Correlation after cutting the sample

Idea. On the whole line, $Z^2$ goes up on both sides, so it has no linear relation with $Z$. Keep only $Z>0$ and $Z^2$ becomes an increasing function of $Z$: the correlation jumps close to 1.

Step 1: moments given $Z>0$. With $m=\sqrt{2/\pi }$:

• $\mathbb{E}[Z\mid Z>0]=m$, $\mathbb{E}[Z^2\mid Z>0]=1$;
• $\mathbb{E}[Z^3\mid Z>0]=2m$, $\mathbb{E}[Z^4\mid Z>0]=3$.

Step 2: covariance and variances. $$\begin{array}{rl}\mathrm{Cov}& =2m-m\cdot 1=m,\\ \mathrm{Var}(Z)& =1-m^2=1-\frac{2}{\pi },\\ \mathrm{Var}(Z^2)& =3-1=2.\end{array}$$

Step 3. $$\begin{array}{rl}\mathrm{corr}& =\frac{m}{\sqrt{2(1-2/\pi )}}=\frac{\sqrt{2/\pi }}{\sqrt{2(\pi -2)/\pi }}\\ & =\frac{1}{\sqrt{\pi -2}}\approx \mathrm{0.936.}\end{array}$$

The lesson. Zero correlation does not mean independence, and correlation depends on the population you look at. Selecting a subsample (only positive returns, only funded companies, only admitted students) can create strong correlations that do not exist in the whole population.