We conducted a number of tests on individual species over a period of several days. Let $x_{k,s,t}$ denote the result of test $k$ for species $s$ at time $t$. In total there are $S$ species, $K$ tests, and $T$ days.
We are interested in how the tests relate to each other. For example: Do species that are good in test $k=1$, tend to be good in test $k=2$ as well? You may think of the $K\times K$ relationships as a network connecting all tests.
We started out calculating Pearson correlations to construct a $K\times K$ correlation matrix. But since simple correlations confound direct and indirect associations, we are thinking about running multiple regressions of the form $$ x_{k,s,t}=\sum_{k=1,k\not =l}^{K}\beta_{lk}x_{l,s,t}+\delta_{k,s}% +\varepsilon_{k,s,t}% $$ where $\delta_{k,s}$ is a test-species fixed-effect and $\varepsilon _{k,s,t}$ the residual. There are $K$ regression that we estimate separately by OLS. We have $K<<T\times S$. An obvious problem here is that the regressors (except for the fixed effects) are endogenous. The use of instrumental variables is not possible here.
Our main question is whether in this case multiple regressions can be justified over the use of simple correlations. Stated differently, is the endogeneity problem large enough to prohibit this approach here?
This is a follow-up question of this question which is about prediction.
