# Joint significance test in a difference-in-difference setting

I ran following difference-in-difference regression in software and have some questions about the interpretation of the resulting coefficients and their (joint) significance:

$$P_{it} = \gamma_i + \lambda_t + \eta T_i + P^{C}_{t} + P^{f_1}_{t} + P^{f_2}_{t} + \delta_1 (T_i \times P^{C}_{t}) + \delta_2 (T_i \times P^{f_1}_{t}) + \delta_3 (T_i \times P^{f_2}_{t}) + \epsilon_{it},$$

where $$P_{it}$$ (e.g. stock return ) is observed for firm $$i$$ on day $$t$$ during the first and second quarters of 2020. The parameters $$\gamma_i$$ and $$\lambda_t$$ denote fixed effects for firms and days, respectively. $$T_i$$ is a treatment dummy which equals 1 for firm $$i$$ if it is in Treatment group, 0 otherwise. The post-treatment indicators $$P^{C}_{t}$$, $$P^{f_1}_{t}$$, $$P^{f_2}_{t}$$ index the different post-treatment epochs.

To be precise:

• $$P^{C}_{t}$$ equals 1 in all firms from February 24th to June 30th, 2020, 0 otherwise
• $$P^{f_1}_{t}$$ equals 1 in all firms from March 18th to June 30th, 2020, 0 otherwise
• $$P^{f_2}_{t}$$ equals 1 in all firms from March 27th to June 30th, 2020, 0 otherwise

Note that $$P^{f_1}_{t}$$ and $$P^{f_2}_{t}$$ partially overlap with the post-treatment indicator $$P^{C}_{t}$$ in order to control for the two subsequent events (introduced March 18th & introduced March 27th). Furthermore, all post-treatment indicators 'turn on' at different points in time, but 'stay on' until June 30th.

The respective, exemplary dataset would look like:

$$\begin{array}{ccc} firm & day & T_i & P^{C}_t & P^{f_1}_t & P^{f_2}_t \\ \hline 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 2 & 0 & 0 & 0 & 0 \\ 1 & 3 & 0 & 0 & 0 & 0 \\ 1 & 4 & 0 & 1 & 0 & 0 \\ 1 & 5 & 0 & 1 & 1 & 0 \\ 1 & 6 & 0 & 1 & 1 & 1 \\ \hline 2 & 1 & 1 & 0 & 0 & 0 \\ 2 & 2 & 1 & 0 & 0 & 0 \\ 2 & 3 & 1 & 0 & 0 & 0 \\ 2 & 4 & 1 & 1 & 0 & 0 \\ 2 & 5 & 1 & 1 & 1 & 0 \\ 2 & 6 & 1 & 1 & 1 & 1 \\ \end{array}$$

Running the regression in software results in following coefficients:

Variable Coefficients (𝛿)
$$T_i$$ -0,000
$$P^{C}_{t}$$ -0.012***
$$T_i$$ * $$P^{C}_{t}$$ 0,004***
$$P^{f_1}_{t}$$ 0,020***
$$T_i$$ * $$P^{f_1}_{t}$$ -0.011***
$$P^{f_2}_{t}$$ -0,007***
$$T_i$$ * $$P^{f_2}_{t}$$ 0.006***

Following exemplary statements can be drawn from the results:

1. $$T_i$$ * $$P^{C}_{t}$$: Firms in the Treatment group earned an average daily return of 0.4% percent relative to firms in the Control-group from February 24th to March 17th. The results are statistically significant at the 1% level.

2. $$T_i$$ * $$P^{f_1}_{t}$$: Firms in the Treatment-group earned an additional average daily return of -1,1% after the imposition of the first event introduced March 18th, 2020 compared to firms in the Control-group. The results are statistically significant at the 1% level. Furthermore, this results to an unique, overall average daily return of (+0,4% -1,1%) = -0,7% of firms in Treatment-group during March 18th and March 26th compared to firms in Control-group.

My questions is related to the last sentence: As stated in the last sentence I also want to make statements about the unique, overall effects within the individual time frames (by adding up subsequent coefficients on the interaction terms, just as I did above). Do I have to test for "joint significance" of the sum of coefficients to make valid statements? In other words, e.g. both coefficients on the first and second interaction term are statistically significant at the 1% level. Is this still the case if I add them up in order to get the unique effect? If not, how do I test this?

Thanks for the help, very much appreciated!

• This is related to a question I answered here. It is crossposted on CV, though arguably better suited for this community. – Thomas Bilach Apr 2 at 6:50