# What does relaxing the iid assumptions mean? Intuitive and technical perspectives.

I believe the most restrictive assumption we can place on a series of observations is that they are iid.

It is possible to relax these assumptions. For example relaxing the independent distribution results in independent heterogeneously distributed random variables. In other words the distribution of each random variables can itself vary. What does this mean? There must be some importance or we would not bother to specify the properties of the distribution. Can the distribution depend on preceding observations or would this break the independent observation restriction that we still have in place?

Can anyone think of any intuitive examples to help explain this (in the same way that tossing a fair coin or rolling a fair die are good examples of iid observations) and any important statistical concepts that arise out of relaxing this assumption? My degree is not in statistics although I am quite interested in this subject. Thank you all so much.

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"relaxing the independent distribution results in independent heterogeneously..." Perhaps you mean "relaxing the identical ..." ? –  leonbloy Aug 29 at 14:20
Relaxing the assumptions doesn't necessarily mean that you go from assuming a normal variable in period $t$, and a binomial in period $t+1$, or something similarly drastic. You are typically dealing with a parametric probability distribution; eg $\mathcal{N}(\mu, \sigma)$, which you would change to $\mathcal{N}(\mu_t, \sigma_t)$, or also, say, to something like $\mathcal{N}(\mu, \sigma_{f(x_0, ..., x_{t-1})})$, among many other variations. The last case is essentially what is done in models very commonly used in financial time series modeling, so-called ARCH models and their variants (c., e.g., http://en.wikipedia.org/wiki/Autoregressive_conditional_heteroskedasticity, or http://www.amazon.com/Time-Analysis-James-Douglas-Hamilton/dp/0691042896). Why this is useful depends on the case. When you look at historic time series, and sticking to this last example, you notice that the variance of certain time series tends to change over time. Hence, such a model would fit the data better than an iid assumption.