The Normality of Joint and Marginal Distributions

I'm often surprised how many people are confused when it comes to joint and marginal normal distributions.

Most students of econometrics are taught that the marginal and conditional distributions associated with a multivariate normal random vector are themselves normal. That is, if

         p(x₁, x₂, ...., x_n) ~ MVN[μ₁, ....., μ_n ; V]              ;      where V = {v_ij}

then
         p(x_i) ~ N[μ_i ; v_ii]  .

Similarly, p(x₁ | x₂, x₃, ...., x_n), and all of the other conditional densities are normal.

However, what they don't seem to get taught is that the converse is not true. That is, if we have several random variables, each with normal marginal distributions, then the joint distribution of these variables is not necessarily normal.

It all depends on what copula is used to construct the joint distribution.


© 2014, David E. Giles