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In econometrics, an independent process means that all values are independent of each other, but does this also mean that all independent processes are white noise processes? and is the reverse true?

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In discrete time

Random variables in a discrete white noise process have 0 expectation and finite variance. Furthermore, all correlations (except with itself) are zero.

So, both directions are wrong:

  • E.g. an independent process with trend would not be considered as white noise.
  • On the other hand, you can construct white noise that is not an independent process (correlation does not imply independence).

See Wiki for much more information.

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The technical definition of white noise is that it has equal intensity at all frequencies. This corresponds to a delta function autocorrelation. This is only possible if there is no correlation between any sequential values. So yes, the independence is true both backwards and forwards.

Note that the actual distribution is irrelevant.

I believe this is correct, but I hope someone will come along and provide a more rigorous answer.

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  • $\begingroup$ First moment has to be zero and second moment has to be finite though. (iid ) White noise is always an independent process but reverse may not be true. $\endgroup$ May 14, 2020 at 18:35
  • $\begingroup$ I am definitely not an expert here. But if the rvs are independent, then that should meet the requirements of delta autocorrelation, and hence flat spectrum. What am I missing there? $\endgroup$
    – abalter
    May 14, 2020 at 19:15

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