Differences between dummy regression event study and regression on residuals from market model

Question

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 1

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 **