Encouragement / Promotion Design

An experimental design that randomises access to or encouragement for a program rather than compulsory treatment, then uses two-stage least squares to estimate the local average treatment effect (LATE) for compliers — those who took up the intervention because they were encouraged.

What it is

An encouragement design is used when the researcher cannot compel take-up of the intervention — only encourage or facilitate it. A randomly selected group of households receives encouragement (an invitation, information, a subsidy, a voucher) to participate in a program; another group does not. Take-up is voluntary: some encouraged households will not participate; some non-encouraged households may find and join the program anyway. The random encouragement is then used as an instrumental variable (IV) to estimate the causal effect of actual participation.

This design identifies the local average treatment effect (LATE) — the average treatment effect for compliers, defined as units who take up the treatment when encouraged and do not when not encouraged. It does not identify the effect for never-takers (who would not participate regardless) or always-takers (who participate regardless of encouragement).

When to use it

Encouragement designs are appropriate when: compulsory treatment assignment is infeasible or unethical; programs have voluntary participation; or the researcher wants to estimate the effect of actual program participation (rather than the intent-to-treat effect). They are common in evaluations of voluntary health interventions (immunisation campaigns, health programmes), education programmes (scholarship lotteries, invitation-based tutoring), and financial inclusion programmes (mobile money adoption, savings accounts).

The design requires a strong first stage: the encouragement must substantially increase take-up probability. A weak first stage (low first-stage F-statistic) produces imprecise and potentially biased IV estimates.

How it works

Intent-to-treat (ITT) effect. The reduced-form effect of encouragement on the outcome — comparing encouraged and non-encouraged groups on the outcome regardless of actual take-up. The ITT is the policy-relevant effect of the encouragement program itself.

First stage. The effect of encouragement on take-up: D_i = α + γ·Z_i + u_i, where Z_i is the encouragement indicator and D_i is the take-up indicator. The F-statistic on γ should exceed 10 as a minimum for reliable IV estimation; values above 20–30 are preferred.

Two-stage least squares (2SLS). The LATE is estimated by dividing the ITT by the first-stage effect:

LATE = ITT / First stage = E[Y|Z=1] − E[Y|Z=0] / E[D|Z=1] − E[D|Z=0]

In regression terms, this is estimated by 2SLS using Z as an instrument for D. The estimate is the average treatment effect for compliers only.

Monotonicity assumption. IV identification requires monotonicity: no one is discouraged from taking up by the encouragement (no “defiers” — units who would take up if not encouraged but not if encouraged). This is usually plausible in development contexts where encouragement takes the form of facilitation (vouchers, invitations) rather than social pressure.

Key decisions

Design of the encouragement. Encouragement must be strong enough to produce a meaningful first stage but not so overwhelming that it induces near-perfect take-up (which would make the LATE approximately equal to the ATE, but requires perfect compliance for identification, which is rarely plausible). A take-up rate differential of 20–50 percentage points between encouraged and non-encouraged is typical.

Two-sided non-compliance. When both encouraged and non-encouraged individuals can access the program (some non-encouraged individuals self-select in), standard IV identifies the LATE for compliers. When the program is genuinely restricted to the encouraged group (one-sided non-compliance — no always-takers), IV identifies the ATT (average treatment effect on the treated), which is a simpler parameter.

Sample size for IV designs. IV estimates are less precise than OLS estimates with the same sample size. The effective sample size for IV is approximately reduced by a factor of 1/F_stage1 (the compliance rate squared). Power calculations for IV designs must account for the compliance rate.

Caveats & common mistakes

Weak instruments. An F-statistic below 10 in the first stage indicates a weak instrument. Weak IV estimates are severely biased toward OLS (which is generally biased when take-up is endogenous) and produce unreliable confidence intervals. With weak instruments, Limited Information Maximum Likelihood (LIML) or Anderson-Rubin confidence sets are more reliable than standard 2SLS.

Exclusion restriction violations. The IV exclusion restriction requires that encouragement affects the outcome only through take-up, not directly. This is violated if: the invitation itself changes behaviour (information effects, Hawthorne effects, stigma); encouraged units differ in their effort or investments in response to the encouragement; or enumerators interact differently with encouraged respondents. The plausibility of the exclusion restriction must be argued from the design.

Interpreting LATE as ATE. The LATE is the effect for compliers — those who respond to the encouragement. Compliers may differ systematically from the population: they may be more motivated, more responsive, or have higher returns to the intervention. Generalising LATE estimates to the population or to other contexts requires acknowledging this selection.

Analysis Guide

import pandas as pd
import numpy as np
import statsmodels.formula.api as smf
from linearmodels.iv import IV2SLS

# 1. First stage: regress take-up on encouragement plus covariates — the
#    F-statistic on the excluded instrument is the key diagnostic; if it falls
#    below 10 the instrument is too weak and 2SLS estimates will be biased toward
#    the OLS estimate of take-up on outcome regardless of sample size
first = smf.ols("takeup ~ encouraged + age + female + hh_size",
              data=df).fit(cov_type="HC1")
f_test = first.f_test("encouraged = 0")
print(f"First-stage F = {float(f_test.statistic):.2f}")

# 2. Intent-to-treat: the reduced-form effect of encouragement on the outcome —
#    this is the policy effect of running the encouragement programme itself and
#    is valid regardless of instrument strength; report alongside the LATE
itt = smf.ols("outcome ~ encouraged + age + female + hh_size",
            data=df).fit(cov_type="HC1")
print(itt.params["encouraged"], itt.bse["encouraged"])

# 3. 2SLS via IV2SLS.from_formula: brackets isolate the endogenous regressor and
#    its instrument; exogenous controls appear outside the brackets; the 2SLS
#    coefficient is the LATE for compliers and equals ITT / first-stage
iv = IV2SLS.from_formula(
  "outcome ~ 1 + age + female + hh_size + [takeup ~ encouraged]",
  data=df).fit(cov_type="robust")
print(iv.summary)

# 4. First-stage diagnostics from the 2SLS object — first_stage reports the
#    excluded-instrument F-statistic after partialling out controls; if it drops
#    below 10 once controls are added, the controls are absorbing the variation
#    that identified the instrument
print(iv.first_stage)

# 5. Manual LATE check: ITT divided by the first-stage coefficient should
#    reproduce the 2SLS coefficient exactly; a mismatch flags a specification
#    inconsistency (e.g., different covariates between the two regressions)
late_manual = itt.params["encouraged"] / first.params["encouraged"]
print(f"LATE (manual) = {late_manual:.4f}")

library(estimatr)
library(ivreg)

# 1. First stage: regress take-up on encouragement plus covariates — the F-statistic
#    on "encouraged" is the key diagnostic; if it falls below 10, the instrument
#    is too weak and 2SLS estimates will be unreliable regardless of sample size
first_stage <- lm_robust(takeup ~ encouraged + age + female + hh_size,
                        data = df, se_type = "HC2")
summary(first_stage)   # check F-statistic

# 2. ITT: the reduced-form effect of encouragement on the outcome — report this
#    alongside the LATE because it is valid even if the instrument is weak and
#    represents the actual effect of the encouragement intervention itself
itt <- lm_robust(outcome ~ encouraged + age + female + hh_size,
                data = df, se_type = "HC2")

# 3. 2SLS via iv_robust: the formula syntax places the endogenous variable (takeup)
#    and its instrument (encouraged) after the pipe; covariates appear on both sides
#    as they instrument themselves; the resulting coefficient is the LATE for compliers
iv_fit <- iv_robust(outcome ~ takeup + age + female + hh_size |
                  encouraged + age + female + hh_size,
                  data = df, se_type = "HC2")
summary(iv_fit)

# 4. Manual LATE check: ITT / first-stage coefficient should reproduce the 2SLS
#    coefficient exactly; if it does not, there is a specification mismatch in the
#    model (e.g., different control variables between the two regressions)
late_manual <- coef(itt)["encouraged"] / coef(first_stage)["encouraged"]
cat("LATE (manual):", round(late_manual, 4))

Reading the output

First-stage F-statistic below 10 signals a weak instrument; 2SLS estimates will be biased toward OLS and confidence intervals unreliable. Values above 20 are preferred for reliable inference.
The ITT coefficient is the intent-to-treat effect — the effect of being offered encouragement regardless of take-up. This is the policy-relevant estimate for the encouragement programme itself.
The 2SLS coefficient is the LATE — the effect of actual take-up for compliers. It will be larger in magnitude than the ITT whenever the compliance rate is below 100%; the ratio ITT / first-stage coefficient should reproduce it manually.
If estat firststage reports an F below 10, consider LIML estimation (ivregress liml) or Anderson-Rubin confidence sets, which are more reliable under weak identification.
Compare the 2SLS coefficient to the OLS regression of outcome on take-up; a large difference confirms endogeneity of take-up and validates the IV design.

References

Angrist, J. D., Imbens, G. W., & Rubin, D. B. (1996). Identification of causal effects using instrumental variables. Journal of the American Statistical Association, 91(434), 444–455. https://doi.org/10.2307/2291629

Duflo, E., & Saez, E. (2003). The role of information and social interactions in retirement plan decisions: Evidence from a randomized experiment. Quarterly Journal of Economics, 118(3), 815–842. https://doi.org/10.1162/00335530360698432

Imbens, G. W., & Angrist, J. D. (1994). Identification and estimation of local average treatment effects. Econometrica, 62(2), 467–475. https://doi.org/10.2307/2951620

Stock, J. H., Wright, J. H., & Yogo, M. (2002). A survey of weak instruments and weak identification in generalized method of moments. Journal of Business & Economic Statistics, 20(4), 518–529. https://doi.org/10.1198/073500102288618658