Why the normal-return model choice changes your abnormal returns

The abnormal return is realised return minus the model's predicted normal return, so every abnormal return inherits the model's assumptions. A model that removes more systematic variation reduces the variance of the abnormal return and raises the power to detect an effect. The benefit scales with the regression $R^2$, which is roughly 20 to 40 percent for daily single-stock returns under the market model and 30 to 50 percent once size and value factors are added (MacKinlay, 1997). For short windows, though, the benchmark choice matters less than it appears, because the event-period return swamps it (Brown & Warner, 1985).

Constant-mean and market-adjusted: the simplest baselines

Constant-mean (comparison-period mean) model. The normal return is the firm's own average return over the estimation window: $E(R_{i,t})=\bar R_i$. No parameters beyond the mean; surprisingly competitive for short daily windows (Brown & Warner, 1985).

Market-adjusted model. The normal return is just the market return: $E(R_{i,t})=R_{m,t}$. This is the market model with $\alpha_i=0$ and $\beta_i=1$ imposed, so nothing is estimated. Useful when the estimation window is too short to fit a regression, for example for IPOs or newly listed firms.

The market model: the default workhorse

$$R_{i,t}=\alpha_i+\beta_i R_{m,t}+\varepsilon_{i,t}.$$

The market model regresses the firm return on a single market factor. It is the standard for a reason: it removes market-wide movement, cutting abnormal-return variance, while remaining simple and stable. Its two assumptions to keep in mind are that $\alpha_i$ and $\beta_i$ are stable across the estimation and event windows, and that a single factor adequately captures systematic risk. For most daily, single-event applications it is the right choice.

CAPM and multi-factor models: when factors matter

CAPM imposes the equilibrium restriction $E(R_{i,t})-R_{f,t}=\beta_i\,(E(R_{m,t})-R_{f,t})$. In practice the market model dominates it for event studies because it does not force the intercept to the risk-free rate, which is a strong and often-rejected restriction.

Fama-French three-factor adds size (SMB) and value (HML) factors: $R_{i,t}-R_{f,t}=\alpha_i+\beta_i(R_{m,t}-R_{f,t})+s_i\,SMB_t+h_i\,HML_t+\varepsilon_{i,t}$. The Carhart four-factor model adds momentum (WML/UMD); the Fama-French five-factor model adds profitability (RMW) and investment (CMA). These matter when the event sample is systematically tilted, for example small-cap or distressed firms, where the market model would misattribute a factor return to the event.

Decision table: from sample to model

Your situation	Recommended model	Why
Short daily window, broad sample	Market model	Standard, low variance, event return dominates
Very short estimation history (IPO, new listing)	Market-adjusted	No parameters to estimate
Quick baseline / robustness check	Constant-mean	Simplest; competitive for short windows
Sample tilted on size or value	Fama-French 3-factor	Removes size/value return the market model misses
Momentum-sorted or trend sample	Carhart 4-factor	Adds the momentum factor
Long-horizon study (months to years)	Fama-French 3/5-factor	Benchmark error accumulates; factors matter more

Estimation-window length, beta stability, thin trading

Use 120 to 250 trading days for daily data. Too short inflates parameter error; too long risks beta drift across the window. For thinly traded stocks, stale prices bias beta toward zero; lengthen the window or use a non-parametric test (see Significance tests). Always leave a gap between the estimation and event windows so pre-event leakage does not enter the normal-return fit.

Sensitivity check: how results shift across models

Best practice is to run the event on two or three models and report whether the CAAR and its significance are stable. If the market model and Fama-French give materially different CAAR, the difference is a factor return your event sample is loaded on, not the event itself, and you should report the factor-adjusted number. If they agree, the market model result is robust and you can lead with it.

Pick a model and run it free in ARC

The ARC calculator implements each model on this page directly. Select it in Step 1, upload your data, and download AR, CAR, CAAR and the test statistics. The model definitions and formulas are on Expected-return models.

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