I used the copula to evaluate the $\mathbb{E}[Y|X]$ and from my experiments on some copulas, I observed that when the random variables have positive correlation coefficient, $\mathbb{E}[Y|X]$ is strictly increasing and when they are negatively correlated $\mathbb{E}[Y|X]$ is strictly decreasing. Also, the conditional expectation plots in this post show the strict behavior too. I'm wondering if this is true and how to prove it.
1 Answer
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Unless you're leaving out some condition, the answer is no.
For example, let $c=\Phi(1)\approx 0.8413$ where $\Phi$ is the standard normal cdf. Consider the bivariate copula that corresponds to $V=1-U$ for $1-c<U<c$ and $V=U$ otherwise, where $U$ is standard uniform. The correlation of $U$ and $V$ is positive (roughly 0.36), but $E(V|U=u)$ is decreasing when $u$ is in that middle section.
Now let $X=\Phi^{-1}(U)$ and $Y=\Phi^{-1}(V)$; $(X,Y)$ has the same copula. Now for $(X,Y)$ the Pearson correlation is higher (around 0.6) but again the conditional expectation function is not strictly increasing.
The value of $c$ is not special; I simply chose it because it was fairly clear that the correlation in the Gaussian margins case would be positive. For the copula itself (uniform margins), any positive $c$ less than about 0.8968 works, for the Gaussian margins case positive $c$ less than about 0.938 works. You could choose a convenient value of $c$ and compute an exact correlation on the copula if you wanted.
