White Noise

Every model in this course is, at bottom, a recipe for turning something unpredictable into something structured. That unpredictable raw material has a name: white noise. It is the irreducible randomness left over once all the predictable memory has been squeezed out — the atom of the whole subject. A good model does not remove the noise (that is impossible); it explains everything except the noise, and leaves behind residuals that look exactly like it.

So white noise plays two roles at once. It is the building block from which we construct models — every AR and MA process is white noise filtered through some memory — and it is the target we check for when we validate them. Learn its precise definition and you have the yardstick against which every fitted model is measured.

The definition

A process \{\varepsilon_t\} is white noise if it is a sequence of uncorrelated, zero-mean, constant-variance shocks — nothing more.

Zero mean: \mathbb{E}[\varepsilon_t] = 0 for all t.
Constant variance: \operatorname{Var}(\varepsilon_t) = \sigma^2 < \infty for all t.
No autocorrelation: \gamma(h) = \operatorname{Cov}(\varepsilon_t, \varepsilon_{t+h}) = 0 for every h \neq 0.

These three conditions make white noise automatically weakly stationary: the mean and variance are constant, and the autocovariance depends on lag alone. Its autocorrelation function is the simplest possible — a single spike and then flat zero:

\rho(h) = \begin{cases} 1 & h = 0 \\ 0 & h \neq 0. \end{cases}

What it looks like

A white-noise series has no texture: no trend, no waves, no clustering — each value is unrelated to its neighbours, so the plot is a structureless flicker around zero. Its correlogram is the giveaway: one spike at lag 0 (a series is always perfectly correlated with itself) and everything else buried inside the bands.

That flat ACF is exactly the "nothing left to model" signature you hunt for in a correlogram of residuals — if the residuals look like the top figure and their ACF looks like the bottom one, your model has captured all the structure there was to capture.

Three flavours, in increasing strength

"White noise" is a family, and the differences matter enormously once we reach non-linear models. From weakest to strongest:

Weak white noise — the definition above: uncorrelated, zero mean, constant variance. Says nothing about higher dependence; values may be uncorrelated yet still tangled together.
iid white noise — the shocks are fully independent and identically distributed, not merely uncorrelated. This rules out any hidden non-linear dependence.
Gaussian white noise — iid and each \varepsilon_t \sim \mathcal{N}(0, \sigma^2). Here uncorrelated \iff independent, so the three notions coincide.

The chain is Gaussian \Rightarrow iid \Rightarrow weak, and none of the arrows reverse in general. Classical linear ARMA theory needs only the weakest version; the finance models at the end of the course exploit precisely the gap between "uncorrelated" and "independent."

Worked example: a filter of white noise is not white noise

To feel why white noise is a building block, build something with it. Let \varepsilon_t be white noise and form X_t = \varepsilon_t + 0.8\,\varepsilon_{t-1}. Then:

\mathbb{E}[X_t] = 0 — still zero mean;
\gamma_X(0) = (1 + 0.8^2)\sigma^2 = 1.64\,\sigma^2;
\gamma_X(1) = 0.8\,\sigma^2 \neq 0 — non-zero lag-1 covariance, because X_t and X_{t+1} share the shock \varepsilon_t.

So X_t is not white noise: it has memory of one step, with \rho_X(1) = 0.8 / 1.64 \approx 0.49. We took the memoryless atom and, by mixing adjacent copies, manufactured dependence. That is the entire idea behind moving-average and autoregressive models: structure is white noise passed through memory.

This is the single most consequential subtlety about white noise. "No autocorrelation" only says the linear relationship between \varepsilon_t and \varepsilon_{t+h} is zero. It does not forbid a non-linear one. A daily stock return is close to weak white noise — you cannot predict tomorrow's return from the sign or size of today's — yet its squared values are strongly autocorrelated: violent days cluster together (a calm week, then a stormy week). The returns are uncorrelated but emphatically not independent; their variance is predictable even when their level is not. This gap is invisible to the ordinary ACF and is the entire reason GARCH models exist. Only for Gaussian white noise are uncorrelated and independent the same thing — outside that special case, never conflate them.

The name is borrowed from optics. White light is an even mixture of all visible frequencies at equal intensity — no colour dominates. White noise is the temporal analogue: because it has no memory, every frequency contributes equally to its spectral density, which is perfectly flat. There is no preferred rhythm, no favoured cycle — the energy is spread uniformly across the spectrum, just as white light spreads evenly across colours. A process that did favour some frequency (a slow AR swing, say) would have a spectrum peaked there and, by analogy, a "colour." So the flat ACF and the flat spectrum are two views of the same emptiness — and the word "white" captures both at once.