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?

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    $\begingroup$ There's simply no "typical" value to use. In some applications 0 makes sense, in others 0 would make no sense at all. In some of those "0 isn't meaningful for a null" cases, some specific null value might exist, but often none does. $\endgroup$ – Glen_b Jul 1 '15 at 12:15

Rote calculation of p-values for tests of uninteresting null hypotheses ought to be discouraged; & much more often than not, tests that intercepts—as well as first-order terms when higher-order (interactions & non-linear) terms are included in the model—are equal to nought are uninteresting. If there's an interesting hypothesis about the intercept in any particular case, by all means test it; but you're not obliged to perform a hypothesis test at all.

Why do we often see p-values reported for tests of uninteresting null hypotheses? Software's to blame for spitting out p-values for every test of $\beta_j=0$ by default. (There may be more to it—see The abundance of P values in absence of a hypothesis.)

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