Dummy Regression and Residual Methods for Event Study Abnormal Returns
Summary
This note compares two ways to estimate an event-window abnormal return. One fits a market model with an event indicator included as a regressor; the other first fits the market model, then regresses its residuals on the indicator. In both formulations, the indicator coefficient represents the event effect after accounting for the market return.
An answer argues that the coefficients are effectively the same and illustrates the comparison with a simulated return series. The reported estimates and significance statistics are very close, supporting the equivalence in that example. The note does not provide a derivation or establish that the methods match under every specification. In particular, the residual regression's equivalence depends on using the same observations and the same market model fit; different samples or estimation choices can affect results.
Key ideas
- A market model can include an event-window dummy directly to estimate an abnormal return.
- A two-stage alternative regresses market-model residuals on the event indicator.
- A simulation in the note gives closely matching event coefficients and significance results.
- The example supports equivalence but does not prove it for every sample or estimation setup.
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# Differences between dummy regression event study and regression on residuals from market model
# Differences between dummy regression event study and regression on residuals from market model
I have two different event study approaches and I wonder if the results are exactly the same.
Model 1 applies a dummy regression market model:
(1) $R_{t}=\beta_{0} + \beta_{1}R_{mt}+\beta_{2}D_{t}+\epsilon_{t}$
where ${R}_{t}$ is the return of a company at time t, $R_{mt}$ is the market return at time t and $D_{t}$ is a dummy variable that equals one in the event window and 0 otherwise. As far as I know: the coefficient $\beta_{2}$ signals the abnormal return of the event.
Model 2 applies a market model and then the dummy regression on its residuals:
(2.1) $R_{t}=\beta_{0} + \beta_{1}R_{mt}+u_{t}$
(2.2) $\hat{u}_{t}=\gamma_{0} + \gamma_{1}D_{t}+\epsilon_{t}$
Here is $D_t$ the measure for the abnormal return.
My question is: Does it make any difference to apply Model 1 or Model 2? Is the interpretation of the abnormal return measures exactly the same in both models?
Thanks for your help!
## Answer by Robert (score 1, accepted)
https://quant.stackexchange.com/a/21155
It is the same. With enough data, you could not reject the null `γ1=β2`.
You could test that with simulation.
See this with `R`:
```
##
set.seed(12456)
ns=500
t=1:ns
D[]=0
D[t>.1*ns&t<.33*ns]=1
rm=rnorm(ns,.01,1.5)
ri=0.01+1.2*rm+.15*D+rnorm(ns,0,.5)
plot(ri~rm,col=D+2)
#Model 1
summary(lm(ri~rm+D))
#Model 2
(m1=lm(ri~rm))
res=resid(m1)
summary(lm(res~D))
```
> `#Model 1 Dependent ri Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.04691 0.02504 1.873 0.06160 . rm 1.19633 0.01433 83.459 < 2e-16 *** D 0.14057 0.05246 2.680 0.00762 ** #Model 2 Dependent ri Coefficients: (Intercept) rm 0.079 1.195 #Dependent u Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -0.03198 0.02500 -1.279 0.20138 D 0.14026 0.05235 2.679 0.00762 ** `
```
#Model 1 Dependent ri
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.04691 0.02504 1.873 0.06160 .
rm 1.19633 0.01433 83.459 < 2e-16 ***
D 0.14057 0.05246 2.680 0.00762 **
#Model 2 Dependent ri
Coefficients:
(Intercept) rm
0.079 1.195
#Dependent u
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.03198 0.02500 -1.279 0.20138
D 0.14026 0.05235 2.679 0.00762 **
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