<p>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 <span class="math-container">$\gamma$</span> coefficient.</p>
<p>OVB can drastically change the value of coefficients, if the true model is given by</p>
<p><span class="math-container">$$Y_{it}=\alpha_i+β_t+\gamma D_{it} + \theta X_{it} + e_{it}$$</span></p>
<p>but you only fit</p>
<p><span class="math-container">$$Y_{it}=\alpha_i+β_t+\gamma D_{it}  + e_{it}$$</span></p>
<p>it can be shown that the parameter estimate will be given:</p>
<p><span class="math-container">$$E[\hat{\gamma}]= \gamma + \theta \frac{COV(D,X)}{VAR(D)}$$</span></p>
<p>So unless <span class="math-container">$COV(D,X)=0$</span>, or unless <span class="math-container">$\theta=0$</span>, if you omit theta your estimate of gamma will biased in a unpredictable way.</p>
<p>For example if the true coefficient of <span class="math-container">$\gamma$</span> is <span class="math-container">$-0.5$</span> and the true coefficient of theta is <span class="math-container">$-2$</span> and variance of <span class="math-container">$D$</span> is <span class="math-container">$1$</span> and covariance of <span class="math-container">$D$</span> and <span class="math-container">$X$</span> is also <span class="math-container">$1$</span> omitting theta would result in the following biased coefficient:</p>
<p><span class="math-container">$$E[\hat{\gamma}] = -0.5 -2 \cdot \frac{1}{1}=-2.5$$</span></p>
<p>of course the larger absolute value of coefficient the higher significance you get (ceteris paribus) since you are testing hypothesis of <span class="math-container">$\gamma \neq 0$</span>.  Thus once you add <span class="math-container">$\theta$</span> and thus remove omitted variable bias from <span class="math-container">$\gamma$</span>, assuming <span class="math-container">$\gamma$</span>'s standard error does not change, you should expect <span class="math-container">$\gamma$</span> to be less significant in the example above. This is basically what is happening to you.</p>

CC BY-SA 4.0
2021-09-05T10:16:57.997Z
2021-09-05T10:16:57.997Z
1muflon1
<p>I am working on DiD setting, the variable of interest is</p>
<p><span class="math-container">$Y_{i,t} = \alpha_i + \beta_t + \gamma D_{i,t}$</span></p>
<p>while <span class="math-container">$i$</span> and <span class="math-container">$t$</span> are unit and time fixed effects. <span class="math-container">$\gamma$</span> is the coefficient of variables of interest D.</p>
<p>In another word, <span class="math-container">$\gamma$</span> tells us how the laws affect the outcome Y.
In my research, the result is <span class="math-container">$\gamma$</span> is negative and significant at 1%. However, when I add one more control variable called <span class="math-container">$\theta$</span> into the regression, the significant levels of <span class="math-container">$\gamma$</span> reduces to 5%, and the coefficient of <span class="math-container">$\gamma$</span> become higher (less negative).</p>
<p>What should I say about the role of <span class="math-container">$\theta$</span> in this situation ?</p>

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

(Asked by: username_0 on 2021-09-05)

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 ?

**Tags:** econometrics

---

## 1 answers

### Answer by username_1 on 2021-09-05. Score: 2 | Accepted answer

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.


---

