Alternative version of linear regression models

Question

I am learning regression this semester and we are just starting out on simple linear regression. The textbook that my lecturer is using teaches that the most basic model is of the form $$Y_i = \beta_0 + \beta_1 X_i + \epsilon_i.$$ The textbook goes on to explain the terms in the equation, which I shall not write here because it is irrelevant to my question.

I understand the above just fine.

Then, the textbook goes on to mention that an alternative form of the model can be written as $$Y_i = \beta_0X_0 + \beta_1 X_i + \epsilon_i$$ where $X_0 \equiv 1.$

The textbook just gives a one-line elaboration for the alternative model, saying that it associates an $X$ variable with each regression coefficient.

I have two questions:

1. I am just unclear as to what context would the alternative model be applicable/relevant.

2. As an aside (just a trivial question, really), why did they use $\equiv$ and not $=$? I understand that $\equiv$ means identically equal and $=$ just means equal and I have also read some posts regarding the two signs but I still have some confusion as to when which is used.

Note also that everything written above is as per the textbook; I have not rephrased anything.

I know that this may be very basic for the geniuses here, but I feel that it is important for me to get my foundations right before moving on, so any explanations will be greatly appreciated! :)

Answer 1

1. In effect, the alternative model is exactly the same as the basic one. It could be helpful for some software out there though, where all $ \beta $ terms have to have an accompanying $ X $.
2. It is not just stating that $ X_0 $ is equal to 1, but rather, it is defining $ X_0 $ as 1.