Test whether the sum of two coefficients is significantly different that zero in indicator regression

Question

I am using indicator regression with an interaction term to compare the slopes to two groups.

Here is the form of the equation:

$Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + + \beta_3 X_1 X_2$

$X_2 $ is my dummy variable (1 or 0).

Based on how I set up the regression:

When $X_2 = 0$, then the $slope_0 = \beta_1 $.

When $X_2 = 1$, then the $slope_1 = \beta_1 + \beta_3 $.

What I would like to determine is whether there is a test to determine whether $slope_1$ is significantly different than zero? I know I can split the dataset into two (based on the dummy variable) and regress separately to determine whether $slope_1$ is significantly different than zero, but could I could determine this without the separate regression? Is there a test for this in R?

The two most obvious alternatives are to reparameterize so that your sum of coefficients in the original formulation becomes a single parameter under the reparameterized formulation (and from there you can just look at the p-value of the resulting coefficient estimate in the subsequent regression output), or you can perform a test of the general linear hypothesis. Both approaches (for a slightly different hypothesis) are discussed here. — Glen_b, Nov 28 '16 at 11:02
Thanks, that seems like what I am looking for. Just to confirm, in the re-parameterization, the coefficients will not have much (physical) meaning, but the p-values will tell you whether the re-parameterized coefficients are significant or not? — ken, Nov 30 '16 at 22:07
Sometimes it's the case that the reparameteried coefficients don't have a direct physical meaning (though they're usually interpretable as sums or differences of coefficients), So sure, you might in some situation say "well, $\beta_1+\beta_3$ doesn't have direct physical meaning" but ... "$\text{slope}_1$ when $X_2=1$" seems like a readily understood concept. — Glen_b, Nov 30 '16 at 22:15
Possible duplicate of Is there a hypothesis test for B1 > B2 in multiple regression? — kjetil b halvorsen, Aug 09 '18 at 17:25
Possible duplicate of Testing whether two regression coefficients are significantly different (in R ideally) — mdewey, Aug 10 '18 at 12:31

Test whether the sum of two coefficients is significantly different that zero in indicator regression

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