<p>Looking at what you wrote, your regression has the form:
<span class="math-container">$$
y_i = \beta_0 + \beta_1 x_{1,i} + \beta_2 x_{1,i} x_{2,i} + \varepsilon_i
$$</span>
Here <span class="math-container">$x_2$</span> is a dummy taking the value <span class="math-container">$1$</span> for a subsample of observations and <span class="math-container">$0$</span> otherwise.</p>
<p>Then:
<span class="math-container">$$
E[y_i|x_{1,i} = x_1, x_{2,i} = 0] = \beta_0 + \beta_1 x_{1}.
$$</span>
and
<span class="math-container">$$
E[y_i| x_{1,i} = x_1, x_{2,i} = 1] = \beta_0 + (\beta_1 + \beta_2) x_{1}.
$$</span></p>
<p>There are various hypothesis that you can test:</p>
<ol>
<li><span class="math-container">$H_0$</span>: there is no (mean) effect of <span class="math-container">$x_1$</span> on <span class="math-container">$y$</span> for the subset where <span class="math-container">$x_{2} = 0$</span>. This amounts to a test of <span class="math-container">$\beta_1 = 0$</span>.</li>
<li><span class="math-container">$H_0$</span>: there is no (mean) effect of <span class="math-container">$x_1$</span> on <span class="math-container">$y$</span> for the subset where <span class="math-container">$x_{2} = 1$</span>. This amounts to a test of <span class="math-container">$\beta_1 + \beta_2 = 0$</span>.</li>
<li><span class="math-container">$H_0$</span>: the (mean) effect of <span class="math-container">$x_1$</span> on <span class="math-container">$y$</span> is the same for the subsets where <span class="math-container">$x_2 = 1$</span> and <span class="math-container">$x_2 = 0$</span>. This amounts to a test of <span class="math-container">$\beta_2 = 0$</span>.</li>
</ol>
<blockquote>
<p>I am getting that the interaction term is significantly different from zero,</p>
</blockquote>
<p>This is a test of point 3. that the (mean) effect is different for the two subgroups.</p>
<blockquote>
<p>when I test whether beta_x1+beta_interaction*[mean of x2]=0, I get that the H0 is not rejected</p>
</blockquote>
<p>You should not multiply <span class="math-container">$\beta_2$</span> with the mean of <span class="math-container">$x_2$</span>. To test for 2, you should conduct the hypothesis test that <span class="math-container">$\beta_1 + \beta_2 = 0$</span>.</p>
<blockquote>
<p>when I test whether beta_x1=0, that is also not rejected.</p>
</blockquote>
<p>This is a test of 1. which indicates that you cannot reject that there is no (mean) effect for the subgroup where <span class="math-container">$x_2 = 0$</span>.</p>
<p><strong>Remark 1</strong>: Hypothesis tests are a bit <em>strange</em> in the sense that they can contradict each other. For example it is possible that you do not reject <span class="math-container">$\beta_1 = 0$</span> and <span class="math-container">$\beta_2 = 0$</span> but you do reject <span class="math-container">$\beta_1 + \beta_2 = 0$</span>.</p>
<p><strong>Remark 2</strong>: related to this, notice that the tests are not independent. So if you test all three hypothesis, you have the <em>mutliple testing</em> issue (cfr <a href="https://en.wikipedia.org/wiki/Multiple_comparisons_problem" rel="nofollow noreferrer">wiki link</a>) in the sense that the probability of rejecting at least one of the three is (likely) bigger than the nominal value of the test.</p>

CC BY-SA 4.0
2021-04-30T06:38:13.380Z
2021-04-30T06:38:13.380Z
tdm
<p>I am trying to find out if some channel is important in the transmission of shock x1 to outcome variable y.
For this, I am running a regression in which y is on the LHS and x1 on the RHS, once alone and once interacted with a variable x2 that differentiates between observations where the channel can exist and where it cannot.
I am getting that the interaction term is significantly different from zero, but when I test whether beta_x1+beta_interaction*[mean of x2]=0, I get that the H0 is not rejected, and when I test whether beta_x1=0, that is also not rejected.
What would you suggest, should I conclude that the channel is important on the basis of this?</p>

<econometrics>
Interaction term significant, joint hypothesis not
lippi
# Interaction term significant, joint hypothesis not

(Asked by: username_0 on 2021-04-30)

I am trying to find out if some channel is important in the transmission of shock x1 to outcome variable y.
For this, I am running a regression in which y is on the LHS and x1 on the RHS, once alone and once interacted with a variable x2 that differentiates between observations where the channel can exist and where it cannot.
I am getting that the interaction term is significantly different from zero, but when I test whether beta\_x1+beta\_interaction\*[mean of x2]=0, I get that the H0 is not rejected, and when I test whether beta\_x1=0, that is also not rejected.
What would you suggest, should I conclude that the channel is important on the basis of this?

**Tags:** econometrics

---

## 1 answers

### Answer by username_1 on 2021-04-30. Score: 3 | Accepted answer

Looking at what you wrote, your regression has the form:
$$
y\_i = \beta\_0 + \beta\_1 x\_{1,i} + \beta\_2 x\_{1,i} x\_{2,i} + \varepsilon\_i
$$
Here $x\_2$ is a dummy taking the value $1$ for a subsample of observations and $0$ otherwise.

Then:
$$
E[y\_i|x\_{1,i} = x\_1, x\_{2,i} = 0] = \beta\_0 + \beta\_1 x\_{1}.
$$
and
$$
E[y\_i| x\_{1,i} = x\_1, x\_{2,i} = 1] = \beta\_0 + (\beta\_1 + \beta\_2) x\_{1}.
$$

There are various hypothesis that you can test:

1. $H\_0$: there is no (mean) effect of $x\_1$ on $y$ for the subset where $x\_{2} = 0$. This amounts to a test of $\beta\_1 = 0$.
2. $H\_0$: there is no (mean) effect of $x\_1$ on $y$ for the subset where $x\_{2} = 1$. This amounts to a test of $\beta\_1 + \beta\_2 = 0$.
3. $H\_0$: the (mean) effect of $x\_1$ on $y$ is the same for the subsets where $x\_2 = 1$ and $x\_2 = 0$. This amounts to a test of $\beta\_2 = 0$.

> I am getting that the interaction term is significantly different from zero,

This is a test of point 3. that the (mean) effect is different for the two subgroups.

> when I test whether beta\_x1+beta\_interaction\*[mean of x2]=0, I get that the H0 is not rejected

You should not multiply $\beta\_2$ with the mean of $x\_2$. To test for 2, you should conduct the hypothesis test that $\beta\_1 + \beta\_2 = 0$.

> when I test whether beta\_x1=0, that is also not rejected.

This is a test of 1. which indicates that you cannot reject that there is no (mean) effect for the subgroup where $x\_2 = 0$.

**Remark 1**: Hypothesis tests are a bit *strange* in the sense that they can contradict each other. For example it is possible that you do not reject $\beta\_1 = 0$ and $\beta\_2 = 0$ but you do reject $\beta\_1 + \beta\_2 = 0$.

**Remark 2**: related to this, notice that the tests are not independent. So if you test all three hypothesis, you have the *mutliple testing* issue (cfr [wiki link](https://en.wikipedia.org/wiki/Multiple_comparisons_problem)) in the sense that the probability of rejecting at least one of the three is (likely) bigger than the nominal value of the test.


---

