How to interpret regression coefficients with dummy explanatory variables?

Question

I am a bit confused about the interpretation of the regression coefficients in a regression model:

$R_{t}=\beta_0+\beta_1R_{mt}+\beta_2D_{t}+\epsilon_t$

where $R_{t}$ is the log return of some stock, which is defined as $log(P_t) - log(P_{t-1})$, $R_{mt}$ is the log return of some market index e.g., SP500) and $D_t$ is a dummy variable ($D_t=1$ if earnings announcements are published on day $t$ and $D_t = 0$ otherwise).

The results are $\beta_1= 0.024$ and $\beta_2= -0.03$. Is the following interpretation correct?

(1) an increase in the market return of 1% leads to an increase of the stock return of 2.4% or 0.024% (as both variables are in logs and thus $\beta_1$ can be interpreted as elasticity)?

(2) And on days with earnings announcements, the return is -3% or -0.03% lower than the average return of the stock (here we have a log dependent and a non-log independent)?

Answer 1

The dummy function is always used to construct non-linear models. In your model, it is interpreted that the announcements have an non-linear effect on the return. So it is incorrect to say it is a linear regression problem, it should be called as a non-linear regression problem. In total, it means the announcements have asymmetric effects in explaining the returns.

Answer 2

To answer you correctly we'd need to see the exact inputs of your regression... and I doubt you can mix easily linear and binary variables like that.

If the market return is 1% at time $t$ do you have $R_{m,t} = 0.01$ or $R_{m,t} = 1$. Same question for $R_t$

Assuming both are using the "0.01" convention, then a move of $1\% = 0.01$ results in a move of $\beta_1 \cdot 0.01 = 0.00024 = 0.024\%$. Same reasoning for the other beta.

You should also make sure that the parameters you fitted are statistically meaningful, by checking their p-values, as a starting point.

Answer 3

Is this for one firm only? Is there positive and negative announcements (ie do the abnormal returns differ in sign)?

As per Binder (1998): $$R_{it}=\alpha _{i} + \beta _{i}R_{mt} + \gamma _{i}D_{i} + u_{it} $$

where the coefficient $\gamma _{i}$ is the abnormal return for security $i$ during period $t$. If the events tend to affect the security prices both positive and negative, a regression such as yours tend not be very powerful. Binder (1998) suggests a multivariate regression model with one equation for each of the $N$ events. $$ \\ R_{1t}=\alpha _{1} + \beta _{1}R_{mt} + \sum_{a=1}^{A} \gamma _{1a}D_{at} + u_{1t} \\ \vdots \\ R_{Nt}=\alpha _{N} + \beta _{N}R_{mt} + \sum_{a=1}^{N} \gamma _{1a}D_{Nt} + u_{1t} $$

Answer 4

What you are doing is to try to construct a variable (Rt) by decomposing the value in some explanatory components. Therefore your interpretation is correct. You need to substitute the values you obtained in your equation, and that gives you the answer to your question:

(1)0.024-> 2.4%

(2)−0.03-> -3%