How can I calculate Slope and Y-intercept in Multiple Regression? What is the formula for Slope and Y-intercept in Multiple Linear Regression? We can easily find Slope and Y-intercept of Linear Regression meaning the data having only one Independent Variable?
Is there any general way to calculate it?
 A: Suppose you have the following regression function: $ y_i = \beta_{0} + \beta_{1} x_{i1} + \cdots + \beta_{p} x_{ip} + \varepsilon_i$, where $\varepsilon_i$ is the random part (white noise). Here you have $p+1$ parameters. To estimate the the parameters $b_0,b_1,\ldots, b_p$ we need the following matrix and vectors.
$\mathbf{y} = \begin{pmatrix} y_1 \\ y_2 \\ \vdots \\ y_n \end{pmatrix},X =  \begin{pmatrix} 1 &  x_{11} & \cdots & x_{1p} \\
 1 & x_{21} & \cdots & x_{2p} \\
 \vdots & \vdots & \ddots & \vdots \\
 1 & x_{n1} & \cdots & x_{np}
 \end{pmatrix}, \textbf β = \begin{pmatrix} \beta_0 \\ \beta_1 \\ \beta_2 \\ \vdots \\ \beta_p \end{pmatrix}, $
Suppose you have the following data

Here is $p=2$ and $n=17$. Then the corresponding vector and matrix is
$\mathbf{y} = \begin{pmatrix} 251.3 \\ 251.3 \\ \vdots \\ 349.0 \end{pmatrix},X =  \begin{pmatrix} 1 &  41.9 & 29.1 \\
 1 & 34.4 & 29.3 \\
 \vdots & \vdots & \vdots \\
 1 & 77.8 & 32.9
 \end{pmatrix}$
The estimated parameters are
$ \begin{pmatrix}  \hat \beta_0 \\  \hat \beta_1 \\  \hat \beta_2 \\ \end{pmatrix}=\hat \beta=(\mathbf{X^{'}X)^{-1}}\cdot \mathbf{X^{'}}\cdot \mathbf{y}$
Here $X^{'}$ denotes the transpose of $X$ and $(X^{'}X)^{-1}$ the inverse of $X^{'}X$
The result is $\begin{pmatrix}  \hat \beta_0 \\  \hat \beta_1 \\  \hat \beta_2 \\ \end{pmatrix}=\begin{pmatrix}  −153.5 \\  1.24 \\  12.08 \\ \end{pmatrix}$
