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Assume two time series $X_1$ and $X_2$, where $X_i=(x_{i,1},...,x_{i,T})$. How can we test the assumption of bivariate normality for these time series? (Assume each of the time series are stationary and normally distributed).

kjetil b halvorsen
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statwoman
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  • did you try $\chi^2$ test? – Aksakal Jun 16 '22 at 19:38
  • Would you elaborate? @Aksakal – statwoman Jun 16 '22 at 19:55
  • That what is bivariate normal, the joint error term? – Dave Jun 16 '22 at 20:11
  • https://www.sciencedirect.com/topics/mathematics/bivariate-normal-distribution @Dave – statwoman Jun 16 '22 at 20:16
  • That each of the time series are normally distributed does not imply bivariate normality. Do you want to test the former or the latter? – Richard Hardy Jun 16 '22 at 20:18
  • Indeed. I want to test the bivariate normality assumption. @RichardHardy – statwoman Jun 16 '22 at 20:28
  • A standard approach is to test the univariate Normality of linear combinations of the variables, according to this characterization. In almost any practical application, though, you shouldn't be formally testing: you should be characterizing the extent to which the data conform to any bivariate normality condition that might be needed to analyze them. That would start with scatterplots, examination of bivariate outliers, etc. – whuber Jun 16 '22 at 21:28
  • @whuber Not sure if I'm following. For instance, how would you check the jointly normal with a scatterplot? – statwoman Jun 16 '22 at 21:36
  • Look at it. The point is that certain behaviors, such as heavy tail behavior, strong skewness in some direction, tied values, etc. are likely to be what matters in your analysis. It will be rare for any analysis to require such perfect conformance with bivariate normality that you can't determine compliance from a good scatterplot. Given that your time series already are Gaussian processes, it's going to be extremely difficult (in practice--there are theoretical exceptions) to depart from bivariate normality anyway. – whuber Jun 16 '22 at 21:43
  • Just to confirm that I understand this correctly; essentially do a QQ plot with x-axis showing $X_1$ and y-axis $X_2$? @whuber – statwoman Jun 16 '22 at 21:49
  • That wouldn't do it: a QQ plot destroys the natural pairing among the variables. Just plot $(X_1,X_2).$ Add some kind of smoother, such as a 2D KDE, to get a sense of the overall contours of density. – whuber Jun 16 '22 at 21:57
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    Is there a reason that you care about the time series themselves instead of, say, the (bivariate) error term? Any recommendation for a test or examination for normality is unhelpful is you test or examine the wrong numbers. – Dave Jun 16 '22 at 22:31
  • @Dave my goal is to construct a correlation coefficient confidence interval and I want to check if they are bivariate normal. Does that make sense? – statwoman Jun 17 '22 at 12:41

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