Sigma squared times identity matrix in normality of errors

Question

In OLS regression, we have the normality of the error terms

$$\varepsilon \sim N(0,\sigma^2I_n)$$

I understand that we want to have a constant variance for homoscedastic errors, but why is $\sigma^2$ multiplied with the identity matrix ($I_n$)? Is it just in order to transform $\sigma^2$ from a scalar into a matrix?

Answer 1

It is a neat way of succinctly saying the following:

1. The error is a multivariate normal random vector with n components (in this case each component is one observation in the regression model.) That is to say: $$ \varepsilon = (\epsilon_1, \cdots, \epsilon_n) $$ where here each $\epsilon_i$ is a random variable that takes values in $\mathbb{R}$.
2. The error has constant variance $\sigma^2$ in each component/observation because the variance of components correspond to the diagonal entries at each point in the covariance matrix. That is to say: $$ \mathrm{var}(\epsilon_i) = \sigma^2 $$ Moreover, the expectation of the errors is $0$.
3. The error components are uncorrelated/orthogonal because covariances/correlations between different observations correspond to off-diagonal entries in the covariance matrix - which are all 0 for multiples of the identity. That is to say, the matrix with real entries: $$ \mathbb{E}[\epsilon \epsilon^T]_{i,j} = \mathrm{Cov}(\epsilon_i,\epsilon_j) = \begin{cases} = \sigma^2 & i = j \\ 0 & i \neq j\end{cases} $$ In particular: $$ \mathrm{cov}(\epsilon_i,\epsilon_j) = \mathrm{corr}(\epsilon_i,\epsilon_j) = 0 $$ for $i \neq j$, whereas: $$ \mathrm{cov}(\epsilon_i,\epsilon_i) = \mathrm{var}(\epsilon_i) = \sigma^2 $$ Because the errors have a multivariate normal distribution (they are an affine transformation of a vector whose components are independent normals), it follows that $\epsilon_i$ is in fact independent of $\epsilon_j$ (and not just uncorrelated).

All this said - putting the errors like this is a succinct notational way of summarizing the OLS assumption - $n$ error terms, all uncorrelated with each other, and all having constant variance.