For a homework exercise, I am asked to find an example of a white noise series that is not a martingale difference series with respect to its natural filtration. Does anyone know an example? I read somewhere you could use the cosine function, but I am not sure how to deploy that.
2026-03-28 04:51:16.1774673476
Example of a white noise series that is not a martingale difference series with respect to its natural filtration
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Consider the non-linear moving average process $$ y_t = \epsilon_t + \epsilon_{t-1} \epsilon_{t-2},$$ where we take $\epsilon_t \sim N(0,\sigma^2)$ to be i.i.d. random variables. We can easily show that this is a white noise process but $$\mathbb{E} [y_t | y_{t-1},y_{t-2},\ldots] = \mathbb{E} [y_t | \epsilon_{t-1},\epsilon_{t-2},\ldots] = \epsilon_{t-1} \epsilon_{t-2} \neq 0,$$ By noticing that $\sigma(y_t,y_{t-1},\ldots)=\sigma(\epsilon_t,\epsilon_{t-1},\ldots)$. Hence $y_t$ is not a martingale difference series.