Expected return models
Every abnormal return is a residual: what a security actually did minus what it should have done absent the event. The expected-return model is how you fill in the second half. This is the rigorous treatment of the full model menu, the assumptions each one buys, and the empirical evidence on which choice actually moves your inference.
New to return models? Plain-language version
The abnormal return identity is \(AR_{it} = R_{it} - E(R_{it}\mid X_t)\); the entire methodological debate is how you model the expected (normal, benchmark) return \(E(R_{it}\mid X_t)\). MacKinlay (1997) splits the candidates into statistical models (constant-mean, market-adjusted, market model, multifactor) and economic models (CAPM, APT) that impose an equilibrium restriction. For short-horizon studies the verdict from both theory and simulation is blunt: use the single-factor market model, reach for multifactor only when your sample is genuinely tilted on size, value, or momentum, and spend the effort you save on the test statistic, not the benchmark.
The identity that organizes everything
An event study measures the price impact of an event by comparing realized returns to a counterfactual. For security \(i\) on day \(t\), the abnormal return is
\[ AR_{it} = R_{it} - E(R_{it}\mid X_t), \]where \(X_t\) is the conditioning information: a market index, a set of priced factors, or simply a historical average. The realized return \(R_{it}\) is data. The expected term \(E(R_{it}\mid X_t)\) is the modeling choice, and it is the only thing the candidate models below disagree about. A cleaner benchmark means lower residual variance, which means a more powerful test, which is the entire reason any of this matters.
Conventions used throughout: \(R_{it}\) is the return on security \(i\) at time \(t\); \(R_{mt}\) the market (index) return; \(R_{ft}\) the risk-free rate; \(\alpha_i,\beta_i\) the OLS intercept and slope; \(\varepsilon_{it}\) a mean-zero disturbance. Parameters are estimated over the estimation window \(t\in[T_0,T_1]\) of length \(L_1 = T_1 - T_0 + 1\), which must not overlap the event window. See significance tests for what happens to these residuals next.
The MacKinlay taxonomy: statistical vs. economic
MacKinlay (1997) draws the dividing line by what the model assumes:
- Statistical models make only statistical assumptions about the joint distribution of returns. They do not claim to describe how prices are set, only how they tend to co-move. The constant-mean, market-adjusted, market, and multifactor models are all statistical.
- Economic models add an equilibrium restriction from asset-pricing theory: the CAPM forces the expected return onto the security market line; the APT onto a factor structure. The restriction can sharpen estimates if it is true, but it inherits the bias of the theory if it is false.
MacKinlay's conclusion, which the rest of this page elaborates, is that the economic restrictions buy little and can hurt, so the statistical market model is the workhorse. The models below run easiest to hardest.
The model menu
Each card gives the estimating equation, the abnormal-return formula, the load-bearing assumption, and the one-line intuition. They are ordered the way a reader meets them, naive baseline first.
1. Mean-adjusted (constant-mean)
\(\displaystyle \bar{R}_i = \tfrac{1}{L_1}\sum_{t=T_0}^{T_1} R_{it}\), then \(AR_{it} = R_{it} - \bar{R}_i\).
Assumes each security's return has a constant mean \(\mu_i\) across estimation and event windows. No index, no systematic-risk control.
Intuition: did the stock beat its own usual day? Subtract the historical average.
2. Market-adjusted
\(AR_{it} = R_{it} - R_{mt}\).
Assumes every stock has \(\beta=1\) and \(\alpha=0\). This is the market model with parameters restricted, so it needs no estimation window. Wrong per stock, but the error washes out across a sample.
Intuition: did the stock beat the market today? Just subtract the index.
3. Market model (single-factor OLS), the default
\(R_{it} = \alpha_i + \beta_i R_{mt} + \varepsilon_{it}\); fit by OLS on the estimation window, then \(AR_{it} = R_{it} - (\hat{\alpha}_i + \hat{\beta}_i R_{mt})\).
Assumes jointly normal i.i.d. returns, stable \(\alpha,\beta\) across windows, one factor sufficient.
Intuition: strip out the part of today's move the whole market explains given how strongly this stock tracks it; the leftover is the news.
4. CAPM (economic)
\(AR_{it} = R_{it} - \big[R_{ft} + \beta_i(R_{mt} - R_{ft})\big]\).
Assumes the Sharpe–Lintner equilibrium: expected excess return proportional to beta alone, intercept forced to \(r_f\).
Intuition: like the market model, but theory dictates the slope-implied expected return, and the theory does not hold well enough to bother.
5. Fama–French 3-factor
\(R_{it}-R_{ft} = \alpha_i + b_i(R_{mt}-R_{ft}) + s_i\,SMB_t + h_i\,HML_t + \varepsilon_{it}\); AR is the residual.
Assumes three priced risks: market, size (SMB), value (HML).
Intuition: also subtract what small-vs-large and cheap-vs-expensive stocks did.
6. Carhart 4-factor
FF3 plus \(u_i\,UMD_t\) (momentum, winners minus losers); AR is the residual.
Assumes the FF3 risks plus momentum.
Intuition: FF3 plus what recent winners did minus recent losers.
7. Scholes–Williams beta
\(\hat{\beta}_i^{SW} = \dfrac{\hat{\beta}_i^{-} + \hat{\beta}_i + \hat{\beta}_i^{+}}{1 + 2\hat{\rho}_m}\), then an ordinary market-model AR using the corrected beta.
Assumes stale prices from non-synchronous trading bias OLS beta toward zero for thin stocks; lead-and-lag removes it.
Intuition: illiquid stocks react a day late, so add yesterday's and tomorrow's market reaction to the beta.
8. GARCH / EGARCH
Keep a market-model mean equation; replace constant residual variance with a time-varying \(\sigma_t^2\).
Assumes persistent, clustering volatility (plus sign-asymmetry for EGARCH).
Intuition: do not assume the stock is equally jumpy every day; let calm and stormy stretches scale the yardstick.
Market model vs. CAPM: why the free intercept wins
The market model and the CAPM look almost identical: both regress the security return on the market. The difference is one restriction. The market model lets \(\alpha_i\) and \(\beta_i\) be free OLS estimates. The CAPM forces the intercept to the risk-free rate and the slope-implied expected return onto the security market line:
\[ E(R_{it}) = R_{ft} + \beta_i\,(E(R_{mt}) - R_{ft}). \]That restriction is the Sharpe (1964)–Lintner (1965) equilibrium, and it is the problem. The restriction is empirically violated: the size and value effects mean that expected returns are not proportional to beta alone. Forcing the benchmark onto a rejected security market line means your abnormal returns inherit that model's bias before you have measured anything. The market model avoids this entirely by never imposing the equilibrium in the first place. It buys the same variance reduction (it still removes the market component of returns) without the straitjacket. MacKinlay (1997) records that the event-study use of the CAPM "has almost ceased," and modern practice agrees.
More factors is not more rigorous. The instinct to stack five factors onto a three-day study because it "looks careful" gets the trade-off backwards. Extra factors only change inference when the sample loads systematically on them. On a factor-balanced sample they add estimation noise and leave the abnormal return essentially where the market model put it. Reaching for FF5 on a balanced short-window study is rigor theater, not better inference; see the multifactor section.
Scholes–Williams: correcting beta for thin trading
When a stock trades infrequently, its last recorded price is stale relative to the index close. The contemporaneous OLS beta is then biased, typically toward zero, because part of the stock's response to today's market move shows up tomorrow. Scholes & Williams (1977) fix this by running three market-model regressions and combining the slopes from the lagged, contemporaneous, and lead market returns:
\[ \hat{\beta}_i^{SW} = \frac{\hat{\beta}_i^{-} + \hat{\beta}_i + \hat{\beta}_i^{+}}{1 + 2\hat{\rho}_m}, \qquad \hat{\alpha}_i^{SW} = \bar{R}_i - \hat{\beta}_i^{SW}\,\bar{R}_m, \]where \(\hat{\beta}_i^{-}\) is the slope on \(R_{m,t-1}\), \(\hat{\beta}_i\) on \(R_{m,t}\), \(\hat{\beta}_i^{+}\) on \(R_{m,t+1}\), and \(\hat{\rho}_m\) is the first-order autocorrelation of the market return. The corrected \(\hat{\beta}_i^{SW}\) and \(\hat{\alpha}_i^{SW}\) then feed an ordinary market-model abnormal return. Use it when the sample contains thinly or infrequently traded stocks: small caps, illiquid names, emerging markets.
GARCH / EGARCH: a moving yardstick for variance
The market model assumes a constant residual variance \(\sigma_{\varepsilon_i}^2\). When volatility clusters, calm stretches followed by stormy ones, that assumption mis-sizes the test: a constant variance over-weights quiet days and under-weights turbulent ones. Conditional-variance models keep the market-model mean equation and let the variance evolve. The GARCH(1,1) of Bollerslev (1986) is
\[ \sigma_t^2 = \omega + \gamma_1\,\varepsilon_{t-1}^2 + \delta_1\,\sigma_{t-1}^2, \qquad \omega > 0,\ \gamma_1,\delta_1 \ge 0. \]The EGARCH(1,1) of Nelson (1991) models the log variance, which removes the non-negativity constraints and admits an asymmetry term:
\[ \ln(\sigma_t^2) = \omega + \alpha\big(|z_{t-1}| - E|z_{t-1}|\big) + \gamma\,z_{t-1} + \beta\ln(\sigma_{t-1}^2), \qquad z_t = \varepsilon_t / \sigma_t. \]Here \(z_t\) is the standardized residual and \(\gamma\) is the leverage parameter: a negative \(\gamma\) lets bad news raise volatility more than equally sized good news. Use these when the security or period shows pronounced volatility clustering, or the event itself plausibly changes variance. This is mostly a single-firm or small-sample concern; across a large cross-section the heteroskedasticity averages out and a constant-variance test is usually fine.
Where the windows sit
The estimation window and the event window are disjoint by construction. Parameters are fit on the former and the abnormal returns are measured on the latter, so the event's own returns never contaminate the normal-return estimate.
Model comparison at a glance
| Model | Core assumption | When to use | Data needed |
|---|---|---|---|
| Mean-adjusted | Constant mean return per security; no index | No reliable index exists, or a deliberate naive baseline | Security returns only |
| Market-adjusted | Every stock has \(\beta=1\), \(\alpha=0\) | Quick estimate, missing or short estimation window (IPOs), robustness | Security + index, no estimation window |
| Market model | One factor sufficient; stable \(\alpha,\beta\); i.i.d. normal returns | Almost always for short horizons (the default) | Security + index, estimation window |
| CAPM | Sharpe–Lintner equilibrium; intercept forced to \(r_f\) | Essentially never; restriction is empirically violated | Security + index + risk-free rate |
| Fama–French 3-factor | Three priced risks: market, size, value | Sample tilted on size or value | Above + SMB, HML factor series |
| Carhart 4-factor | FF3 risks plus momentum | Sample correlated with recent momentum (e.g. selected on prior performance) | Above + UMD momentum series |
| Scholes–Williams | Non-synchronous trading biases OLS beta | Thinly or infrequently traded stocks (small caps, emerging markets) | Security + index (lead/lag/contemporaneous) |
| GARCH / EGARCH | Persistent, clustering (asymmetric) volatility | Pronounced volatility clustering; event changes variance; single-firm studies | Market-model inputs + conditional-variance fit |
Why the market model is the default
The market model wins on a cost/benefit basis, sandwiched between two failure modes. Against the constant-mean model it removes the market component of returns, cutting residual variance and so raising test power, the single most valuable thing a benchmark can do for a short-window study. Against the CAPM and multifactor models it adds no fragile restriction: it lets \(\alpha_i\) and \(\beta_i\) be free OLS estimates rather than imposing an equilibrium the data reject. MacKinlay (1997) and Campbell, Lo & MacKinlay (1997, ch. 4) both land here, and surveyed practice agrees. The market model is used in the large majority of published short-horizon event studies, with the market-adjusted model a distant second and multifactor or CAPM specifications rare.
Brown & Warner: for short windows the benchmark barely matters
Brown & Warner (1980, monthly data; 1985, daily data) ran the canonical simulations: insert a known artificial return into real data and test which methodology best detects it. Their core finding is the one to foreground here. For short event windows, methodologies based on the market model and even simpler models perform similarly, and beyond a one-factor market model there is no evidence that more complicated methodologies convey any benefit. Daily data work well despite non-normality, and procedures are generally well-specified even when the special characteristics of daily returns are ignored.
Spend effort on the test statistic, not the benchmark. For a three-day window, pick the market model and move on; the real gains in a short-horizon study come from handling event clustering, cross-correlation, and variance estimation correctly, not from agonizing over mean-adjusted vs. market vs. multifactor. See significance tests.
When multifactor genuinely changes inference
Adding SMB, HML, or UMD helps only when the event sample loads systematically on those factors. If your event firms are, on average, disproportionately small, value, or momentum-tilted, the market model will hand the event credit (or blame) for what is really a factor return, biasing the cumulative abnormal return. There FF3 or Carhart genuinely change the answer. If the sample is factor-balanced, which random event timing across a broad cross-section tends to produce, the extra factors mostly add estimation noise and leave inference unchanged.
The decision rule
| Situation | Recommended benchmark |
|---|---|
| Short horizon, factor-balanced sample | Market model |
| Sample tilted on size or value | Fama–French 3-factor |
| Sample selected on prior performance / momentum | Carhart 4-factor |
| Thinly traded stocks | Market model with Scholes–Williams beta |
| Pronounced volatility clustering, single firm | GARCH / EGARCH mean-variance model |
| Long horizon | Multifactor or characteristic benchmark (idiosyncratic risk compounds) |
Estimation-window length, and why
Standard practice estimates parameters over roughly 120 to 250 trading days before the event (Campbell, Lo & MacKinlay use 250; MacKinlay's worked example uses 120). The window has to satisfy four competing pressures:
- Long enough that \(\hat{\alpha}_i, \hat{\beta}_i\) are precise and the residual variance is well estimated. Too short means a noisy beta and weak power.
- Short enough that the stable-parameter assumption holds. Betas drift over multi-year spans.
- Non-overlapping with the event window, so event returns do not contaminate the normal-return estimate.
- Gapped: usually leave a small gap before the event window so pre-event leakage does not bleed into the estimation period.
For daily data, about 250 days (one trading year) is the common choice and about 120 days is the practical floor. Monthly data needs a longer calendar span to reach a comparable observation count.
Typical R² and what it actually buys
For a single stock's daily returns the market-model regression \(R^2\) is typically about 0.02 to 0.10 for small and mid caps, rising to roughly 0.20 to 0.40 for large, liquid index-tracking names. Adding the Fama–French or Carhart factors lifts it modestly, often into the 0.30 to 0.50 range for well-explained names. Diversified portfolios sit much higher because idiosyncratic risk averages away.
The point of a higher \(R^2\) here is not fit for its own sake. It is lower residual variance, hence a tighter benchmark and a more powerful abnormal-return test. EST's own calculator reports roughly 20 to 40 percent for single daily stocks and 30 to 50 percent with additional factors, consistent with the figures above. Every model on this page is traced to its primary reference in the references list below.
Factor return series (SMB, HML, RMW, CMA, momentum) with regional coverage are available free from the Kenneth R. French Data Library. Ready to compute these benchmarks on your own data? The Abnormal Returns Calculator implements the market model, market-adjusted, mean-adjusted, multifactor, and Scholes–Williams specifications directly.
The one idea behind every model
An event study asks a simple question: when news hits a stock, how much did the price move because of that news, and not because of everything else going on in the market? To answer it, you first need a guess for what the stock would have done on that day if nothing had happened. That guess is what an expected-return model produces. It is just a prediction of the stock's normal return on a given day.
The piece you actually care about is the abnormal return: the part of the move the model could not explain. You get it by subtracting the prediction from what really happened.
Abnormal return = actual return minus expected (predicted) return. Every model on this page is just a different way of predicting the expected part. The abnormal return is the leftover, and the leftover is the news.
In symbols, for a stock \(i\) on day \(t\):
\[ AR_{it} = R_{it} - E(R_{it}) \]Here \(AR_{it}\) is the abnormal return, \(R_{it}\) is the stock's actual return on that day, and \(E(R_{it})\) is the expected (predicted) return. Every model below is a recipe for that last term.
The plain steer: most people use the market model
If you remember one thing, remember this: for short events (a few days around the news), most researchers use the market model, and that is almost always the right call. The market model predicts a stock's return from the return of the overall market, measured by an index such as the S&P 500.
The reason it wins is practical. The market moves every day for reasons that have nothing to do with your event. The market model removes that shared movement, so the leftover abnormal return is cleaner and your test is more likely to detect a real effect. It does this without forcing the data to obey any economic theory: it simply lets the numbers fit.
Two careful studies by Brown and Warner (1980, 1985) tested these models by hiding a known fake return inside real data and checking which model found it. Their verdict for short windows: the market model works well, and nothing more complicated reliably does better. So pick the market model and move on, and spend your effort on the test instead.
The models, simplest first
Below is the family of models, from the most basic to the most involved. Each gets one plain-language idea. Every symbol is named in words so you never have to decode notation.
Mean-adjusted model. The simplest of all. Take the stock's own average return over a calm stretch before the event, and treat that average as the prediction. The question it asks: did the stock beat its own usual day? Use it only when no good market index exists, or as a deliberately plain baseline.
Market-adjusted model. Predict that the stock moves exactly with the market, then subtract the index return. The question: did the stock beat the market today? It needs no calm-period estimation at all, so it is handy for brand-new listings (IPOs) with no history, or as a quick check.
Market model. The workhorse. It learns, from the calm period, how strongly this particular stock tends to track the market, then strips that shared part out. The leftover is the news. This is the default, and the default in the ARC calculator.
For the market-adjusted and market models, the prediction is the index return \(R_{mt}\) (the market's return on day \(t\)). The mean-adjusted model instead uses \(\bar R_i\), the stock's own average return over the calm period. In every case the abnormal return is still just actual minus predicted.
When you need more than the market
Two further models add more reference points than the single market index. You will not need their formulas here, only the idea of when they help.
The richer models in one line each
| Model | What it adds | When it matters |
|---|---|---|
| CAPM | Like the market model, but a classic theory dictates the prediction instead of letting the data fit freely. | Almost never used today. The theory does not hold up well, so most researchers skip it. |
| Fama-French | Adds two extra yardsticks beyond the market: how small companies did versus large ones, and how cheap companies did versus expensive ones. A later version (Carhart) adds a third: how recent winners did versus losers. | Only when your set of event stocks leans heavily small, cheap, or recently hot. If your stocks are a balanced mix, the extra factors add noise and change nothing. |
More factors look more rigorous, but they only sharpen the answer when your sample is genuinely tilted toward small, value, or momentum stocks. On a balanced, short-window study, piling on factors is rigor for show, not better results. When in doubt, the market model is the safe default.
Beyond these, a few specialist tools exist for awkward data: a beta correction for stocks that trade rarely, and volatility models for stocks whose price swings come in calm and stormy stretches. These are edge cases. The comprehensive version of this page covers them with the full math.
Ready for the formal treatment? Switch to the comprehensive version using the toggle at the top of the page. It carries every formula, the estimation-window rules, the evidence behind the market-model default, and the full reference list. Or jump straight in: the free ARC calculator fits every one of these models for you, no install required.
References
Each model on this page is traced to its primary reference; every citation was verified for year, journal, volume, and page range.
- MacKinlay, A. Craig. 1997. "Event Studies in Economics and Finance." Journal of Economic Literature 35 (1): 13-39. https://www.jstor.org/stable/2729691
- Campbell, J. Y., A. W. Lo, and A. C. MacKinlay. 1997. The Econometrics of Financial Markets. Princeton University Press. Ch. 4, 149-180. press.princeton.edu
- Brown, S. J., and J. B. Warner. 1980. "Measuring Security Price Performance." Journal of Financial Economics 8 (3): 205-258. https://doi.org/10.1016/0304-405X(80)90002-1
- Brown, S. J., and J. B. Warner. 1985. "Using Daily Stock Returns: The Case of Event Studies." Journal of Financial Economics 14 (1): 3-31. https://doi.org/10.1016/0304-405X(85)90042-X
- Sharpe, W. F. 1964. "Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk." The Journal of Finance 19 (3): 425-442. https://doi.org/10.1111/j.1540-6261.1964.tb02865.x
- Lintner, J. 1965. "The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets." The Review of Economics and Statistics 47 (1): 13-37. https://doi.org/10.2307/1924119
- Scholes, M., and J. Williams. 1977. "Estimating Betas from Nonsynchronous Data." Journal of Financial Economics 5 (3): 309-327. https://doi.org/10.1016/0304-405X(77)90041-1
- Fama, E. F., and K. R. French. 1993. "Common Risk Factors in the Returns on Stocks and Bonds." Journal of Financial Economics 33 (1): 3-56. https://doi.org/10.1016/0304-405X(93)90023-5
- Carhart, M. M. 1997. "On Persistence in Mutual Fund Performance." The Journal of Finance 52 (1): 57-82. https://doi.org/10.1111/j.1540-6261.1997.tb03808.x
- Bollerslev, T. 1986. "Generalized Autoregressive Conditional Heteroskedasticity." Journal of Econometrics 31 (3): 307-327. https://doi.org/10.1016/0304-4076(86)90063-1
- Nelson, D. B. 1991. "Conditional Heteroskedasticity in Asset Returns: A New Approach." Econometrica 59 (2): 347-370. https://doi.org/10.2307/2938260
Factor return series (SMB, HML, RMW, CMA, momentum) with regional coverage are available free from the Kenneth R. French Data Library. See the full bibliography for all sources cited across the site.
Compute these benchmarks on your own data
Apply this to your own data, free. The ARC calculator fits the market model, market-adjusted, mean-adjusted, multifactor, and Scholes–Williams specifications from a CSV upload and returns AR, CAR and CAAR, no install required.
Run it free in ARC →See also Significance tests for the parametric and non-parametric battery that decides whether the abnormal returns these models produce are different from zero. For a guided walkthrough of the model decision on a concrete dataset, follow the tutorial Choosing an expected return model; the calculation mechanics that consume these benchmarks are in How to calculate abnormal and cumulative abnormal returns.
Last reviewed: June 30, 2026. Maintained by EventStudyTools since 2014.