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I am new to time series analysis and would like to test a multivariate time series (12 components) for normality. I found several straightforward normality tests and some multivariate normality tests.

My problem is that I do not understand the need for multivariate normality tests in the first place. Can't I just test each of the 12 components for normality separately and conclude that the multivariate time series is normal if all of its components are normal?

Alice Schwarze
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    Re the notion that marginal normality implies joint normality -- there are many counterexamples on site. e.g. See [here](https://stats.stackexchange.com/a/30205), [here](https://stats.stackexchange.com/questions/25890/same-joint-distribution-different-conditional-and-marginal-distribution/25919#25919), and [here](https://stats.stackexchange.com/questions/120861/example-of-two-correlated-normal-variables-whose-sum-is-not-normal) – Glen_b Jan 30 '18 at 00:02

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No, the fact that the marginal distributions are normal does not lead to the joint distribution being normal. You can have all kinds of joint distributions that lead to normal marginals. Only if your variables are independent you get this link.

Having said that, one rarely needs normality tests, yet alone multivariate ones outside the classroom

Aksakal
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  • Thanks! Why does one only rarely need normality tests? I learned that OLS only works if errors have a normal distribution. – Alice Schwarze Jan 29 '18 at 20:12
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    No, OLS works fine without normal distributed errors. There's a ton of discussion on this subject. It's good to have normality, but it's not required for the most of things you do with OLS – Aksakal Jan 29 '18 at 20:31
  • I want to use an uniformly minimum variance unbiased estimator for the entropy of a normal distribution (http://ieeexplore.ieee.org/document/30996/) to compute an entropy for my time series. I don't quite understand how big or how small an error I would have in my estimates if my time series isn't normal. (This is perhaps too specific for stack exchange, but I just wanted to mention it for completeness.) – Alice Schwarze Jan 29 '18 at 21:24
  • Ok, for something like an entropy of the distribution my gut feeling tells me that the exact distribution matters. However, unless you have a ton of data, multivarite normality tests are going to be weak in power if not useless – Aksakal Jan 29 '18 at 22:41