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

Can anyone explain to me why A white noise process (εt) and An IID process (εt) are martingales?

Contents

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.)

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