# Interpretation of t-test in event study with dummy regression

I am not sure about my interpretation of the t-ratios in dummy regression models for event studies. I have the results for two different groups of models examining the impact of news on stock returns and I want to compare them.

The first group applies the following model:

(1) $R_{t}=\beta_{0} + \beta_{1}R_{mt}+\beta_{2}D_{Gt}+\beta_{3}D_{Bt}+\epsilon_{t}$

where ${R}_{t}$ is the return of a company at time t, $R_{mt}$ is the market return at time t, $D_{Gt}$ is a dummy variable that equals one in the event window of Good News occurring and $D_{Bt}$ is a similar dummy indicating the occurrence of bad news. Thus, the coefficient $\beta_{2}$ ($\beta_{3}$) signals the abnormal returns after good (bad) news.

The second group includes just a dummy for good news:

(2) $R_{t}=\beta_{0} + \beta_{1}R_{mt}+\beta_{4}D_{Gt}+\epsilon_{t}$

here $D_{Gt}$ is equal to one if good news occur and 0 if bad news occur. Thus, $\beta_{4}$ shows the difference in the returns after good news in comparison to bad news.

My question is: How to get the absolute abnormal returns for good and bad news from model type 2? Is the abnormal return after good news $\beta_{0}+\beta_{4}$? And if $\beta_{4}$ has a t-value of 3.00, can I say that the t-value of $\beta_{0}+\beta_{4}$ is also 3.00 and thus the abnormal returns after good news are statistically significant?

As far as your second model concerned:

1. Abnormal returns for good news is $\beta_4$
2. The t-value of 3 tells it is significantly different from 0
3. The model does not account for effect of bad news so the effect of bad news will mostly be found in spikes in residuals around time of bad news releases.
4. $\beta_0$ is return when all other factors in the model (market return, good news) are 0, i.e. more or less "risk-free" return.

For the t ratio, you should re-parameterise your equation so that $(β_0+β_4)$ is treated as one coefficient, say $\gamma$ or a coeff of a single variable. you cannot use one's t ratio for making inference on the other. $β_0$ is the average return on this stock, the coef on dummy variable absorbs the abnormal returns. You could do this: $$R_t=\beta_0+\beta_{0}D_{Gt}-\beta_{0}D_{Gt}+\beta_{1}R_{mt}+\beta_{4}D_{Gt}+\epsilon$$ then, collecting the common terms you will have $$R_t=\beta_0(1-D_{Gt})+\beta_{1}R_{mt}+(\beta_{0}+\beta_{4})D_{Gt}+\epsilon$$

here, $\gamma=(\beta_{0}+\beta_{4})$, and $D=(1-D_{Gt})$ is a new variable which you can create in excel. you should now regress $R_t$ as follows $$R_t=\beta_0D+\beta_{1}R_{mt}+\gamma D_{Gt}+\epsilon$$, the t value of $\gamma$ can be used for inference on $(\beta_{0}+\beta_{4})$.

I am not quit understand what do you mean by absolute abnormal returns. I can understand you want to test the asymmetric effects of good news and bad news. There is a similar test named sign bias test by Engle, exactly, which is applied in the GARCH model, you can check it.

My understanding, if you want to test the difference effects of good news and bad news, your equation 1 is enough. In your equation 2, I assume $D_{Gt}$ is different from it is equation 1 according to your explanation, then $\beta_4$ is fine to explain that the abnormal returns for good and bad news (in total) affect the return of the company.