How to calculate the standard error of multiple linear regression coefficient

Question

I'm learning linear regression. I'm stuck on how to calculate the standard error of the coefficients. I know that in the simple linear regression provided by this equation $ y = \beta_0 + \beta_1x + \varepsilon$ is:
\begin{align} SE(\hat\beta_0)^2 &= \sigma^2 \left( \frac{1}{n} + \frac{\bar{x}^2}{\sum^{i = 1}_{n} (x_i - \bar{x})^2} \right ) \\[5pt] SE(\hat\beta_1)^2 &= \frac{\sigma^2}{\sum^{i = 1}_n (x_i - \bar{x} ) ^2} \end{align}

Given that $\sigma^2$ is the variance of the error term ($\,\varepsilon\,$)

But when the model is extended to $k$ variables ($y = \beta_0 + \beta_1x_1 + \beta_2x_2 + ... + \beta_kx_k + \varepsilon$), can we have a general formula to calculate each $SE(\hat\beta_i)$ (given that all the model assumption about the data is correct)?

Answer 1

To get a general formula for a multiple regression coefficient standard error, you need to use linear (matrix) algebra. The formula for the variance-covariance matrix of the betas is:
$$ VCOV(\hat{\beta}) = s^2{\bf (X'X)^{-1}} $$ From there, you take the positive square root of the $j^{\rm th}$ diagonal element to get the standard error of that beta.