What value does one set, in general, for the null hypothesis of $\beta_0$ (“Y-intercept”)?

Question

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?

Answer 1

If you simply want to estimate the values of Y corresponding to particular values of X, you can calculate the regression to obtain estimates of $\beta_0$ and $\beta_1$, and need not be concerned with hypothesis testing.

If you want to test one or more hypotheses, then the choice of hypotheses should be driven by the purpose of your research. A common aim is to assess the relevance of an independent variable by testing the hypothesis that its parameter equals zero, but that is just one choice. In this case you might want to test the hypothesis that $\beta_1=0$, and you can do so without testing any hypothesis about $\beta_0$.

Where matters possibly become confusing is that regression software often, by default, generates t-values for all the independent variables and the constant. If you want to test the hypotheses that $\beta_0=0$ or $\beta_1=0$, these t-values are useful. But you can also ignore them if you are not interested in testing those hypotheses. The estimated parameters don't depend on these hypotheses or t-values, so you can still use them in estimating values of Y.

Answer 2

It would vary from case to case. To clarify: You can set whatever you want as a null hypothesis, even $B_0 = \pi$. You have to choose your null based on what you are testing for.

An example:

Suppose I flip a 1 euro coin once everyday and you win it if it comes out heads. Let $Y_i$ denote your winnings on day $i$ and let $X_i$ be the average temperature of day $i$. If you gathered data for a year you could probably not reject $B_1 = 0$.
If the coin is fair (50% chance of heads) then you can also not reject $B_0 = \frac{1}{2}$.
(This makes $u$ a random variable with mean 0.)
But suppose it is a trick coin that never comes out heads. Then $B_0$ is 0. So if you want to test fairness, you would say in your null hypothesis that $B_0 = \frac{1}{2}$. If you reject this then you assume the coin is not fair. But remember that the unit of measurement of $B_0$ here is not probability, but euros. If I had a 2 euro coin instead of a 1 euro coin your null to measure fairness should be $B_0 = 1$.

Thus even when testing for the same thing, the value of $B_0$ in your null would vary from case to case.

Answer 3

'When testing your sample's regression model ' --> Testing for significance you mean?

I think it depends on what X and Y are. Sometimes testing $B_0 = 0$ is meaningless. Iirc, one replaces zero with some other number depending on the data or the variables.

Check out the section 'An α-level hypothesis test for intercept parameter β0'.

This seems to disagree with me. Not really sure. I think you'll get faster and better answers in Stats SE.

Answer 4

The "constant term" in the regression specification may be a consequence of the theoretical model that is estimated. For example, if we assume a production function of the form

$$Q = AK^aL^b$$ then taking logarithms we obtain the linear-in-logarithms specification

$$\ln Q = \ln A + a\ln K + b\ln L$$

and it appears that the constant term in the econometric regression specification will be estimating the logarithm of the production shifter (technology) $A$.

...but also, the constant term guards to a degree against misspecification: if we wrongly assume that the error term that will appear in the stochastic specification has a zero mean, then the inclusion of the constant term, even if it is not provided by the theory, effectively "takes on" this non-zero value, permitting to treat the now transformed stochastic error as indeed zero-mean. This is important for one of the dominant methods of estimation in econometrics, Least-squares: this method produces unavoidably residuals that have zero mean. And we want to be able to assume that the unknown error is also zero-mean, so that we can treat the residuals as a good estimator of the unknown error.

Now if one puts these two together, one realizes that interpreting the constant term is problematic: does it represent the constant postulated by the theory, or the non-zero mean of the error term? And if both, what meaning could we give to its magnitude, so as to subsequently test this magnitude?

...this is why, in many econometrics papers, the constant term is not discussed at all, while in others, it is not even reported.