White Noise

A white noise is a simple type of stochastic process. Precise definitions vary. One simple definition is that a white noise is a (univariate or multivariate) discrete-time stochastic process whose terms are independent and identically distributed (IID), all with zero mean. While this definition captures the spirit of what constitutes a white noise, the IID requirement is often too restrictive for applications. Typically the IID requirement is replaced with a requirement that terms have constant second moments, zero autocorrelations and zero means. Let's formalize this.

If you have not already done so, see the notation conventions documentation. A one-dimensional stochastic process

..., t–2W, t–1W, tW, t+1W, ...

[1]

is said to be white noise if unconditional means, standard deviations and autocorrelations satisfy

E(tW) = 0	[2]
std(tW) =	[3]
cor(tW, t+nW) = 0	[4]

for some constant and any integer n. To distinguish this definition of white noise from that which requires IID terms, we call the latter an independent white noise or strong white noise. Note that the definition of white noise is more restrictive than that of independent white noise in just one respect. With a white noise, means, standard deviations and autocorrelations must exist. For independent white noise, they need not.

While the definition of independent white noise is otherwise more restrictive than that of white noise, it is also simpler. An independent white noise is necessarily a very simple process. Conditions [2] through [4], which define a white noise, can accommodate more complicated processes. For example, conditions [2] and [3] apply only to unconditional moments. There is nothing to stop a white noise from being conditionally heteroskedastic. That is impossible with an independent white noise.

An independent white noise whose terms are all normally distributed is called a Gaussian white noise. A realization of a univariate Gaussian white noise with variance 1 is graphed in Exhibit 1.

Univariate Gaussian White Noise Exhibit 1

A realization of a univariate Gaussian white noise with variance 1.

All these concepts generalize to multivariate processes. An n-dimensional stochastic process

[5]

is said to be white noise if unconditional expectations satisfy

	[6]
	[7]

for some constant covariance matrix . Condition [7] does not require that the be independent. If we make this stronger assumption, the process is called independent white noise. If we further assume the are joint normal, it is called Gaussian white noise.

White noises are important in time series analysis because more complicated stochastic processes are generally defined in terms of white noises.