Long-Run Event Studies

CAR, cumulative abnormal return

The monthly abnormal return is \( AR_{it} = R_{it} - E[R_{it}] \), where the benchmark \( E[R_{it}] \) is a reference portfolio or matched control return. The cumulative abnormal return over event months \( \tau_1 \) to \( \tau_2 \) is the arithmetic sum:

\[ CAR_{i}(\tau_1,\tau_2) = \sum_{t=\tau_1}^{\tau_2} AR_{it} \]

Measures: an additive, implicitly monthly-rebalanced abnormal return. Benchmark: reference portfolio or matched control. Less skewed than BHAR and better behaved statistically, but it does not represent the return to a realistic buy-and-hold investor.

BHAR, buy-and-hold abnormal return

BHAR compounds the event firm's return and subtracts the compounded return of a matched benchmark:

\[ BHAR_{i}(\tau_1,\tau_2) = \prod_{t=\tau_1}^{\tau_2}\bigl(1+R_{it}\bigr) - \prod_{t=\tau_1}^{\tau_2}\bigl(1+E[R_{it}]\bigr) \]

Measures: what an investor would have earned holding the event firm instead of an otherwise-similar firm. Benchmark: a single matched control firm, per Barber and Lyon (1997), matched on size (market cap between 70% and 130% of the event firm) then, among those, the closest book-to-market. Matching this way removes the new-listing, rebalancing, and skewness biases that contaminate reference-portfolio BHARs. Economically the most intuitive estimator, but the most skewed and the most exposed to cross-correlation.

Calendar-time alpha (Jensen's alpha)

Instead of aligning returns in event time, align them in calendar time. In each calendar month, form a portfolio of every firm that experienced the event within the prior holding horizon, compute the portfolio excess return \( R_{pt} - R_{ft} \), and run one time-series regression on a factor model, for example Fama-French three-factor:

\[ R_{pt} - R_{ft} = \alpha_p + \beta_p\,(R_{mt}-R_{ft}) + s_p\,SMB_t + h_p\,HML_t + \varepsilon_{pt} \]

Measures: the average monthly abnormal return, read directly off the intercept \( \alpha_p \). Benchmark: the factor model itself. Because every month is a single portfolio observation, cross-sectional correlation among event firms is absorbed into the portfolio variance automatically, so the alpha standard error is free of the cross-firm dependence that breaks event-time tests. It is not automatically free of time-series problems (the firm count varies month to month, so residual variance is heteroskedastic), which is why the calendar-time alpha still needs White or WLS standard errors, covered in the testing section below. Fama (1998) calls this the preferred method: forming portfolios in calendar time handles the cross-correlation of event-firm returns without estimating any covariances.

BHAR: skewness-adjusted t, then bootstrap

Because BHARs are right-skewed, the ordinary t is biased. Barber and Lyon (1997) and Lyon, Barber and Tsai (1999) use the Johnson (1978) skewness-adjusted statistic. With mean \( \overline{BHAR} \), cross-sectional standard deviation \( \sigma \), sample size \( n \), estimated skewness \( \hat\gamma \), and \( S = \overline{BHAR}/\sigma \):

\[ t_{sa} = \sqrt{n}\left( S + \tfrac{1}{3}\hat\gamma\,S^{2} + \tfrac{1}{6n}\hat\gamma \right) \]

where  gamma_hat = ( (1/n) * sum_i (BHAR_i - mean_BHAR)^3 ) / sigma^3

step 1:  compute t_sa on the observed sample
step 2:  bootstrap. draw many resamples, recompute t_sa in each,
         build the empirical distribution of t_sa
step 3:  compare the observed t_sa to bootstrapped critical values,
         not to normal critical values

Lyon, Barber and Tsai (1999) show the skewness adjustment alone still over-rejects, so the bootstrapped distribution of \( t_{sa} \) is the version that survives their simulations in random samples.