The classical linear regression assumptions (Gauss-Markov conditions) are:

1. Linearity: $E[Y_i | \mathbf{x}_i] = \mathbf{x}_i^T \boldsymbol{\beta}$ (correct functional form)
2. Independence: errors $\epsilon_i$ are independent
3. Homoscedasticity: $\operatorname{Var}(\epsilon_i) = \sigma^2$ (constant variance)
4. Normality (for inference): $\epsilon_i \sim N(0, \sigma^2)$

Assumption 4 is needed for exact $t$-tests and $F$-tests; assumptions 1-3 suffice for the Gauss-Markov theorem.

Key residual diagnostics:

1. Residuals vs. fitted values: plots $e_i$ vs. $\hat{y}_i$. Patterns indicate nonlinearity or heteroscedasticity.
2. Normal Q-Q plot: plots ordered residuals vs. theoretical normal quantiles. Departures from the line indicate non-normality.
3. Scale-location plot: plots $\sqrt{|e_i|}$ vs. $\hat{y}_i$. A trend indicates heteroscedasticity.
4. Leverage plot: the leverage $h_{ii} = [\mathbf{H}]_{ii}$ measures how far $\mathbf{x}_i$ is from the center of the predictor space. High leverage points have $h_{ii} > 2p/n$.