# Interaction term significant, joint hypothesis not

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?

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) in the sense that the probability of rejecting at least one of the three is (likely) bigger than the nominal value of the test.