# How can we explain when adding a variable to a regression, the coefficient and significant levels of variable of interest decrease?

I am working on DiD setting, the variable of interest is

$$Y_{i,t} = \alpha_i + \beta_t + \gamma D_{i,t}$$

while $$i$$ and $$t$$ are unit and time fixed effects. $$\gamma$$ is the coefficient of variables of interest D.

In another word, $$\gamma$$ tells us how the laws affect the outcome Y. In my research, the result is $$\gamma$$ is negative and significant at 1%. However, when I add one more control variable called $$\theta$$ into the regression, the significant levels of $$\gamma$$ reduces to 5%, and the coefficient of $$\gamma$$ become higher (less negative).

What should I say about the role of $$\theta$$ in this situation ?

This is because the first coefficient estimate was estimated in the presence of omitted variable bias (OVB), and the effect of omitted variable just previously loaded onto the the $$\gamma$$ coefficient.

OVB can drastically change the value of coefficients, if the true model is given by

$$Y_{it}=\alpha_i+β_t+\gamma D_{it} + \theta X_{it} + e_{it}$$

but you only fit

$$Y_{it}=\alpha_i+β_t+\gamma D_{it} + e_{it}$$

it can be shown that the parameter estimate will be given:

$$E[\hat{\gamma}]= \gamma + \theta \frac{COV(D,X)}{VAR(D)}$$

So unless $$COV(D,X)=0$$, or unless $$\theta=0$$, if you omit theta your estimate of gamma will biased in a unpredictable way.

For example if the true coefficient of $$\gamma$$ is $$-0.5$$ and the true coefficient of theta is $$-2$$ and variance of $$D$$ is $$1$$ and covariance of $$D$$ and $$X$$ is also $$1$$ omitting theta would result in the following biased coefficient:

$$E[\hat{\gamma}] = -0.5 -2 \cdot \frac{1}{1}=-2.5$$

of course the larger absolute value of coefficient the higher significance you get (ceteris paribus) since you are testing hypothesis of $$\gamma \neq 0$$. Thus once you add $$\theta$$ and thus remove omitted variable bias from $$\gamma$$, assuming $$\gamma$$'s standard error does not change, you should expect $$\gamma$$ to be less significant in the example above. This is basically what is happening to you.