White Noise

Study guide: Exam MAS-II

White noise is a Time Series of uncorrelated, mean-zero, constant-variance terms. It is the input every ARIMA model is built from, and the thing a fitted model's residuals must look like if the model has captured everything.

E[\varepsilon_t] = 0, \quad \mathrm{Var}(\varepsilon_t) = \sigma^{2}, \quad \mathrm{Cov}(\varepsilon_t, \varepsilon_s) = 0 \ (t \neq s)

\rho_k = 0 \ \text{ for all } k \geq 1

• White noise is stationary by construction and completely unforecastable — the best prediction of \varepsilon_{t+1} is 0, whatever the history
• Gaussian white noise additionally assumes normality, which is what prediction intervals need; uncorrelated does not imply independent without it
• The diagnostic use is the important one: after fitting, the residual ACF should show no spike outside the \pm 1.96/\sqrt{n} bands. A surviving spike means structure the model has missed
• The Ljung-Box portmanteau test checks several lags at once: Q = n(n+2)\sum_{k=1}^{h}\frac{\hat\rho_k^2}{n-k} \sim \chi^2_{h-p-q}
• About 1 lag in 20 will breach the bands by chance — a single mid-range spike is not evidence of misfit

Are These Residuals White Noise? After fitting an ARIMA(1,1,0) to 100 observations, the residual ACF shows \hat\rho_1 = 0.05, \hat\rho_2 = -0.31, \hat\rho_3 = 0.08. Is the fit adequate?

Answer The bands are \pm 1.96/\sqrt{100} = \pm 0.196. \hat\rho_1 and \hat\rho_3 sit well inside. \hat\rho_2 = -0.31 is outside — and at lag 2, not at some random high lag where a chance breach would be unsurprising. Not white noise. There is structure left at lag 2; try adding an MA(2) or AR(2) term and re-check.