2 Five Estimands on One DGP

library(ggplot2)
library(dplyr)
library(gridExtra)

Before choosing an estimator, we need to know what we are estimating. The applied causal-inference literature has accumulated a zoo of estimands — ATE, ATT, LATE, CATE, QTE — and beginners often treat them as competing answers to one question. They are not. Each is a separate target that the data may or may not pin down. The estimator choice flows from which estimand you want.

This chapter constructs one data-generating process where all five estimands are well-defined and computable. We then compute each one and discuss what the differences tell us.

2.1 The data-generating process

Each individual $i$ has a baseline covariate $X_i$ (think: years of schooling), a latent type $U_i$ (think: ability), and a binary treatment $D_i$ (think: enrolling in a job-training programme). The instrument $Z_i$ is a randomly offered programme slot — it shifts treatment but does not directly affect the outcome.

set.seed(42)
n <- 20000

X <- runif(n, 0, 1)        # observed covariate, in [0,1]
U <- rnorm(n)              # latent type, unobserved
Z <- rbinom(n, 1, 0.5)    # random instrument

# Treatment assignment: depends on Z (the instrument) and U (selection on type)
# Higher U → more likely to take treatment regardless of Z (always-takers)
# Lower  U → less likely to take treatment regardless of Z (never-takers)
# Middle U → responsive to Z (compliers)
pD0 <- pmin(pmax(0.10 + 0.20 * U,        0), 1)  # P(D=1 | Z=0, U)
pD1 <- pmin(pmax(0.10 + 0.20 * U + 0.50, 0), 1)  # P(D=1 | Z=1, U) — adds 0.50

D0 <- rbinom(n, 1, pD0)   # potential treatment if not offered slot
D1 <- rbinom(n, 1, pD1)   # potential treatment if offered slot
D  <- ifelse(Z == 1, D1, D0)

# Heterogeneous treatment effect: depends on X
# Y(d) = baseline + d * tau(X) + 0.3 * U + noise
tau_fn <- function(x) 1.0 + 2.0 * x   # the true individual treatment effect

Y0 <- 0.5 * X + 0.3 * U + rnorm(n)
Y1 <- Y0 + tau_fn(X)
Y  <- ifelse(D == 1, Y1, Y0)

df <- data.frame(X = X, Z = Z, D = D, Y = Y)
head(df)

          X Z D           Y
1 0.9148060 1 1  2.40361930
2 0.9370754 0 0  2.21392002
3 0.2861395 1 1  0.78122406
4 0.8304476 1 0 -1.81205205
5 0.6417455 0 0  0.84444499
6 0.5190959 0 0  0.04843539

Notice that we have full access to the potential outcomes Y0 and Y1 — a luxury only available in simulation. In real data we observe only one of them per unit, which is the fundamental problem of causal inference.

2.2 The five estimands

2.2.1 ATE — average treatment effect

\[ \text{ATE} = \mathbb{E}[Y(1) - Y(0)] \]

The expected effect of treatment if everyone in the population were treated vs. if everyone were not.

ate_true <- mean(Y1 - Y0)
cat(sprintf("ATE (population mean of Y1 - Y0) = %.3f\n", ate_true))

ATE (population mean of Y1 - Y0) = 1.996

# Theoretical ATE: integral of (1 + 2x) over Uniform[0,1] = 1 + 1 = 2
cat("Theoretical ATE = integral(1 + 2x)dx on [0,1] = 2.000\n")

Theoretical ATE = integral(1 + 2x)dx on [0,1] = 2.000

2.2.2 ATT — average treatment effect on the treated

\[ \text{ATT} = \mathbb{E}[Y(1) - Y(0) \mid D = 1] \]

The expected effect of treatment among those who actually took treatment. Differs from ATE when treatment uptake is selective.

att_true <- mean(Y1[D == 1] - Y0[D == 1])
cat(sprintf("ATT (mean of Y1-Y0 conditional on D=1) = %.3f\n", att_true))

ATT (mean of Y1-Y0 conditional on D=1) = 1.998

The ATT here is similar to the ATE because the heterogeneity in $\tau$ depends on $X$ (uniform), and treatment selection depends on $U$ which is independent of $X$. If $X$ were correlated with $U$, ATT would diverge from ATE.

2.2.3 LATE — local average treatment effect (Imbens-Angrist)

\[ \text{LATE} = \mathbb{E}[Y(1) - Y(0) \mid D(1) > D(0)] \]

The effect among compliers — individuals whose treatment status changes with the instrument. The Wald estimator identifies LATE under the standard IV assumptions (exclusion, monotonicity, relevance).

compliers <- (D1 == 1) & (D0 == 0)
late_true <- mean(Y1[compliers] - Y0[compliers])
cat(sprintf("LATE (mean of Y1-Y0 conditional on complier status) = %.3f\n", late_true))

LATE (mean of Y1-Y0 conditional on complier status) = 1.994

cat(sprintf("Share of compliers: %.3f\n", mean(compliers)))

Share of compliers: 0.486

# Wald estimator from the data alone
wald <- (mean(Y[Z == 1]) - mean(Y[Z == 0])) /
        (mean(D[Z == 1]) - mean(D[Z == 0]))
cat(sprintf("Wald IV estimate (should match LATE): %.3f\n", wald))

Wald IV estimate (should match LATE): 2.044

LATE is what an IV regression actually estimates under heterogeneous effects — not the ATE. The Wald estimator recovers LATE because the instrument only shifts the compliers’ treatment status, so the IV “averages” the effect over that subpopulation.

2.2.4 CATE — conditional average treatment effect

\[ \text{CATE}(x) = \mathbb{E}[Y(1) - Y(0) \mid X = x] \]

The effect as a function of the covariate $X$. By construction in our DGP, CATE($x$) = $\tau(x) = 1 + 2x$.

nbins <- 20
breaks <- seq(0, 1, length.out = nbins + 1)
centers <- (breaks[-1] + breaks[-(nbins + 1)]) / 2
bin_idx <- cut(X, breaks = breaks, include.lowest = TRUE, labels = FALSE)

cate_est <- tapply(Y1 - Y0, bin_idx, mean)

cat(sprintf("CATE at x=0.2: estimated %.2f, true %.2f\n",
            cate_est[4], tau_fn(0.2)))

CATE at x=0.2: estimated 1.35, true 1.40

cat(sprintf("CATE at x=0.8: estimated %.2f, true %.2f\n",
            cate_est[16], tau_fn(0.8)))

CATE at x=0.8: estimated 2.55, true 2.60

2.2.5 QTE — quantile treatment effect

\[ \text{QTE}(\tau) = F^{-1}_{Y(1)}(\tau) - F^{-1}_{Y(0)}(\tau) \]

The difference between the $\tau$-th quantile of the treated marginal distribution and the $\tau$-th quantile of the control marginal. Note that QTE compares distributions (one quantile to one quantile) — it does not track individuals across the two potential outcomes.

qte_grid <- seq(0.05, 0.95, by = 0.05)
qte_est  <- quantile(Y1, qte_grid) - quantile(Y0, qte_grid)

cat(sprintf("QTE(0.10) = %.3f\n", quantile(Y1, 0.10) - quantile(Y0, 0.10)))

QTE(0.10) = 1.706

cat(sprintf("QTE(0.50) = %.3f\n", quantile(Y1, 0.50) - quantile(Y0, 0.50)))

QTE(0.50) = 1.992

cat(sprintf("QTE(0.90) = %.3f\n", quantile(Y1, 0.90) - quantile(Y0, 0.90)))

QTE(0.90) = 2.294

2.3 All five estimands on one plot

df_cate <- data.frame(x = centers, cate = as.numeric(cate_est),
                      true_tau = tau_fn(centers))
df_qte  <- data.frame(tau = qte_grid, qte = as.numeric(qte_est))

p1 <- ggplot(df_cate, aes(x = x)) +
  geom_line(aes(y = cate, colour = "CATE(x)"), linewidth = 1.2) +
  geom_line(aes(y = true_tau, colour = "True τ(x)"),
            linetype = "dotted", linewidth = 1) +
  geom_hline(aes(yintercept = ate_true,  colour = "ATE"),  linetype = "dashed") +
  geom_hline(aes(yintercept = att_true,  colour = "ATT"),  linetype = "dashed") +
  geom_hline(aes(yintercept = late_true, colour = "LATE"), linetype = "dashed") +
  scale_colour_manual(values = c(
    "CATE(x)" = "#c0392b", "True τ(x)" = "#e74c3c",
    "ATE" = "#2c3e50", "ATT" = "#2980b9", "LATE" = "#27ae60"
  )) +
  labs(x = "X (covariate)", y = "Effect", colour = NULL,
       title = "CATE(x) vs scalar estimands") +
  theme_minimal() +
  theme(legend.position = "bottom")

p2 <- ggplot(df_qte, aes(x = tau, y = qte)) +
  geom_line(colour = "#8e44ad", linewidth = 1.2) +
  geom_hline(yintercept = ate_true, linetype = "dashed", colour = "#2c3e50") +
  annotate("text", x = 0.07, y = ate_true + 0.06, label = "ATE",
           colour = "#2c3e50", size = 3) +
  labs(x = "Quantile τ", y = "QTE(τ)",
       title = "QTE — distribution-level effect") +
  theme_minimal()

grid.arrange(p1, p2, ncol = 2)

Five views of the same DGP. The horizontal lines for ATE/ATT/LATE collapse the effect to a single scalar. CATE(x) shows how the effect varies with X. QTE(τ) shows how the effect varies across the outcome distribution. Each curve answers a different question.

The left panel makes the conceptual point clearly. ATE, ATT, and LATE each collapse the heterogeneous $\tau(x)$ curve into a single number — but they collapse it differently. ATE averages over the full population’s $X$ distribution. ATT averages over the treated subpopulation. LATE averages over compliers.

CATE keeps the heterogeneity along $X$ explicit. QTE (right panel) keeps heterogeneity along the outcome distribution explicit. Neither nests the others: CATE and QTE answer fundamentally different questions about heterogeneity (covariate-level vs distribution-level).

2.4 When the estimands differ

In this DGP all four scalar estimands are similar because:

$X$ is uniform on [0, 1] (so ATE = average of τ over uniform $X$ = 2)
Treatment selection is on $U$ (unobserved type), not $X$ (which drives τ), so ATT ≈ ATE
The instrument shifts compliers uniformly across $X$, so LATE ≈ ATE

Change any of these assumptions and the estimands diverge. To see this, modify the DGP so that high-$X$ individuals are more likely to take treatment:

set.seed(7)
n2 <- 20000
X2 <- runif(n2, 0, 1)
U2 <- rnorm(n2)
Z2 <- rbinom(n2, 1, 0.5)

# High X → much more likely to take treatment (selection on X, the modifier)
pD0_2 <- pmin(pmax(0.10 + 0.20 * U2 + 0.6 * X2, 0), 1)
pD1_2 <- pmin(pmax(pD0_2 + 0.50,                 0), 1)
D0_2  <- rbinom(n2, 1, pD0_2)
D1_2  <- rbinom(n2, 1, pD1_2)
D2    <- ifelse(Z2 == 1, D1_2, D0_2)
Y0_2  <- 0.5 * X2 + 0.3 * U2 + rnorm(n2)
Y1_2  <- Y0_2 + tau_fn(X2)
Y2    <- ifelse(D2 == 1, Y1_2, Y0_2)

ate2  <- mean(Y1_2 - Y0_2)
att2  <- mean(Y1_2[D2 == 1] - Y0_2[D2 == 1])
compliers2 <- (D1_2 == 1) & (D0_2 == 0)
late2 <- mean(Y1_2[compliers2] - Y0_2[compliers2])

cat("Selection-on-X DGP:\n")

Selection-on-X DGP:

cat(sprintf("  ATE  = %.3f\n", ate2))

  ATE  = 2.001

cat(sprintf("  ATT  = %.3f  (now higher: treated have higher X → larger τ)\n", att2))

  ATT  = 2.124  (now higher: treated have higher X → larger τ)

cat(sprintf("  LATE = %.3f  (compliers' mean X drives this)\n", late2))

  LATE = 1.921  (compliers' mean X drives this)

Now ATT exceeds ATE because the treated subpopulation has higher $X$ and therefore larger treatment effects. The choice between reporting ATE and ATT is no longer cosmetic — it changes what the policy implication is.

2.5 Which estimand should you choose?

The right estimand depends on the policy question:

Policy question	Right estimand
“What if we treated everyone?”	ATE
“What did treating the currently-treated achieve?”	ATT
“What can the instrument tell us?”	LATE
“Who benefits most?”	CATE($x$)
“Does the effect vary across the outcome distribution?”	QTE($\tau$)
“What is the distribution of individual effects?”	Often unidentifiable; bounds required

The same applied paper can defensibly report multiple estimands. A regression with heterogeneous effects can yield ATE, ATT, and CATE($x$) jointly. The complementary direction is to keep the estimator fixed and vary the estimand target — each is a different research question.

2.6 Summary

ATE / ATT / LATE are different scalar averages of the underlying heterogeneous treatment effect. They coincide only under strong homogeneity or specific selection structure.
CATE($x$) and QTE($\tau$) preserve the heterogeneity, along different dimensions: covariates vs the outcome distribution.
IV regressions estimate LATE, not ATE. Reporting an IV coefficient as “the” causal effect is a category error when effects are heterogeneous.
Picking the right estimand is the substantive step. The estimator question — OLS, IV, matching, AIPW, TMLE — only makes sense once the estimand is fixed.

# Five Estimands on One DGP ```{r} #| include: false library(ggplot2) library(dplyr) library(gridExtra) ``` ```{r} #| eval: false library(ggplot2) library(dplyr) library(gridExtra) ``` Before choosing an estimator, we need to know what we are estimating. The applied causal-inference literature has accumulated a zoo of estimands — ATE, ATT, LATE, CATE, QTE — and beginners often treat them as competing answers to one question. They are not. Each is a separate target that the data may or may not pin down. The estimator choice flows from which estimand you want. This chapter constructs one data-generating process where all five estimands are well-defined and computable. We then compute each one and discuss what the differences tell us. ## The data-generating process Each individual $i$ has a baseline covariate $X_i$ (think: years of schooling), a latent type $U_i$ (think: ability), and a binary treatment $D_i$ (think: enrolling in a job-training programme). The instrument $Z_i$ is a randomly offered programme slot — it shifts treatment but does not directly affect the outcome. ```{r} #| label: dgp #| cache: true set.seed(42) n <- 20000 X <- runif(n, 0, 1) # observed covariate, in [0,1] U <- rnorm(n) # latent type, unobserved Z <- rbinom(n, 1, 0.5) # random instrument # Treatment assignment: depends on Z (the instrument) and U (selection on type) # Higher U → more likely to take treatment regardless of Z (always-takers) # Lower U → less likely to take treatment regardless of Z (never-takers) # Middle U → responsive to Z (compliers) pD0 <- pmin(pmax(0.10 + 0.20 * U, 0), 1) # P(D=1 | Z=0, U) pD1 <- pmin(pmax(0.10 + 0.20 * U + 0.50, 0), 1) # P(D=1 | Z=1, U) — adds 0.50 D0 <- rbinom(n, 1, pD0) # potential treatment if not offered slot D1 <- rbinom(n, 1, pD1) # potential treatment if offered slot D <- ifelse(Z == 1, D1, D0) # Heterogeneous treatment effect: depends on X # Y(d) = baseline + d * tau(X) + 0.3 * U + noise tau_fn <- function(x) 1.0 + 2.0 * x # the true individual treatment effect Y0 <- 0.5 * X + 0.3 * U + rnorm(n) Y1 <- Y0 + tau_fn(X) Y <- ifelse(D == 1, Y1, Y0) df <- data.frame(X = X, Z = Z, D = D, Y = Y) head(df) ``` Notice that we have full access to the potential outcomes `Y0` and `Y1` — a luxury only available in simulation. In real data we observe only one of them per unit, which is the fundamental problem of causal inference. ## The five estimands ### ATE — average treatment effect $$ \text{ATE} = \mathbb{E}[Y(1) - Y(0)] $$ The expected effect of treatment if everyone in the population were treated vs. if everyone were not. ```{r} #| label: ate #| cache: true ate_true <- mean(Y1 - Y0) cat(sprintf("ATE (population mean of Y1 - Y0) = %.3f\n", ate_true)) # Theoretical ATE: integral of (1 + 2x) over Uniform[0,1] = 1 + 1 = 2 cat("Theoretical ATE = integral(1 + 2x)dx on [0,1] = 2.000\n") ``` ### ATT — average treatment effect on the treated $$ \text{ATT} = \mathbb{E}[Y(1) - Y(0) \mid D = 1] $$ The expected effect of treatment among those who actually took treatment. Differs from ATE when treatment uptake is selective. ```{r} #| label: att #| cache: true att_true <- mean(Y1[D == 1] - Y0[D == 1]) cat(sprintf("ATT (mean of Y1-Y0 conditional on D=1) = %.3f\n", att_true)) ``` The ATT here is similar to the ATE because the heterogeneity in $\tau$ depends on $X$ (uniform), and treatment selection depends on $U$ which is independent of $X$. If $X$ were correlated with $U$, ATT would diverge from ATE. ### LATE — local average treatment effect (Imbens-Angrist) $$ \text{LATE} = \mathbb{E}[Y(1) - Y(0) \mid D(1) > D(0)] $$ The effect among **compliers** — individuals whose treatment status changes with the instrument. The Wald estimator identifies LATE under the standard IV assumptions (exclusion, monotonicity, relevance). ```{r} #| label: late #| cache: true compliers <- (D1 == 1) & (D0 == 0) late_true <- mean(Y1[compliers] - Y0[compliers]) cat(sprintf("LATE (mean of Y1-Y0 conditional on complier status) = %.3f\n", late_true)) cat(sprintf("Share of compliers: %.3f\n", mean(compliers))) # Wald estimator from the data alone wald <- (mean(Y[Z == 1]) - mean(Y[Z == 0])) / (mean(D[Z == 1]) - mean(D[Z == 0])) cat(sprintf("Wald IV estimate (should match LATE): %.3f\n", wald)) ``` LATE is what an IV regression actually estimates under heterogeneous effects — **not** the ATE. The Wald estimator recovers LATE because the instrument only shifts the compliers' treatment status, so the IV "averages" the effect over that subpopulation. ### CATE — conditional average treatment effect $$ \text{CATE}(x) = \mathbb{E}[Y(1) - Y(0) \mid X = x] $$ The effect as a function of the covariate $X$. By construction in our DGP, CATE($x$) = $\tau(x) = 1 + 2x$. ```{r} #| label: cate #| cache: true nbins <- 20 breaks <- seq(0, 1, length.out = nbins + 1) centers <- (breaks[-1] + breaks[-(nbins + 1)]) / 2 bin_idx <- cut(X, breaks = breaks, include.lowest = TRUE, labels = FALSE) cate_est <- tapply(Y1 - Y0, bin_idx, mean) cat(sprintf("CATE at x=0.2: estimated %.2f, true %.2f\n", cate_est[4], tau_fn(0.2))) cat(sprintf("CATE at x=0.8: estimated %.2f, true %.2f\n", cate_est[16], tau_fn(0.8))) ``` ### QTE — quantile treatment effect $$ \text{QTE}(\tau) = F^{-1}_{Y(1)}(\tau) - F^{-1}_{Y(0)}(\tau) $$ The difference between the $\tau$-th quantile of the treated marginal distribution and the $\tau$-th quantile of the control marginal. Note that QTE compares **distributions** (one quantile to one quantile) — it does not track individuals across the two potential outcomes. ```{r} #| label: qte #| cache: true qte_grid <- seq(0.05, 0.95, by = 0.05) qte_est <- quantile(Y1, qte_grid) - quantile(Y0, qte_grid) cat(sprintf("QTE(0.10) = %.3f\n", quantile(Y1, 0.10) - quantile(Y0, 0.10))) cat(sprintf("QTE(0.50) = %.3f\n", quantile(Y1, 0.50) - quantile(Y0, 0.50))) cat(sprintf("QTE(0.90) = %.3f\n", quantile(Y1, 0.90) - quantile(Y0, 0.90))) ``` ## All five estimands on one plot ```{r} #| label: estimand-plot #| cache: true #| fig-cap: "Five views of the same DGP. The horizontal lines for ATE/ATT/LATE collapse the effect to a single scalar. CATE(x) shows how the effect varies with X. QTE(τ) shows how the effect varies across the outcome distribution. Each curve answers a different question." #| fig-width: 10 #| fig-height: 4 df_cate <- data.frame(x = centers, cate = as.numeric(cate_est), true_tau = tau_fn(centers)) df_qte <- data.frame(tau = qte_grid, qte = as.numeric(qte_est)) p1 <- ggplot(df_cate, aes(x = x)) + geom_line(aes(y = cate, colour = "CATE(x)"), linewidth = 1.2) + geom_line(aes(y = true_tau, colour = "True τ(x)"), linetype = "dotted", linewidth = 1) + geom_hline(aes(yintercept = ate_true, colour = "ATE"), linetype = "dashed") + geom_hline(aes(yintercept = att_true, colour = "ATT"), linetype = "dashed") + geom_hline(aes(yintercept = late_true, colour = "LATE"), linetype = "dashed") + scale_colour_manual(values = c( "CATE(x)" = "#c0392b", "True τ(x)" = "#e74c3c", "ATE" = "#2c3e50", "ATT" = "#2980b9", "LATE" = "#27ae60" )) + labs(x = "X (covariate)", y = "Effect", colour = NULL, title = "CATE(x) vs scalar estimands") + theme_minimal() + theme(legend.position = "bottom") p2 <- ggplot(df_qte, aes(x = tau, y = qte)) + geom_line(colour = "#8e44ad", linewidth = 1.2) + geom_hline(yintercept = ate_true, linetype = "dashed", colour = "#2c3e50") + annotate("text", x = 0.07, y = ate_true + 0.06, label = "ATE", colour = "#2c3e50", size = 3) + labs(x = "Quantile τ", y = "QTE(τ)", title = "QTE — distribution-level effect") + theme_minimal() grid.arrange(p1, p2, ncol = 2) ``` The left panel makes the conceptual point clearly. **ATE**, **ATT**, and **LATE** each collapse the heterogeneous $\tau(x)$ curve into a single number — but they collapse it differently. ATE averages over the full population's $X$ distribution. ATT averages over the treated subpopulation. LATE averages over compliers. **CATE** keeps the heterogeneity along $X$ explicit. **QTE** (right panel) keeps heterogeneity along the outcome distribution explicit. Neither nests the others: CATE and QTE answer fundamentally different questions about heterogeneity (covariate-level vs distribution-level). ## When the estimands differ In this DGP all four scalar estimands are similar because: - $X$ is uniform on [0, 1] (so ATE = average of τ over uniform $X$ = 2) - Treatment selection is on $U$ (unobserved type), not $X$ (which drives τ), so ATT ≈ ATE - The instrument shifts compliers uniformly across $X$, so LATE ≈ ATE Change any of these assumptions and the estimands diverge. To see this, modify the DGP so that high-$X$ individuals are more likely to take treatment: ```{r} #| label: dgp-selective #| cache: true set.seed(7) n2 <- 20000 X2 <- runif(n2, 0, 1) U2 <- rnorm(n2) Z2 <- rbinom(n2, 1, 0.5) # High X → much more likely to take treatment (selection on X, the modifier) pD0_2 <- pmin(pmax(0.10 + 0.20 * U2 + 0.6 * X2, 0), 1) pD1_2 <- pmin(pmax(pD0_2 + 0.50, 0), 1) D0_2 <- rbinom(n2, 1, pD0_2) D1_2 <- rbinom(n2, 1, pD1_2) D2 <- ifelse(Z2 == 1, D1_2, D0_2) Y0_2 <- 0.5 * X2 + 0.3 * U2 + rnorm(n2) Y1_2 <- Y0_2 + tau_fn(X2) Y2 <- ifelse(D2 == 1, Y1_2, Y0_2) ate2 <- mean(Y1_2 - Y0_2) att2 <- mean(Y1_2[D2 == 1] - Y0_2[D2 == 1]) compliers2 <- (D1_2 == 1) & (D0_2 == 0) late2 <- mean(Y1_2[compliers2] - Y0_2[compliers2]) cat("Selection-on-X DGP:\n") cat(sprintf(" ATE = %.3f\n", ate2)) cat(sprintf(" ATT = %.3f (now higher: treated have higher X → larger τ)\n", att2)) cat(sprintf(" LATE = %.3f (compliers' mean X drives this)\n", late2)) ``` Now ATT exceeds ATE because the treated subpopulation has higher $X$ and therefore larger treatment effects. The choice between reporting ATE and ATT is no longer cosmetic — it changes what the policy implication is. ## Which estimand should you choose? The right estimand depends on the policy question: | Policy question | Right estimand | |---|---| | "What if we treated everyone?" | ATE | | "What did treating the currently-treated achieve?" | ATT | | "What can the instrument tell us?" | LATE | | "Who benefits most?" | CATE($x$) | | "Does the effect vary across the outcome distribution?" | QTE($\tau$) | | "What is the distribution of individual effects?" | Often unidentifiable; bounds required | The same applied paper can defensibly report multiple estimands. A regression with heterogeneous effects can yield ATE, ATT, and CATE($x$) jointly. The complementary direction is to keep the estimator fixed and vary the estimand target — each is a different research question. ## Summary - **ATE / ATT / LATE** are different scalar averages of the underlying heterogeneous treatment effect. They coincide only under strong homogeneity or specific selection structure. - **CATE($x$)** and **QTE($\tau$)** preserve the heterogeneity, along different dimensions: covariates vs the outcome distribution. - IV regressions estimate LATE, not ATE. Reporting an IV coefficient as "the" causal effect is a category error when effects are heterogeneous. - Picking the right estimand is the substantive step. The estimator question — OLS, IV, matching, AIPW, TMLE — only makes sense once the estimand is fixed.