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

Question

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 ?

Answer 1

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.