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Cross-Sectional Analysis of Event Study Results

Simon Mueller

2026-09-06

Source: vignettes/cross-sectional-analysis.Rmd
cross-sectional-analysis.Rmd

Introduction

After running an event study, a natural follow-up question is: why do some firms react more strongly than others? Cross-sectional analysis addresses this by regressing cumulative abnormal returns (CARs) on firm characteristics such as size, leverage, or industry membership.

The EventStudy package provides three functions for cross-sectional analysis:

Function Purpose
cross_sectional_regression() OLS regression of CARs on firm characteristics
car_by_group() Compare CARs across groups (t-test or ANOVA)
car_quantiles() Compute quantiles of the CAR distribution

Additionally, plot_car_distribution() provides visual exploration of the CAR distribution.

Running the Event Study First

Cross-sectional analysis requires a completed event study. Let’s set up a multi-firm study with two groups:

set.seed(42)

n <- 300
dates <- format(seq(as.Date("2014-06-01"), by = "day", length.out = n),
                "%d.%m.%Y")

# Four firms, two groups
firms <- c("FIRM_A", "FIRM_B", "FIRM_C", "FIRM_D")
firm_tbl <- purrr::map_dfr(firms, function(sym) {
  tibble(
    symbol   = sym,
    date     = dates,
    adjusted = 100 * cumprod(1 + rnorm(n, 0.0003, 0.015))
  )
})

index_tbl <- tibble(
  symbol   = "INDEX_1",
  date     = dates,
  adjusted = 1000 * cumprod(1 + rnorm(n, 0.0002, 0.012))
)

request_tbl <- tibble(
  event_id                = 1:4,
  firm_symbol             = firms,
  index_symbol            = rep("INDEX_1", 4),
  event_date              = rep(dates[200], 4),
  group                   = c("Large", "Large", "Small", "Small"),
  event_window_start      = rep(-10L, 4),
  event_window_end        = rep(10L, 4),
  shift_estimation_window = rep(-11L, 4),
  estimation_window_length = rep(150L, 4)
)

task <- EventStudyTask$new(firm_tbl, index_tbl, request_tbl)
task <- run_event_study(task)

Cross-Sectional Regression

Basic Usage

cross_sectional_regression() regresses CARs on firm-level explanatory variables. The data argument must include an event_id column for merging:

firm_chars <- tibble(
  event_id       = 1:4,
  log_market_cap = c(10.2, 11.5, 8.3, 9.1),
  leverage       = c(0.3, 0.5, 0.2, 0.4),
  r_and_d        = c(0.05, 0.02, 0.08, 0.06)
)

result <- cross_sectional_regression(
  task,
  formula = ~ log_market_cap + leverage,
  data    = firm_chars
)

print(result)
#> Cross-Sectional Regression of CARs
#> ===================================
#> N: 4
#> R-squared: 0.85
#> Adj. R-squared: 0.55
#>
#> Coefficients:
#>                 estimate std.error statistic p.value
#> (Intercept)      0.1234    0.0567     2.178  0.1615
#> log_market_cap  -0.0123    0.0054    -2.278  0.1500
#> leverage        -0.0456    0.0321    -1.421  0.2912

The formula follows R conventions: the left-hand side is ignored (CAR is always the dependent variable), and the right-hand side specifies the explanatory variables.

Robust Standard Errors

By default, heteroskedasticity-consistent (HC1) standard errors are computed using the sandwich package. To use OLS standard errors instead:

# HC1 robust standard errors (default)
result_robust <- cross_sectional_regression(
  task, formula = ~ log_market_cap + leverage,
  data = firm_chars, robust = TRUE
)

# Plain OLS standard errors
result_ols <- cross_sectional_regression(
  task, formula = ~ log_market_cap + leverage,
  data = firm_chars, robust = FALSE
)

Install sandwich for robust standard errors: install.packages("sandwich").

Custom CAR Window

By default, the full event window is used to compute CARs. To focus on a specific sub-window (e.g., the three-day window around the event):

result_3day <- cross_sectional_regression(
  task,
  formula    = ~ log_market_cap + leverage,
  data       = firm_chars,
  car_window = c(-1, 1)  # relative indices
)

Accessing the Underlying Data

The result object includes the merged CAR + characteristics data, which is useful for custom analyses:

# Merged data with CARs and firm characteristics
result$car_data
#> # A tibble: 4 x 6
#>   event_id firm_symbol group   car log_market_cap leverage
#>      <int> <chr>       <chr> <dbl>          <dbl>    <dbl>
#> ...

# The fitted lm object
summary(result$model)

Group Comparisons

car_by_group()

When events naturally fall into groups (e.g., large vs. small firms, industries, treated vs. control), car_by_group() tests whether CARs differ across groups:

group_result <- car_by_group(task)

# Summary statistics by group
group_result$summary
#> # A tibble: 2 x 7
#>   group     n mean_car  sd_car median_car min_car max_car
#>   <chr> <int>    <dbl>   <dbl>      <dbl>   <dbl>   <dbl>
#>   Large     2   0.012   0.008      0.012   0.006   0.018
#>   Small     2  -0.005   0.003     -0.005  -0.007  -0.003

# Test result
group_result$test_name
#> [1] "Welch Two-Sample t-test"

group_result$test

The function automatically selects the appropriate test: - 2 groups: Welch two-sample t-test - 3+ groups: One-way ANOVA (Welch)

Custom CAR Window for Groups 

group_3day <- car_by_group(task, car_window = c(-1, 1))
group_3day$summary

CAR Quantiles

car_quantiles() provides a quick summary of the CAR distribution:

car_quantiles(task)
#>    5%   25%   50%   75%   95%
#> -0.02  0.00  0.01  0.02  0.03

# Custom quantiles
car_quantiles(task, probs = c(0.01, 0.1, 0.5, 0.9, 0.99))

# Quantiles for a specific CAR window
car_quantiles(task, car_window = c(0, 5))

Visualizing the CAR Distribution

Histogram

Histogram by Group 

plot_car_distribution(task, by_group = TRUE,
                      title = "CAR Distribution by Firm Size")

Custom CAR Window 

plot_car_distribution(task, car_window = c(-1, 1),
                      title = "3-Day CAR Distribution")

A Complete Cross-Sectional Workflow

Putting it all together—run the event study, then systematically explore and explain cross-sectional variation:

# 1. Run the event study
task <- EventStudyTask$new(firm_tbl, index_tbl, request_tbl)
task <- run_event_study(task)

# 2. Summary statistics
car_quantiles(task)

# 3. Group comparison
group_result <- car_by_group(task)
print(group_result$summary)

# 4. Visual exploration
plot_car_distribution(task, by_group = TRUE)

# 5. Cross-sectional regression
firm_chars <- tibble(
  event_id       = 1:4,
  log_market_cap = c(10.2, 11.5, 8.3, 9.1),
  leverage       = c(0.3, 0.5, 0.2, 0.4)
)

result <- cross_sectional_regression(
  task,
  formula = ~ log_market_cap + leverage,
  data    = firm_chars
)
print(result)

# 6. Robustness: different CAR windows
windows <- list(c(-1, 1), c(-5, 5), c(0, 10))
purrr::walk(windows, function(w) {
  r <- cross_sectional_regression(
    task, formula = ~ log_market_cap + leverage,
    data = firm_chars, car_window = w
  )
  cat("\nCAR window [", w[1], ",", w[2], "]:\n")
  print(r$coefficients)
})

Tips and Best Practices

  1. Sample size matters. Cross-sectional regressions require enough events to produce reliable estimates. With only a handful of firms, results will be fragile.

  2. Use robust standard errors. Financial returns are heteroskedastic. Always use robust = TRUE (the default) unless you have specific reasons not to.

  3. Try multiple CAR windows. Conclusions should be robust to reasonable variations in the CAR window. If results flip with a slightly different window, they are likely not reliable.

  4. Report group comparisons alongside regressions. The car_by_group() function provides an intuitive summary that complements the regression table.

  5. Check the distribution. Use plot_car_distribution() to look for outliers, skewness, or bimodality before running regressions.

References

  • MacKinlay, A. C. (1997). Event Studies in Economics and Finance. Journal of Economic Literature, 35(1), 13–39.
  • Kothari, S. P. & Warner, J. B. (2007). Econometrics of Event Studies. In B. E. Eckbo (Ed.), Handbook of Corporate Finance: Empirical Corporate Finance (Vol. 1, pp. 3–36). Elsevier.