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)?