Multiple linear regression $b_0 = 0$

Question

I am trying to calculate the coefficients $b_1,b_2, ...$ of a multiple linear regression, with the condition that $b_0=0$. In Excel this can be done using the RGP Function and setting the constant to FALSE.

How can this be done with a simple Formular?

Thank you in Advance!

Answer 1

Can you give me an example, how to use the formula with multiple regression. I was expecting a variation of the least square method: $\beta_1=(X^TX)^{-1}X^Ty$

You have the regression function $y_i=\sum\limits_{i=1}^nb_1x_i$. In this case the matrix $\textbf X$ and the vector $\textbf y$ are $$\textbf X=\left(\begin{array}{} x_{11} & x_{12} &\ldots &x_{1n} \\ x_{21} & x_{22} &\ldots &x_{2n} \\ \vdots & \vdots & \vdots &\vdots \\ x_{m1} & x_{m2} &\ldots &x_{mn} \end{array}\right) \textrm{ and } \textbf y=\left(\begin{array}{} y_1 \\ y_2 \\ \vdots \\ y_n \end{array}\right)$$

In the case where the constant $b_0\neq 0$ the matrix $\textbf X$ is

$$\textbf X=\left(\begin{array}{}\color{blue} 1& x_{11} & x_{12} &\ldots &x_{1n} \\ \color{blue} 1& x_{21} & x_{22} &\ldots &x_{2n} \\ \vdots & \vdots & \vdots &\vdots&\vdots \\ \color{blue} 1& x_{m1} & x_{m2} &\ldots & x_{mn} \end{array}\right)$$