Say there is a linear regression model to estimate Y, that is:
$Y_i = B_0 + B_1X_i + u$
When testing the Betas of our sample's regression model for significance the null hypothesis for $B_1$ would naturally be set as zero (assuming X has no impact on Y).
However, what would the null hypothesis for $B_0$ be set as? If it is set as zero, is that not assuming that a certain value is taken by Y in the absence of X, i.e. zero? In general is there any non-arbitrary setting for $B_0$ we would test against?
Or does the value of $B_0$ vary from case to case?
I am aware that a sampling test is conducted on $B_0$ and as such am curious as to what the exact purpose of this test would be? Is there any actual null hypothesis $B_0$ is being tested against or is the data presented only to illustrate the confidence intervals and other statistics, such as variance?