Residual Diagnostics and Model Checking

Every exact $t$-interval, $F$-test, and confidence band this topic has built rested on a specific, checkable list of assumptions: linearity, constant error variance, uncorrelated errors, and (for the exact small-sample distributions) normality. None of these were ever verified against the data — they were simply assumed so the machinery could run. This closing lesson turns the machinery around: given a fitted model, how does a practitioner check whether the assumptions that licensed it actually hold? The residuals $\hat{\boldsymbol\varepsilon}=\mathbf y-\hat{\mathbf y}$ carry almost all the available evidence, and this note builds the standard toolkit for reading it — standardized and leverage-adjusted residuals, the residual-vs-fitted plot for non-linearity and heteroscedasticity, Cook's distance for influential points, the Durbin-Watson statistic for autocorrelation, and the QQ-plot for normality — closing the topic exactly where a working quant analysis of any regression or ANOVA model must always end: is this fit trustworthy, or does it only look that way?

Learning objectives. By the end of this note you should be able to: