Solved – Why white noise process and IID process are considered martingale

A stochastic process ${X_t}$ is called a martingale if

$$ operatorname{E}[X_{t+1} mid X_{t}, ldots, X_1} = X_t $$

That is, the expectation of the future conditional on the past is the present.

Fumio Hayashi's Econometrics defines a process ${Z_t}$ as white noise if $operatorname{E}[Z_t] = 0$ and for any $j neq 0$ $operatorname{E}[Z_tZ_{t+j}] = 0$.

Let process ${Y_t}$ be a series of independent flips of a fair coin where $Y_t = 1$ if heads and $Y_t = -1$ if tails. Observe that:

• ${Y_t}$ is white noise
• ${Y_t}$ is NOT a martingale. If we flip a coin heads, we don't expect the next flip to be heads! The conditional expectation of $Y_t$ is always zero, not $Y_{t-1}$.

Perhaps what you're thinking? (or what your Prof is leading you to…)

A process $Delta_t$ is called a martingale difference sequence if the conditional expectation of $Delta_t$ given past information $mathcal{F}_{t-1}$ is zero, that is, $operatorname{E}[Delta_t mid mathcal{F}_{t-1}] = 0$. Consequently a white noise process is a martingale difference sequence. Why is $Delta_t$ called a martingale difference sequence? Define $X_t = X_{t-1} + Delta_t$. Then $X_t$ is a martingale.

(Note also that a martingale difference sequence need not be white noise.)