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I was reading this passage in a book and I am confused about the last line. As far as I understand, the white noise process does not allow for conditional heteroskedasticity due to its i.i.d. property. However, can somebody explain why a martingale difference sequence allows for conditional heteroskedasticity?

Martingale: The stochastic process $v_t$ is called a martingale sequence if \begin{align*} E(v_t|v_{t−1}, v_{t−2}, . . .) = v_{t−1} \;\forall t. \end{align*} Martingale Difference: $u_t \equiv \triangle v_t$ is called a martingale difference if \begin{align*} E(u_t|v_{t−1}, v_{t−2}, . . .) = 0 \;\forall t \end{align*} Unlike an iid white noise process, a white noise process that is a martingale difference sequence allows for conditional heteroskedasticity.

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This is just saying that a white $w_t$ noise satisfies

$$Var(w_t|v_{t−1}, v_{t−2}, . . .) = \sigma \;\forall t$$

but a martingale difference $u_t$ does not necessarily satisfy

$$Var(u_t|v_{t−1}, v_{t−2}, . . .) = \sigma \;\forall t$$.

For example, $u_t \sim N(0, t)$, $v_t = \sum_1^t u_t$ is a martingale difference and associated martingale where the variance changes over time. If $u_t \sim N(0, abs(v_t))$, $v_t = \sum_1^t u_t$, then you get a martingale difference and associated martingale where the variance changes according to its present value.

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