Which Effect Does the Regression Estimate? Controlling for a confounder and the ATE, ATT and ATC

Steps

Pumps raise harvests by $6,000 for rich farmers and do nothing for poor farmers. We will control for wealth by fitting this regression to all 100 farmers:

Harvest_i = β₀ + β₁·Pump_i + β₂·Rich_i + ε_i

Pump and Rich are 0/1 indicators. There is no interaction term, so the model has a single pump coefficient, β₁, for rich and poor farmers alike. In R: lm(Y_obs ~ treat + rich).

Before fitting it, predict: where will β₁ land?

Your prediction

ATE ATT ATC Somewhere else

Compare the regression's β₁ (blue diamond) with the ATE, ATT and ATC on the scale. Which is it closest to?

Your answer

Read the weights panel. What share of the regression's weight goes to rich farmers? What share of all farmers, and of pump owners, are rich?

Your answer

Move the pump rate among poor farmers to 90%, the same as the rich, then down to 10%. What happens to the regression's weight on the rich, and to the coefficient?

Your answer

Reset, then set the pump's effect for poor farmers to +6. Why do the ATE, ATT, ATC and the regression now agree, while the naïve comparison still does not?

Your answer

Reset, then press “Give every rich farmer a pump.” How much does the regression now learn from rich farmers? Could any regression on these data estimate the ATE?

Your answer

Who has a pump?

Rich farmers

Poor farmers

Rich farmers (of 100)

Pump rate among rich farmers

Pump rate among poor farmers

The truth and the estimates

Pump's effect for rich farmers ($1,000s)

Pump's effect for poor farmers ($1,000s)

β₁ is an average of the two groups’ effects, with weights of its own

Each row averages the pump's effect for rich and poor farmers. Only the weights differ. The number on the right is the average each row produces.

To get the ATE, ATT or ATC back, let the effect differ by wealth. Then predict every farmer’s harvest with and without a pump, and average the difference over the farmers you care about:

fit <- lm(Y_obs ~ treat * rich, data = df)

# Each farmer's predicted harvest with and without a pump
Y1_hat <- predict(fit, newdata = transform(df, treat = 1))
Y0_hat <- predict(fit, newdata = transform(df, treat = 0))

ATE_hat <- mean(Y1_hat - Y0_hat)
ATT_hat <- mean(Y1_hat[df$treat == 1] - Y0_hat[df$treat == 1])
ATC_hat <- mean(Y1_hat[df$treat == 0] - Y0_hat[df$treat == 0])

Estimand	Truth	From lm(Y ~ pump * rich)

What this page assumes

There are 100 farmers, and you choose how many are rich (default 50).
Harvest without a pump: Y0 = 10 + 5·rich + u, with u ~ Normal(0, 1) drawn from a seeded generator. Units are $1,000s.
Harvest with a pump: Y1 = Y0 + the effect for the farmer's group (defaults +6 rich, +0 poor). Every farmer in a group has the same effect.
Pumps go to exactly round(rate × group size) farmers in each group, so pump ownership depends on wealth and nothing else. Wealth is the only confounder.
The true ATE, ATT and ATC use both potential outcomes. Every estimate uses only the observed harvest.
The regression is OLS of observed harvest on pump and rich, with conventional standard errors. The blue bar shows ±1.96 SE.
Regression weight for a group = n × p × (1 − p), where p is that group's pump rate, normalized to sum to 1.
The noise is generated in the browser, so estimates differ slightly from the R workbook (set.seed(104)). The true ATE, ATT and ATC match it exactly.

Left out on purpose: standard errors for the interaction-model ATE, ATT and ATC, and confidence versus prediction intervals. Both are covered in the workbook.