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Are both terms equivalent? They seem very similar to me. Or does one imply the other?

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No, they are not equivalent. In fact, they are quite unrelated.

Heteroskedasticity is when variance differs between "situations". For instance, in a regression task, the variance of the residuals may be larger or smaller, depending on whether a particular predictor has a certain value or not.

Overdispersion is when variance is greater than the expectation.

For instance, in OLS we assume homoskedasticity (and if we suspect heteroskedasticity, we should use special tools). However, we make no assumptions on equi- or overdispersion. In fact, since we assume that all residuals have the same variance (per the homoskedasticity), while the conditional mean can be any value at all, we might have all of under-, equi- or overdispersion for different observations if they just differ in their conditional means enough.

Conversely, a Poisson regression assumes equidispersion: each observation's conditional variance should be equal to its conditional expectation. But since we are in a regression framework, the expectations will vary with the predictors... and therefore, so will the variances. Thus, in a Poisson regression, we will usually have heteroskedasticity. But it's not a problem here, because Poisson regression handles this automatically.

I work in retail forecasting. Retail time series usually exhibit heteroskedasticity (sales increase during promotions, but so does the variance of sales), as well as overdispersion. (In terms of theory, the negative binomial distribution should be a good description.) However, there are a few products that vary very little and move very fast, like milk, where we might have underdispersion (and still heteroskedasticity, e.g., because of weekday patterns). Or if products habitually go out of stock - maybe the baker consciously plans to go out of stock so they don't have stock left over at the end of the day: censored demand - we will often again have underdispersion.

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