Lets compare the coverage of two confidence intervals - one made using Normal approximation (`equation 3.6` or HC2 robust standard errors), the other under the constant treatment effects assumption.

```
library(dplyr)
library(randomizr)
library(knitr)
set.seed(100)
data <- data_frame(Y0 = rnorm(500, mean = 0, sd = 8), Y1 = Y0 + rnorm(500, mean = 5,
sd = 4), X1 = as.numeric(Y0 >= 0), X2 = randomizr::complete_ra(N = 500,
m = 250), C1 = ifelse(Y0 <= -1, 1, ifelse((Y0 > -1) & (Y0 <= 0), 2, ifelse((Y0 >
0) & (Y0 <= 1), 3, 4))), C2 = sample(x = c(1, 2, 3, 4), size = 500, replace = T,
prob = c(0.25, 0.25, 0.25, 0.25)))
kable(head(data), caption = "Dataset for Confidence Interval Converage Test",
row.names = T)
```

Y0 | Y1 | X1 | X2 | C1 | C2 | |
---|---|---|---|---|---|---|

1 | -4.0175388 | -4.802689 | 0 | 1 | 1 | 2 |

2 | 1.0522493 | 7.315672 | 1 | 1 | 4 | 1 |

3 | -0.6313367 | 2.997673 | 0 | 1 | 2 | 1 |

4 | 7.0942785 | 4.368866 | 1 | 0 | 4 | 2 |

5 | 0.9357702 | 6.907054 | 1 | 1 | 3 | 4 |

6 | 2.5490407 | 6.097969 | 1 | 1 | 4 | 4 |

```
true_ate <- mean(data$Y1 - data$Y0) # True ATE
true_ate
```

`## [1] 5.28487`

```
declaration <- declare_ra(N = nrow(data), m = 250) # Complete random assignment, m=250, N=500
Z <- conduct_ra(declaration)
# Get the sampling distribution of all ATE estimates
D <- obtain_permutation_matrix(declaration)
ate_sims <- rep(NA, ncol(D))
for (i in 1:ncol(D)) {
Z_temp <- D[, i]
Y <- data$Y1 * Z_temp + data$Y0 * (1 - Z_temp)
ate_sims[i] <- mean(Y[Z_temp == 1]) - mean(Y[Z_temp == 0])
}
hist(ate_sims, breaks = 100, main = "Sampling Distribution of ATE Estimates")
```

To get the coverage of confidence intervals constructed using equation 3.6 I run a simulation in which for different samples of the population (\(N_{pop}=500\) and \(n_{sample}=200\)), I compute the ate estimate, standard error estimate, and check if the resulting confidence interval contains the true ATE (which is known from the above dataset: \(E[\tau_i] = 5.28487\)).

```
library(commarobust)
# Define output vector
coverage_normal <- rep(NA, 1000)
# For a random sample, create a declaration in randomizr
sample_declaration <- declare_rs(N = 500, n = 200)
for (i in 1:1000) {
# Generate a random sample
data$selected <- draw_rs(sample_declaration)
sample <- data %>% filter(selected == 1)
# Conduct complete random assignment
Z_temp <- complete_ra(N = 200, m = 100)
# Switching equation
Y <- sample$Y1 * Z_temp + sample$Y0 * (1 - Z_temp)
model <- lm(Y ~ Z_temp)
# ATE estimate
ate_estimate <- commarobust(model)[2, 1]
# SE estimate (HC2 robust SE)
se_estimate <- commarobust(model)[2, 2]
# Construct CI
ci_lower <- ate_estimate - 1.96 * se_estimate
ci_upper <- ate_estimate + 1.96 * se_estimate
# Check if this CI contains true ate
coverage_normal[i] <- as.numeric((true_ate >= ci_lower) & (true_ate <= ci_upper))
}
# Coverage of CI
mean(coverage_normal)
```

`## [1] 0.951`

To get the coverage of confidence intervals constructed under the constant treatment effects assumption, I run a simulation in which for different samples (\(n_{sample} = 200\)) of the population (\(N_{pop} = 500\)), I compute the ate estimate, use that to get a hypothetical schedule of potential outcomes under the fixed effects assumption, and check if the resulting confidence interval contains the true ATE (5.28487)

```
coverage_gerbgreen <- rep(NA, 1000)
sample_declaration <- declare_rs(N = 500, n = 200)
set.seed(32)
for (i in 1:1000) {
# Generate a random sample
data$selected <- draw_rs(sample_declaration)
sample <- data %>% filter(selected == 1)
# Conduct complete random assignment
Z_temp <- complete_ra(N = 200, m = 100)
# Switching equation
Y <- sample$Y1 * Z_temp + sample$Y0 * (1 - Z_temp)
model <- lm(Y ~ Z_temp)
# ATE estimate
ate_estimate <- commarobust(model)[2, 1]
# SE estimate (under constant effects) Schedule of potential outcomes, under
# fixed effects
Y0_temp <- Y - ate_estimate * Z_temp
Y1_temp <- Y0_temp + ate_estimate
# Create permutation matrix
declaration <- declare_ra(N = length(Y), m = length(Y)/2)
D <- obtain_permutation_matrix(declaration, maximum_permutations = 1000)
# Loop to get sampling distribution
ate_sims <- rep(NA, ncol(D))
for (j in 1:ncol(D)) {
z_sim <- D[, j]
# Compute ATE
ate_sims[j] <- mean(Y1_temp[z_sim == 1]) - mean(Y0_temp[z_sim == 0])
}
# Construct CI
ci_lower <- quantile(ate_sims, probs = 0.025)
ci_upper <- quantile(ate_sims, probs = 0.975)
# Check if this CI contains true ate
coverage_gerbgreen[i] <- as.numeric((true_ate >= ci_lower) & (true_ate <=
ci_upper))
}
# Coverage of CI
mean(coverage_gerbgreen)
```

`## [1] 0.956`

I write a function that computes coverage for different sample sizes: \(n_{sample} \in \{10,50,100,250\}\).

Sample.Size | Coverage.NormalApprox | Coverage.GerberGreen |
---|---|---|

10 | 0.912 | 0.896 |

50 | 0.944 | 0.932 |

100 | 0.961 | 0.951 |

250 | 0.949 | 0.947 |