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I am fitting the following model to my data $y = \beta_0 + \beta_1 x_1 + \beta_2 x_2$ and the claim I want to be able to make is that $\beta_1 = 0$ and $\beta_2 \neq 0$. I am going to use bootstrapping for this purpose. However, I don't know how to set up a test statistic for my claim since I have $\beta_1 \neq 0$ in my null hypothesis.

On a higher level what I want to show is that my output $y$ is not affected by $x_1$ whereas the variable $x_2$ helps me predict this output.

My apologies in advance if my question is ill-posed or silly. I am still in the process of coming up with a precise formulation of my problem. So any constructive feedback is appreciated.

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  • $\begingroup$ The two formulations you make of what you are trying to infer are the same. Claiming that $\beta_1 = 0$ is the same as showing that $y$ is not effected by $x_1$. Why do you pose a different hypothesis for both variables? You can also claim $\beta_1 = 0$ and $\beta_2 = 0$ and see if with bootstrapping you'll reject the second and keep the first. $\endgroup$
    – Ivo
    Jun 26, 2015 at 8:59
  • $\begingroup$ It doesn't make sense to have a null of $\neq 0$; any amount of sample variation will always leave your sample more consistent with the null than the alternative. You should probably explain what you're trying to achieve. $\endgroup$
    – Glen_b
    Jun 26, 2015 at 9:01

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One reasonable way to approach something like that would be to abandon the notion of a null of the real line minus a mere single point (i.e. a point alternative, which would yield not chance of rejection) and instead to do equivalence testing on the second parameter, perhaps along with an ordinary hypothesis test on the first parameter.

There are numerous posts on site relating to equivalence testing.

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