# Sigma squared times identity matrix in normality of errors

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?

• It is a neat way of succinctly saying the following: 1. The error is a multivariate random vector with $n$ components (in this case 1 component is one observation) in the regression model. 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. 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. Sep 27, 2020 at 16:51
• @rubikscube09 Could you make this an answer? Sep 27, 2020 at 17:11
• It is a widely-repeated myth that OLS requires normality of errors though that is not true. We only assume normality for hypothesis testing -- such as deciding if coefficient estimates are significant. Use the bootstrap and you need not even make that assumption. Sep 28, 2020 at 17:27

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.