estimate coefficients of $y = \alpha x + \beta y + \gamma z + \epsilon$

Question

I know how to find $m$ and $b$ for $y= mx +b$, which is : $m= \frac{\bar{x}\bar{y}- \bar{xy}}{(\bar{x})^2 - \bar{x^2}}$ and $b= \bar{y} - m\bar{x}$

How can we estimate $\alpha, \beta, \gamma$ and $\epsilon$ for $y = \alpha x + \beta y + \gamma z + \epsilon$

Answer 1

Let $Y = X\beta +\epsilon$ be a multilinear regression model, where $Y = n\times 1$, $\beta=k\times 1$, $X = n\times k$ and $\epsilon$ is an error term. Then the least squares estimator, which minimizes the squared errors is $\hat\beta = (X^TX)^{-1}X^TY$. One can show that the maximum likelihood estimator is the same.