Multivariate regression (multi-target regression) dataset where some regression coefficients are known to be zero

Question

I'm looking for a sample dataset for multivariate linear regression - also known as multi-target or multi-output. Preferably with more than 10 inputs and more than 10 outputs. There don't seem to be many good examples. Ideally, I'd also like the dataset to be such that some regression coefficients are known to be zero, i.e. we know certain inputs don't affect certain outputs. I have created a simulated dataset, but I'm hoping for a real example set of data.

Suppose there are $p$ input variables and $q$ response variables, and the regression model is $Y = XC + \epsilon$.
$Y$ is an $n \times q$ matrix, $X$ is an $n \times p$ matrix, and $C$ is an $p \times q$ matrix. I'm looking for a set of data where we know certain entries of $C$ are 0, but not whole rows of $C$. In other words, we don't have some input variables that don't affect any outputs, just some inputs variables that don't affect a few output variables.

I can't seem to be many good datasets for multi-target, so I'm hoping someone can point me in the right direction.

Thanks

Answer 1

This work on multi-target regression provides detailed information about the used datasets

Edit: Ignore the above. This is the referenced paper on multi-target regression. The datasets used are made publicly available at this link: http://mulan.sourceforge.net/datasets-mtr.html

Answer 2

You could easily collect some data like this. Gather some friends and have them write down their height and the last two digits of their id or credit card number. Let $Y_1$ be height, $Y_2$ be the two digit ID, and $X_1$ be the first of the two digits. Then $X_1$ is unrelated to $Y_1$ but strongly related to $Y_2$.