_did_simple_demo.Rmd

```{r}
# Load required libraries
library(dplyr)
library(ggplot2)
set.seed(1)

# Simulated dataset for illustration
data <- data.frame(
  time = rep(c(0, 1), each = 50),  # Pre (0) and Post (1)
  treated = rep(c(0, 1), times = 50), # Control (0) and Treated (1)
  error = rnorm(100)
)

# Generate outcome variable
data$outcome <- 5 + 3 * data$treated + 2 * data$time + 4 * data$treated * data$time + data$error

# Compute averages for 2x2 table
table_means <- data %>%
  group_by(treated, time) %>%
  summarize(mean_outcome = mean(outcome), .groups = "drop") %>%
  mutate(
    group = paste0(ifelse(treated == 1, "Treated", "Control"), ", ", 
                   ifelse(time == 1, "Post", "Pre"))
  )

# Display the 2x2 table
table_2x2 <- table_means %>%
  select(group, mean_outcome) %>%
  tidyr::spread(key = group, value = mean_outcome)

print("2x2 Table of Mean Outcomes:")
print(table_2x2)

# Calculate Diff-in-Diff manually
Y11 <- table_means$mean_outcome[table_means$group == "Treated, Post"]  # Treated, Post
Y10 <- table_means$mean_outcome[table_means$group == "Treated, Pre"]   # Treated, Pre
Y01 <- table_means$mean_outcome[table_means$group == "Control, Post"]  # Control, Post
Y00 <- table_means$mean_outcome[table_means$group == "Control, Pre"]   # Control, Pre

diff_in_diff_formula <- (Y11 - Y10) - (Y01 - Y00)

# Estimate DID using OLS
model <- lm(outcome ~ treated * time, data = data)
ols_estimate <- coef(model)["treated:time"]

# Print results
results <- data.frame(
  Method = c("Diff-in-Diff Formula", "OLS Estimate"),
  Estimate = c(diff_in_diff_formula, ols_estimate)
)

print("Comparison of DID Estimates:")
print(results)

# Visualization
ggplot(data, aes(x = as.factor(time), y = outcome, color = as.factor(treated), group = treated)) +
  stat_summary(fun = mean, geom = "point", size = 3) +
  stat_summary(fun = mean, geom = "line", linetype = "dashed") +
  labs(
    title = "Difference-in-Differences Visualization",
    x = "Time (0 = Pre, 1 = Post)",
    y = "Outcome",
    color = "Group"
  ) +
  scale_color_manual(labels = c("Control", "Treated"), values = c("blue", "red")) +
  theme_minimal()
```

|              | Control (0)        | Treated (1)         |
|--------------|--------------------|---------------------|
| **Pre (0)**  | $\bar{Y}_{00} = 5$ | $\bar{Y}_{10} = 8$  |
| **Post (1)** | $\bar{Y}_{01} = 7$ | $\bar{Y}_{11} = 14$ |

The table organizes the mean outcomes into four cells:

1.  Control Group, Pre-period ($\bar{Y}_{00}$): Mean outcome for the control group before the intervention.

2.  Control Group, Post-period ($\bar{Y}_{01}$): Mean outcome for the control group after the intervention.

3.  Treated Group, Pre-period ($\bar{Y}_{10}$): Mean outcome for the treated group before the intervention.

4.  Treated Group, Post-period ($\bar{Y}_{11}$): Mean outcome for the treated group after the intervention.

The treatment effect is calculated as:

$\text{DID} = (\bar{Y}_{11} - \bar{Y}_{10}) - (\bar{Y}_{01} - \bar{Y}_{00})$

Using the simulated table:

$\text{DID} = (14 - 8) - (7 - 5) = 6 - 2 = 4$

This matches the **interaction term coefficient** ($\beta_3 = 4$) from the OLS regression.

**Understanding Diff-in-Diff: The Formula and OLS**

Did you know the Difference-in-Differences (DID) treatment effect calculated from the simple formula of averages is identical to the estimate from an OLS regression with an interaction term?

Here's a detailed breakdown with R code and visuals to prove it.

Compute manually:

$(\bar{Y}_{11} - \bar{Y}_{10}) - (\bar{Y}_{01} - \bar{Y}_{00})$

Use OLS regression:

$Y_{it} = \beta_0 + \beta_1 \text{treated}_i + \beta_2 \text{time}_t + \beta_3 (\text{treated}_i \cdot \text{time}_t) + \epsilon_{it}$

The interaction term coefficient ($\beta_3$) is the treatment effect.

Both methods give the same result!