Say there is a linear regression model to estimate Y, that is:
$Y_i = B_0 + B_1X_i + u$
When testing the significance of your sample regression model 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? A lot of sources tend to set the null hypothesis as zero but 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, for no a priori reason? In general is there any non-arbitrary setting for $B_0$ we would test against?
Or does the "sensible" value to test $B_0$ for significance against 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 that is being disproved or is the data presented only to illustrate the confidence intervals and other statistics, such as variance?