How do I calculate the y-intercept for multiple linear regression?

Question

I'm new to much of this and find that a good way to wrap my head around some of these concepts is to calculate them by hand. After working through linear regression, I thought multiple regression would be straightforward because I read that multiple regression is the linear combination of the independent variables. But I can't seem to learn how to calculate the y-intercept ($b_0$) for multiple regression. Maybe I'm using the wrong terminology because I'm finding this difficult to google.

For simple linear regression, the y-intercept is:

$b_0 = \bar y -b_1\bar x$

The equation for multiple regression is:

$ \hat y = b_0 + b_1x_1 + b_2x_2 +b_3x_3... $

I don't understand where the $b_0$ comes from in the multiple regression equation because wouldn't we have a different $b_0$ for every independent variable? How do they turn into a single y-intercept?

Thanks in advance!

Answer 1

The intercept is just a free scalar parameter that is optimized to minimize the squared error loss (i.e. find the best fit line).

Note that optimizing $b_0$ in the fit: $$\hat{y} = b_0 + b_1 x_1 + b_2 x_2 + ...$$ is actually just as flexible as optimizing $b_{0,1}, b_{0,2}, ...$ in the fit: $$\hat{y} = b_{0,1} + b_1 x_1 + b_{0,2} + b_2 x_2 + ...$$

because we could always simply define: $b_0 = b_{0,1} + b_{0,2} + ...$.

So there is no additional benefit that would come from having a different intercept for each independent variable, the resulting squared error would not be any lower.