## Expected Return Models

There are a couple of aspects of the event study methodology which deserve discussion. Among other topics, finance scholars have discussed researcher discretion in choosing expected return models and the appropriate length of the estimation and event windows. With regard to the assumptions underlying the methodology, several issues were discussed, such as infrequent trading of the underlying stocks, and event clustering/confounding events (i.e., structural breaks in beta values) were discussed (Cable and Holland, 1999). Please find below a short summary of these issues.

(1) Expected return models

Finance literature discusses different expected return models. Each of them predicts the benchmark values that are deducted from the actual stock returns for calculating  'abnormal returns'. Based on their underlying assumptions, expected return models can be grouped into statistical (models 1-5 below) and economic models (models 6 and 7); with the latter being associated with more comprehensive theories of market pricing. All of these models are implemented in our abnormal effect calculators and described in detail with formulas at the bottom of this page.

1. Market Model (Abbr.: mm): The 'market model' considers the focal firm's individual CAPM risk by multiplying the market return with the firm individual $\beta$ factor: $\alpha_i+\beta_i R_{m,t}$. Although the 'market model' is widely accepted as the standard model, there is also some criticism. The model assumes that the risk-free interest rate included in the $\alpha$ factor is constant, which conflicts with the presumption that market returns vary over time.
2. Market Adjusted Model (Abbr.: mam): Using the actual market return is the simplest way to 'control' for potential effects of the event on the general market, yet it does not adjust for basic CAPM risk and thus abstracts from the focal firm's distinct systematic risk profile.
3. Comparison Period Mean Adjusted Model (Abbr.: cpmam):
4. Market Model with Scholes-Williams beta estimation (Abbr.: mm-sw):
5. Market Model with GARCH and EGARCH error estimation (Abbr.: garch / egarch):
6. Fama-French 3 Factor Model (Abbr.: ff3f): 'Multi-factor models', as the Fama-French three-factor model or APT suggest to mitigate the issues related with the CAPM model and thus provide better estimates for benchmark returns in event studies.
7. Fama-French-Momentum 4 Factor Model (Abbr.: ffm4f): 'Multi-factor models', as the Fama-French three-factor model or APT suggest to mitigate the issues related with the CAPM model and thus provide better estimates for benchmark returns in event studies.

Additional expected return models we plan to implement:
1. Matched firm model: $R_{mf,t}$: Instead of referring to the market, scholars may also turn to the performance of a comparable firm's stock when seeking a proxy for a distinct firm's expected returns.
2. $R_{f,t}+\alpha_i+\beta_i (R_{m,t}-R_{f,t})$: The 'CAPM model' includes the specific-risk free rate in the estimation and therefore represents a more granular approach than the market model. This prediction model has been criticized: The $\beta$ factor only provides an imperfect correlation to market risks, it was found inferior to e.g., the Fama-French three-factor model. And also the $\alpha$ parameter as a post-event estimator is deemed biased for post-event predictions.

The choice of expected return models is generally discussed in the context of sample selection biases (e.g., Ahern, 2009). Depending on the specific sampling bias, each of the above models may imply slightly different biased results. For example, if a study's sample largely consists of small firms, the CAPM model was found to predict too low returns (Banz, 1981), leading to inflated abnormal returns in the event study. Multi-factor models try to circumvent this problem by considering the factors driving the biased results. As abnormal return biases tend to be small in most instances, research concerned with the economic substance of individual event types, rather than methodological discussions, however, still builds on the 'market model'.

Recent meta research (Holler, 2014) finds that the market model is the predominantly used model for predicted normal returns. From a sample of 400 analyzed event studies, 79.1% of the studies used the 'market model', 13.3% drew on the 'market adjusted return model', 3.3% the 'constant mean return model', 3.6% 'multi-factor models', and only 0.7% deployed the CAPM model (general overview of factor models).

(2) Choices on the estimation and event windows

Allocating the estimation and event windows poses two major challenges to scholars. First, they must identify the correct event date as an 'anchor' for the whole analysis, and second, they need to specify the lengths and positions of the estimation and event windows.

Identifying the event date is not always a simple task. In the analysis of M&A transactions, for example, initial rumors about the transaction are typically followed by an official announcement and a closing of the transaction. On each of these events, information is released, posing the question which date represents the correct event date to be analyzed. Scholars investigating this issue found the information content of the first official announcement being highest and therefore representing the correct event date in the context of M&A studies (Dodd, 1980). Similar questions also arise when studying other event types.

Further, there is no definite rule on the length of an event study's estimation and event windows. Researchers have discretionary choice when deciding about these parameters. The challenge researchers have to solve, with regard to the estimation window, is to find the balance in the tradeoff between improved estimation accuracy and potential parameter shifts. Longer estimation windows promise greater accuracy, as they imply larger samples of returns, but they also bear the risk of covering structural breaks (e.g., due to confounding events) of the $\alpha$ and $\beta$ factors, which will lead to biased estimators. A similar challenge exists with the event window, as one needs to specify over which period the studied event impacted the respective stock.  On the one hand, information leakage and longer information processing periods favor longer event windows, and on the other hand, confounding events suggest shorter event windows.

Recent meta research reviewing 400 event studies finds that estimation window lengths spread out between 30 and 750 days (Holler, 2014). Studies investigating the sensitivity of results (e.g., the predicted return on the event date) suggest that results are not sensitive to varying estimation window lengths as long as the window lenghts exceed 100 days (Armitage, 1995, Park, 2004). Event windows typically range in their length between 1 and 11 days and center symmetrically around the event day (Holler, 2014). The most common choice of event window length in a recent paper by Oler, Harrison, and Allen (2007) is 5 days, representing 76.3% of the reviewed studies.

Another issue of event study methodology relates to the trading of the analyzed firm's stock and the market chosen as a reference index. Infrequent trading of the firm's stock, or a mismatch of trading days between the stock and the reference market, may lead to problems in deriving the estimation parameters $\alpha$ and $\beta$. Specifically, mismatches in the time series of returns in the stock and market returns throughout the estimation window may lead to overall shorter estimation periods and potentially biased parameters. Therefore, mismatches within the event window will lead to failure in calculating individual abnormal returns and thus lead to incomplete cumulative abnormal returns.

(4) Event clustering/Confounding events

If multiple significant events –no matter of which type affect a single firm's stock in close succession, issues arise from the overlapping of estimation and event windows. Specifically, problems of cross-correlation, implying biased estimators, may arise. This potential problem particularly pertains to small sample studies, where flawed results do not become sufficiently 'corrected' by creating mean values over large numbers of observations (i.e., by calculating [cumulative] average abnormal returns).

The following expected return models can be used through our abnormal effect calculators.

Statistical Expected Return Models

[1] Market Model (Abbr.: mm)

In the market model we assume that the return follows a single factor market model

$$R_{it} = \alpha_i + \beta_i \cdot R_{mt} + \varepsilon_{it},$$

where $R_{it}$ is the return of the stock of observation $i$ (e.g. firm) on day $t$, $R_{mt}$ is the return of the reference market on day $t$, $\varepsilon_{it}$ is the error term (a random variable) with expectation zero and finite variance. It is assumed that $\varepsilon_{it}$ is uncorrelated to the market return $R_{mt}$ and firm return $R_{jt}$ with $i \neq j$, not autocorrelated, and homoskedastic. The regression coefficient $\beta_i$ is a measure of the sensitivity of $R_{it}$ on the reference market. The abnormal return is then calculated as follows:

$$AR_{it} = R_{it} - (\alpha_i + \beta_i \cdot R_{mt}).$$

[2] Market Adjusted Model (Abbr.: mam)

In the market adjusted model, the observed return of the reference market on day $t$ $R_{mt}$ is substracted from the return $R_{it}$ of the observation $i$ on day $t$. We get for the abnormal return:

$$AR_{it} = R_{it} - R_{mt}.$$

[3] Comparison Period Mean Adjusted Model (Abbr.: cpmam)

In the comparison period mean model the abnormal return in the event window is the return of observation $i$ on day $t$ minus the average return of the observation $i$ in the estimation window:

$$AR_{it} = R_{it} - \bar{R}_{i},$$

where $\bar{R}_{i} = \frac{1}{T_1 - T_0}\sum\limits_{t\in [T_0, T_1]}R_{it}.$

[4] Market Model with Scholes-Williams beta estimation (Abbr.: mm-sw)

For non-synchronus trading you may choose the market model with Scholes-Williams beta estimation. The betas are defined as

$$\beta^{SW}_i = \frac{\beta_i^- + \beta_i + \beta^+_i}{1 + 2 \cdot \rho_M},$$

where $\beta^-_i$ is the regression coefficient of $R_{it}$ on $R_{m,t-1}$, $\beta^+_i$ is the regression coefficient of $R_{it}$ on $R_{m,t+1}$, and $\rho_M$ is the first-order autocorrelation of $R_m$. The intercept $\alpha^{SW}_t$ is estimated through the sample mean

$$\alpha^{SW}_i = \bar{R}_{i, EST} - \beta^{SW}_i \cdot \bar{R}_{M, EST},$$

where $\bar{R}_{i, EST}$ is the mean of returns of observation $i$ in the estimation window and $\bar{R}_{M, EST}$ is the mean of the returns of the reference market in the estimation window.

[5] Market Model with GARCH and EGARCH error estimation (Abbr.: garch / egarch)

If you choose the GARCH option on our EST API interface a single factor market model with GARCH(1, 1) errors is estimated, namely

$$R_{it} = \alpha_i + \beta_i \cdot R_{mt} + \varepsilon_{it}.$$

The conditional variance (Bollerslev (1986)) may be written as:

$$\sigma^2_{t} = \omega + \gamma_1 \cdot \varepsilon^2_{t-1} + \delta_1 \cdot \sigma^2_{t-1}$$

with $\sigma^2_t$ denoting the conditional variance, $\omega$ the intercept, and $\varepsilon^2_t$ the residuals from the mean filtration process. Parameters are estimated by maximum likelihood (a non-linear solver is used for the optimization problem).

Economic Expected Return Models

[6] Fama-French 3 Factor Model (Abbr.: ff3f)

coming soon

[7] Fama-French-Momentum 4 Factor Model (Abbr.: ffm4f)

coming soon

References and further studies

K. Ahern 2009. 16 (3): 466-482
S. Armitage 1995. 8 (4): 25-52
R.W. Banz 1981. 9 (1): 3-18
S. Brown; J. Warner 1980. 8 (3): 205-258
J. Cable; K. Holland 1999. 5 (4): 331-341
R. Chandra; B. Balachandran 1990. 6 611-640
D.W. Collins; W.T. Dent 1984. 22 48-84
A. Coutts; T.C. Mills; J. Roberts 1995. 2 163-165
P. Dodd 1980. 8 (2): 105-137
P. Draper; K. Paudyal 1995. 22 157-177
D. Oler; J. Harrison; M. Allen 2007.
N. Park 2004. 25 (7): 655-668
J.N. Patell 1976. 14 246-276
N. Strong 1992. 19 533-553