Event Study Methodology

One of the key advantages of using Event Study Metrics as a software solution is its ease of use and simplicity. The computations below are preconfigured in the software and running the event study is pretty much automated, letting you concentrate on the economics (and not the maths!) of your study. Nevertheless, it is important for you to understand what happens behind the scenes. The explanations below will help you make informed decisions on the model and test statistics you chose in the software.

Abnormal Returns

Abnormal Returns are the crucial measure to assess the impact of an event. The general idea of this measure is to isolate the effect of the event from other general market movements. The abnormal return of firm i and event date^\tau is defined as the difference of the realized return and the expected return given the absence of the event:

AR_{i,\tau}=R_{i,\tau}-E[R_{i,\tau}\vert\Omega_{i,\tau}]

The expected return (henceforth referred to as normal return) is unconditional on the event but conditional on a separate information set. Dependent on the definition of the information set (e.g., past asset returns) and the functional form there exist various models for the normal return. Those models are extensively discussed in the following section.

Event Study Metrics offers two different measures of aggregated abnormal returns that are commonly used in event study analyses:

Cumulating abnormal returns across time yields the cumulative abnormal return measure:

 CAR_{i(\tau_{1},\tau_{2})}=\sum_{t=\tau_{1}}^{\tau_{2}} AR_{i,t}

The second measure, the buy-and-hold abnormal return (BHAR), is defined as the difference between the realized buy-and-hold return and the normal buy-and-hold return:

BHAR_{i(\tau_{1},\tau_{2})}=\prod_{t=\tau_{1}}^{\tau_{2}}(1+R_{i,t})- \prod_{t=\tau_{1}}^{\tau_{2}} (1+E[R_{i\tau}\vert\Omega_{i\tau}])

Statistical tests of abnormal returns are commonly based on the cross-average of each measure. For cumulative abnormal returns the cross-sectional average (CAAR) is:

 CAAR_{(\tau_{1},\tau_{2})}=\frac{1}N \sum_{i=1}^{N} CAR_{i(\tau_{1},\tau_{2})}

Whereas, the mean buy-and hold abnormal return is:

 \overline{BHAR}_{(\tau_{1},\tau_{2})}=\frac{1}N \sum_{i=1}^{N} BHAR_{i(\tau_{1},\tau_{2})}

For a detailed discussion of the difference between the two measures you may consult Barber and Lyon (1997) or Ritter (1991).

Normal Return Models

A substantial feature of an event study is the choice of an appropriate normal return model. Some models contain parameters that need to be estimated (constant mean return model, market model, CAPM, and multifactor models). The time period over which parameters are estimated is commonly denoted as the estimation window. Since the normal return is the expected return in absence of the event, overlapping event and estimation windows should be avoided. Otherwise normal return model parameters are estimated from returns affected by the event. Event Study Metrics applies the common approach by restricting the estimation window (L1) to the time period prior to the event window (L2).

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In the following, the most common choices for normal return models are going to be presented.

Constant Mean Return Model

Assume that expected asset returns can differ by company, but are constant over time. Then the constant mean return model is:

R_{i,\tau}=\mu_{i}+\epsilon_{i,\tau}

with

 E[\epsilon_{i,\tau}]=0

and

 VAR[\epsilon_{i,\tau}]=\sigma^2_{\epsilon i}

The parameter\mu is estimated by the arithmetic mean of estimation-window returns:

 \hat{\mu_{i}}=\frac{1}M_i \sum_{i=T_{0+1}}^{T_1} R_{i,\tau}

where M_i  is the number of non-missing returns over the estimation window. Please note that  M_{i} \leq L_{i}.

Even though the constant mean return model is simple and highly restrictive compared to other models, Brown and Warner (1980, 1985) show that results based on this model do not systematically deviate from results based on more sophisticated models. Please note that Brown and Warner (1980, 1985) only analyze short-term event studies. The selection of the benchmark models is crucial when performing a long-term event study.

Market Return Model

Abnormal returns are calculated by subtracting the contemporaneous return of a market index:

 AR_{i,\tau}=R_{i,\tau}-R_{M,\tau}

where R_{M,\tau} is the return of a market index (e.g., S&P 500).

This model can be viewed as a restricted market model with alpha equal to zero and beta equal to one for each stock (see MacKinlay (1997)).

Since the parameters are predefined, a separate estimation window is not necessary. Thus, Event Study Metrics will ignore any settings specifying the estimation window when you select Market Return as normal return model.

Market Model

The market model is based on the assumption of a constant and linear relation between individual asset returns and the return of a market index:

 R_{i,\tau}=\alpha_i+\beta_i R_{M,\tau}+\epsilon_{i,\tau}

with

 E[\epsilon_{i,\tau}]=0

and

 VAR[\epsilon_{i,\tau}]=\sigma^2_{\epsilon i}

Event Study Metrics estimates the model parameters by ordinary least squares regressions based on estimation-window observations.

Alternatively, you may choose the Scholes/Williams option from the Estimation Method menu. Instead of ordinary least squares, Event Study Metrics will then apply the method proposed by Scholes and Williams (1977) to account for non-synchronous trading. Event Study Metrics calculates the market model parameters applying the Scholes/Williams approach by:

 \hat{\beta}_{i,SW}=\frac{\hat{\beta}_{i,lag}+\hat{\beta_i}+\hat{\beta}_{i,lead}}{1+2\hat{\rho}_M}

and

 \hat{\alpha}_{i,SW}=\frac{1}{L_1-2} \sum_{t=T_0+2}^{T_1-1}(R_{i,t})-\hat{\beta}_{i,SW}\sum_{t=T_0+1}^{T_1}(R_{M,t})

where \hat{\beta}_{i,  lag}, \hat{\beta_i}, \hat{\beta}_{i,  lead} are the OLS estimates from the regression of R_M,\tau-_1R_M,\tau   and R_M, \tau+_1 on R_i, _\tau and \hat\rho_M is the first-order autocorrelation of R_{M,\tau}.

Some financial databases use adjusted betas following Blume (1975). By choosing the option Blume Adjustment in the settings menu, betas are adjusted according to Blume (1975):

 \hat{\beta}_{i,Blume}=0.33+0.67\hat{\beta}_i

CAPM

According to the capital asset pricing model, the expected excess return of asset i is given by:

 E[R_i-r_f]=\alpha_{i}+\beta_{i}[r_m-r_f]+\epsilon_{i,\tau}

where r_f is the risk-free return.

Event Study Metrics estimates the model parameters of the capital asset pricing model by a time-series regression based on realized returns:

 (R_{i,\tau}-r_{f,\tau})=\alpha_i+\beta_i(R_{M,\tau}-r_{f,\tau})+\epsilon_{i,\tau}

with

 E[\epsilon_{i,\tau}]=0

 and

 VAR[\epsilon_{i,\tau}]=\sigma^2_{\epsilon i}

Please make sure that the time-series of risk-free returns is not annualized, but instead matches your data frequency.

Multi-factor Models

Event Study Metrics offers the possibility to apply a multi-factor model to measure normal returns. You can choose a multifactor model based on three or four factors. The best known approach is the three-factor model developed by Fama and French (1993). Based on their empirical findings two additional factors are added to the CAPM that should increase the explanatory power of the model:

 (R_{i,\tau}-r_{f,\tau})=\alpha_i+\beta_{i,M}(R_{M,\tau}-r_{f,\tau})+\beta_{i,SMB}SMB_{\tau}+\beta_{i,HML}HML_{\tau}+\epsilon_{i,\tau}

where SMB_\tau stands for ‘small minus big’ and HML_\tau stands for ‘high minus low’. The SMB_\tau factor should capture the excess return of small over big stocks (measured by market cap). The HML_\tau factor should capture the excess return of stock with a high market-to-book ratio over stocks with a low market-to-book ratio. For a detailed description of the factors and the construction of the underlying portfolios you might refer to Fama and French (1993). You may obtain time-series data from Kenneth French’s website.

A common extension of the three-factor model is the four factor model that additionally contains the momentum factor MOM_\tau as introduced by Carhart (1997):

formel1

where MOM_\tau is a factor that should capture the excess return of past winning over past losing stocks. For a detailed description of the momentum factor and the construction of the underlying portfolios refer to Carhart (1997).

Matched Firms (Portfolios)

Lyon, Barber and Tsai (1999) propose the use of a portfolio matched by size and market-to-book ratio as measure of normal returns for each event. They claim that this measure is free of the new-listing and rebalancing bias and propose to draw statistical evidence applying a bootstrapped version of the skewness-adjusted t-test. However, the authors cast doubt whether this approach yields well-specified test statistics in non-random samples.

To apply the matched firms (portfolio) approach you need to specify a reference firm (portfolio) to each event. The Event List contains additional fields for each event that allow you to match a firm (portfolio). Event Study Metric treats the asset pricing data of matched firms (portfolios) equal to the data of event firms. Therefore, you can simply add the asset pricing data of your matched firms to the common Dataset.

Event Study Metrics will calculate abnormal returns by subtracting the contemporaneous return of the individually matched firm (portfolio):

AR_{i,\tau}=R_{i,\tau}-R_{MP_i,\tau}

where  R_{i,\tau}-R_{MP_i,\tau} is the contemporaneous return of the individually matched firm (portfolio).

Bonds (Matched Portfolios)

Bessembinder et al. (2009) discuss different methods to measure abnormal bond performance. Since some firms might have multiple bonds outstanding, they propose to conduct a bond event study on firm-level portfolios.  To apply this approach you need to select the Bonds (Matched Portfolios) option from the Normal Return Model menu. Event Study Metrics will then automatically create firm-level portfolios.  Each portfolio consists of all assets that share the same event date and firm identifier based on the entries of the Event List.

Event Study Metrics allows you to utilize rating equivalent reference portfolios (or portfolios matched by maturity, etc.) as measure of normal returns for each event. For a detailed discussion of feasible reference portfolios you may refer to Bessembinder et al. (2008).

Event Study Metrics will calculate abnormal returns by subtracting the contemporaneous return of the individually matched firm (portfolio):

 AR_{i,\tau}=R_{i,\tau}-R_{MP_i,\tau}

where  R_{i,\tau}-R_{MP_i,\tau} is the contemporaneous return of the individually matched firm (portfolio).

Some test statistics require an estimation window. Event Study Metrics allows you to conduct the aforementioned matching approach with an estimation window. The estimation window has no influence on the normal return measure itself, but is solely used to calculate test statistics. To apply this approach you need to select the Bonds (Matched Portfolios) Est option from the Normal Return Model menu.

Calendar-Time Portfolio Regressions

The Calendar-Time Portfolio method allows you to assess if event firms persistently earn abnormal returns. The general idea is to form a portfolio of event firms and to test if this portfolio exhibits any abnormal return not captured by common risk factors.

Suppose you want to assess abnormal returns over a three-year period. For each month in calendar-time the portfolio is constructed by all firms that had an event in the three years prior to the calendar month. By default, Event Study Metrics forms equal weighted portfolio. You may select the Use Weights option to form value weighted portfolios.

You can employ the Fama-French three-factor model or a factor model that additionally contains the Carhart (1997) momentum factor to analyze returns of calendar-time portfolios.

formel2

Under the null hypothesis of no abnormal return, the estimate of  should be not statistically different from zero.

Lyon, Barber and Tsai (1999) suggest the error term in calendar-time portfolio regressions may be heteroscedastic, since the number of securities varies over time. They propose to employ a weighted least squares regression, where the weighting factor is based on the number of assets in the portfolio. You may follow their proposal by selecting the WLS option from the Estimation Method menu. Additionally, Event Study Metrics reports t-statistics based on White (1980) robust standard errors.

Significance Tests

Cross-Sectional t-Test

The cross-sectional t-test is defined as:

 T_{cross}=\frac{CAAR(\tau_1,\tau_2)}{\hat{\sigma}_{CAAR_{(\tau_1,\tau_2)}}}

Under the null hypothesis, the cumulative average abnormal return is equal to zero. The variance estimator of this statistic is based on the cross-section of abnormal returns.

 \hat{\sigma^2}_{CAAR(\tau_1,\tau_2)}=\frac{1}{N(N-d)} \sum_{i=1}^{N}[CAR_i(\tau_1,\tau_2)-CAAR_i(\tau_1,\tau_2)]^2

Brown and Warner (1980) show that the cross-sectional t-test is robust to an event-induced variance increase. However, Boehmer, Musumeci and Poulsen (1991) provide evidence that their standardized cross-sectional test (requiring an estimation window) exhibits a comparable size, but is more powerful.

Standardized Residual Test

The standardized residual test, developed by Patell (1976), tests the null hypothesis that the cumulative average abnormal return is equal to zero. Under the assumption that abnormal returns are uncorrelated and variance is constant over time, each abnormal return is standardized by its estimated standard deviation:

 SAR_{i,\tau}=\frac{AR_{i,\tau}}{S(AR_i)}

The standard deviation is estimated from the time-series of abnormal returns of the estimation window:

 \hat{\sigma^2}_{AR_i}=\frac{1}{M_i-d} \sum_{t=Est_{min}}^{Est_{max}}(AR_{i,t})^2

where M_i is the number of non-missing returns and d the degrees of freedom (e.g., market model d=2 ). To account for the fact that the event-window abnormal returns are an out-of-sample prediction, the standard error is adjusted by the forecast error:

 S(AR_i)=\hat{\sigma}_{AR_i} \sqrt{1+\frac{1}{M_i}+\frac{(R_{m,\tau}-\overline{R}_{m,Est})^2}{\sum_{Est_{min}}^{Est_{max}}(R_{m,\tau}-\overline{R}_{M,Est})^2}}

As simple abnormal returns, the standardized version can be cumulated over time:

 CSAR_i(\tau_1,\tau_2)=\sum_{t=\tau_1}^{\tau_2}\frac{AR_{i,t}}{S(AR_i)}

Under the null hypothesis the distribution of SAR_{i} is a Student’s t-distribution with M_{i}-d degrees of freedom (for a further discussion see Campbell, Lo and MacKinlay (1997) pp. 160). It directly follows that the expected value of CSAR_{i} is zero and the standard deviation is:

 S(CSAR_i)=\sqrt{(\tau_2-\tau_1+1)\frac{M_i-d}{M_i-2d}}

The test statistic for the null hypothesis, that the cumulative average abnormal return is equal to zero, is:

 T_{Patell}=\frac{1}{\sqrt{N}} \sum_{i=1}^{N}\frac{CSAR_i(\tau_1,\tau_2)}{S(CSAR_i)}

The standardized residual test is robust to heteroscedastic event-window abnormal returns. By standardizing abnormal returns before forming portfolios, the standardized residuals test assigns a lower weight to abnormal returns of securities with large variances than a simple time-series t-test.

Boehmer, Musumeci and Poulsen (1991) show that under the absence of an event-induced variance increase, the standardized residuals test is well specified and has appropriate power. If the variance of stock returns increases around the event date, the standardized residuals test rejects the null hypothesis too often.

Standardized Cross-Sectional Test

Boehmer, Musumeci and Poulsen (1991) combine the standardized residuals test with an empirical variance estimate based on the cross section of event-window abnormal returns to construct a test that is robust to event-induced variance increases of stock returns.

Initially, abnormal returns are standardized as described in the previous section. Then the cross-sectional average of  CSAR_i(\tau_1,\tau_2) is calculated:

 \overline{CSAR}(\tau_1,\tau_2)=\frac{1}{N} \sum_{i=1}^{N}{CSAR_i(\tau_1,\tau_2)}

The standard deviation of  \overline{CSAR}(\tau_1,\tau_2) is estimated from the cross section of event-window abnormal returns:

  S(\overline{CSAR})=\sqrt{\frac{1}{N(N-1)} \sum_{i=1}^{N}{[CSAR_i(\tau_1,\tau_2)-\overline{CSAR}(\tau_1,\tau_2)]^2}}

The standardized cross-sectional test statistic for the null hypothesis that the cumulative average abnormal return is equal to zero is:

 T_{Boehmer}=\frac{\overline{CSAR}(\tau_1,\tau_2)}{S(\overline{CSAR})}

Event Study Metrics allows you to optionally use an adjusted version of the standardized cross-sectional test following Kolari and Pynnönen (2010), which accounts for cross-correlation.

The adjusted standardized cross-sectional test statistic for the null hypothesis that the cumulative average abnormal return is equal to zero is:

 T_{Boehmer_{adj.}}=T_{Boehmer}\sqrt{\frac{1-\overline{\rho}}{1+(n-1)\overline{\rho}}}

where  \overline{\rho} denotes the average cross-correlation among abnormal returns.

Corrado Rank Test

The non-parametric rank test proposed by Corrado (1989) tests the null hypothesis that the average abnormal return is equal to zero. Initially, abnormal returns are transformed into ranks. This is done asset by asset for the joint time period consisting of the estimation window and the event window:

 K_{i,\tau}=rank(AR_{i,\tau})

Tied ranks are treated by the method of midranks (see Corrado (1989) footnote 5). Corrado and Zivney (1992) propose a uniform transformation of ranks to adjust for missing values:

 U_{i,\tau}=\frac{K_{i,\tau}}{(1+M_i)}

where M_{i} is the number of non-missing returns for each asset.

The single day test statistic is defined as:

 T_{Corrado}=\frac{1}{\sqrt{N}} \sum_{i=1}^{N}(U_{i,\tau}-0.5)/S(U)

The estimated standard deviation is defined as:

 S(U)=\sqrt{\frac{1}{L_1+L_2} \sum\nolimits_{\tau}\left[\frac{1}{\sqrt{N_{\tau}}}\sum_{i=1}^{N_{\tau}}(U_{i,\tau}-0.5) \right]^2}

where N_{\tau} is the number of non-missing returns (cross-section) at \tau=t. A multiday version can be achieved by taking the average of single day statistics multiplied by the inverse of the square root of the period’s length.

Generalized Sign Test

The generalized sign test proposed by Cowan (1992) is based on the ratio of positive cumulative abnormal returns p^+_0 over the event window.  Under the null hypothesis this ratio should not systematically deviate from the ratio of positive cumulative abnormal returns over the estimation window p^+_{Est}. Since the ratio of positive cumulative abnormal returns is a binominal random variable, the follow test statistic is used:

 t_{GS}=\frac{p^+_0-p^+_{Est}}{\sqrt{p^+_{Est}(1-p^+_{Est})/N}}

Under the null hypothesis that the average cumulative abnormal return is not statistically different from zero, the test statistic approximately follows a normal distribution.

Skewness-Adjusted t-Test

Buy-and-hold abnormal returns are positively skewed (e.g., Barber and Lyon (1997), Kothari and Warner (1997)). The skewness-adjusted t-test, originally developed by Johnson (1978), is a transformed version of the usual t-test to eliminate the skewness bias. The test statistic for the null hypothesis that the mean buy-and-hold abnormal return is equal to zero is:

 T_{Skewness-Ajusted}=\sqrt{N}\left[S+\frac{1}{3}\hat{\gamma}S^2+\frac{1}{6N}\hat{\gamma}\right]

where

 S=\frac{\overline{BHAR}(\tau_1,\tau_2)}{\hat{\sigma}_{BHAR}}

and

 \hat{\gamma}=\frac{\sum_{i=1}^{n}[BHAR_i(\tau_1,\tau_2)-\overline{BHAR}(\tau_1,\tau_2)]^3}{N\hat{\sigma}_{BHAR}^{3}}

Since \sqrt{N}S  is the usual t-statistic the estimated standard deviation is defined by:

 \hat{\sigma}_{BHAR}=\sqrt{\frac{1}{N-1} \sum_{i=1}^{N}[BHAR_i(\tau_1,\tau_2)-\overline{BHAR}(\tau_1,\tau_2)]^2}

Lyon, Barber and Tsai (1999) recommend the use of a bootstrapped version of the skewness-adjusted t-test that yields well-specified test statistics. You can enable the bootstrapped version in the Bootstrap section of the Settings form. Additionally, you can specify the number and size of resamples. Event Study Metrics will report bootstrapped p-values and critical values for the 5% significance level (two-sided).