# Is there any case that the coefficient of variable of interest in DiD more significant when adding more variables?

Considering a Difference-in-Differences equation with a dummy variable:

$$y_i = \alpha + \beta D_i + Xit + \varepsilon_i.$$ While $$Xit$$ is a set of covariates. Today when I add more covariates, the p-value decrease. Is it normal?

And if it is a problem, what should I deal with that?

• It can happen if $D$ and $X$ are not much correlated each other and X is important for explaining $y$, when X decreases the variance of the (remaining) error and the variability in D is not much affected by partialing out X, resulting in a reduced standard error. Sep 22 at 14:37

First of all, let me tell you that it is bad statistics to compare $$p$$-values across different specification. What you certainly should not do is to pick the specification based on the $$p$$-values that you obtain.
Having said this. Assume that the right specification is given by: $$y_i = \alpha + \beta D_i + \gamma X_{i} + \varepsilon_i$$ Assume you wrongly estimate: $$y_i = \alpha + \beta D_i + \delta_i$$ where now $$\delta_i = \varepsilon_i + \gamma X_i$$ captures both the random error $$\varepsilon_i$$ and the effect of $$X_i$$.
If you would estimate the second specification, you actually identify the following: \begin{align*} \hat \beta &= \mathbb{E}(y_i|D_i = 1) - \mathbb{E}(y_i|D_i = 0),\\ &= \beta + \mathbb{E}(\delta_i|D_i = 1) - \mathbb{E}(\delta_i|D_i = 0),\\ &= \beta + \gamma\left[\mathbb{E}(X_i|D_i = 1) - \mathbb{E}(X_i|D_i = 0)\right] \end{align*} Depending on the second term, $$\hat \beta$$ might be lower or higher than $$\beta$$. This will depend on the sign of $$\gamma$$ and the direction of correlation between $$X_i$$ and $$D_i$$. For example if $$\gamma > 0$$ and $$D_i = 1$$ is associated with higher values of $$X_i$$ then $$\hat \beta > \beta$$, so you will tend to overestimate the effect of $$D_i$$ on $$y_i$$ (as you are also capturing part of the effect of $$X_i$$ in your estimate).
Whether you should include $$X_i$$ or not into the regression is not something that can be answered using statistics alone. If $$X_i$$ captures confounders (i.e. something that affects both $$y_i$$ and $$D_i$$), then yes you should add it as you otherwise do not capture the causal effect of $$D_i$$ on $$y_i$$ but also part of the effect of $$X_i$$ on $$y_i$$.
When you add covariates, $$\beta$$ is interpreted as "the effect of $$D$$ on $$y$$ holding fixed $$X$$. Adding variables to $$X$$ can either increase or decrease the p-value depending on the relationship of $$X, D,$$ and $$y$$.