You have fitted a model and it produced forecasts. Before you stake a decision on it, one question towers over the rest: did the model actually capture the structure in the data, or is there signal it missed? The answer lives in the residuals — the one-step forecast errors \hat\varepsilon_t = x_t - \hat x_t left over after the fit. The governing idea is simple and powerful: a model has extracted all the predictable structure exactly when nothing predictable is left in the residuals — that is, when they look like white noise. Residual diagnostics are the battery of checks that put that claim on trial.

What "good residuals" means

White noise has three defining traits, and each becomes a thing to check:

Any residual pattern is the model talking to you: it is telling you what it left on the table. Structure in the residual mean says the AR/MA orders are wrong; structure in the residual variance says something else is going on entirely.

The residual correlogram — the first thing to plot

Plot the residual autocorrelations r_k against lag and overlay the \pm 1.96/\sqrt{T} significance bands (for white noise, each r_k is approximately normal with standard deviation 1/\sqrt{T}). The verdict is visual: all the stems should sit inside the bands, with no more than the occasional chance excursion. The correlogram below is a clean bill of health — every lag is comfortably within \pm 1.96/\sqrt{T}. A single spike poking out at a meaningful lag (lag 1, or a seasonal lag like 12) is the tell-tale of a missed term.

Eyeballing individual lags has a flaw, though: with 20 lags plotted you expect about one to breach a 95% band by pure chance. That is why we need a single test that judges all the lags together.

The Ljung–Box portmanteau test

A portmanteau test bundles many lags into one number. The Ljung–Box statistic pools the first h squared residual autocorrelations, each weighted to correct for small-sample bias:

This is a proper hypothesis test, so read it the usual way: compare Q to the \chi^2_{h-m} critical value (or just look at the p-value). We want to fail to reject here — a comfortably small Q is the model passing.

Worked example — computing Q

Take T = 100 observations and the first three residual autocorrelations r_1 = 0.08,\ r_2 = -0.05,\ r_3 = 0.11. Then T(T+2) = 100 \cdot 102 = 10200, and

so Q \approx 10200 \times 2.15\times 10^{-4} \approx 2.19. If the model was an AR(1) (m = 1 parameter), compare against \chi^2_{3-1} = \chi^2_{2}, whose 5% critical value is 5.99. Since 2.19 < 5.99 we do not reject: no evidence of leftover autocorrelation. The model passes this check.

Beyond correlation: normality and changing variance

Two more checks round out the diagnosis:

• Normality — a Q–Q plot of the residuals against normal quantiles should track the diagonal; systematic bends in the tails reveal skew or heavy tails, which quietly invalidate the Gaussian prediction intervals. A Jarque–Bera test formalises it.
• Constant variance — plot the residuals over time and look for volatility clustering: calm stretches punctuated by bursts of large swings. The autocorrelations of the squared residuals catch this even when the raw residuals look uncorrelated. Such ARCH effects mean the mean model is fine but the variance is time-varying — the cue to reach for a variance model rather than re-tinkering with the AR/MA orders.