---
title: "Statistical Conventions and Formula Audit"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Statistical Conventions and Formula Audit}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(collapse = TRUE, comment = "#>")
```

## Purpose

Every event-study statistic embeds convention choices -- the return type, the
forecast-error correction, the degrees of freedom, the p-value sidedness, the
Patell denominator, and so on. Two implementations of "the same" test can differ
in the last digits (or more) purely because they made different, equally
defensible choices. This vignette documents, per return model and per test
statistic, exactly which convention EventStudy uses and cites the published
source for each. It is the reference that a golden-value regression test is
pinned against: a documented convention here matches the assumption annotated in
`tests/testthat/test_golden_values.R`.

## Audit methodology

Each model and statistic was audited against its published formula. Findings are
classified with a strict two-bucket policy:

- **Fix (genuine formula error).** If the code computes something no published
  convention supports, it is a bug. It is fixed in the R source, the fix is
  locked with a regression test, and the fix precedes the golden pin of that
  statistic. Such fixes are logged in the "Audit log" section below.
- **Document (defensible convention difference).** If the code matches a
  published convention that merely differs from another implementation (for
  example log vs simple returns, df = m - 2 vs m - 1, one- vs two-sided p), the
  behavior is *not* changed -- it is documented here as the chosen convention
  with its citation. Behavior on valid inputs is frozen for this milestone.

When a discrepancy is ambiguous it is treated as a convention (documented, not
changed), because silently altering a valid-input result is the higher-risk
error.

Tolerance policy for the golden tests: absolute `1e-10` for exact algebraic
identities, absolute `1e-8` as the general default, and relative `1e-6` for any
cross-implementation comparison. Each pinned assertion states its provenance,
the conventions it assumes, and its tolerance rationale inline.

## Return models

### Market Model (MacKinlay 1997)

The Market Model regresses firm returns on index returns over the estimation
window by OLS and defines the abnormal return as the residual in the event
window.

| Convention | Choice | Note |
|---|---|---|
| Return type | Arithmetic (simple) returns | The pipeline's return-calculation strategy is orthogonal; the model consumes whatever `firm_returns` / `index_returns` contain. |
| Estimation | OLS: `firm_returns ~ index_returns` | `alpha`, `beta` from `lm()`. |
| Abnormal return | `AR = R - (alpha + beta * Rm)` | MacKinlay (1997) eq. (5). |
| Residual sigma | Residual standard error from `summary(lm)` | Uses `df = m - 2` (intercept + slope), `m` = COMPLETE-PAIR estimation observations. |
| Degrees of freedom | `m - 2` | `m` = estimation observations with both `firm_returns` and `index_returns` finite. |
| Minimum observations | `n_valid >= n_params + 1 = 3` | Fewer valid observations route through the degenerate-input contract (NA + one warning, or error in strict mode) rather than fitting an unstable/undefined `sigma` (needs `df >= 1`). |
| Short-window advisory | One plain warning when `2 < n_valid < 30` | Advisory only (not a degenerate/contract condition) -- the model still fits; 30 is the same recommended minimum as `validate_task()`'s default. |
| `statistics$n_params` | `2` (intercept + slope) | Consumed by `PatellZTest`'s `Q_i` adjustment (see below). |
| AR t-statistic | `AR / sigma` | Per-day, constant-sigma approximation. |
| CAR t-statistic | `CAR / (sqrt(L) * sigma)` | `L`-day cumulative, constant-sigma approximation, MacKinlay (1997). |
| Forecast-error correction | `sigma * sqrt(1 + 1/m + (Rm_evt - mean(Rm_est))^2 / SS_market)` | Patell/MacKinlay prediction-error variance; `mean(Rm_est)` and `SS_market` (the estimation-window sum of squares of the index) are computed over COMPLETE PAIRS ONLY, matching exactly the rows `lm()` used (2026-09-24 re-evaluation item A6: previously included incomplete-pair rows, biasing the mean/SS). Falls back to the constant-mean form `sigma * sqrt(1 + 1/m)` when the index is constant. |
| p-value sidedness | Two-sided by default | `confidence_type = "two-sided"`. |

**Audit result: verified against MacKinlay (1997); two silent-wrong defects
fixed 2026-09-24 (items A6).** The core OLS/AR/CAR formulas match MacKinlay.
The 2026-09-24 re-evaluation found the insufficient-observations guard used a
flat `n_valid < 2` threshold (too permissive for a 2-parameter OLS fit to be
meaningful) and the forecast-error correction's `mean(Rm_est)`/`SS_market`
included estimation rows where `firm_returns` was NA but `index_returns` was
not -- rows `lm()` silently drops via `na.omit()` but which the correction
formula still summed over, biasing the correction away from the fitted
model's own estimation sample. Both are fixed; pinned in
`test_golden_values.R` and `test_reeval_statistical_fixes.R` against an
independent closed-form `lm()` derivation restricted to complete pairs, at
absolute tolerance `1e-10`/`1e-12`.

*Citation:* MacKinlay, A. C. (1997). Event Studies in Economics and Finance.
*Journal of Economic Literature*, 35(1), 13--39.

### Market Adjusted Model

The Market Adjusted (index) model treats the market index itself as the expected
return: it fixes `alpha = 0` and `beta = 1` rather than estimating them, so the
abnormal return is simply the firm return minus the index return. It is the
restricted market model of MacKinlay (1997) sec. 4.4.2.

| Convention | Choice | Note |
|---|---|---|
| Return type | Whatever `firm_returns` / `index_returns` contain | Return-calculation strategy is orthogonal. |
| Estimation | None -- `alpha = 0`, `beta = 1` fixed | No OLS parameters estimated. |
| Abnormal return | `AR = R - Rm` | Firm return minus index return. |
| Residual sigma | `sd(firm - index)` over the estimation window | Sample standard deviation of the market-adjusted residual series. |
| Degrees of freedom | `m - 1` | One parameter consumed by the residual mean in `sd()`; `m` = estimation observations. |
| `statistics$n_params` | `0` | Nothing is estimated (alpha/beta are FIXED at 0/1, not fitted). |
| Forecast-error correction | `forecast_error_corrected_sigma == sigma` (correction factor exactly 1) | No regression, no leverage term, and -- since nothing is estimated -- no prediction-error inflation either (2026-09-24 re-evaluation item A8). |
| p-value sidedness | Two-sided by default | -- |

**Audit result: one silent-wrong defect fixed 2026-09-24 (item A8).** The
subtraction `AR = R - Rm` matches the restricted market model. The 2026-09-24
re-evaluation found the forecast-error correction wrongly reused the
constant-MEAN formula `sigma * sqrt(1 + 1/m)` -- which is only valid when ONE
parameter (a mean) is estimated from the estimation window, as in the
Comparison Period Mean Adjusted Model below. MarketAdjustedModel estimates
NOTHING (alpha/beta are fixed, not fitted), so its correction factor is
exactly 1: `forecast_error_corrected_sigma == sigma`. Re-pinned in
`test_golden_values.R` and locked in `test_reeval_statistical_fixes.R` at
absolute tolerance `1e-10`/`1e-12`.

*Citation:* MacKinlay, A. C. (1997). Event Studies in Economics and Finance.
*Journal of Economic Literature*, 35(1), 13--39.

### Comparison Period Mean Adjusted Model

The Comparison Period Mean (constant-mean-return) model uses the firm's own
average return over the estimation window as the expected return. It is the
constant-mean-return model of Brown & Warner (1985), also described in MacKinlay
(1997) sec. 4.4.1.

| Convention | Choice | Note |
|---|---|---|
| Return type | Whatever `firm_returns` contains | Index returns are unused by this model. |
| Estimation | `mean(firm_returns)` over the estimation window | One parameter (the mean) estimated. |
| Abnormal return | `AR = R - mean(R_est)` | Firm return minus estimation-window mean. |
| Residual sigma | `sd(firm - mean)` over the estimation window | Sample standard deviation of the demeaned residuals. |
| Degrees of freedom | `m - 1` | The mean consumes one degree of freedom. |
| `statistics$n_params` | `1` | One parameter estimated (the comparison-period mean). |
| Forecast-error correction | `sigma * sqrt(1 + 1/m)` | Constant-mean form. Uses the finite value count (NA rows do not inflate `m`). Unlike MarketAdjustedModel (A8), this model DOES estimate one parameter, so the constant-mean correction is the correct form here. |
| p-value sidedness | Two-sided by default | -- |

**Audit result: verified, no change.** AR, sigma, df, and FEC match the
constant-mean-return model. Pinned in `test_golden_values.R` against a
closed-form subtraction against the fixed estimation mean at absolute tolerance
`1e-10`.

*Citation:* Brown, S. J., & Warner, J. B. (1985). Using daily stock returns:
The case of event studies. *Journal of Financial Economics*, 14(1), 3--31.
MacKinlay (1997).

### Fama-French Three-Factor Model

Regresses firm excess returns on the three Fama-French factors -- market excess
(`market_excess`), size (`smb`), and value (`hml`) -- by OLS over the estimation
window; the abnormal return is the residual in the event window. All factor
models share the `LinearFactorModel` OLS core.

| Convention | Choice | Note |
|---|---|---|
| Return type | Excess returns (`excess_return` = firm minus risk-free) | Regressed on `market_excess + smb + hml`. |
| Estimation | OLS via `lm(excess_return ~ market_excess + smb + hml)` | `k = 3` factors plus intercept. |
| Abnormal return | `AR = excess_return - predicted` | `predicted` from the fitted OLS coefficients. |
| Residual sigma | Residual standard error from `summary(lm)` | -- |
| Degrees of freedom | `m - (k + 1) = m - 4` | Intercept plus three factor slopes. |
| Minimum observations | `n_valid >= n_params + 1 = 5` | `n_params = length(all.vars(formula)) = 4` (response + 3 factors, i.e. intercept + 3 slopes); fewer valid observations route through the degenerate-input contract (2026-09-24 item A6). |
| Short-window advisory | One plain warning when `4 < n_valid < 30` | Same advisory convention as the Market Model. |
| `statistics$n_params` | `4` | Consumed by `PatellZTest`'s `Q_i` adjustment. |
| Forecast-error correction | `sigma * sqrt(1 + h_t)`, `h_t = x_t' (X'X)^{-1} x_t` | Full multi-factor hat-value (leverage) form using the estimation design matrix. |
| p-value sidedness | Two-sided by default | -- |

**Audit result: verified, no change (guard strengthened 2026-09-24, item A6).**
The OLS fit and hat-value FEC match
Fama-French (1993). Pinned in `test_golden_values.R` against an independent
`lm()`/`predict()` derivation at absolute tolerance `1e-10`.

*Citation:* Fama, E. F., & French, K. R. (1993). Common risk factors in the
returns on stocks and bonds. *Journal of Financial Economics*, 33(1), 3--56.

### Fama-French Five-Factor Model

Extends the three-factor model with profitability (`rmw`) and investment (`cma`)
factors -- Fama & French (2015). Same OLS core, five factors.

| Convention | Choice | Note |
|---|---|---|
| Return type | Excess returns (`excess_return`) | Regressed on `market_excess + smb + hml + rmw + cma`. |
| Estimation | OLS on five factors plus intercept | `k = 5`. |
| Abnormal return | `AR = excess_return - predicted` | -- |
| Residual sigma | Residual standard error | -- |
| Degrees of freedom | `m - (k + 1) = m - 6` | -- |
| Minimum observations | `n_valid >= n_params + 1 = 7` | `n_params = 6` (intercept + 5 factors); 2026-09-24 item A6. |
| `statistics$n_params` | `6` | Consumed by `PatellZTest`'s `Q_i` adjustment. |
| Forecast-error correction | `sigma * sqrt(1 + h_t)` | Multi-factor hat-value form. |
| p-value sidedness | Two-sided by default | -- |

**Audit result: verified, no change (guard strengthened 2026-09-24, item A6).**
Pinned in `test_golden_values.R` against
an independent `lm()`/`predict()` derivation at absolute tolerance `1e-10`.

*Citation:* Fama, E. F., & French, K. R. (2015). A five-factor asset pricing
model. *Journal of Financial Economics*, 116(1), 1--22.

### Carhart Four-Factor Model

Extends the three-factor model with a momentum factor (`mom`) -- Carhart (1997).
Same OLS core, four factors.

| Convention | Choice | Note |
|---|---|---|
| Return type | Excess returns (`excess_return`) | Regressed on `market_excess + smb + hml + mom`. |
| Estimation | OLS on four factors plus intercept | `k = 4`. |
| Abnormal return | `AR = excess_return - predicted` | -- |
| Residual sigma | Residual standard error | -- |
| Degrees of freedom | `m - (k + 1) = m - 5` | -- |
| Minimum observations | `n_valid >= n_params + 1 = 6` | `n_params = 5` (intercept + 4 factors); 2026-09-24 item A6. |
| `statistics$n_params` | `5` | Consumed by `PatellZTest`'s `Q_i` adjustment. |
| Forecast-error correction | `sigma * sqrt(1 + h_t)` | Multi-factor hat-value form. |
| p-value sidedness | Two-sided by default | -- |

**Audit result: verified, no change (guard strengthened 2026-09-24, item A6).**
Pinned in `test_golden_values.R` against
an independent `lm()`/`predict()` derivation at absolute tolerance `1e-10`.

*Citation:* Carhart, M. M. (1997). On persistence in mutual fund performance.
*Journal of Finance*, 52(1), 57--82.

### Custom Model

The Custom Model inherits the Market Model's OLS estimation but defines the
abnormal return as `firm_returns - predict(user_model)`, plus an optional
event-date adjustment `loss_market_cap` added only on the event date. There is
no single published closed-form here -- the convention is deliberately
"whatever the supplied model predicts."

| Convention | Choice | Note |
|---|---|---|
| Return type | Whatever the supplied model consumes | User-defined. |
| Estimation | Inherited from the Market Model (OLS) | The fitted object is user-supplied via the pipeline. |
| Abnormal return | `AR = R - predict(model)` (+`loss_market_cap` on the event date) | The event-date term lets a user inject a known shock. |
| Residual sigma / df | Inherited from the underlying OLS fit (`df = m - 2`) | -- |
| p-value sidedness | Two-sided by default | -- |

**Audit result: documented convention (no closed-form pin possible).** With
`loss_market_cap = 0` the Custom Model reduces exactly to the Market Model; that
identity is what `test_golden_values.R` pins (absolute `1e-10`). A general
user-supplied prediction has no published reference value to pin against.

*Citation:* Convention is user-defined; the reduction case cites MacKinlay
(1997).

### BHAR Model (Barber-Lyon 1997)

The Buy-and-Hold Abnormal Return (BHAR) model is for long-horizon studies where
compounding matters. Instead of summing daily abnormal returns it compounds firm
and benchmark returns separately over the event window and takes the difference.

| Convention | Choice | Note |
|---|---|---|
| Return type | Arithmetic returns compounded multiplicatively | `cumprod(1 + R)`. |
| Estimation | None estimated for the AR itself; the benchmark is the index | sigma is estimated from the estimation window for the t-test. |
| Abnormal return | `BHAR = cumprod(1 + firm) - cumprod(1 + index)` | Barber & Lyon (1997) buy-and-hold definition; compounded within each event window separately. |
| Residual sigma | `sd(firm - index)` over the estimation window | Sample SD of the one-period market-adjusted residual series. |
| Degrees of freedom | `m - 1` | Finite-pair count via `.finite_residual_df()` (NA rows do not inflate `m`). |
| BHAR standard error | `sigma * sqrt(n)` | `n` is the day index within the event window -- the random-walk sqrt-of-horizon scaling of Lyon, Barber & Tsai (1999). |
| BHAR t-statistic | `BHAR / (sigma * sqrt(n))` | Returns NA when `sigma == 0` (degenerate guard), never Inf/NaN. |
| p-value sidedness | Two-sided by default | -- |

**Audit result: verified, no change.** The compounded BHAR, the `sigma * sqrt(n)`
scaling, and the constant-mean forecast-error correction match Barber-Lyon
(1997) / Lyon-Barber-Tsai (1999). Pinned in `test_golden_values.R` against an
independent closed-form compounding at absolute tolerance `1e-10`.

*Citation:* Barber, B. M., & Lyon, J. D. (1997). Detecting long-run abnormal
stock returns: The empirical power and specification of test statistics.
*Journal of Financial Economics*, 43(3), 341--372. Lyon, J. D., Barber, B. M., &
Tsai, C.-L. (1999). Improved methods for tests of long-run abnormal stock
returns. *Journal of Finance*, 54(1), 165--201.

### Volume Model

The Volume model runs an event study on trading volume rather than returns.
Expected volume is the estimation-window mean of (optionally log-transformed)
volume, and abnormal volume is the deviation from that mean. It writes abnormal
volume into the `abnormal_returns` column so the standard test statistics apply.

| Convention | Choice | Note |
|---|---|---|
| Measure | `log(firm_volume + 1)` by default (`log_transform = TRUE`) | The `+1` avoids `log(0)`; set `log_transform = FALSE` for raw volume. |
| Estimation | `mean(log(volume + 1))` over the estimation window | One parameter (the mean). |
| Abnormal measure | `AR = log(volume + 1) - expected` | Deviation from the estimation-window log-mean. |
| Residual sigma | `sd(residuals)` over the estimation window | -- |
| Degrees of freedom | `m - 1` | Finite value count (NA/non-finite rows excluded). |
| p-value sidedness | Two-sided by default | -- |

**Audit result: verified, no change.** The log-mean expected volume, the
deviation-from-mean abnormal volume, and `df = m - 1` match the standard
abnormal-trading-volume convention. Pinned in `test_golden_values.R` against a
direct log-mean subtraction at absolute tolerance `1e-10`.

*Citation:* Convention follows the abnormal-trading-volume literature (e.g.
Campbell & Wasley 1996, log relative volume); the mean-adjusted form is the
volume analogue of the constant-mean-return model of Brown & Warner (1985).

### Volatility Model

The Volatility model runs an event study on return variance. Expected variance
is the estimation-window sample variance, and abnormal volatility is the ratio of
each event-window squared return to that variance, minus one (so zero means
"as expected"). It writes the measure into `abnormal_returns` for compatibility.

| Convention | Choice | Note |
|---|---|---|
| Measure | Squared return relative to estimation variance | -- |
| Estimation | `est_var = var(estimation firm_returns)` | Sample variance, `df = m - 1`. |
| Abnormal measure | `AR = firm_returns^2 / est_var - 1` | Zero when the squared return equals the expected variance. |
| Residual sigma | `sd(residuals)` of the ratio series over the estimation window | -- |
| Degrees of freedom | `m - 1` | Finite value count. |
| p-value sidedness | Two-sided by default | -- |

**Audit result: verified, no change.** The ratio-to-expected-variance abnormal
measure and `df = m - 1` match the abnormal-volatility convention. Pinned in
`test_golden_values.R` against a direct ratio computation at absolute tolerance
`1e-10`.

*Citation:* The squared-return / expected-variance ratio follows the
event-induced-variance literature (Beaver 1968; Patell & Wolfson 1979), adapted
to the constant-variance benchmark of the estimation window.

### Rolling-Window Model

The Rolling-Window model captures parameter instability by estimating the market
model over a rolling OLS window and using the LAST window's parameters to predict
the event window. This yields a time-varying beta without a full GARCH model.

| Convention | Choice | Note |
|---|---|---|
| Estimation | Rolling OLS `firm ~ index` over windows of `window_size` (default 60), requiring `min_obs` (default 30) finite pairs | Last window's `alpha`, `beta` used for prediction. |
| Effective window | `ws = min(window_size, n_est)` | When `n_est <= window_size` there is a single window equal to the full sample; `n_est` is the estimation window's ROW span (not a valid-pair count). |
| Within-window pairs | Each `ws`-row window restricts `x_bar`/`y_bar`/`ss_xx`/`ss_xy`/residuals/sigma to COMPLETE PAIRS (both firm and index finite) | 2026-09-24 re-evaluation item A1: previously `x_bar`/`ss_xx` were computed from ALL index values in the window, including rows where the firm return was NA -- if those NA-firm rows are not missing-at-random in the index (e.g. an index outlier co-occurring with a missing firm observation), this silently biases beta away from what `lm()` on the complete-pair subset would produce. |
| Abnormal return | `AR = firm - (alpha_last + beta_last * index)` | Uses the most recent (last) rolling parameters. |
| Residual sigma | Last-window residual SE, `denom = max(n_complete_last - 2, 1)` | `n_complete_last` = complete pairs in the LAST window (not a freshly recomputed valid-count that could disagree with the actual last-window row span). |
| Degrees of freedom | `max(n_complete_last - 2, 1)` | `fit()` and `calculate_statistics()` share the SAME complete-pair count (stored in `private$.rolling_params$n_completes`), so the reported df always matches the sigma that was actually estimated. |
| `statistics$n_params` | `2` (alpha, beta) | Consumed by `PatellZTest`'s `Q_i` adjustment. |
| p-value sidedness | Two-sided by default | -- |

**Audit result: one silent-wrong defect fixed 2026-09-24 (item A1).** The
rolling OLS, last-window prediction, and `df = n_complete - 2` are internally
consistent once restricted to complete pairs. Pinned in `test_golden_values.R`
on a 30-observation fixture where `ws = min(60, 30) = 30` collapses to a single
closed-form OLS, and in `test_reeval_statistical_fixes.R` on a 150-day window
with firm-return NAs co-occurring with index outliers every 3rd day (beta,
alpha, sigma, and df all match `lm()` on the complete-pair subset within
`1e-10`).

*Citation:* Rolling-window / time-varying market-model estimation is standard
practice for capturing beta instability (e.g. Fama & MacBeth 1973 rolling
regressions); the underlying market model is MacKinlay (1997).

### GARCH Model

The GARCH model fits a GARCH(1,1) mean-variance model to firm returns with the
index return as an external regressor in the mean equation (via the optional
**rugarch** package). The abnormal return is the mean-equation residual.

| Convention | Choice | Note |
|---|---|---|
| Backend | `rugarch::ugarchfit`, `sGARCH`, `garchOrder` default `c(1, 1)`, normal errors | Optional dependency, guarded by `requireNamespace`. |
| Mean equation | `firm = mu + mxreg1 * index + e_t` | Index return as an external regressor. |
| Abnormal return | `AR = firm - (mu + mxreg1 * index)` | The fitted mean-equation residual. |
| Residual sigma | Mean conditional sigma from the GARCH fit | Falls back to `sd(residuals)` if non-finite. |
| Degrees of freedom | `max(length(cond_sigma) - length(coefs), 1)` | -- |
| p-value sidedness | Two-sided by default | -- |

**Audit result: verified, no change (identity-pinned).** Because rugarch is an
optional dependency and its fitted coefficients vary across package and solver
versions, pinning a raw fitted constant would be fragile (threat T-26-02). The
documented golden-value choice is therefore to pin the STABLE ALGEBRAIC IDENTITY
`AR = firm - (mu + mxreg1 * index)`, which holds for any fitted coefficients, in
`test_golden_values.R` guarded by `skip_if_not_installed("rugarch")` -- locking
the AR formula without a version-fragile number.

*Citation:* Engle, R. F. (1982). Autoregressive conditional heteroscedasticity
with estimates of the variance of United Kingdom inflation. *Econometrica*,
50(4), 987--1007. Bollerslev, T. (1986). Generalized autoregressive conditional
heteroskedasticity. *Journal of Econometrics*, 31(3), 307--327.

### DCC-GARCH Model

The DCC-GARCH model (via the optional **rmgarch** package) fits a bivariate
Dynamic Conditional Correlation GARCH to firm and market returns, yielding a
time-varying beta `beta_t = Cov(firm, mkt)_t / Var(mkt)_t`. The last conditional
beta predicts the event window.

| Convention | Choice | Note |
|---|---|---|
| Backend | `rmgarch::dccfit`, univariate `sGARCH(1,1)`, DCC order default `c(1, 1)`, multivariate normal | Optional dependency, guarded by `requireNamespace`. |
| Time-varying beta | `beta_t = H[firm, mkt]_t / H[mkt, mkt]_t` | From the conditional covariance array; non-finite / zero-variance entries yield NA and fall back to the last valid beta. |
| Intercept | `alpha_last = mean(firm - beta_last * index)` over the estimation window | Mean-equation intercept implied by the last beta. |
| Abnormal return | `AR = firm - (alpha_last + beta_last * index)` | Uses the last conditional beta. |
| Residual sigma | `mean(conditional firm sigma)` | -- |
| Degrees of freedom | `max(n_t - 4, 1)` | -- |
| p-value sidedness | Two-sided by default | -- |

**Audit result: verified, no change (identity-pinned).** As with GARCH, rmgarch
is optional and non-deterministic across versions, so the documented golden-value
choice is to pin the stable identity `AR = firm - (alpha_last + beta_last *
index)` in `test_golden_values.R` guarded by `skip_if_not_installed("rmgarch")` /
`("rugarch")`, rather than a fragile fitted constant.

*Citation:* Bollerslev, T. (1990). Modelling the coherence in short-run nominal
exchange rates: A multivariate generalized ARCH model. *Review of Economics and
Statistics*, 72(3), 498--505. Engle, R. F. (2002). Dynamic conditional
correlation. *Journal of Business & Economic Statistics*, 20(3), 339--350.

## Test statistics

### Multi-event all-NA exclusion (2026-09-24 re-evaluation, item A2)

Every CAR-based multi-event statistic (`CSectTTest`, `PatellZTest`, `SignTest`,
`GeneralizedSignTest`, `BMPTest`, `KolariPynnonenTest`) shares one exclusion
rule, implemented once in `R/contract.R` (`.exclude_all_na_events()`) and
applied centrally by `calculate_statistics()` before any configured statistic
runs (so N statistics in one group produce AT MOST ONE warning, not N), and
defensively again at the top of each `compute()` for direct/standalone calls:
an event with **zero finite event-window abnormal returns** is EXCLUDED from
the group entirely -- not NA-propagated and coalesced to 0 -- because
cumulating a fully-degenerate event's abnormal returns as 0 would silently
understate the true cross-sectional dispersion (an entirely different failure
mode from a normal 0 abnormal return). Excluded events whose model already
emitted the one contract warning guaranteed by CONTRACT-04 (an unfitted
`ModelBase` instance) are dropped SILENTLY; any other excluded event is
reported once, naming the `event_id`(s) (strict mode errors instead). This
exclusion leaves the pre-existing STATS-03 convention for a PARTIAL gap
unchanged: an event with at least one finite event-window abnormal return, but
a missing value on some individual day, still contributes 0 to THAT EVENT's
own CAR on the missing day(s) via `coalesce(abnormal_returns, 0)`.

### AR / CAR t-tests (ARTTest, CARTTest)

Documented above with the Market Model, since the tracer anchors them together.
The AR t-test divides each event-day abnormal return by the model sigma; the CAR
t-test divides the cumulative abnormal return by `sqrt(L) * sigma`. Both are
distributed as Student's t with the model's residual degrees of freedom and are
two-sided by default. Every return model also computes
`forecast_error_corrected_sigma_car` (the FEC's per-event covariance/hat-value
term), but `ARTTest`/`CARTTest` INTENTIONALLY do not use it -- they keep the
simpler constant-sigma approximation (item A11, documented convention, no
formula change); `forecast_error_corrected_sigma_car` exists for consumers that
need the full prediction-error covariance structure. *Citation:* MacKinlay
(1997).

### BHAR t-test (BHARTTest)

The single-event buy-and-hold abnormal-return t-test. Documented above with the
BHAR Model, since the model and its t-test share the same compounded definition
and `sigma * sqrt(n)` scaling. In its general form the statistic is
`bhar_t = BHAR / (sigma * sqrt(n))`, where `BHAR = cumprod(1 + firm) -
cumprod(1 + index)` over the event window, `sigma = sd(firm - index)` over the
estimation window, and `n` is the day index within the event window. It returns
NA (never Inf/NaN) when `sigma == 0`. Two-sided by default. *Citation:* Barber &
Lyon (1997); Lyon, Barber & Tsai (1999).

### Cross-Sectional t-test (CSectTTest)

The cross-sectional t-test evaluates, at each relative event day, whether the
average abnormal return across events is zero, using the cross-event dispersion
of abnormal returns as the standard error. It is the workhorse parametric test
of Brown & Warner (1985).

| Convention | Choice | Note |
|---|---|---|
| Point statistic | `aar_t = sqrt(N) * aar / sd(AR)` | `aar` = mean AR across events; `sd(AR)` = cross-event sample SD (denominator `N - 1`). |
| Cumulative statistic | `caar_t = sqrt(N) * caar / sd(CAR)` | `caar` = mean of per-event cumulative ARs; `sd(CAR)` = cross-event sample SD of the per-event CARs. |
| Sample size | `N` = number of valid (non-NA) events at that day | NA abnormal returns are excluded from both the mean and the SD. |
| Distribution | Student's t with `N - 1` degrees of freedom | -- |
| Standardization | None -- raw abnormal returns | Unlike Patell/BMP, ARs are not divided by a per-event sigma; the cross-event SD is the only scale. |
| Positive count | `n_pos = sum(AR > 0)` -- STRICT inequality | 2026-09-24 item A9: an AR exactly 0 counts as NON-positive (`n_neg = sum(AR <= 0)`), matching the `SignTest` convention. |
| All-NA event exclusion | An event with ZERO finite event-window ARs is excluded from every multi-event statistic (once per group, one contract warning/error) | 2026-09-24 item A2; see "Multi-event all-NA exclusion" below. A PARTIAL gap (one missing AR mid-window in an otherwise-valid event) still contributes 0 to that event's own CAR (STATS-03, unchanged). |
| p-value sidedness | Two-sided by default | -- |

**Audit result: verified against Brown & Warner (1985); two silent-wrong
defects fixed 2026-09-24 (items A2, A9).** The core
`sqrt(n_valid_events) * aar / sd(AR)` / `caar / sd(CAR)` construction matches
Brown & Warner. The 2026-09-24 re-evaluation found `n_pos` used `>= 0` (so a
zero AR silently counted as "positive") and that a fully-degenerate event (all
event-window ARs NA, e.g. an unfitted model) was not excluded from the
cumulative CAR-based statistics -- its `coalesce(AR, 0)` silently understated
cross-event dispersion instead of being excluded. Both are fixed; pinned in
`test_golden_values.R` and `test_reeval_statistical_fixes.R` against an
independent closed-form cross-sectional computation at absolute tolerance
`1e-10`/`1e-12`.

*Citation:* Brown, S. J., & Warner, J. B. (1985). Using daily stock returns: The
case of event studies. *Journal of Financial Economics*, 14(1), 3--31.

### Patell Z-test (PatellZTest)

The Patell (standardized-residual) test standardizes each event's abnormal
return by its own forecast-error-corrected standard deviation before aggregating,
which accounts for the fact that event-window prediction errors have larger
variance than estimation-window residuals. The aggregate is approximately
standard normal.

| Convention | Choice | Note |
|---|---|---|
| Standardization | `SAR = AR / forecast-error-corrected sigma` | Uses the per-event, per-day FEC sigma, not the raw model sigma (this is the defining feature vs. BMP). |
| Patell denominator | `Q_total = sqrt(sum_i Q_i)`, `Q_i = (m - k) / (m - k - 2)` | `m` = number of FINITE estimation-window abnormal returns for event `i` (2026-09-24 item A6: NOT the raw row count -- NA rows in the estimation window must not inflate `m`). `k` = event `i`'s model's `statistics$n_params` (MarketModel 2, FF3 4, Carhart4 5, FF5 6, CPMA 1, MarketAdjusted 0), read explicitly rather than derived as `length(resid) - df` (the pre-2026-09-24 derivation silently produced the WRONG `k` whenever `df` already reflected an unrelated correction, e.g. HAC). `Q_i` inflates the variance of the standardized sum for the estimation-error in each sigma. |
| Q_i fallback | `Q_i = 1` when `m <= k + 2` | Insufficient observations for the adjustment; avoids a negative/undefined denominator. |
| AAR statistic | `aar_z = sum_i SAR_i / Q_total` | Sum (not mean) of standardized abnormal returns over the Patell denominator. |
| CAAR statistic | `caar_z = (1 / sqrt(N)) * sum_i (cumsum(SAR_i) / sqrt(n * Q_i))` | Per-event cumulative SAR scaled by `sqrt(n * Q_i)` where `n` is the day count, then averaged. |
| Positive count | `n_pos = sum(AR > 0)` -- STRICT inequality | 2026-09-24 item A9: matches `SignTest`/`CSectTTest`. |
| Degenerate-event handling | Events with all-NA FEC sigma are excluded from `Q_total` (WR-01); events with ZERO finite event-window ARs are excluded from the WHOLE statistic (2026-09-24 item A2) | Prevents a `Q_i = 1` fallback (or a coalesced-to-0 CAR contribution) from a degenerate event distorting valid-event z-scores. |
| N >= 2 guard | `aar_z` / `caar_z` are NA when `n_valid_events <= 1` (STATS-04) | A single event gives a statistically invalid z; NA is returned rather than a finite-but-wrong number. |
| Distribution | Approximately `N(0, 1)` | -- |
| p-value sidedness | Two-sided by default | -- |

**Audit result: verified against Patell (1976); three silent-wrong defects
fixed 2026-09-24 (items A2, A6, A9).** The standardization by the
forecast-error-corrected sigma and the `Q_total = sqrt(sum Q_i)` aggregation
match Patell. The 2026-09-24 re-evaluation found: (1) `k` was derived as
`length(resid) - df` instead of read from each model's own `n_params`, silently
using the WRONG `k` whenever `df` reflected an unrelated adjustment; (2) `m`
used the raw estimation-window row count instead of the finite-observation
count; (3) `n_pos` used `>= 0`; and (4) an all-NA event was not excluded from
the statistic. All four are fixed; re-pinned in `test_golden_values.R` and
locked in `test_reeval_statistical_fixes.R` against an independent closed-form
Patell computation at absolute tolerance `1e-10`, plus a guard test confirming
NA for a single-event group.

*Citation:* Patell, J. M. (1976). Corporate forecasts of earnings per share and
stock price behavior: Empirical tests. *Journal of Accounting Research*, 14(2),
246--276.

### BMP test (BMPTest)

The Boehmer, Musumeci & Poulsen (1991) standardized cross-sectional test
combines Patell-style standardization with a cross-sectional variance estimate,
making it robust to event-induced increases in return variance (which the plain
Patell test does not accommodate). It standardizes abnormal returns by the model
sigma, then applies a cross-sectional t-test to the standardized values.

| Convention | Choice | Note |
|---|---|---|
| Standardization | `SAR = AR / model sigma` | Uses the per-event MODEL sigma (contrast: Patell uses the forecast-error-corrected sigma). |
| Point statistic | `bmp_t = sqrt(N) * mean(SAR) / sd(SAR)` | Cross-sectional t of the standardized abnormal returns; `sd(SAR)` is the cross-event sample SD (this cross-sectional SD is what makes BMP robust to event-induced variance). |
| Cumulative statistic | `cbmp_t = sqrt(N) * mean(cumsum(SAR)) / sd(cumsum(SAR))` | Same cross-sectional t applied to per-event cumulative SARs. |
| Join key | `event_id` (one sigma per event) | A `firm_symbol` join would be many-to-many when a firm recurs across events. |
| All-NA event exclusion | An event with ZERO finite event-window ARs is excluded (2026-09-24 item A2) | See "Multi-event all-NA exclusion" below. |
| Distribution | Approximately `t_{N-1}` | -- |
| p-value sidedness | Two-sided by default | -- |

**Audit result: verified, no change (all-NA exclusion added 2026-09-24, item
A2; roxygen documentation corrected, item A11).** The standardization by MODEL
sigma (not the forecast-error-corrected sigma -- this is the defining
contrast with `PatellZTest`/`KolariPynnonenTest`) and the cross-sectional t
`sqrt(N) * mean(SAR) / sd(SAR)` match Boehmer, Musumeci & Poulsen (1991). The
2026-09-24 re-evaluation found the roxygen documentation itself INCORRECTLY
described the standardization as using the forecast-error-corrected sigma; the
formula was always right, only the docstring was fixed (item A11, docs-only).
Pinned in `test_golden_values.R` against an independent closed-form BMP
computation on a fixed three-event fixture at absolute tolerance `1e-10`.

*Citation:* Boehmer, E., Musumeci, J., & Poulsen, A. B. (1991). Event-study
methodology under conditions of event-induced variance. *Journal of Financial
Economics*, 30(2), 253--272.

### Sign test (SignTest)

The sign test is a nonparametric test of whether positive abnormal returns occur
more often than the 0.5 that would be expected by chance. Under the null the
count of positive abnormal returns is Binomial(N, 0.5), approximated by a normal.

| Convention | Choice | Note |
|---|---|---|
| Positive count | `n_pos = sum(AR > 0)` -- STRICT inequality | A zero abnormal return counts as NON-positive (`n_neg = sum(AR <= 0)`). This is the `> 0` (not `>= 0`) convention; it is the conservative choice, treating an exact zero as evidence against a positive effect. |
| Point statistic | `sign_z = (n_pos - 0.5 N) / (0.5 sqrt(N))` | Binomial-normal approximation with `p = 0.5`; `N` = valid events at that day. |
| Cumulative statistic | `csign_z = (n_pos_car - 0.5 N) / (0.5 sqrt(N))` | Same form applied to the count of positive per-event CARs. |
| N >= 2 guard | `sign_z` / `csign_z` are NA when `n_valid_events < 2` (STATS-04, WR-02) | A single observation gives a finite-but-meaningless z; NA is returned instead. |
| All-NA event exclusion | An event with ZERO finite event-window ARs is excluded (2026-09-24 item A2) | See "Multi-event all-NA exclusion" below. |
| Distribution | Approximately `N(0, 1)` | -- |
| p-value sidedness | Two-sided by default | -- |

**Audit result: verified, no change (all-NA exclusion added 2026-09-24, item
A2).** The `(n_pos - 0.5 N)/(0.5 sqrt(N))`
statistic, the strict `> 0` positive-count convention, and the `N >= 2` guard
match the standard sign test. Pinned in `test_golden_values.R` against a direct
binomial-normal computation on the fixed three-event fixture at absolute
tolerance `1e-10`, plus a guard test confirming NA for a single-event group.

*Citation:* The binomial sign test for event studies is standard; see Brown &
Warner (1980, 1985) and Cowan (1992) for its use alongside the generalized form.

### Generalized Sign test (GeneralizedSignTest)

The generalized sign test (Cowan 1992) relaxes the ordinary sign test's implicit
`p = 0.5` by estimating the expected fraction of positive abnormal returns from
the estimation window, which corrects for asymmetry (skewness) in the return
distribution.

| Convention | Choice | Note |
|---|---|---|
| Expected proportion | `p_hat = mean over EVENTS of mean(estimation AR > 0)` | Estimated PER EVENT from the estimation window, then averaged (2026-09-24 item A4: previously grouped by `firm_symbol`, which silently pooled a recurring firm's estimation windows ACROSS its distinct events -- each event is fit independently, so this mixed unrelated data). Replaces the ordinary test's fixed 0.5. |
| Positive count | `n_pos = sum(AR > 0)` | Same strict `> 0` convention as the sign test. |
| Point statistic | `gsign_z = (n_pos - N p_hat) / sqrt(N p_hat (1 - p_hat))` | Cowan (1992) generalized sign statistic. |
| Cumulative statistic | `cgsign_z = (n_pos_car - N p_hat) / sqrt(N p_hat (1 - p_hat))` | Same form on positive per-event CAR counts. |
| Degenerate guard | NA when the denominator is non-finite or zero (`p_hat` in {0, 1}) | -- |
| All-NA event exclusion | An event with ZERO finite event-window ARs is excluded (2026-09-24 item A2) | See "Multi-event all-NA exclusion" below. |
| Distribution | Approximately `N(0, 1)` | -- |
| p-value sidedness | Two-sided by default | -- |

**Audit result: one silent-wrong defect fixed 2026-09-24 (item A4; all-NA
exclusion added, item A2).** The
`(n_pos - N p_hat)/sqrt(N p_hat (1 - p_hat))` statistic and the degenerate-
denominator guard match Cowan (1992). The 2026-09-24 re-evaluation found
`p_hat` was estimated by grouping the estimation window by `firm_symbol`
rather than `event_id`: when a firm recurs across multiple events, its
estimation-window abnormal returns from DIFFERENT events were silently pooled
into one `p_hat`, instead of estimating one `p_hat` per event and averaging
those. Fixed to group by `event_id`; pinned in `test_golden_values.R` against
an independent closed-form computation on the three-event fixture (where the
balanced estimation residuals give `p_hat = 0.5`, so the generalized test
reduces to the ordinary sign test) at absolute tolerance `1e-10`, and locked
in `test_reeval_statistical_fixes.R` (identical results whether a firm recurs
across events or every event has a distinct firm).

*Citation:* Cowan, A. R. (1992). Nonparametric event study tests. *Review of
Quantitative Finance and Accounting*, 2(4), 343--358.

### Rank test (RankTest)

The Corrado (1989) rank test is a nonparametric test that ranks each firm's
abnormal returns across the combined estimation and event windows, then tests
whether the event-window ranks are unusually high or low. Ranking makes it robust
to non-normality and to event-induced variance.

| Convention | Choice | Note |
|---|---|---|
| Ranking scope | Ranks computed within each EVENT over the COMBINED estimation + event window | `ar_rank = rank(AR)`, `T` = combined observations for that event (2026-09-24 item A4: previously grouped by `firm_symbol`, silently pooling a recurring firm's ranks ACROSS its distinct events). |
| Centered rank | `K = rank / (T + 1) - 0.5` | Corrado's mean-zero centered rank; `na.last = "keep"` so missing ARs do not receive a rank. |
| Rank SD | `S_rank = sd(per-day mean centered rank across ALL combined days)` | The scale is estimated from the full combined window, not just the event window. |
| Point statistic | `rank_z = mean_rank_day / S_rank` | Per event day, the cross-firm mean centered rank divided by `S_rank`. |
| Degenerate guard | `rank_z` is NA when `S_rank` is non-finite or zero | Guarded once on the scalar `S_rank`, then the per-day vector is divided directly. |
| Distribution | Approximately `N(0, 1)` | -- |
| p-value sidedness | Two-sided by default | -- |

**Audit result: FIXED (broadcast bug), then verified.** The centered-rank
construction and the `mean_rank / S_rank` statistic match Corrado (1989). The
audit found a genuine implementation error: `rank_z` was computed with base
`ifelse(is.finite(S_rank) & S_rank > 0, mean_rank / S_rank, NA_real_)`. Because
`S_rank` is a scalar, `ifelse` returned a length-1 result (the length of its
condition), collapsing the per-day `mean_rank / S_rank` vector to its first
element, which `dplyr::mutate` then recycled across every event day -- so all
event days reported the day-0 z. No published convention makes the rank z
identical on every day, so this is a bug, not a convention. It was fixed to guard
the scalar `S_rank` with a plain `if()` and divide the vector directly; a
regression test (`test_golden_values.R`) locks the corrected per-day values
(`+1.586, -0.952, +1.269`) and asserts they are distinct, preventing the
broadcast from silently returning. The fix precedes the golden pin.

**2026-09-24 re-evaluation (item A4):** ranking/centering was also found to
group by `firm_symbol` instead of `event_id`, silently pooling a recurring
firm's ranks across its distinct events. Fixed to group by `event_id`; locked
in `test_reeval_statistical_fixes.R` (identical results whether a firm recurs
across events or every event has a distinct firm).

*Citation:* Corrado, C. J. (1989). A nonparametric test for abnormal
security-price performance in event studies. *Journal of Financial Economics*,
23(2), 385--395.

### Calendar-Time Portfolio test (CalendarTimePortfolioTest)

The calendar-time portfolio approach forms, for each relative event day, an
equal-weighted portfolio of all event firms' abnormal returns and tests whether
the portfolio's mean return differs from zero. Aggregating into a portfolio
naturally absorbs the cross-sectional dependence that arises when events cluster
in calendar time.

| Convention | Choice | Note |
|---|---|---|
| Portfolio | Equal-weighted mean abnormal return per relative day (`aar`) | -- |
| Time-series scale | `ts_sd = sd(aar)` over the ESTIMATION-window days | 2026-09-24 item A7 (Brown-Warner crude-dependence adjustment): `ts_sd` is the standard deviation of the cross-event AAR series computed OVER THE ESTIMATION WINDOW, not over the (much shorter, event-specific) event window. This makes the denominator independent of the event window and of any event-day shock magnitude. |
| Estimation-window AAR days | `n_estimation_days` = number of FINITE estimation-window AAR observations | Exposed as `attr(result, "caltime_df")`. |
| Point statistic | `caltime_t = aar / ts_sd` | Time-series t of the portfolio AAR under `H0: E[AAR] = 0`; `aar` is still the EVENT-window per-day portfolio mean. |
| Cumulative statistic | `ccaltime_t = caar / (ts_sd sqrt(L))` | `L` = day index within the event window; random-walk sqrt-of-horizon scaling of the cumulative portfolio return. |
| Degrees of freedom | `n_estimation_days - 1` | Brown-Warner time-series df, NOT a cross-sectional `N - 1`; `adjust_p_values()` and `tidy()` both read `attr(result, "caltime_df")` rather than deriving a df of their own. |
| Positive count | `n_pos = sum(AR > 0)` -- STRICT inequality | 2026-09-24 item A9: matches `SignTest`/`CSectTTest`. |
| Degenerate guard | Both t's are NA when `ts_sd` is non-finite/zero or `n_estimation_days < 2` | One contract warning/error (not a silent NA) when the estimation-window AAR series cannot support a standard deviation. |
| p-value sidedness | Two-sided by default | -- |

**Audit result: FIXED (broadcast bug; then Brown-Warner denominator
redefinition, item A7; then A9).** The equal-weight portfolio construction
matches the calendar-time portfolio approach. Three issues were found across
two audit rounds:

1. **Broadcast bug (original audit).** `caltime_t` / `ccaltime_t` used base
   `ifelse(is.finite(ts_sd) & ts_sd > 0, ...)` with the scalar `ts_sd` as the
   condition, collapsing the per-day t vectors to their first element and
   recycling it across every event day. Fixed by guarding the scalar once with
   a plain `if()` and dividing the vectors directly.
2. **Self-referential denominator (2026-09-24 re-evaluation, item A7).**
   `ts_sd` was computed from the EVENT-WINDOW portfolio AAR series itself --
   the same series `aar` that appears in the numerator -- so a larger
   event-day shock inflated BOTH the numerator and (via the window's own
   dispersion) the denominator, and the statistic was not independent of the
   event window's length or contents. This is REDEFINED (not merely a bug fix
   in the narrow sense) to the Brown-Warner (1980, 1985) crude-dependence
   adjustment: `ts_sd` is now the ESTIMATION-window AAR series's standard
   deviation, with `df = n_estimation_days - 1` exposed via
   `attr(result, "caltime_df")` and threaded through to `adjust_p_values()`
   and `tidy()`. This is a genuine convention change (not a silent-input-only
   fix) -- valid-input `caltime_t`/`ccaltime_t` values differ from the prior
   release; see NEWS.md. Locked in `test_reeval_statistical_fixes.R`: doubling
   a pure event-day shock exactly doubles `caltime_t` on that day (proving the
   denominator no longer depends on the event window).
3. **Sign convention (item A9).** `n_pos` used `>= 0`; fixed to strict `> 0`.

A regression test locks the corrected per-day broadcast values and asserts
they are distinct; the golden values were RE-PINNED for the A7 redefinition
(see `test_golden_values.R`, "re-pinned 2026-09-24: A7").

*Citation:* Brown, S. J., & Warner, J. B. (1980). Measuring security price
performance. *Journal of Financial Economics*, 8(3), 205--258. Brown, S. J., &
Warner, J. B. (1985). Using daily stock returns: The case of event studies.
*Journal of Financial Economics*, 14(1), 3--31. The calendar-time portfolio
approach is also due to Jaffe (1974) and
Mandelker (1974), popularized for long-horizon studies by Fama (1998) and Mitchell
& Stafford (2000) as a control for cross-sectional dependence.

### Kolari-Pynnonen test (KolariPynnonenTest)

The Kolari & Pynnonen (2010) test corrects the BMP standardized cross-sectional
test for cross-sectional correlation of abnormal returns. When events share
calendar time, their abnormal returns are positively correlated, which biases the
BMP variance downward and over-rejects the null; the KP adjustment scales the BMP
statistic down by a factor derived from the average pairwise correlation.

| Convention | Choice | Note |
|---|---|---|
| Base statistic | BMP cross-sectional t (`SAR = AR / model sigma`, `bmp_t = sqrt(N) mean(SAR)/sd(SAR)`) | Identical to `BMPTest`; KP scales it. |
| Usable events | An event contributes to `r_bar` only if its estimation-window SAR series has `>= 2` finite values, a finite model sigma, AND a strictly positive (non-constant) SAR standard deviation | 2026-09-24 item A3: a NON-usable event is excluded (once, with a contract warning/error naming it) -- never silently folded into the correlation matrix as a degenerate (constant or all-NA) series. |
| Average correlation | `r_bar` = average off-diagonal pairwise correlation of estimation-window SARs, over the USABLE events only | `r_bar = (sum(cor) - n_usable) / (n_usable (n_usable - 1))`; events (not firms) are the cross-sectional units, so the SAR matrix is pivoted by `event_id`. Numerically identical to the pre-2026-09-24 formula when every event is usable. |
| Adjustment factor | `kp_adj_t = sqrt((1 - r_bar) / (1 + (n_t - 1) r_bar))` PER DAY | Kolari-Pynnonen (2010) correction; `n_t` is THAT DAY's `n_valid_events` (2026-09-24 item A3: previously a single scalar `n <- aar_stats$n_valid_events[1]` -- the FIRST day's count -- was applied uniformly to every day, silently wrong whenever the per-day valid-event count varies). `kp_adj < 1` when `r_bar > 0`, shrinking the statistic toward zero. |
| Point statistic | `kp_t = bmp_t * kp_adj_t` | -- |
| Cumulative statistic | `ckp_t = cbmp_t * kp_adj_t'` | `kp_adj_t'` uses the cumulative SAR's own per-day `n_valid` (from `cum_sar`), not the AAR-level `n_valid_events`. |
| Degenerate guard | `kp_t`/`ckp_t` are NA (NEVER `kp_adj = 1`) when fewer than 2 usable events remain, an off-diagonal correlation is non-finite, or the adjustment's numerator/denominator is invalid | 2026-09-24 item A3: the pre-fix fallback silently substituted `kp_adj = 1` (i.e. the UNADJUSTED BMP value) whenever the correction was undefined -- indistinguishable from "no correlation correction needed" when it actually meant "the correction could not be computed". Exactly one warning/error per `compute()` call covers both the usable-event exclusion and this NA substitution (never two). |
| All-NA event exclusion | An event with ZERO finite event-window ARs is excluded from the whole statistic (2026-09-24 item A2) | See "Multi-event all-NA exclusion" below; distinct from the estimation-window "usable events" check above (A3) -- an event can have valid event-window ARs but a non-usable estimation-window SAR series, or vice versa. |
| Distribution | Approximately `t_{N-1}` | -- |
| p-value sidedness | Two-sided by default | -- |

**Audit result: verified against Kolari & Pynnonen (2010); two silent-wrong
defects fixed 2026-09-24 (items A2, A3).** The BMP base statistic and the
`sqrt((1 - r_bar)/(1 + (n_t - 1) r_bar))` adjustment formula match
Kolari-Pynnonen. The 2026-09-24 re-evaluation found: (1) a degenerate
(all-NA or constant) event's estimation-window SAR series was silently folded
into the `r_bar` correlation matrix instead of being excluded, and when the
adjustment was undefined the code fell back to `kp_adj = 1` -- silently
reporting the UNADJUSTED BMP value as if it were KP-corrected; and (2) the
per-day window-size term used only the FIRST day's `n_valid_events`, applied
uniformly to every day, instead of that day's own count. Both are fixed;
re-pinned in `test_golden_values.R` against the package pipeline's own output
on the three-event fixture (where the events' estimation SARs are highly
correlated by construction, `r_bar ~ 0.983`, so `kp_adj ~ 0.076` shrinks the
BMP t sharply -- unaffected here since all three events remain usable and
`n_valid_events` is constant across days in this fixture), and locked in
`test_reeval_statistical_fixes.R` against a 6-event fixture with one
constant-estimation-window (non-usable) event and a per-day varying
`n_valid_events`, at absolute tolerance `1e-8`.

*Citation:* Kolari, J. W., & Pynnonen, S. (2010). Event study testing with
cross-sectional correlation of abnormal returns. *Review of Financial Studies*,
23(11), 3996--4025.

### PermutationTest (unwired stub)

`PermutationTest` is exported but its `compute()` body is empty (returns `NULL`)
and it is not wired into any statistics set or parameter set -- it is not on any
pipeline path. It is recorded here for completeness; a decision to implement or
remove it is deferred (see the audit log).

### TestStatisticBase `confidence_type` (2026-09-24 re-evaluation, item A10)

Every `TestStatisticBase` subclass accepts `confidence_level` and
`confidence_type` at construction, but `confidence_type` was never read by any
`compute()` method -- every p-value computed anywhere in EventStudy is
two-sided. `initialize()` now validates `confidence_type` with
`match.arg(c("two-sided", "less", "greater"))` (an invalid value errors) and
emits ONE warning at construction when a non-default value ("less" or
"greater") is supplied, stating that it is currently ignored. The value is
still stored (for forward compatibility) but has no effect; one-sided p-values
are NOT implemented by this fix. This is a documented convention (no p-value
formula changed) with an added validation/warning surface.

### Cross-sectional regression (2026-09-24 re-evaluation, item A5)

`cross_sectional_regression()` regresses per-event CARs on user-supplied firm
characteristics. Prior to 2026-09-24 it silently tolerated three sample-
integrity hazards:

| Hazard | Behavior now |
|---|---|
| Duplicated `event_id` in the `data` argument | Errors (`eventstudy_error_bad_argument`) naming the duplicated id(s) -- a duplicate would otherwise inflate the join many-to-many. |
| `car_window` extending outside the event window of any event | Errors (`eventstudy_error_bad_argument`) naming the window, the available range, and the offending event(s), instead of silently summing over rows that do not exist for that event. |
| A task event with no matching row in `data` | One warning listing the count/ids, then dropped (unchanged: an empty match set still errors). |
| A missing abnormal return inside `car_window` for an otherwise-matched event | That event's CAR is `NA` (`.extract_cars()` no longer uses `na.rm = TRUE`, so ANY missing AR in the window makes the whole-window CAR `NA`, not the sum of the finite subset) and it is excluded from the regression with one warning listing the event(s). `car_by_group()`, `car_quantiles()`, and `plot_car_distribution()` inherit the same NA-CAR semantics from the shared `.extract_cars()` helper WITHOUT an additional warning (their own `na.rm = TRUE` summary statistics already tolerate NA rows silently, as documented). |

This is a sample-integrity fix (T-sfl-02 in the threat register), not a
formula change: the OLS regression itself, robust-SE handling, and CAR
extraction for a fully-matched, non-degenerate window are unchanged.

### Wild bootstrap inference (2026-09-24 re-evaluation, item A12)

`bootstrap_test()` computes bootstrap p-values for AAR/CAAR test statistics.
Two silent-wrong defects were found and fixed:

- **A12a: observed-vs-draw construction mismatch.** `observed_caar` /
  `caar_t` were previously built as `cumsum()` of the cross-event DAILY mean
  AR (`na.rm = TRUE`), while each bootstrap draw built its CAAR as the MEAN
  over events of each event's own `cumsum(coalesce(AR, 0))`. The two
  constructions coincide only when every event has a fully-observed event
  window; with a partial gap in ONE event (but not all), they diverge, and
  only the per-event-CAR-mean construction is the correct STATS-03-consistent
  comparison for what the bootstrap draws compute. Fixed so `observed_caar`
  and `caar_t` use the SAME per-event `coalesce`-to-0 CAR construction as the
  draws; on NA-free input this is numerically identical to the old
  `cumsum(aar)` form.
- **A12b: NA draws counted as "did not exceed".** A bootstrap draw whose
  cross-sectional statistic is `NA` (e.g. a Rademacher sign-flip combination
  that zeroes the cross-sectional SD) was previously counted as "did not
  exceed the observed statistic" while still inflating the `n_boot + 1`
  denominator -- silently biasing the p-value toward significance (too small)
  whenever NA draws are common. Fixed to track the number of VALID (finite)
  draws per day and use `p = (exceed + 1) / (n_valid_draws + 1)`, with `NA`
  when the observed statistic is `NA` or no draw produced a finite statistic.
- **A12c: bootstrap clustering unit (REVISED, no change).** The bootstrap
  weights are clustered by `firm_symbol` (one weight per unique firm per
  draw, shared across that firm's events) -- a deliberate prior fix; firm-level
  clusters are the coarser, correlation-robust choice when a firm recurs
  across multiple events. The 2026-09-24 re-evaluation confirmed this is
  intentional and made NO change; it is locked with a hand-replication test
  (`test_bootstrap.R`).

## Audit log

| Component | Result | Citation |
|---|---|---|
| Market Model + AR/CAR t | **FIXED 2026-09-24** (A6: insufficient-obs guard now `n_params`-based; FEC restricted to complete pairs) | MacKinlay (1997) |
| Market Adjusted Model | **FIXED 2026-09-24** (A8: FEC == sigma exactly, correction factor 1) | MacKinlay (1997) |
| Comparison Period Mean Adjusted Model | verified, no change | Brown-Warner (1985); MacKinlay (1997) |
| Fama-French Three-Factor Model | **FIXED 2026-09-24** (A6: insufficient-obs guard strengthened) | Fama-French (1993) |
| Fama-French Five-Factor Model | **FIXED 2026-09-24** (A6: insufficient-obs guard strengthened) | Fama-French (2015) |
| Carhart Four-Factor Model | **FIXED 2026-09-24** (A6: insufficient-obs guard strengthened) | Carhart (1997) |
| Custom Model | documented convention (user-supplied prediction; reduces to Market Model when `loss_market_cap = 0`) | MacKinlay (1997) |
| BHAR Model + BHARTTest | verified, no change | Barber-Lyon (1997); Lyon-Barber-Tsai (1999) |
| Volume Model | verified, no change | Campbell-Wasley (1996); Brown-Warner (1985) |
| Volatility Model | verified, no change | Beaver (1968); Patell-Wolfson (1979) |
| Rolling-Window Model | **FIXED 2026-09-24** (A1: complete-pair restriction inside each rolling window) | Fama-MacBeth (1973); MacKinlay (1997) |
| GARCH Model | verified, no change (identity-pinned, skip-guarded on rugarch) | Engle (1982); Bollerslev (1986) |
| DCC-GARCH Model | verified, no change (identity-pinned, skip-guarded on rmgarch) | Bollerslev (1990); Engle (2002) |
| AR / CAR t-tests (ARTTest, CARTTest) | verified, no change (documented: `forecast_error_corrected_sigma_car` intentionally unused, A11) | MacKinlay (1997) |
| BHAR t-test (BHARTTest) | verified, no change | Barber-Lyon (1997); Lyon-Barber-Tsai (1999) |
| Multi-event all-NA exclusion (shared mechanism) | **FIXED 2026-09-24** (A2: all-NA events excluded from every CAR-based multi-event statistic, once per group) | -- |
| Cross-Sectional t-test (CSectTTest) | **FIXED 2026-09-24** (A2 exclusion; A9: strict `n_pos > 0`) | Brown-Warner (1985) |
| Patell Z-test (PatellZTest) | **FIXED 2026-09-24** (A2 exclusion; A6: `m`/`k` redefinition; A9: strict `n_pos > 0`) | Patell (1976) |
| BMP test (BMPTest) | **FIXED 2026-09-24** (A2 exclusion; A11: roxygen corrected, docs-only) | Boehmer-Musumeci-Poulsen (1991) |
| Sign test (SignTest) | **FIXED 2026-09-24** (A2 exclusion) | Brown-Warner (1985); Cowan (1992) |
| Generalized Sign test (GeneralizedSignTest) | **FIXED 2026-09-24** (A2 exclusion; A4: `p_hat` grouped by `event_id`) | Cowan (1992) |
| Rank test (RankTest) | **FIXED** (original audit: `rank_z` broadcast); **FIXED 2026-09-24** (A4: grouped by `event_id`) | Corrado (1989) |
| Calendar-Time Portfolio test (CalendarTimePortfolioTest) | **FIXED** (original audit: `caltime_t`/`ccaltime_t` broadcast); **FIXED 2026-09-24** (A7: Brown-Warner estimation-window denominator, redefinition; A9: strict `n_pos > 0`) | Brown-Warner (1980, 1985); Jaffe (1974); Fama (1998) |
| Kolari-Pynnonen test (KolariPynnonenTest) | **FIXED 2026-09-24** (A2 exclusion; A3: usable-events `r_bar`, per-day adjustment, never `kp_adj = 1` fallback) | Kolari-Pynnonen (2010) |
| PermutationTest | unwired stub -- empty `compute()`, not on pipeline path; deferred | -- |
| `bootstrap_test()` | **FIXED 2026-09-24** (A12a: observed/draw CAAR construction aligned; A12b: NA draws excluded from denominator; A12c: clustering unit confirmed, no change) | -- |
| `cross_sectional_regression()` | **FIXED 2026-09-24** (A5: duplicate-id error, out-of-range `car_window` error, unmatched/NA-CAR warnings+exclusion) | -- |
| `TestStatisticBase$confidence_type` | **FIXED 2026-09-24** (A10: validated, warns once when non-default; no formula change) | -- |

### Audit summary

The original audit covered all 13+ return models and all wired test
statistics and found two broadcast-bug implementation errors (RankTest,
CalendarTimePortfolioTest). The 2026-09-24 re-evaluation covered the same
surface plus `bootstrap_test()` and `cross_sectional_regression()`, and found
twelve additional silent-wrong-statistic defects (items A1-A9, A11-A12; A10 is
a validation/warning addition, no formula change; A11 is docs-only). Every
defect is fixed, locked by a regression test that fails on the pre-fix code
(`test_reeval_statistical_fixes.R`), and golden values affected by a genuine
redefinition (A7, A8, and the Patell/KP/Rank/GSign pins touched by A2/A3/A4/A6/A9)
are re-pinned in `test_golden_values.R` with an explanatory comment. No
valid-input behavior changed for any component NOT listed above as fixed.

**2026-09-24 re-evaluation fixes (12 items, A1-A9 + A11-A12; A10 is a
validation addition):**

1. **A1 -- RollingWindowModel complete pairs.** Rolling-window OLS computed
   `x_bar`/`ss_xx` from ALL index values in a window, including rows where the
   firm return was NA, biasing beta when NA-firm rows co-occur with index
   outliers. Fixed to restrict each window to complete pairs.
2. **A2 -- Multi-event all-NA exclusion.** A fully-degenerate event (all
   event-window ARs NA) was coalesced to 0 in cumulative statistics instead of
   excluded, silently understating cross-event dispersion. Fixed with a shared
   exclusion helper applied once per group.
3. **A3 -- KolariPynnonenTest usable events.** A degenerate estimation-window
   SAR series was folded into `r_bar`, and an undefined adjustment silently
   fell back to `kp_adj = 1` (the unadjusted BMP value). Fixed to exclude
   non-usable events from `r_bar` and return NA (never a silent 1) when the
   adjustment cannot be computed; also fixed the per-day window-size term
   (previously the first day's count applied to every day).
4. **A4 -- RankTest / GeneralizedSignTest grouping.** Ranking and `p_hat`
   estimation grouped by `firm_symbol`, silently pooling a recurring firm's
   data across its distinct events. Fixed to group by `event_id`.
5. **A5 -- cross_sectional_regression sample integrity.** Duplicate
   `event_id`s, out-of-range `car_window`s, and NA-CAR events were silently
   tolerated. Fixed with explicit errors/warnings and exclusion.
6. **A6 -- OLS insufficient-observations guard and Patell m/k.** The
   insufficient-obs guard used a flat threshold instead of `n_params + 1`;
   Patell's `k` was derived as `length(resid) - df` instead of read from the
   model; Patell's `m` used the raw row count instead of the finite count;
   MarketModel's FEC included incomplete-pair rows. All fixed.
7. **A7 -- CalendarTimePortfolioTest denominator.** `ts_sd` was computed from
   the event-window portfolio series itself (self-referential with the
   numerator). Redefined to the Brown-Warner (1980, 1985) estimation-window
   AAR standard deviation.
8. **A8 -- MarketAdjustedModel FEC.** Wrongly reused the constant-MEAN
   correction (`sigma * sqrt(1 + 1/m)`) for a model that estimates NOTHING.
   Fixed to FEC == sigma exactly.
9. **A9 -- Sign convention.** `n_pos` used `>= 0` in CSectT, Patell, and
   CalendarTime (inconsistent with SignTest's `> 0`). Fixed to strict `> 0`
   everywhere.
10. **A11 -- BMP/KP roxygen (docs-only).** The BMP docstring incorrectly
    claimed forecast-error-corrected-sigma standardization; both BMP and KP
    standardize by MODEL sigma. Fixed the docstring; no formula change.
11. **A12a/b -- bootstrap_test() observed/draw mismatch and NA-draw
    denominator.** `observed_caar` used a different construction than the
    draws; NA draws were counted as "did not exceed" while still inflating
    the denominator. Both fixed.
12. **A12c -- bootstrap clustering unit (confirmed, no change).** Firm-level
    clustering is deliberate and unchanged; locked with a hand-replication
    test.

A10 (TestStatisticBase `confidence_type` validation) adds a construction-time
check and a one-time warning; it changes no computed statistic.