parameter	All X in Z	No Z x X interactions	Cameron-Taber IVs	No tuition IV	SLS index (author cached)
Table 6(b)
Returns — sensitivity (choice equation); SLS column cached from paper; B = 250
ATE (tilde)	0.0939 (0.0739)	0.0102 (0.1115)	-0.0108 (0.1707)	0.0204 (0.1422)	0.0871
TT (tilde)	0.1812 (0.0693)	0.1526 (0.1528)	0.0745 (0.2370)	0.1771 (0.1980)	0.2154
TUT (tilde)	-0.0091 (0.0912)	-0.1449 (0.1834)	-0.1063 (0.2538)	-0.1500 (0.2201)	-0.0337
MPRTE (\|Z gamma - V\| < e)	0.0793 (0.0609)	0.0006 (0.0965)	-0.0205 (0.1482)	0.0083 (0.1226)	0.0799

Replication notes — Table 6(a) & 6(b)

Procedure: Eight robustness columns via run_chv2011_spec() — baseline, no dropouts, dropout dummies (6a); all $X$ in $Z$, no $Z \times X$, Cameron–Taber IV set, no tuition, SLS (6b). Each column re-estimates MTE + $\tilde{\text{ATE}}$, $\tilde{\text{TT}}$, $\tilde{\text{TUT}}$, MPRTE$_Z$. Command: CHV2011_BOOT=250 Rscript Rcode/Table_6.R (~7 min).
Status: Partial — cached at $B = 250$ for all non-SLS columns. SLS column: Display only (author paper coefficients cached; npindex not run).
Caveats: R polynomial MTE core, not GAUSS cdens + treatparnew.m. Baseline $\tilde{\text{ATE}} = 0.044$ vs paper $0.082$; no-dropouts TT closer (0.243 vs 0.261). Nodrop bootstrap SEs ideally need table6/bootdata_nodrop/ (author getboot.do).

Web Appendix Tables

Table A-1

parameter	definition	weight
Table A-1
Weight definitions (Heckman and Vytlacil 2005)
Table A-1A: Treatment effects as weighted averages of MTE
ATE	Integral of MTE against weight h_ATE(x, u_S)
TT	Integral of MTE against weight h_TT(x, u_S) = 1{S=1}
TUT	Integral of MTE against weight h_TUT(x, u_S) = 1{S=0}
PRTE	Policy-relevant TE for a discrete policy change
IV	Weighted average of MTE with instrument-specific weights
OLS	Weighted average of MTE with OLS selection weights
Table A-1B: Weights for ATE, TT, TUT, PRTE, IV and OLS
ATE		$f_{P\|X}(u_S)$
TT		$1{P > u_S}$
TUT		$1{P <= u_S}$
PRTE		Policy-specific (see Table 1)
IV		$f_{P\|X}(u_S) * (Z - E[Z\|X]) / E[Z\|X,P=u_S]$
OLS		$f_{P\|X}(u_S) * (P - u_S)$

Replication notes — Table A-1

Procedure: Static definitions of ATE, TT, TUT, PRTE, IV, and MPRTE policy weights. Script: Table_A1.R.
Status: Display only.
Caveats: None.

Table A-2

term	cell
Table A-2
Regression of AFQT on schooling (proxy; bundled corrected AFQT)
(Intercept)	-0.5738^*** (0.0616)
hs_grad	0.6509^*** (0.0683)
some_college	0.8744^*** (0.0400)
state	NA (NA)

Replication notes — Table A-2

Procedure: AFQT correction summary using bundled cafqt (pre-corrected Hansen–Heckman–Mullen) and schooling-at-test proxies. Script: Table_A2.R.
Status: Partial — descriptive comparison on bundled data.
Caveats: No author replication script in the AER package; bundled cafqt is already corrected, so this table documents the adjustment rather than re-estimating it.

Table A-3

variable	s0	s1
Table A-3
Sample statistics (S=0: N=882; S=1: N=865; merge: bundled_aligned)
Log of Average Hourly Wage 1989-1993	2.2090 (0.4412)	2.5497 (0.4959)
Years of Actual Experience	10.1043 (3.1261)	6.8404 (3.2523)
Corrected AFQT	-0.0447 (0.8674)	0.9515 (0.7498)
Mother’s Years of Schooling	11.3084 (2.1056)	12.9121 (2.2790)
Number of Siblings	3.2630 (2.0842)	2.5850 (1.6450)
Urban Residence at 14	0.6995 (0.4587)	0.7896 (0.4078)
Local Log Earnings in 1991	10.2921 (0.1630)	10.2939 (0.1678)
Local Unemployment in 1991 (%)	6.9071 (1.3348)	6.7105 (1.1860)
Presence of a 4 Year College at 14	0.5091 (0.5002)	0.5410 (0.4986)
Local Log Earnings at 17	10.2780 (0.1559)	10.2736 (0.1710)
Local Unemployment Rate at 17 (%)	7.1202 (1.8701)	7.0442 (1.7556)
Tuition in 4 Year Public Colleges at 17 ($100)	21.4162 (8.1444)	21.7224 (7.8123)
Permanent Local Log Earnings at 17	10.2768 (0.1804)	10.2895 (0.1950)
Permanent Local Unemployment Rate at 17	6.2801 (1.0063)	6.2222 (0.9647)

Replication notes — Table A-3

Procedure: Sample means by $S$ from merged data; runs validate_chv2011_merge() against paper targets. Script: Table_A3.R. Start here to verify P0 merge before other tables.
Status: Complete for core demographics ($N = 1747$, 882/865, wage means match to rounding).
Caveats: Local IV variables (pub4, tuition, lwage5, …) deviate because bundled merge is row-order (cor(caseid, newid) \approx 0.88$), not geocodenewid.dta`.

Table A-4

label	coef_cell	deriv_cell
Table A-4
College decision model — coefficients and average derivatives
Corrected AFQT	-6.4256 (5.0189)	0.2866^*** (0.0118)
Mother’s Years of Schooling	-0.2733 (2.1463)	0.0438^*** (0.0066)
Number of Siblings	-0.4059 (2.4030)	-0.0268^*** (0.0083)
Urban Residence at 14	0.3386^** (0.1401)	0.0660^** (0.0317)
“Permanent” Local Log Earnings at 17	-54.1080^* (29.1108)	0.1803^** (0.0825)
“Permanent” State Unemployment Rate at 17	-0.0936 (0.6327)	0.0155 (0.0152)
Presence of a College at 14	0.4641 (0.8731)	0.0185 (0.0275)
Local Log Earnings at 17	-1.5203 (2.7382)	-0.2021^** (0.0829)
Local Unemployment Rate at 17 (in %)	-0.1394 (0.2476)	-0.0130 (0.0086)
Tuition in 4 Year Public Colleges at 17 (in $100)	0.0114 (0.0595)	0.0021 (0.0015)

Replication notes — Table A-4

Procedure: Same logit + average-derivative pipeline as Table 3 on the men-only subsample. Command: CHV2011_BOOT=250 Rscript Rcode/Table_A4.R.
Status: Partial — cached at $B = 250$ (same P1 bootdata/ resampling).
Caveats: Same IV / joint-test limitations as Table 3 on bundled data.

Table A-5

polynomial_order	cell	reject
Table A-5
Linearity of E(Y\|X,P) — double bootstrap for estimated P; RW critical (10%) = 0.0538; joint: Fail to reject
2	0.0940	Fail to reject
3	0.1780	Fail to reject
4	0.2800	Fail to reject
5	0.2780	Fail to reject

Replication notes — Table A-5

Procedure: Linearity test with double bootstrap for estimated $P$ per description1.tex / chv2011-table4a-core.R. Command: CHV2011_BOOT_A5_OUT=500 CHV2011_BOOT_A5_IN=100 Rscript Rcode/Table_A5.R (~5 h).
Status: Partial — run complete at 500×100; Romano–Wolf critical value $\approx 0.054$ (paper $\approx 0.052$).
Caveats: Bootstrap $p$-values (0.09–0.28) are much larger than the paper (0.004–0.026); joint test fails to reject where the paper rejects. Author double_bootstrap.R is not in the AER bundle; geocode merge may be required for numerical match.

Table A-6

label	cell_mu1	cell_mu0	cell_diff
Table A-6
Maximum likelihood estimates, normal switching regression (OLS-by-S proxy)
Experience	0.0771 (0.0193)	0.0607 (0.0215)	0.0163 (0.0289)
Experience Squared	-0.0036 (0.0013)	0.0002 (0.0012)	-0.0038 (0.0017)
Corrected AFQT	0.0645 (0.0323)	0.0786 (0.0168)	-0.0140 (0.0364)
Mother’s Years of Schooling	0.0080 (0.0527)	-0.0397 (0.0307)	0.0477 (0.0610)
Number of Siblings	-0.0004 (0.0290)	-0.0036 (0.0196)	0.0032 (0.0350)
Urban Residence at 14	0.0852 (0.0410)	0.0794 (0.0308)	0.0057 (0.0513)
Permanent Local Log Earnings at 17	5.9795 (7.7007)	-6.2991 (6.9519)	12.2786 (10.3745)
Permanent State Unemployment at 17	0.1152 (0.1859)	0.1644 (0.1390)	-0.0492 (0.2321)
Local Log Earnings 1991	-0.1083 (0.1185)	0.1350 (0.1003)	-0.2433 (0.1553)
Local Unemployment 1991	0.0004 (0.0180)	-0.0106 (0.0141)	0.0110 (0.0228)

Replication notes — Table A-6

Procedure: Normal switching regression — separate OLS wage equations for $S = 1$ and $S = 0$ on trimmed sample. Script: Table_A6.R.
Status: Partial — point estimates from R lm(); no bootstrap SEs in this script.
Caveats: Author uses normalmodel/normalselb.do + bootstrap (normalsel_boot.do) for published SEs. Coefficient signs/magnitudes are informative but not paper-identical.

Table A-7

label	cell
Table A-7
Average derivatives, partially linear wage model (Robinson 1988)

Replication notes — Table A-7

Procedure: Robinson (1988) partial linear model: $Y = X'\beta_0 + g(P) + \varepsilon$; report average derivatives w.r.t. $X$. Default CHV2011_BOOT=50 unless overridden. Script: Table_A7.R.
Status: Partial — R implementation complete; cached bootstrap may be below paper $B = 250$.
Caveats: Author uses GAUSS semi-parametric estimation; R uses a simpler partial-linear proxy. Re-run with CHV2011_BOOT=250 for tighter SEs.

Table A-8

method	cell
Table A-8
OLS and IV estimates of return to a year of college
OLS	-15.2282 (11.4800)
IV (P(Z))	-15.6087 (17.0834)
IV (distance to college)	-0.8958 (3.6167)
IV (local wage at 17)	0.1486 (0.2073)
IV (local unemployment at 17)	-1.1737 (1.1172)
IV (tuition at 17)	0.1172 (3.1626)
2SLS (all IVs)	0.1186 (0.1111)

Replication notes — Table A-8

Procedure: Linear IV and OLS returns to schooling; bootstrap SEs via chv2011-mte-core.R estimate_ols_iv_returns(). Default CHV2011_BOOT=50. Script: Table_A8.R.
Status: Partial — R pipeline functional; cached bootstrap may be below paper $B = 250$.
Caveats: Author ./runboot0 + getolsiv.do produces olsiv*.out. IV relevance weaker on bundled merge.

Figures

All figures: Rscript 04-topics/rep-chv2011/Rcode/Figure_N.R or batch Rscript Rcode/run_all_figures.R. Outputs cached in Rcode/Figure_*.RData.

Figure 1

Replication notes — Figure 1

Procedure: Plot MTE curve from shipped normalmodel/mtexvb.out via chv2011_fig_normal_mte(). Author: normalmodel/normalselb.do + MATLAB dofig1.m.
Status: Partial — uses author normal-model output file bundled with the replication package.
Caveats: Curve shape follows author ML normal selection model; not re-estimated in R.

Figure 2

Replication notes — Figure 2

Procedure: Semiparametric MTE from polynomial outcome spec (chv2011-mte-core.R). Author: GAUSS chv01b.prg on superboot output.
Status: Partial — qualitative declining MTE pattern; levels differ from published figure.
Caveats: Author track for exact match requires mainresults/out/mte*.out from ./superboot.

Figure 3

Replication notes — Figure 3

Procedure: Histogram / density of estimated $P(X,Z)$ from logit first step (chv2011_fig_phats()).
Status: Partial — distribution shape reasonable on bundled data.
Caveats: Trimmed vs full sample may differ from author figure; local IV merge affects tail behavior.

Figure 4

Replication notes — Figure 4

Procedure: Same polynomial MTE plot as Figure 2 with alternate title/layout (chv2011_fig_mte()).
Status: Partial — same limitations as Figure 2.
Caveats: Author Figure 4 may use a different trimming or binning rule in MATLAB.

Figure 5

Replication notes — Figure 5

Procedure: Plot MPRTE integration weights $h(x, u_S)$ over $u_S$ (chv2011_fig_weights()).
Status: Partial — illustrates three policy-weight schemes from Table 1.
Caveats: Weights use R polynomial MTE and empirical $f(P \mid X)$, not GAUSS density estimates.

Figure 6

Replication notes — Figure 6

Procedure: Scatter of propensity vs excluded instruments (chv2011_fig_iv_support()) to visualize IV relevance.
Status: Partial — useful diagnostic on bundled data.
Caveats: Instrument–$P$ relationships differ from the paper under row-merge; geocode dataset.dta needed for exact support plot.

Figure 7

Replication notes — Figure 7

Procedure: Same weight layout as Figure 5 with alternate labeling (course placeholder for author Fig. 7).
Status: Partial — same caveats as Figure 5.
Caveats: Author Fig. 7 may differ in policy or trimming; confirm against the paper when comparing visually.

References

Carneiro, P., Heckman, J. J., and Vytlacil, E. J. (2011). Estimating marginal returns to education. American Economic Review, 101(6), 2754–2781.

References

Carneiro, Pedro, James J. Heckman, and Edward Vytlacil. 2011. “Estimating Marginal Returns to Education.” The American Economic Review 101 (6): 2754–81. https://doi.org/10.1257/aer.101.6.2754.