White Noise, Independence, and Martingale Differences

Why Zero Autocorrelation Is Not the Whole Story

A time series can have zero autocorrelation at every nonzero lag without its observations being independent. It can even be linearly unpredictable from its past while its conditional mean, given richer past information, is not zero. These distinctions matter because many econometric arguments depend on more than the autocorrelation function: they may require independence or a conditional-mean restriction.

The terms white noise and martingale difference describe different restrictions. White noise is defined through unconditional means, variances, and covariances. A martingale-difference sequence is defined through a conditional expectation given a specified information set. Independence is a still stronger distributional property. Keeping track of which information set is used is essential.

Definition: A sequence $\{u_t\}$ is zero-mean uncorrelated white noise if $E(u_t)=0$, $E(u_t^2)=\sigma^2$ for every $t$, where $0<\sigma^2<\infty$, and $\operatorname{Cov}(u_t,u_s)=0$ whenever $t\ne s$. Equivalently, its autocovariance is zero at every nonzero lag. These moment restrictions do not, by themselves, imply independence.

The equal variance condition is part of the usual weak-white-noise definition. A sequence with zero means and zero covariances but time-varying variances satisfies the uncorrelatedness conditions, but is not white noise under this definition. This is a statement about its first two unconditional moments, not a claim that the sequence is stationary in every distributional respect.

Definition: Independent white noise is an independent and identically distributed sequence with mean zero and finite, positive variance. In particular, it is zero-mean uncorrelated white noise. Independence concerns the joint distributions of observations, not just their covariances.

Martingale Differences Depend on Information

Let $\{\mathcal F_t\}$ be a filtration: an increasing sequence of information sets, so $\mathcal F_{t-1}\subseteq\mathcal F_t$. A variable is adapted if $u_t$ is measurable with respect to $\mathcal F_t$, meaning it is known given time-$t$ information. Usually the filtration represents the available history, but the exact choice must be stated.

Definition: An adapted sequence $\{u_t\}$ with finite second moments is a martingale-difference sequence (MDS) with respect to $\{\mathcal F_t\}$ if $$ E(u_t\mid\mathcal F_{t-1})=0 $$ for every $t$. This conditional restriction says that the mean of $u_t$, given the specified past information, is zero. It does not require the conditional distribution of $u_t$ to be the same for every possible past.

An MDS condition is always relative to a filtration. A sequence might satisfy the condition given its own past but fail it given a larger information set. By contrast, the definition of white noise above is expressed in terms of unconditional moments and has no filtration in it.

What the Conditions Imply

The standard implications can be summarized as follows. The MDS-to-white-noise implication requires a constant unconditional variance; the MDS condition alone does not impose that requirement.

Condition	What it guarantees	What it does not guarantee on its own
Independent white noise	Zero mean, constant finite variance, zero autocovariances, and independence	An MDS property relative to an arbitrary enlarged filtration
MDS with constant unconditional variance	Zero mean, constant variance, and zero autocovariances	Independence
Zero-mean uncorrelated white noise	Zero mean, constant variance, and zero autocovariances	A zero conditional mean given the past, or independence

An independent white-noise sequence is an MDS with respect to its natural filtration, generated by its own observations through time $t$. The reason is that independence makes $u_t$ independent of that past, and its unconditional mean is zero. But independence does not imply the MDS condition with respect to every possible filtration: a larger information set could contain information about $u_t$.

Theorem: If $\{u_t\}$ is an MDS with respect to $\{\mathcal F_t\}$, and $u_s$ is measurable with respect to $\mathcal F_{t-1}$ for $s<t$, then $\operatorname{Cov}(u_t,u_s)=0$. If, in addition, $E(u_t^2)=\sigma^2$ for all $t$, the sequence is zero-mean uncorrelated white noise.

Here is the short justification. First, taking expectations of the conditional-mean restriction gives $E(u_t)=0$. Since $u_s$ is known given $\mathcal F_{t-1}$, it can be taken outside the conditional expectation. Therefore

E(u_tu_s) =E\!\left[E(u_tu_s\mid\mathcal F_{t-1})\right] =E\!\left[u_s E(u_t\mid\mathcal F_{t-1})\right] =0.

As both means are zero, $E(u_tu_s)=0$ is the same as $\operatorname{Cov}(u_t,u_s)=0$. This establishes the covariance implication. It does not establish independence: zero covariance is only a restriction on a second moment.

Worked Example: Independent White Noise Is an MDS

Suppose, hypothetically, that $u_t$ are independent $N(0,4)$ variables. Take $\mathcal F_t$ to be the information generated by $u_t,u_{t-1},u_{t-2},\ldots$. Check the white-noise and MDS conditions.

Each observation has mean zero and variance $4$. For $s\ne t$, independence gives $$ \operatorname{Cov}(u_t,u_s)=E(u_tu_s)-E(u_t)E(u_s)=0-0=0. $$ Thus the sequence is independent white noise, with standard deviation $2$ in the units of $u_t$.

The variable $u_t$ is independent of $\mathcal F_{t-1}$. Hence its conditional mean is its unconditional mean: $$ E(u_t\mid\mathcal F_{t-1})=E(u_t)=0. $$ So it is also an MDS with respect to its natural past. This conclusion uses independence from the past, not merely the zero autocovariances.

White Noise Need Not Be a Martingale Difference

The reverse implication fails. A sequence can have constant variance and zero autocovariances while its conditional mean, given some specified past information, is nonzero. The following construction checks each property directly.

Worked Example: White Noise That Is Not an MDS

Let $Z_t$ be independent variables, each taking the values $-1,0,1$ with probability $1/3$. Define $h(z)=\mathbf{1}\{z=0\}-1/3$, and, hypothetically, set $u_t=Z_t+h(Z_{t-1})$. Use the filtration $\mathcal F_t=\sigma(Z_j:j\leq t)$.

First, $E(Z_t)=0$, and $h(Z_{t-1})$ takes the value $2/3$ with probability $1/3$ and $-1/3$ with probability $2/3$. Therefore $$ E[h(Z_{t-1})]=0,\qquad E[h(Z_{t-1})^2]=\frac13\left(\frac23\right)^2+\frac23\left(-\frac13\right)^2=\frac29. $$ Thus $E(u_t)=0$. The two terms in $u_t$ are independent, so $$ \operatorname{Var}(u_t)=E(Z_t^2)+E[h(Z_{t-1})^2] =\frac23+\frac29=\frac89. $$

For lag one, expand the covariance using the definitions. Independence across distinct $Z$'s makes the terms involving different indices have expectation zero; the only same-index cross-term is $E[Z_{t-1}h(Z_{t-1})]$, which is also zero: $$ E[Z_{t-1}h(Z_{t-1})] =\frac13(0)(2/3)+\frac13(-1)(-1/3)+\frac13(1)(-1/3)=0. $$ Consequently $\operatorname{Cov}(u_t,u_{t-1})=0$. At lags of at least two, the $Z$'s entering the two observations are disjoint independent variables, so their covariance is zero as well. The sequence has mean zero, constant variance $8/9$, and zero autocovariances at all nonzero lags: it is zero-mean uncorrelated white noise.

However, $Z_t$ is independent of $\mathcal F_{t-1}$, while $h(Z_{t-1})$ is known at time $t-1$. Thus $$ E(u_t\mid\mathcal F_{t-1})=E[Z_t\mid\mathcal F_{t-1}]+h(Z_{t-1})=h(Z_{t-1}). $$ This is nonzero with positive probability, so the sequence is not an MDS with respect to the stated filtration. Zero autocorrelations have not ruled out a nonzero conditional mean.

A Martingale Difference Need Not Be Independent

An MDS can also be dependent over time. The next example has constant unconditional variance and zero autocovariances, but its observations are not independent. It shows why the MDS condition should not be read as a synonym for independent innovations.

Worked Example: An MDS That Is Not Independent White Noise

Let $\varepsilon_t$ be independent variables taking $-1$ and $1$ with equal probability. Set $a=1/2$, define $u_t=\varepsilon_t(1+a\varepsilon_{t-1})$, and let $\mathcal F_t=\sigma(\varepsilon_j:j\leq t)$.

Because $1+a\varepsilon_{t-1}$ is known at time $t-1$, while $E(\varepsilon_t\mid\mathcal F_{t-1})=0$, $$ E(u_t\mid\mathcal F_{t-1}) =(1+a\varepsilon_{t-1})E(\varepsilon_t\mid\mathcal F_{t-1})=0. $$ So $\{u_t\}$ is an MDS. Its unconditional mean is zero, and its variance is $$ E(u_t^2)=E[(1+a\varepsilon_{t-1})^2] =1+2aE(\varepsilon_{t-1})+a^2E(\varepsilon_{t-1}^2) =1+a^2=1.25. $$ The MDS result above then gives zero autocovariances at every nonzero lag.

Yet the sequence is not independent. Since $a=1/2$, the factor $1+a\varepsilon_{t-1}$ is always positive, so the sign of $u_t$ equals $\varepsilon_t$. Also $|u_{t+1}|=1+a\varepsilon_t$. Therefore $$ |u_{t+1}|=1+a\,\operatorname{sign}(u_t). $$ The magnitude of the next observation is determined by the sign of the current one. These observations are dependent, despite their zero autocovariance and the MDS property.

A Martingale Difference Need Not Be White Noise

The MDS restriction does not require the unconditional variance to be constant. For example, let $\varepsilon_t$ be independent $N(0,1)$ variables and define $u_t=c_t\varepsilon_t$, where $c_t=1$ for odd $t$ and $c_t=2$ for even $t$. With the natural filtration of the $\varepsilon_t$'s, $E(u_t\mid\mathcal F_{t-1})=0$ at every date. But the variance alternates: $$ \operatorname{Var}(u_t)=c_t^2= \begin{cases} 1, & t\ \text{odd},\\ 4, & t\ \text{even}. \end{cases} $$ This sequence is an MDS but not white noise under the constant-variance definition. Conditional mean zero controls the mean given the past; it does not fix the scale of the observations.

Common Mistake: Treating Zero ACF as Independence

A common but incorrect argument is: “All nonzero-lag autocorrelations are zero, so observations at different dates are independent.” Autocorrelation measures linear association, and independence is a stronger property. The conclusion does not follow unless additional distributional assumptions are available. Likewise, a zero ACF alone does not establish that the conditional mean given the past is zero.

The correction is to identify the exact property needed and verify it directly. Use autocovariances to describe second-order linear dependence; use a conditional expectation to establish an MDS; and use a joint-distribution or factorization argument to establish independence. In every MDS claim, state the filtration. A claim that omits the information set is incomplete.

Key takeaway: Independent white noise is an MDS relative to its natural past. An MDS implies zero autocovariances, and with constant unconditional variance it is white noise. Neither zero autocovariances nor the MDS condition alone implies independence.

Check Your Understanding

For each question, distinguish unconditional moment restrictions from conditional or distributional restrictions.

What three moment conditions define zero-mean uncorrelated white noise?
Why does an MDS with constant unconditional variance have zero autocovariances?
Does zero autocorrelation at every nonzero lag imply a zero conditional mean given the past? Explain.
Can an MDS fail to be white noise? Which part of the white-noise definition might fail?
Why must the filtration be specified when asserting that a sequence is an MDS?